Direction of Arrival Estimation Using a
Multiple-Input-Multiple-Output Radar with
Applications to Automobiles
von der Fakultät Informatik, Elektrotechnik und Informationstechnik
der Universität Stuttgart
zur Erlangung der Würde eines Doktor-Ingenieurs (Dr.-Ing.)
genehmigte Abhandlung
vorgelegt von
Kilian Rambach
aus Schweinfurt
Hauptberichter: Prof. Dr.-Ing. Bin Yang
Mitberichter: Prof. Dr.-Ing. Johann F. Böhme
Tag der mündlichen Prüfung: 20.06.2016
Institut für Signalverarbeitung und Systemtheorie
der Universität Stuttgart
2017

Vorwort
Die vorliegende Arbeit entstand während meiner Forschungsarbeit am Institut für Signalverarbeiung
und Systemtheorie (ISS) an der Universität Stuttgart. Ich bin froh darüber, Prof. Dr.-Ing. Bin
Yang als meinen Doktorvater zu haben. Ich möchte mich recht herzlich bei ihm bedanken, für
die Möglichkeit an seinem Institut zu promovieren, für die konstruktiven Diskussionen, die Frei-
heit eigene Ideen umzusetzen, sowie seine lockere und muntere Art. Das genaue Nachrechnen und
Überprüfen meiner Berechnungen schätze ich sehr. Prof. Dr.-Ing. Johann Böhme danke ich für die
Übernahme des Mitberichts.
Allen Kollegen am ISS danke ich für das gute Arbeitsklima, das freundliche Miteinander (inklusive
Wasserschlachten) und den vielen fachlichen und auch nicht fachlichen Diskussionen. Mein beson-
derer Dank gilt Thomas Küstner, Dr. Benedikt Lösch und Lukas Mauch für das Korrekturlesen dieser
Arbeit, auch in letzter Minute.
Ich bedanke mich bei Patrick Häcker, der mir insbesondere zu Beginn meiner Tätigkeit mit vielen
fachlichen Gesprächen zur Seite gestanden hat. Danken möchte ich auch Dr. Christof Zeile für das
Korrekturlesen und dass er dazu beigetragen hat, diesen Satz zu schreiben.
Weiterhin bedanke ich mich bei den Kollegen von Robert Bosch in Leonberg in der Abteilung
ECR3, insbesondere Dr. Götz Kühnle, Dr. André Treptow, Volker Groß, Dr. Michael Schoor und
Dr. Benedikt Lösch. Sie haben mit fruchtbaren Diskussionen und mit vielen Anregungen aus der
Praxis zum Gelingen meiner Arbeit beigetragen. Ich bedanke mich bei Michael Schoor für das
Korrekturlesen der Arbeit.
Mein besonderer Dank gilt meiner Partnerin Eva Schopf, die mich immer unterstützt und ermutigt
hat. Ich bedanke mich auch für ihre Geduld insbesondere während des Zusammenschreibens der
Arbeit.
Stuttgart, November 2015 Kilian Rambach

– 5 –
Contents
Notation and Abbreviations 9
Abstract 15
Kurzfassung 17
1. Introduction 19
2. Radar Basics 23
2.1. Working Principle of a Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1. Pulse Doppler Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.2. Linear Frequency Modulated Continuous Wave Radar . . . . . . . . . . . . 25
2.1.3. Chirp Sequence Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2. Parameter Estimation and Performance Measures . . . . . . . . . . . . . . . . . . . 27
2.2.1. Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.2. Accuracy and Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.3. Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3. MIMO Radar Concepts and State of the Art 33
3.1. Types and Properties of MIMO Radars . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1. MIMO Radar with Widely Separated Antennas . . . . . . . . . . . . . . . . 34
3.1.2. Colocated MIMO Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2. Multiplexing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1. Time Division Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2. Frequency Division Multiplexing . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.3. Code Division Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.3.1. CDM in Fast Time . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.3.2. CDM in Slow Time . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3. MIMO Radars for Automotive Applications and Relation to our Work . . . . . . . . 45
4. DOA Estimation of Stationary Targets 47
4.1. Radar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2. SIMO Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.1. Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.2. Cramer-Rao Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.2.1. Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.2.2. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3. MIMO Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1. Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.2. Cramer-Rao Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
– 6 –
4.4. Comparison of the SIMO and MIMO Radar . . . . . . . . . . . . . . . . . . . . . . 55
4.4.1. SNR Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4.2. Comparison of the SNRs for SIMO and MIMO Radar . . . . . . . . . . . . 59
4.4.3. Comparison of the CRBs and Discussion . . . . . . . . . . . . . . . . . . . 60
4.4.3.1. Without Tx Beampattern, B(u) = 1 . . . . . . . . . . . . . . . . . 60
4.4.3.2. With Tx Beampattern . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.4. Notes on CRBu,SIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.5. Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5. Comparison of Code and Time Division Multiplexing . . . . . . . . . . . . . . . . . 65
4.5.1. Code Division Multiplexing using Orthogonal Codes . . . . . . . . . . . . . 68
4.5.2. Time Division Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5. DOA Estimation of Moving Targets 73
5.1. Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1.1. Baseband Signal Model in the Time Domain . . . . . . . . . . . . . . . . . 74
5.1.1.1. Baseband Signal of a Pulse Doppler Radar . . . . . . . . . . . . . 74
5.1.1.2. Baseband Signal of an FMCW and Chirp Sequence Radar . . . . . 75
5.1.1.3. Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . 81
5.1.2. Signal Model of a TDM MIMO Radar for DOA estimation . . . . . . . . . . 82
5.1.2.1. Planar Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1.2.2. Notes and Discussion of the Model . . . . . . . . . . . . . . . . . 87
5.1.2.3. Linear Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.2.4. TDM Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2. One Moving Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2.1. Cramer Rao Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2.1.1. CRB for a Planar Array . . . . . . . . . . . . . . . . . . . . . . . 90
5.2.1.2. CRB for a Linear Array . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.2. Properties of the CRB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.2.1. Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.2.2. Comparison to CRB for a Stationary Target and to CRB for a
SIMO Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.3. Decoupling of Electrical Angles from the Doppler Frequency . . . . . . . . 99
5.2.3.1. Theorem for Decoupling of the Electrical Angles from the
Doppler Frequency . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.3.2. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2.3.3. Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.4. Optimal TDM Schemes for Linear Arrays . . . . . . . . . . . . . . . . . . . 108
5.2.4.1. Optimal TDM Schemes Under Limited Transmission Energy and
Aperture Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.4.2. Optimal TDM Schemes with Constant Transmission Energy and
Equidistant Transmission Times . . . . . . . . . . . . . . . . . . . 111
5.3. Two Moving Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.3.1. Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3.2. Cramer Rao Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3.2.1. Derivation of the CRB . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3.2.2. Notes on the CRB . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3.3. Decoupling of Electrical Angles and Doppler Frequencies . . . . . . . . . . 124
– 7 –
5.3.4. Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.3.5. Angular Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4. Arbitrary Number of Moving Targets . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.5. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6. Conclusions and Future Work 135
6.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2. Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A. Sample Statistics 139
A.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.2. Lemmas of the Sample Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
A.2.1. Lemmas of ES,CorrS,CovS . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.2.2. Lemmas of EWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.2.3. Lemmas of CorrWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.2.4. Lemmas of CovWS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
A.2.5. Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.2.5.1. Proofs of Lemmas of CorrWS . . . . . . . . . . . . . . . . . . . . 143
A.2.5.2. Proofs of Lemmas of CovWS . . . . . . . . . . . . . . . . . . . . 144
B. Derivation of the Cramer-Rao Bound for a SIMO Radar 145
C. Maximal Beamforming Gain 149
D. Computation of the SNR after a Discrete Fourier Transform 151
E. Maximum Likelihood Estimator for an NTargets Target Model 153
F. Asymptotic Properties of Maximum Likelihood Estimators 157
G. Baseband Signal Model for a Moving Target in the Time Domain 159
G.1. Computation of the Received Signal . . . . . . . . . . . . . . . . . . . . . . . . . . 159
G.2. Baseband Signal After the IQ-Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . 162
G.3. Influence of the Phase Error on the Error in the Electrical Angle . . . . . . . . . . . 163
G.4. Phase Difference Between Two Chirps . . . . . . . . . . . . . . . . . . . . . . . . . 164
G.5. Simplification of ∆ϕ1k for a Chirp Sequence Radar . . . . . . . . . . . . . . . . . . 167
G.6. Estimation of the Error in ∆ϕcorrected,1k . . . . . . . . . . . . . . . . . . . . . . . . . 167
H. Derivation of the Cramer-Rao Bound for a Planar TDM MIMO Radar with a Non-
Stationary Target 169
I. Proof of Theorem 1 173
J. Proof of Theorem 2: Weighted Mean Transmit Times and Decoupling of Electrical
Angles from the Doppler Frequency 175
J.1. Lemmas of the Weighted Sample Covariance of Clustered Values . . . . . . . . . . . 175
J.2. Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
– 8 –
K. Proof of Theorem 3: Optimal TDM Schemes for Constrained Transmission Energy and
Constrained Aperture Size 181
K.1. Missing Part of the Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . 181
K.2. Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
L. Proof of Theorem 4: Optimal TDM Schemes Under Additional Constraints 189
M. Derivation of the Cramer-Rao Bound for a TDM MIMO Radar with Two Non-
Stationary Targets 191
N. Proof of Theorem 6: Decoupling of Electrical Angles and Doppler Frequencies for Ar-
bitrary Number of Moving Targets 201
Bibliography 207
– 9 –
Notation and Abbreviations
Notation
x scalar
x column vector
X matrix
x∗, x∗,X∗ complex conjugate of a scalar, vector or matrix
xH,XH conjugate transpose of a vector or matrix
xT ,XT transpose of a vector or matrix
X−1 inverse of a matrix
X+ Moore–Penrose pseudo-inverse of a matrix
xi or [x]i i-th element of vector x
[X]i, j element of matrix X at i-th row and j-th column
0K×M 0K×M ∈ RK×M, matrix with all elements equal 0
0 shortcut for 0K×M with the dimension following from the context
1K 1K ∈ RK , vector of length K with all elements equal 1
1 shortcut for 1K with the dimension following from the context
1K×M 1K×M ∈ RK×M, matrix with all elements equal 1
1 shortcut for 1K×M with the dimension following from the context
IK×K IK×K ∈ RK×K , identity matrix
I identity matrix with the dimension following from the context
Z,R,C set of integer, real, and complex numbers
Mathematical Operations
¬A negation “not A”
A =⇒ B implication, i. e. if A is true then B is true. Stated in other words, A is sufficient for
B.
A ⇐⇒ B equivalence, i. e. A is true if and only if B is true. Stated in other words, A is
necessary and sufficient for B.
– 10 –
‖.‖ Euclidean norm of a vector or matrix
 entrywise Hadamard product, i. e. x = y  z means xi = yi · zi ∀i
⊗ Kronecker tensor product
CorrS(.) sample cross-correlation, defined in Appendix A
CorrWS(.) weighted sample cross-correlation, defined in Appendix A
CovS(.) sample cross-covariance, defined in Appendix A
CovWS(.) weighted sample cross-covariance, defined in Appendix A
diag(x) diagonal matrix consisting of the elements of x
E(.) expectation operator
ES(.) sample mean, defined in Appendix A
EWS(.) weighted sample mean, defined in Appendix A
exp(x) or e x exponential function of each element of vector x, i. e. y = exp(x) means yi =
exp(xi) ∀i
max(x) the element with the maximal value of the real vector x
min(x) the element with the minimal value of the real vector x
rx autocorrelation of x
Re(.) real part of a complex number or matrix
√x square root of each element of vector x, i. e. y = √x means yi = √xi ∀i
Tr(X) trace of a matrix
Symbols
Latin Symbols
∝ proportional to
.ˆ estimated value of a parameter, e. g. Θˆ is the estimated value of Θ
aRx, aTx, aPulse, aVirt steering vector of the Rx and Tx array, the array defined by dPulse, and virtual
array, respectively
ARx, ATx steering matrix of the Rx and Tx array, respectively
b(u, ω) steering vector of a TDM MIMO radar as a function of the electrical angle
u and Doppler frequency ω
B(U, ω) steering matrix of a TDM MIMO radar as a function of the electrical angles
U and Doppler frequencies ω of several targets
bΘˆ, bΘˆ bias of Θˆ and Θˆ, respectively
B transmit beampattern
c speed of light
Cn noise covariance matrix
– 11 –
CΘˆ, CΘˆ variance of Θˆ and covariance matrix of Θˆ, respectively
CRBΘ CRB of a scalar parameter Θ
CRBΘ CRB of a parameter vector Θ
d distance of the target
d0, dM distance of the target at the beginning and at the middle of the transmission
of a chirp, respectively
dPulse,DPulse positions of the Tx antennas in the sequence in which they transmit, divided
by λ2pi , for a linear and planar array, respectively
dRx,DRx positions of the Rx antennas, divided by λ2pi , for a linear and planar array,
respectively
dTx,DTx positions of the Tx antennas, divided by λ2pi , for a linear and planar array,
respectively
dVirt,DVirt positions of the virtual antennas, divided by λ2pi , for a linear and planar array,
respectively
∆dRx,∆dTx geometrical size of the Rx and Tx array, respectively
f0 carrier frequency
fd Doppler frequency
fdiff difference frequency for a chirp or FMCW radar
fs sampling frequency
fRx, fTx received and transmitted frequency of a chirp, respectively
F, Fω bandwidth of a chirp in terms of frequency or angular frequency
j imaginary unit j =
√−1
J Fisher Information Matrix
L number of measurement cycles
MΘˆ, MΘˆ mean square error of Θˆ and Θˆ, respectively
n dependent on the context: scalar noise or discrete sampling point in time
n, N noise
NPulse number of transmitted chirps or pulses
NRx number of Rx antennas
Nsamples number of samples per chirp
NTx number of Tx antennas
NVirt number of virtual antenna elements. Depending on the context
NVirt = NRxNTx or NVirt = NRxNPulse
NTargets number of targets
p, p transmission power of one Tx antenna or of several pulses
pmax maximal transmission power
PD, PFA probability of detection and false alarm, respectively
– 12 –
RΦ correlation matrix of the transmitted codes
RMSEΘˆ root mean square error of Θˆ
s, s complex signal strength of one and several targets, respectively
S matrix containing the complex signal strengths of several targets
sTx(t) transmitted signal
S total SNR
t continuous time
tstart start time of the transmission of a chirp
tM time in the middle of a chirp
∆tM,1k difference between the time in the middle of chirp k and chirp 1
∆t depending no the context: sampling time ∆t = 1/ fs, difference between the
transmission time of two successive pulses, or the duration of one pulse
∆t ∈ RNPulse durations of the transmitted pulses
t ∈ RNPulse transmission times of the transmitted pulses
t(k) transmission times of the pulses transmitted by the k-th Tx antenna
T chirp duration
Tcycle time of one measurement cycle, i. e. one coherent processing interval
Ttotal total measurement time
u electrical angle of a target
u electrical angles of one target, when estimating two DOAs using a planar
array
U ∈ RNTargets electrical angles of NTargets targets
∆U difference of the electrical angles of two targets
USIMO,UMIMO sample variance of the Rx antenna positions in a SIMO array and the virtual
antenna positions in a MIMO array, respectively
v relative radial velocity of the target
x depending on the context: signal after range processing, used for the DOA
estimation, or the baseband signal
X signal after range compression for code division multiplexing
Greek Symbols
β radial velocity over speed of light β = vc
δl,m Kronecker delta, i. e. δl,m = 1 if l = m and δl,m = 0 otherwise
δRL, δARL resolution and angular resolution, respectively
λ carrier wavelength
ϕ dependent on the context: Elevation angle in the case of a planar antenna array or
phase of the baseband signal
– 13 –
ϕRx(t), ϕTx(t) time dependent phase of the received and transmitted signal, respectively
∆ϕ1k phase difference between the first and k-th pulse
ϕbase(t) time dependent phase of the baseband signal
Φ transmitted codes in CDM MIMO radar
ψTDM parameters which characterize a TDM scheme
ρs energy per sample
ρ energy of the transmitted pulses
ρ(k) energy of the pulses transmitted by the k-th Tx antenna
σΘˆ standard deviation of Θˆ
σ2, σ2n variance of the noise
σ2s signal strength
ϑ direction of arrival, the azimuth
Θ unknown scalar parameter
Θ unknown parameter vector
Θˆ, Θˆ estimated parameter or parameter estimator for a scalar and vector parameter, re-
spectively
τ time delay between transmitted and received signal (round trip time). Precise
definition depends on the context.
τ0, τM time delay (round trip time) at the beginning and at the middle of the transmission
of a chirp, respectively
χ f , χ chirp rate of the frequency and angular frequency, respectively
ω0 angular carrier frequency
ω0M angular carrier frequency in the middle of a chirp
ωDoppler angular Doppler frequency
ω dependent on the context: general angular frequency or short notation for the an-
gular Doppler frequency of a target
ω angular Doppler frequency of several targets
∆ω difference of the angular Doppler frequencies of two targets
ω(t) time dependent angular frequency of a frequency modulated chirp
ωdiff angular difference frequency for a chirp or FMCW radar
Abbreviations
ADC analog to digital converter
CDM code division multiplexing
CPI coherent processing interval
– 14 –
CRB Cramer-Rao bound
DFT discrete Fourier transform
DML deterministic maximum likelihood
DOA direction of arrival
DTFT discrete time Fourier transform
FDM frequency division multiplexing
FFT fast Fourier transform
FIM Fisher Information Matrix
FMCW frequency modulated continuous wave
GLRT generalized likelihood ratio test
GMTI ground moving target indication
iff if and only if
MC Monte-Carlo
MIMO multiple input multiple output
ML maximum likelihood
MSE mean squared error
pdf probability density function
RCS radar cross section
RMSE root-mean-square error
R-TDMA randomized time division multiple access
Rx receiver or receiving
SIMO single input multiple output
SNR signal-to-noise ratio
SINR signal-to-noise-and-interference ratio
STAP space time adaptive processing
SCR signal-to-clutter ratio
TDM time division multiplex
TDMA time division multiple access
Tx transmitter or transmitting
ULA uniform linear array
w. l. o. g. without loss of generality
w. r. t. with respect to
– 15 –
Abstract
The thesis at hand investigates the direction of arrival (DOA) estimation using a Multiple-Input-
Multiple-Output (MIMO) radar system. The application of MIMO radars in automobiles is studied.
A MIMO radar consists of several transmitting (Tx) and receiving (Rx) antennas. We focus on a time
division multiplexed (TDM) MIMO radar with colocated Tx and Rx antennas. The motivation is the
use of a radar as a security system in automotive applications, e. g. to identify a dangerous situation
and react automatically. Security systems must be very reliable. Hence, besides a good estimation of
the distance and velocity, a high performance in DOA estimation is necessary. This is a demanding
task, since only a small number of antennas is used and the radar is limited to a small geometrical
size. Compared to the corresponding Single-Input-Multiple-Output (SIMO) radar, a MIMO radar
with colocated antennas can achieve a higher accuracy in DOA estimation due to its larger virtual
aperture. Therefore it is a promising technique for the use in automobiles. The obtained results of
this thesis enable us to find optimal TDM schemes which yield a very high DOA accuracy for targets
which are stationary as well as for targets which are moving relative to the radar system. The results
are not confined to MIMO radars in automobiles, but can be used in other applications as well.
In Chapter 2 we introduce the basics of radar systems, which are important for this thesis. First we
present three different kinds of radars, the pulse Doppler radar, the frequency modulated continuous
wave (FMCW) radar, and the chirp sequence radar. After that, the performance measures of a radar
system, which are used in the following chapters, are introduced. For that purpose, the concept
of parameter estimators are presented. Thereafter, the accuracy is defined and a definition for the
resolution is given.
Chapter 3 gives an overview on the two types of MIMO radars: The MIMO radar with widely
separated antennas and the colocated MIMO radar. A survey on the research already done on MIMO
radars is presented. Thereafter, the different multiplexing techniques are discussed, which can be
used in a MIMO radar to separate the signals of the different Tx antennas. The last part of Chapter 3
focuses on the application of MIMO radars in automobiles. It shows why TDM MIMO radars are
suitable for the use in automotive applications.
In Chapter 4 we study the DOA estimation for stationary targets, i. e. the targets are not moving
relative to the radar. We investigate SIMO and MIMO radars and compute the deterministic Cramer-
Rao Bound (CRB) of the DOA for both types of systems. With these results we can determine
the maximal achievable DOA accuracy. We compare both CRBs and show under which conditions
– 16 –
the MIMO radar achieves a lower CRB. Furthermore, we analyze the DOA performance of a code
division multiplexed (CDM) MIMO radar. If the total transmission power of the radar is limited by
the maximal transmission power of one Tx antenna, then the TDM and CDM MIMO radar yield
the same CRB for the DOA. If certain properties of the noise covariance matrix are satisfied, we
can show with the concept of sufficient statistics that both radars even yield the same amount of
information on the DOA. Hence, not only the CRB but all statistical properties like the ambiguities,
sidelobe levels, and resolution are the same for the TDM and CDM MIMO radar.
Chapter 5 contains the main contribution of the thesis. In this chapter targets which are moving
relative to the radar are studied. To estimate the DOA using a SIMO radar, the phases of the received
baseband signals are analyzed. In a MIMO radar the phase relation between the Tx antennas is
processed as well. A target which is moving relative to the radar induces an additional phase shift
in the baseband signal due to the Doppler effect. In general, this decreases the DOA estimation
accuracy, since the Doppler frequency has to be estimated in addition. We study this scenario. After
a precise derivation of the signal model, the deterministic CRB is computed for a general TDM
scheme for one moving target. This enables to investigate the maximal achievable accuracy of the
DOA estimation of a TDM MIMO radar. Different TDM MIMO radars are compared. Moreover
we derive TDM schemes which result in a decoupling of the DOA and Doppler frequency in the
CRB. Hence, using those TDM schemes, the CRB for the DOA is as small as for a stationary target.
Under different practical boundary conditions, optimal TDM schemes are derived, which achieve
the highest possible DOA accuracy. Thereafter, we investigate the DOA estimation of two and more
targets. Conditions on the TDM schemes are derived which guarantee the decoupling of the DOAs
from the Doppler frequencies for certain scenarios. Besides the DOA accuracy, the resolution is
investigated for different TDM schemes.
Chapter 6 summarizes the thesis and gives an outlook on possible future work.
– 17 –
Kurzfassung
In der vorliegenden Arbeit wird die Winkelschätzung (auch Einfallsrichtung genannt, engl. Direc-
tion of Arrival (DOA)) mit Hilfe eines Multiple-Input-Multiple-Output (MIMO) Radars untersucht.
Darüber hinaus wird die Verwendung eines MIMO Radars in automobilien Anwendungen betra-
chtet. Ein MIMO Radar besteht aus mehreren Sende- (Tx) und Empfangsantennen (Rx). Wir be-
trachten insbesondere MIMO Radare die im Zeitmultiplexverfahren (engl. time division multiplex
(TDM)) betrieben werden und geometrisch nahe beieinander liegende Antennen (engl. colocated)
besitzen. Die Motivation dieser Untersuchungen ist die Verwendung von Radarsystemen als Sicher-
heitssysteme in Fahrzeugen, z.B. um eine gefährliche Situation zu detektieren und darauf automa-
tisch zu reagieren. Sicherheitssysteme müssen sehr zuverlässig sein. Daher ist neben einer genauen
Abstands- und Geschwindigkeitsschätzung auch eine hohe Performance in der Winkelschätzung
nötig. Dies ist eine anspruchsvolle Aufgabe, da nur eine geringe Anzahl an Antennen zur Verfü-
gung steht und das Radarsystem nur eine kleine geometrische Größe aufweisen darf. Im Vergleich
zu einem entsprechenden Single-Input-Multiple-Output (SIMO) Radar kann ein colocated MIMO
Radar aufgrund seiner größeren virtuellen Apertur eine höhere Winkelgenauigkeit erreichen. Da-
her ist es eine vielversprechende Technik für die Anwendung in Fahrzeugen. Die Ergebnisse dieser
Arbeit ermöglichen uns optimale Zeitmultiplexverfahren zu finden, welche sowohl für stationäre
Objekte als auch für Objekte die sich relativ zum Radar bewegen, eine hohe Winkelgenauigkeit er-
reichen. Die Ergebnisse beschränken sich nicht nur auf Radare in Fahrzeugen, sondern können auch
in anderen Anwendungen verwendet werden.
In Kapitel 2 führen wir die Grundlagen der Radarsysteme ein, welche für die vorliegende Arbeit
wichtig sind. Zuerst stellen wir drei verschiedene Typen von Radaren vor: Das Puls-Doppler-Radar,
das frequenzmodulierte Dauerstichradar (engl. frequency modulated continuous wave (FMCW)
radar) und das Radar mit einer langen Folge von chirps (engl. chirp sequence radar). Daraufhin
werden die Performancemaße eines Radars eingeführt, die in den folgenden Kapiteln benutzt wer-
den. Für diesen Zweck wird das Konzept der Parameterschätzung vorgestellt. Danach definieren wir
die Genauigkeit und Trennfähigkeit (auch Auflösung genannt).
In Kapitel 3 geben wir einen Überblick über die zwei Typen von MIMO Radaren: MIMO Radare mit
räumlich weit getrennten Antennen (engl. widely separated antennas) und colocated MIMO Radare.
Wir stellen den Stand der Technik zu MIMO Radaren vor. Daraufhin diskutieren wir verschiedene
Multiplexverfahren die in einem MIMO Radar verwendet werden können, um die Signale der ver-
schiedenen Sender zu unterscheiden. Der letzte Teil von Kapitel 3 behandelt die Anwendung von
– 18 –
MIMO Radaren in Fahrzeugen. Es wird gezeigt warum TDM MIMO Radare für die Anwendung in
Automobilen geeignet sind.
In Kapitel 4 untersuchen wir die Winkelschätzung stationärer Ziele, d.h. das Ziel bewegt sich nicht
relativ zum Radar. Wir betrachten SIMO und MIMO Radare und berechnen die deterministische
Cramer-Rao Schranke des Winkels für beide Systeme (engl. Cramer-Rao Bound (CRB)). Mit dem
Ergebnis können wir die maximal mögliche Winkelgenauigkeit bestimmen. Wir vergleichen beide
CRBs und zeigen unter welchen Bedingungen das MIMO Radar eine kleinere CRB erreicht. Darüber
hinaus analysieren wir die Winkelperformance eines MIMO Radars welches mit code division mul-
tiplex (CDM) betrieben wird. Wenn die gesamte Sendeleistung des Radars durch die maximale
Sendeleistung einer Antenne beschränkt ist, dann besitzen das TDM und CDM MIMO Radar die
gleiche CRB des Winkels. Erfüllt die Kovarianzmatrix des Rauschens bestimmte Voraussetzungen,
dann können wir mit Hilfe der sufficient statistics zeigen, dass beide Radarsysteme sogar die gleiche
Information über den Winkel liefern. Daher ist dann nicht nur die CRB gleich, sondern alle statistis-
chen Eigenschaften, wie z.B. die Mehrdeutigkeiten, die Höhe der Nebenkeulen und die Auflösung,
sind für das TDM und CDM MIMO Radar identisch.
Kapitel 5 entählt den Hauptbeitrag der vorliegenden Arbeit. In diesem Kapitel werden Ziele unter-
sucht, welche sich relativ zum Radar bewegen. Um den Winkel mit einem SIMO Radar zu schätzen
werden die Phasen des empfangenen Basisbandsignals analysiert. In einem MIMO Radar wird
die Phasenbeziehung zwischen den Sendeantennen ebenfalls verarbeitet. Ein Ziel, das sich rela-
tive zum Radar bewegt, bewirkt einen zusätzliche Phasenverschiebung aufgrund des Dopplereffekts.
Im Allgemeinen wird dadurch die Winkelgenauigkeit schlechter, da die Dopplerfrequenz zusätzlich
geschätzt werden muss. Wir untersuchen dieses Szenario. Nach einer genauen Herleitung des Sig-
nalmodells berechnen wir die deterministische CRB für ein allgemeines Zeitmultiplexschema für
ein bewegtes Ziel. Dadurch können wir die maximal erreichbare Winkelgenauigkeit eines TDM
MIMO Radars untersuchen. Verschiedene TDM MIMO Radare werden miteinander verglichen.
Außerdem leiten wir Zeitmultiplexschemata her, welche zu einem Entkoppeln des Winkels und der
Dopplerfrequenz in der CRB führen. Wenn man also solche Zeitmultiplexschemata benutzt, dann
ist die CRB des Winkels genauso groß wie für ein stationäres Ziel. Für verschiedene praktische
Randbedingungen werden optimale Zeitmultiplexschemata hergeleitet, welche die höchst mögliche
Winkelgenauigkeit erreichen. Weiterhin untersuchen wir die Winkelschätzung für zwei und mehr
Ziele. Wir leiten Bedingungen an die Zeitmultiplexschemata her, welche für bestimmte Szenarien
ein Entkoppeln der Winkel von der Dopplerfrequenz garantieren. Neben der Genauigkeit wird auch
die Trennfähigkeit für verschiedene Zeitmultiplexschemata untersucht.
In Kapitel 6 fassen wir die vorliegende Arbeit zusammen und geben einen Ausblick auf weitere
mögliche Arbeiten.
– 19 –
1. Introduction
The history of radar started with the invention of Christian Hulsmeyer in 1904, who developed the
first form of a radar. In 1935 Sir Robert Watson-Watt demonstrated a working radar system and in
world war II the research and development of radars increased further [Li and Stoica, 2009, Ch. 2].
Nowadays radar systems are not only used for air surveillance or military applications, but also in
completely different topics. They are used e.g. in breast cancer imaging [Klemm et al., 2010] or in
automobiles, in order to offer different security and comfort functions [Winner, 2012].
With the availability of more and more computer capabilities, more complex systems like Multiple-
Input-Multiple-Output (MIMO) radars are built. MIMO systems consist of several transmitting (Tx)
and receiving (Rx) antennas. In general, every transmitter can transmit an arbitrary and independent
waveform. In communications, MIMO systems gained much attention in the 1990s [Li and Stoica,
2009, Ch. 3]. One goal is the optimization of these systems to achieve a high transfer rate [Sayeed
and Raghavan, 2007]. There are several similarities between MIMO communication systems and
MIMO radar systems, but the radar systems are designed to achieve a different goal. They are used
to detect and track targets and estimate target parameters, e.g. the distance, the direction of arrival
(DOA), the speed or the radar cross section. To the best of our knowledge, the first colocated MIMO
radar was invented at the end of the 80s [Lesturgie, 2011], under the name “Synthetic Antenna and
Impulse Radar” (RIAS) [Dorey et al., 1989; Luce et al., 1992]. A similar radar, called “Sparse-
array Synthetic Impulse and Aperture Radar” (SIAR) was developed later [Baixiao et al., 2001]. By
using digital beamforming, this radar system is able to track several targets at the same time. Many
additional interesting aspects are investigated nowadays, e.g. the performance gain of a MIMO radar
compared to the Single-Input-Multiple-Output (SIMO) counterpart [Friedlander, 2009], the design
of the transmitted waveforms [Li et al., 2008], and the development of new estimation techniques
[Chen, 2009].
The motivation for our work is the use of MIMO radars in automotive applications. Radars in
automobiles offer comfort functions, e. g. adaptive cruise control, and even more important, security
functions [Winner, 2012]. If a radar identifies a possible accident, e. g. two cars are moving too
fast in the direction of each other, it can warn the driver or react automatically. In many situations,
automatic systems can react faster than human beings. Hence, radar systems can help to prevent
injuries or even deaths of humans. In order to be used as a security system, the radar has to be
reliable and offer good estimation performance of the target’s parameters. The DOA estimation
is a difficult task and still has to be improved. Consider a car at a large distance. To decide if it
– 20 –
is on the left or right driving lane, an accurate DOA estimation is required. Usually only a small
number of antennas can be used in an automotive radar and their positions are constrained to a small
geometrical size. A MIMO radar is a promising approach to improve the DOA estimation under
these conditions since it offers a larger virtual aperture than the corresponding SIMO radar [Chen,
2009].
We are especially interested in the DOA accuracy. The Cramer-Rao bound (CRB) is a lower bound
on the covariance matrix of any unbiased estimator [Kay, 1993]. Therefore we can use it to examine
the maximum DOA accuracy of a radar system, independent of the used estimation algorithm. The
CRB has been derived for different settings and radar systems. The CRB for a SIMO radar is
computed in [Stoica and Nehorai, 1989]. In [Ward, 1995; Dogandzic and Nehorai, 2001], the CRB
for non-stationary targets is studied. The DOA estimation for a SIMO radar with only one Rx channel
is investigated in [Lee et al., 2004]. The authors of [Bekkerman and Tabrikian, 2006; Forsythe and
Bliss, 2005] derived the CRB for the DOA estimation of a stationary target using a MIMO radar with
a general transmit signal. A MIMO radar with simultaneously transmitting antennas is considered
in [Boyer, 2011; Chong et al., 2011] and the CRB for the DOA and Doppler estimation is computed.
A MIMO radar can be realized by different multiplexing techniques like frequency, code, or time di-
vision multiplexing (TDM). All multiplexing techniques come with different advantages and draw-
backs. In this thesis, we focus on TDM MIMO radars. In general, they have lower hardware re-
quirements than other multiplexing techniques. Furthermore, the transmit signal can be kept as in
a SIMO radar, since the antennas do not transmit simultaneously. Hence, waveforms which have
been already optimized for distance and velocity estimation can still be used, e. g. linear frequency
modulated continuous waves (LFMCW). TDM MIMO radars have been investigated experimentally
[Feger et al., 2009; Jahn et al., 2012; Schmid et al., 2012; Guetlein et al., 2012, 2013a], but quantita-
tive comparisons with different radar systems, especially for the DOA estimation of moving targets,
are still missing.
We fill this gap by studying the DOA estimation of stationary and non-stationary targets using SIMO
and TDM MIMO radars. Note that the Tx antennas in a TDM MIMO radar transmit successively
and not simultaneously. Therefore the signal model has a different structure than that in [Boyer,
2011; Chong et al., 2011], where simultaneously transmitting Tx antennas are assumed.
We introduce the basics of radar systems in Chapter 2 and give an overview of the working principle
of MIMO radars and the current state of the art in Chapter 3. The DOA estimation of stationary
targets is studied in Chapter 4 and a comparison between SIMO and MIMO radars is made with
respect to the achievable DOA accuracy. Moreover, we compare TDM and CDM MIMO radars. We
extend the studies to moving targets in Chapter 5. In DOA estimation, the phases of the baseband
signals are processed. A MIMO radar analyzes the phase relation between the Tx channels as well.
A moving target induces an additional phase shift in the baseband signals due to the Doppler effect.
Thus, the Doppler frequency has to be estimated in addition to the DOA. In general, this decreases
the DOA accuracy. We investigate this scenario and derive the CRB for the DOA for linear and
– 21 –
planar antenna arrays and study different TDM schemes. Optimal TDM schemes are presented,
which lead to a decoupling of the DOAs from the Doppler frequency. Thus, a DOA accuracy can
be achieved which is as high as for a stationary target, despite the unknown Doppler frequency.
Chapter 6 concludes the thesis and gives an outlook on possible future work.
Chapter 4 and 5 contain the contributions to the research on MIMO radars. Chapter 5 contains the
main results of the thesis for TDM MIMO radars.

– 23 –
2. Radar Basics
In this chapter, we give a brief overview of the working principle of different radar systems which are
considered in the context of the MIMO radar in later chapters. Moreover, we present some important
performance measures. For more details see e. g. [Skolnik, 2001].
2.1. Working Principle of a Radar
A radar transmits an electromagnetic wave. The wave propagates, is reflected at different objects,
and a part of it travels back to the radar. This is the signal which the radar receives. This signal
can be analyzed to determine different parameters of the objects. The parameters which can be
measured and are important for automotive applications are the distance d of a target to the radar, its
relative radial velocity v and the angle ϑ under which the target appears, see Fig. 2.1. The angle ϑ
is also called direction of arrival (DOA). The distance can be determined by the time delay between
the transmitted and received signal. Due to the Doppler frequency, the velocity can be estimated.
One possibility to measure the DOA is the use of an array of receiving (Rx) antennas, i. e. several
antennas at different positions. In Fig. 2.2 an antenna array and the propagation direction of the
reflected signal are depicted for a target which is far away compared to the geometrical size of the
antenna array. Hence, the reflected wave can be approximated by a plane wave. For ϑ , 0, the time
at which the signal is received is different for each Rx antenna. The time differences depend on ϑ
and can therefore be used to estimate ϑ.
In the following, we briefly present three different radar types.
2.1.1. Pulse Doppler Radar
A pulse Doppler radar transmits several short pulses repetitively. The pulses have a constant trans-
mission frequency f0. The time delay τ between the transmitted and received echo pulse is measured
to determine the distance d. The relation between τ and d is
τ =
2 d
c
, (2.1)
– 24 –
d
v
ϑ
vrelative
Figure 2.1.: The target’s parameters d, v, and ϑ. vrelative is the total relative velocity and v the radial
part of it. [Openclipart, 2014, graphical parts used and adapted]
Tx Rx1 Rx2 Rx3
ϑ
Figure 2.2.: Antenna array with several Rx antennas. The blue arrow indicates the propagation di-
rection of the reflected plane wave.
where c is the speed of light. The factor 2 arises from the fact that the pulse has to travel forth and
back. Compared to the transmitted signal, the received signal is shifted in frequency by the Doppler
frequency fd with [Skolnik, 2001]
fd ≈ 2 vc f0. (2.2)
Here, the velocity v is defined to be positive, if the target moves away from the radar. The approxi-
mation is valid for |v|  c. The relation between the received frequency fRx, the transmitted one f0
and the Doppler frequency is
fd = f0 − fRx. (2.3)
This radar type is described in more detail in [Skolnik, 2001; Bühren, 2008].
– 25 –
2.1.2. Linear Frequency Modulated Continuous Wave Radar
A linear frequency modulated continuous wave (LFMCW, often just FMCW) radar transmits pulses
with a linearly changing frequency. These pulses are also called chirps. The transmitted (Tx) time
dependent frequency fTx(t) of one pulse, depicted in Fig. 2.3, can be written as
fTx(t) = f0 + χ f (t − tM), t ∈ [tstart, tstart + T ] (2.4)
where t is the time, f0 the frequency at time tM = tstart + T2 in the middle of the chirp and χ f the
chirp rate. tstart is the start of the transmission time and T the duration of the transmission. The
received signal is mixed with the transmitted signal with an IQ-mixer. Afterwards a low pass filter
is applied. This is depicted in Fig. 2.4. Hence, the baseband signal contains the difference frequency
fdiff, which is the difference between the transmitted frequency fTx(t) and received frequency fRx(t).
The received frequency differs from the transmitted frequency because of the round trip delay τ and
the Doppler effect, cf. Fig. 2.5. fdiff is constant and given by [Stove, 1992; Reiher, 2012]
fdiff = fTx(t) − fRx(t) ≈ fd + χ f τ. (2.5)
Thus, the baseband signal is a complex oscillation exp ( j (2pi fdiff t + ϕ)) with constant frequency fdiff
and phase offset ϕ. In an FMCW radar, fdiff is usually determined by a fast Fourier transform (FFT).
Due to (2.1) and (2.2) we can write
fdiff ≈ 2 vc f0 + χ f
2 d
c
. (2.6)
Thus, fdiff depends both on the unknown velocity v and the unknown distance d. Hence we have
one equation but two unknowns. Because of this, at least two measurements are necessary using
chirps with different chirp rates χ f , which can also be negative. For several targets, more chirps have
to be transmitted and processed. Then, in general, each received baseband signal contains several
difference frequencies. The difference frequencies in each baseband signal, which correspond to the
same target, have to be matched in order to estimate d and v of the target. This process is called
frequency matching which is investigated in detail in [Reiher, 2012]. As an example for the different
t
tstart tM
fTx(t)
f0
T
χ f
Figure 2.3.: Linear modulated frequency
– 26 –
Figure 2.4.: Structure of an FMCW or chirp sequence radar system with an IQ mixer
radar parameters, we consider a typical FMCW radar which is used in automotive applications. It
transmits 4 chirps which have a duration T of 2 to 10 ms and a bandwidth F of a few hundred MHz.
The chirps have all different chirp rates χ f , some of them positive and some of them negative.
A detailed derivation of the received baseband signal is presented in Section 5.1.1.
The Fourier transformed baseband signal is also called the compressed pulse, because the energy of
the signal is not distributed in all time domain samples, but only in a few frequency bins. Mixing
down the received signal with the transmitted one and taking the Fourier transform of it is similar
to the correlation of the received signal with the transmitted one. This technique is called pulse
compression [Skolnik, 2001]. For an array with NRx Rx antennas, we have NRx received baseband
signals. For each received baseband signal, the complex value of the Fourier transform at the differ-
ence frequency fdiff of the target is taken. These values, denoted by x ∈ CNRx , are used to estimate the
target’s DOA [Van Trees, 2002; Schoor, 2010]. Thus, x is the signal after pulse compression. The
principle of DOA estimation is explained in more detail in the context of colocated MIMO radars in
Section 3.1.2.
A thorough treatment of the FMCW radar is presented in [Reiher, 2012; Lübbert, 2005].
– 27 –
t
fTx(t), fRx(t)
fTx
fRx
τ
fd
fdiff
Figure 2.5.: Relation between Tx and Rx chirp
2.1.3. Chirp Sequence Radar
Another type of radar transmits many equal chirps of short duration. Such a radar is called a chirp
sequence radar [Winner, 2012]. The structure is the same as the FMCW radar, cf. Fig. 2.4. Exem-
plary values are 512 chirps of a duration of 10 to 100 µs and a bandwidth of approximately 1 GHz.
Thus, the chirp rate χ f is much higher than that in an FMCW radar. As a consequence, the term χ f τ
in (2.5) is usually much larger than the Doppler frequency fd. Hence the difference frequency fdiff
depends approximately only on the distance [Winner, 2012]
fdiff ≈ χ f τ = χ f 2 dc . (2.7)
Therefore, the distance d can be determined from fdiff. Compared to the FMCW radar, in the case of
several targets no matching algorithm, which relates several difference frequencies to one target, is
necessary. This is a big advantage. As in the FMCW radar, fdiff is usually determined by a FFT. The
Doppler effect becomes perceivable by a phase shift between the successive received chirps. Hence
the velocity v can be determined by analyzing the phase change from chirp to chirp. This is usually
done by another FFT. For each bin of the first FFT, the second FFT is computed. This corresponds
to a two dimensional FFT. For a SIMO radar with an array of NRx Rx antennas, the values of the two
dimensional Fourier transform at the estimated values dˆ, vˆ for each Rx antenna are used as the signal
x ∈ CNRx for the DOA estimation. Hence, x is gained by pulse compressing the baseband signal two
times. For a detailed derivation of the baseband signal of a chirp sequence radar, see Section 5.1.1.
2.2. Parameter Estimation and Performance Measures
In a radar system, different unknown parameters of a target, e. g. its distance, velocity, and DOA,
have to be estimated from the measured data. First we introduce the notation of a general parameter
– 28 –
estimator. After that we take a look at the performance measures to characterize a radar system.
There are many different ones, e. g. the accuracy and resolution of certain parameters, ambiguities
and sidelobe levels, the robustness of the radar against different perturbations, the number of de-
tectable targets, and the clutter rejection ability. In this thesis, we focus on the accuracy and the
resolution of the DOA.
2.2.1. Parameter Estimation
We want to determine the unknown, deterministic parameter vector Θ ∈ RM from a measurement
x ∈ RK . The statistical signal model is defined by a random vector X with its probability density
function fX. The measurement x is an observed value, i. e. a realization of X. The estimation function
g is used to estimate the unknown parameter Θ from the measurement x
g : RK → RM,
x 7→ g(x). (2.8)
For one measurement x, the estimate of Θ is the value ϑˆ = g(x). With the random vector X, we can
define the new random vector Θˆ = g ◦ X. Θˆ is called the estimator. Often the estimation function is
also denoted by Θˆ. It is clear from the context what Θˆ denotes.
In some applications, the measurements are complex values x ∈ CK . This is for example the case,
if a radar system with an IQ-mixer is investigated, which results in a complex baseband signal. This
can be mapped to the real case by considering the real and imaginary part separately.
2.2.2. Accuracy and Lower Bounds
To define the accuracy, we first define the bias, the covariance, and the mean square error (MSE) of
an estimator.
Bias, Covariance and MSE The estimator’s bias b
Θˆ
, covariance matrix CΘˆ and mean square error
(MSE) matrix MΘˆ are defined as
b
Θˆ
= E
(
Θˆ − Θ
)
, (2.9)
CΘˆ = E
([
Θˆ − E
(
Θˆ
)] [
Θˆ − E
(
Θˆ
)]T )
, (2.10)
MΘˆ = E
([
Θˆ − Θ
] [
Θˆ − Θ
]T )
, (2.11)
– 29 –
respectively. Here, E(.) denotes the expectation operator. It can easily be shown that
MΘˆ = bΘˆ b
T
Θˆ
+ CΘˆ. (2.12)
An estimator is called unbiased if b
Θˆ
= 0. For a scalar parameter Θ, the standard deviation σΘˆ and
root mean square error (RMSE) of the estimator Θˆ are defined as the square root of the variance CΘˆ
and MSE MΘˆ, respectively:
σΘˆ =
√
CΘˆ, (2.13)
RMSEΘˆ =
√
MΘˆ. (2.14)
Accuracy For a scalar unknown parameter Θ ∈ R, the accuracy of an estimator is defined as the
root mean square error (RMSE) of the estimator [Skolnik, 2001]. Sometimes the accuracy is defined
to be proportional to the RMSE, where the proportional constant depends on the definition. For
an unknown parameter vector Θ ∈ RK , the MSE matrix MΘˆ can be used in order to specify how
accurate the parameter vector is estimated.
If an estimator is unbiased, or the bias is negligible compared to the covariance, then MΘˆ ≈ CΘˆ. In
that case the covariance matrix can be used to specify the accuracy.
Given a signal model of the radar and the target, there exist different lower bounds on the MSE or
the covariance matrix. These bounds depend only on the signal model and are independent of the
used estimator. They are usually valid for all estimators which satisfy certain conditions. Hence
they can be used to determine the maximal achievable accuracy of a radar system. This is a big
advantage compared to the investigation of the accuracy of a specific estimator. Moreover, for certain
bounds an analytical expression can be derived. This enables to optimize different parameters of
the radar system, e. g. the positions of the Tx and Rx antennas or the transmission times of the
pulses. It is important that the bound under investigation is tight, i. e. a certain estimator actually
achieves the bound. This can be verified by numerical simulations. If the bound is not tight, the
optimization could be useless. Some well known lower bounds are the Cramer-Rao bound (CRB)
[Cramer, 1945; Rao, 1945] and the Weiss Weinstein bound [Weiss and Weinstein, 1985]. A brief
overview of different lower bounds is presented in [Uhlich, 2012]. We focus on the CRB, since it is
easier to compute than many other bounds and analytical results are feasible in many cases. For an
introduction to the CRB, see [Kay, 1993].
The CRB has the following property: Under some regularity conditions on the likelihood function,
the CRB matrix CRBΘ is a lower bound for the covariance matrix of any unbiased estimator [Kay,
1993]
CΘˆ  CRBΘ, (2.15)
where  means that the difference CΘˆ − CRBΘ is a positive semidefinite matrix.
– 30 –
2.2.3. Resolution
The resolution describes the ability of a system to distinguish between two or more targets. Roughly
speaking, the resolution is the smallest possible distance between two objects such that they can be
distinguished, i. e. the system detects two and not only one target. Here the distance is the difference
of one or more physical quantities, depending on the considered system, e. g. the frequency, the
DOA or the geometrical position. Resolution is an important performance criterion, not only in radar
systems. In astronomy, the resolution of a telescope describes how far two point like objects (e. g.
stars) have to be separated in the angle in order to be resolved. There are many different definitions
for the resolution, see a survey in [den Dekker and van den Bos, 1997]. Some of them cannot be
justified thoroughly. One classical, often used definition for angular resolution or the resolution in
frequency is the Rayleigh criterion: Two targets can be resolved if they are separated such that the
first minimum of the main lobe of the first target is at the position of the maximum of the main
lobe of the second target [den Dekker and van den Bos, 1997]. This definition does not depend on
the signal-to-noise-ratio (SNR). But it is obvious that the ability to distinguish between two targets
clearly depends on the SNR of both targets, and therefore a meaningful definition of resolution
should incorporate this. In [Clark, 1995], the resolution is defined using ellipsoidal confidence
regions. But in this approach the cross correlation between the parameter estimates of different
targets is neglected.
We are interested in the angular resolution. A promising definition of the angular resolution is
presented in [Liu and Nehorai, 2007]. They study a signal model of two closely spaced sources.
To decide, if one or two targets are present, a binary hypothesis test is considered. They use the
generalized likelihood ratio test (GLRT). Since its distribution is difficult to obtain in general, its
asymptotic distribution is computed. The dependence of the GLRT on the separation distance of the
targets is used to define the resolution. The authors of [El Korso et al., 2011] extend this resolution
definition to multiple parameters. Below we present the definition for one parameter of interest ϕi
per target, i = 1, 2. The parameter of interest could be the DOA for example. Then, the resolution
δRL is defined implicitly as the solution of the following equation [El Korso et al., 2011]
δRL = η
√
CRBδ(δRL). (2.16)
Here, η is a factor which depends on the probability of false alarm PFA and probability of detection
PD
η = η(PFA, PD) (2.17)
and CRBδ is the CRB of the parameter δ = |∆ϕ| where ∆ϕ = ϕ2 − ϕ1 is the difference of the para-
meters of interest of the two targets. In general, CRBδ is a function of δ, i. e. CRBδ = CRBδ(δ).
Assuming η = 1, then the resolution δRL is the distance between the two targets at which the estima-
tion uncertainty
√
CRBδ is as large as the distance δRL, as depicted in Fig. 2.6. The factor η includes
– 31 –
the fact that the detection process depends on PFA and PD. Fig. 2.7 shows the dependence of η on
PFA and PD: For increasing PD and/or decreasing PFA, η increases. Fig. 2.6 shows that a larger η
leads to a larger δRL. We can interpret this in the following way: For a higher detection probability
PD with the same PFA, the targets must have a larger separation. This makes sense, since targets with
a larger separation can be detected easier. Also, for a smaller PFA the separation has to be larger,
such that fewer false detections happen. It is interesting to note that this resolution definition relates
detection to estimation theory: CRBδ is a result of estimation theory and equation (2.16) is derived
by means of detection theory using the asymptotic distribution of the GLRT.
δδRL
δ, η
√ C
R
B
δ
(δ
)
δ
η
√
CRBδ(δ)
Figure 2.6.: Implicit definition of the resolution δRL
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2
4
6
PD
η
PFA = 10−1
PFA = 10−2
PFA = 10−3
Figure 2.7.: η as a function of PFA, PD. For increasing PD and decreasing PFA, η increases. The
computation of η is described in [Liu and Nehorai, 2007; El Korso et al., 2011].

– 33 –
3. MIMO Radar Concepts and State of the
Art
In this chapter, we give an overview of the different aspects of a MIMO radar and their current state
of the art.
MIMO radars have several Tx and Rx antenna elements which can be used independently. Hence
every Tx antenna can transmit an arbitrary waveform and the signal received by a Rx antenna can
be recorded independently of the other Rx antennas. We denote the number of Tx antennas by NTx
and the number of Rx antennas by NRx. Thus there are NTx · NRx different antenna combinations
which are used in the signal processing. In general, each combination receives a different signal.
As a rough rule of thumb, the hardware complexity grows as NTx + NRx and the number of different
signals as NTx · NRx, which is a promising property of a MIMO radar. We present different types of
MIMO radars and their properties in Section 3.1. Different multiplexing techniques which can be
used in a MIMO radar are discussed in Section 3.2. The application of MIMO radars in automobiles
and the relation to our research is described in Section 3.3.
3.1. Types and Properties of MIMO Radars
MIMO radars can be divided into two different regimes: MIMO radars with widely separated anten-
nas [Haimovich et al., 2008] and MIMO radars with colocated antennas [Li and Stoica, 2007; Boyer,
2011]. The term widely separated antennas means that the geometrical distance between some or
all of the antennas is comparable or larger than the distance between an antenna and the target. An
example is depicted in Fig. 3.1a. Here we can see that in general the angle under which the target
appears is different for the different antennas. On the other hand, if the distance between all antennas
is much smaller than the target’s distance, the radar is called colocated MIMO radar. In such a radar
system, the target’s DOA is approximately the same for all antennas, cf. Fig. 3.1b. We give some
important aspects on MIMO radars with widely separated antennas and then focus on MIMO radars
with colocated antennas.
– 34 –
Txs,
Rxs
ϑ1
Txs,
Rxs
ϑ2
target
(a) MIMO radar with widely separated antennas,
ϑ1 , ϑ2
Txs,
target
Rxs
Txs,
Rxs
ϑ1 ϑ2
(b) colocated MIMO radar, ϑ1 = ϑ2
Figure 3.1.: Example of two different types of MIMO radars
3.1.1. MIMO Radar with Widely Separated Antennas
The Tx and/or Rx array elements are widely separated such that each Tx-Rx combination gives an
independent scattering response of the target [Li and Stoica, 2009, Ch. 2]. Therefore, this kind of
radar is also called statistical MIMO radar or, due to the different geometrical positions, multistatic
MIMO radar. A survey of this radar type is presented in [Haimovich et al., 2008]. In [Lehmann
et al., 2007] and [Fishler et al., 2004], a model is derived which states the conditions under which
independent scattering responses are observed. Statistical MIMO radars offer several advantages,
which we present briefly in the following.
Usually the radar cross section (RCS) of a target is strongly angle dependent [Skolnik, 2001]. Hence
it is possible that from a particular direction a target’s signal is weak, whereas from another direction
at the same distance to the target, it is much stronger. A MIMO radar with widely separated antennas
can mitigate these angle-dependent fluctuations of the radar cross section and thus offer a higher
resolution in target localization [Haimovich et al., 2008]. Moreover, this can also lead to a higher
accuracy in estimating the DOA [Lehmann et al., 2007; Fishler et al., 2004], and slowly moving
targets can be handled better by using Doppler estimates from different directions [Haimovich et al.,
2008]. The larger number of degrees of freedom of a MIMO radar compared to a SIMO radar can
result in a larger number of parameters which can be estimated uniquely [Haimovich et al., 2008].
The benefits of a such a MIMO radar come at the cost of a more complex hardware since the Tx
antennas have to be controlled independently. Moreover, one can use a system consisting of widely
separated Tx antennas and colocated Rx antennas. This allows coherent signal processing for DOA
estimation, but requires the phase of all Tx antenna elements to be kept synchronized despite the
– 35 –
Table 3.1.: Advantages and disadvantages of MIMO radars with widely separated antennas com-
pared to SIMO radars
advantages mitigation of RCS fluctuations
high resolution in target localization
better detection of slowly moving targets
estimation of a larger number of parameters
disadvantages phase synchronization over large distances
more complex hardware
more demanding signal processing algorithms
higher computational burden
large distance, or it has to be estimated as well [Fletcher and Robey, 2003]. Moreover, the signal
processing algorithms become more complex and in general the computational effort is higher.
The mentioned advantages and disadvantages of a MIMO radar with widely separated antennas
compared to a SIMO radar are summarized in Tab. 3.1.
3.1.2. Colocated MIMO Radar
In this kind of MIMO radar all antenna elements are colocated, i.e. their geometrical distance is
small compared to the distance to the target. Therefore, the target’s RCS is the same for every
Tx-Rx combination and the target appears under the same angle for every antenna. The signals
can be processed coherently. This is the reason why this kind of radar is sometimes also called
coherent MIMO radar. An overview on these radars can be found in [Li and Stoica, 2007] and
in the introduction part of [Boyer, 2011]. Several different aspects of colocated MIMO radars are
investigated in [Bekkerman and Tabrikian, 2006]. The properties of colocated MIMO radars and
important investigations, already done by different authors, are presented in the following.
Virtual Array First we present the signal model for DOA estimation of a SIMO radar. Afterwards
we introduce the signal model for a MIMO radar from which the concept of the virtual array follows.
We consider a SIMO radar with one Tx antenna and a linear array of NRx receiving antennas. All
antennas are assumed to have isotropic transmitting and receiving characteristics. The target is in
the far field, i. e. the antenna array receives a plane wave. This is depicted in Fig. 3.2. The target’s
DOA ϑ is measured perpendicular to the antenna array. The sign of ϑ is defined such that ϑ > 0 in
Fig. 3.2. The time delay between the receiving of the signal at the different Rx antennas results in
phase changes of the baseband signal. The DOA can be estimated by analyzing the phase relation
of the NRx receiving antennas. The positions of the Rx antennas normalized by λ2pi are given in
– 36 –
ϑ
Tx1 Rx1 Rx2 Rx3
position
Figure 3.2.: Antenna array of a SIMO radar. The blue arrow indicates the propagation direction of
the wave front.
dRx ∈ RNRx . Here, λ denotes the carrier wavelength. We divide by λ2pi because it allows a convenient
notation in the following. As an example, dRxi = pi means that the i-th Rx antenna is at position
pi · λ2pi = λ2 . Hence, dRx has the unit λ2pi . The complex signal xSIMO after pulse compression, i. e. after
Fourier transform and range and velocity estimation, can be modeled as [Krim and Viberg, 1996;
Schoor, 2010; Van Trees, 2002]
xSIMO = a
Rx(u) s + n ∈ CNRx . (3.1)
Here, s ∈ C is the complex signal strength of the target. It includes several quantities, e. g. the
radar cross section (RCS) of the target as well as the damping of the electromagnetic wave during
propagation. n ∈ CNRx denotes the measurement noise and aRx(u) is the so called steering vector
defined as
aRx(u) = exp( j dRxu) ∈ CNRx . (3.2)
We use j as the symbol for the imaginary unit, i. e. j2 = −1. For any vector x ∈ CK , the ex-
pression exp(x) is to be understood as an element-by-element operation, i. e. y = exp(x) means
yi = exp(xi)∀i = 1, . . . ,K. The electrical angle u is defined as
u = sin(ϑ). (3.3)
The steering vector aRx(u) describes the phase change of the Rx antennas due to their different
positions. The signal model (3.1) requires that the narrow band assumption for an array is fulfilled
[Krim and Viberg, 1996]: The geometrical size of the Rx array ∆dRx, the bandwidth F of the signal
and the propagation speed c have to satisfy ∆dRxF  c. For an automotive radar, ∆dRx ≤ 50 mm
and c = 3 · 108 ms is the speed of light. Hence F has to satisfy F  6 GHz. This condition is fulfilled
for usual automotive radars.
Now we present the signal model for a MIMO radar with NTx Tx and NRx Rx antennas and a station-
ary target. An example of the MIMO radar is depicted in Fig. 3.3. Let every Tx antenna transmit
once. Similar to the SIMO radar we denote the complex signal after pulse compression gained by
– 37 –
Tx1 Rx1 Rx2 Rx3
ϑ
position
Tx2
Figure 3.3.: Antenna array of a MIMO radar. The blue arrow indicates the propagation direction of
the wave front.
the transmitted signal of the i-th Tx antenna by xTx iSIMO ∈ CNRx . Since every Tx antenna transmits
once, we have NTx complex signals xTx iSIMO, i = 1, . . . , NTx. We stack these vectors to define the new
vector
xMIMO =

xTx1SIMO
...
xTxNTxSIMO
 ∈ CNTx·NRx . (3.4)
In addition to the phase change due to the different Rx antenna positions, we get a phase change due
to the different Tx antenna positions as well. Therefore, we introduce the steering vector of the Tx
array
aTx(u) = exp( j dTx u) ∈ CNTx , (3.5)
where dTx ∈ RNTx contains the positions of the Tx antennas in unit of λ2pi . Then, the complex signal
xMIMO after pulse compression is given as [Chen, 2009] [Li and Stoica, 2009, Ch. 2]
xMIMO = a
Tx(u) ⊗ aRx(u) s + n ∈ CNTx·NRx . (3.6)
Here, ⊗ is the Kronecker tensor product or outer vector product. For vectors y ∈ CK , z ∈ CM it is
defined as
x = y ⊗ z =

y1 · z
...
yK · z
 ∈ CK·M. (3.7)
We define the new steering vector aVirt(u) = aTx(u) ⊗ aRx(u) which contains all Tx-Rx combinations.
– 38 –
It can be written as
aVirt(u) = aTx(u) ⊗ aRx(u)
= exp( j dTx u) ⊗ exp( j dRx u)
= exp( j dVirt u) ∈ CNVirt , (3.8)
NVirt = NTx · NRx, (3.9)
with
dVirt = dTx ⊗ 1NRx + 1NTx ⊗ dRx ∈ RNVirt . (3.10)
Here, 1K ∈ RK denotes a vector with all elements equal 1. dVirt are the positions of a new array,
called virtual array, and aVirt(u) is the corresponding steering vector. Using (3.8) we can write
xMIMO = a
Virt(u) s + n ∈ CNVirt . (3.11)
Let us interpret this: The signal of the MIMO radar xMIMO (3.11) can be written in the same way as
that of the SIMO radar xSIMO (3.1) with the Rx array replaced by the virtual array. Thus the MIMO
radar with NTx Tx and NRx Rx antennas is mathematical equivalent to a SIMO radar with 1 Tx and
NVirt = NTx · NRx Rx antennas. An example of a virtual array is depicted in Fig. 3.4. Descriptively
speaking, the virtual array can be constructed by taking the Rx array and putting it at the position of
the first Tx antenna, at the position of the second Tx antenna and so on. Mathematically, this is the
convolution between the Tx and Rx array. The concept of the virtual array can be easily generalized
to two-dimensional arrays, i. e. planar arrays [Chen, 2009]. An example is depicted in Fig. 3.5.
The virtual array of the MIMO radar has a larger aperture than the receiving or transmitting array
alone. This yields a narrower beam for DOA estimation which results in a higher angular accuracy
and a higher angular resolution. That is one important benefit of a colocated MIMO radar. The
geometric positions of the Tx and Rx antenna elements can be optimized in several ways. For
Tx array Rx arry
Rx1 Rx2 Rx3Tx2Tx1
Virtual array
Tx Rx1 Rx2 Rx3 Rx1 Rx2 Rx3
∆dTx ∆dRx
∆dTx + ∆dRx
Figure 3.4.: Example of the construction of a linear virtual array
– 39 –
x
y
Tx1
Tx3
Tx2
x
y
x
y
Rx1
Rx2
Rx3
Tx array Rx array Virtual array
Figure 3.5.: Example of the construction of a planar virtual array
example, they can be optimized to result in a virtual array with a) a large aperture and b) low sidelobe
levels. Some considerations and examples for such optimizations are given in [Li and Stoica, 2009,
Ch. 2]. Moreover, it is possible to construct a virtual array, in which the spacing between two antenna
elements is smaller than the real geometrical spacing. Hence sparse Tx or Rx arrays can be used to
decrease the mutual coupling between antenna elements, and therefore increase the performance of
the radar system [Feger et al., 2009]. Moreover, the antenna gain can be increased by using larger
antenna patches, without violating the sampling theorem, which would result in ambiguities in angle
estimation (grating lobes) [Feger et al., 2008, 2009].
In addition, a MIMO radar has a larger number of received signals compared to a SIMO array,
NTx ·NRx for a MIMO and NRx for a SIMO array. This leads to a larger number of targets that can be
detected uniquely [Bekkerman and Tabrikian, 2006; Li and Stoica, 2007].
General Comparison of SIMO and MIMO Radars In order to compare the performance of a
SIMO and MIMO radar, it is important to know under which conditions the radars receive the same
signals or the information in the signals is exactly the same. The author of [Friedlander, 2009]
shows which requirements have to be fulfilled such that a MIMO radar and an appropriate SIMO
radar receive the same information. In these cases, the theoretical knowledge, e.g. performance
bounds and algorithms, can be easily transferred from a SIMO to a MIMO radar. In [Vaidyanathan
and Pal, 2009], the signal-to-noise (SNR), signal-to-clutter (SCR), and signal-to-interference ratios
(SIR) are compared for SIMO and MIMO radars and a third radar system, called interpolated finite
impulse response (IFIR) radar.
DOA Estimation Accuracy and Design of Transmit Beampatterns To investigate the achievable
DOA accuracy, the Cramer Rao Bound (CRB) has been computed for different settings as presented
in Chapter 1.
The Weiss-Weinstein bound is a lower bound on the MSE in parameter estimation [Weiss and Wein-
stein, 1985]. It requires no bias assumption, like the CRB, and can be used to analyze the threshold
point of estimators. It is tighter than the CRB and the Bobrovsky-Zakai bound, but more complicated
to compute. In [Tran et al., 2012], the Weiss-Weinstein bound is applied to colocated MIMO radars
– 40 –
to predict the SNR threshold point. Since the computation is difficult, an analytical approximation
is derived. This bound can be used to help the design of the array geometry.
In a MIMO radar, Tx antennas can be used which transmit an isotropic or a very broad beam, because
the beam forming of the Tx antennas is done digitally in the signal processing. Hence the coverage,
i. e. the area in which targets can be detected, is larger compared to a SIMO radar [Bekkerman and
Tabrikian, 2006]. Moreover, the spatial transmitted peak power density can be smaller, which is
important in radar systems which should not be detected, e. g. in military applications. On the other
hand, the larger number of transmit antennas of a MIMO radar offers a larger number of degrees
of freedom to realize a given transmit beampattern [Li and Stoica, 2007]. This property can be
used for adaptive transmit beampattern design: First, the target’s DOA is estimated roughly. Then
a transmit beampattern is formed which steers a certain beam in the direction of the target. Thus
the target’s DOA can be estimated more precisely. The disadvantage of such a beam is a smaller
coverage area. In [Forsythe and Bliss, 2005; Li et al., 2008], the CRB for DOA estimation of several
stationary targets is computed as a function of the covariance matrix of the transmitted signals. Then
the transmitted signals are optimized in order to achieve a small CRB in a certain sense under the
constraint of a constant total transmitting power. The optimal signals depend on the number and
positions of the Tx and Rx antenna elements as well as on the optimization criterion. For the one
target case, the solutions are a sum beam which steers the transmit power in the direction of the
target, a difference beam which places a null in the direction of the target, or orthogonal signals.
There is a trade-off between the sum beam and orthogonal signals, since the sum beam provides a
higher SNR and the orthogonal signals lead to a larger virtual aperture. It depends on the geometrical
arrangement of the antenna positions which beam is better. For the two target case, the optimization
is even more complicated, but the results suggest that the waveform diversity of a MIMO radar is
even more important in the two target case than in the one target case. In [Li and Stoica, 2009, Ch.
2], the one target case is examined further. The authors show that mixing a difference beam with
orthogonal signals leads to a good performance in DOA estimation.
Friedlander shows in [Friedlander, 2012] the effects of a mismatch between the signal model and the
reality in MIMO radar. The numerical studies suggest that, especially for null steering, i.e. placing a
null at the target’s DOA, model mismatch can be crucial. Beamforming seems not to be so sensitive
to model errors.
The authors of [Hassanien and Vorobyov, 2010a,b] study the DOA estimation performance of a so
called phased MIMO radar. The MIMO radar has the benefit of a large virtual aperture, but a lower
SNR than the SIMO radar, since the latter uses transmit beamforming to focus more transmitted
power on the target. The phased MIMO radar is a mix of a SIMO and MIMO radar. The transmit
array is partitioned into several subarrays. The transmitters belonging to the same subarray transmit
the same waveform coherently as in a SIMO radar. This results in a transmit beam gain. The
waveforms of the subarrays are orthogonal to each other, a typical property of a MIMO radar. Since
a transmitting antenna can belong to several subarrays, the transmitted waveforms become more
– 41 –
complex. This idea corresponds to transforming the transmit element space to the transmit beam
space [Hassanien and Vorobyov, 2010b]. For the comparison of these different radar systems, the
authors investigate especially the different beampatterns and SNRs. If the target’s DOA is known
approximately a priori, a coherent processing gain can be achieved by steering the transmit beam
in this direction. In automotive applications, the location of several targets is estimated with one
measurement. In general, they have different DOAs. Hence, in these applications it is not possible
to steer the transmit beam in one certain direction in order to achieve a processing gain for the whole
estimation process.
If we want to transmit a given spatial beampattern using a MIMO radar, we must be able to com-
pute the signals to transmit which yield the desired beampattern. For that purpose, the authors of
[Fuhrmann and San Antonio, 2004] derive an algorithm to compute the cross-correlation matrix of
the transmitted signals such that the true beampattern achieves a given spatial beampattern. More-
over, they present in [Fuhrmann and San Antonio, 2008] an algorithm to compute the code sequences
of coded binary phase shift keyed (BPSK) waveforms which yield the desired cross-correlation ma-
trix.
Ambiguity Function With the help of the CRB, local properties of estimators like the DOA ac-
curacy can be investigated. In order to examine not only local but also global properties of a radar
system, the ambiguity function was invented, which is contributed to Ville and Woodward [Li and
Stoica, 2009, Ch. 3]. It enables to examine the accuracy, resolution as well as ambiguities of a radar
system. A new ambiguity function for more general parameter estimation problems was derived
by the authors of [Rendas and Moura, 1998]. Woodward’s ambiguity function is a special case of
this one for narrowband radar systems. An ambiguity function specially for MIMO radar systems is
derived in [Li and Stoica, 2009, Ch. 3] and [San Antonio et al., 2007]. The authors show how the
ambiguity function depends on the covariance matrix of the transmitted waveforms. Hence it can be
used as a tool to optimize the transmitted signals to achieve a high accuracy, a good resolution and
few ambiguities.
Space Time Adaptive Processing and Ground Moving Target Indication Space Time Adaptive
Processing (STAP) tries to maximize the signal-to-interference-and-noise ratio (SINR) by linearly
combining the received signals [Brennan and Reed, 1973; Melvin, 2004]. It can be used to suppress
clutter and interference and is applied e.g. in airborne radar systems for ground moving target
indication (GMTI). The application of STAP in MIMO radars is an interesting topic and active
research area due to the large number of degrees of freedom [Chen, 2009]. The CRB for STAP is
derived by the authors of [Chong et al., 2011], and multipath clutter mitigation is investigated in
[Mecca et al., 2006]. Different multiplexing schemes in slow-time for moving targets are compared
in the context of GMTI in [Xue et al., 2009, 2010], where both the angle and Doppler frequency
– 42 –
Table 3.2.: Advantages and disadvantages of colocated MIMO radars compared to SIMO radars
advantages larger virtual array
higher angular accuracy
higher angular resolution
lower sidelobe levels in DOA estimation
larger coverage area
smaller spatial transmitted peak power density
use of sparse real arrays which result in a non-sparse virtual array
larger number of detectable targets
flexible transmit beampattern design
larger degrees of freedom especially for STAP
disadvantages more complex hardware
SNR loss compared to a focused transmit beam
more demanding signal processing algorithms
higher computational burden
more complicated calibration of the Tx antennas
are estimated. Furthermore, [Li and Stoica, 2007] reports a higher detection rate for slowly moving
targets.
General Disadvantages of MIMO Radars Many advantages and disadvantages depend on the
precise implementation and application of the MIMO radar. But in general, MIMO radars need a
more demanding and complicated hardware compared to SIMO radars [Bekkerman and Tabrikian,
2006]. Moreover, the signal models are more involved which also leads to more demanding signal
processing algorithms. This results in a higher computational burden. The use of several colocated
Tx antennas can lead to electromagnetic coupling problems [Feger et al., 2009]. This also reduces the
power efficiency and impacts the decoding and signal processing. Moreover, it makes the calibration
of the Tx antennas more complicated.
The mentioned advantages and drawbacks of colocated MIMO radars compared to conventional
SIMO radars are summarized in Tab. 3.2. The relation of the presented state of the art of MIMO
radars to our work is discussed in Section 3.3.
3.2. Multiplexing Techniques
In order to use all Tx-Rx combinations in the signal processing, one has to distinguish which part
of the received signal corresponds to which Tx antenna. There are several multiplexing techniques
which can be used for that purpose. If the transmitted signals are mutually orthogonal, then in
– 43 –
principle every portion of the received signal can be assigned perfectly to one Tx antenna. But it is
also possible that some transmitters send the same signal, but phase shifted, which can be used for
beamforming. We discuss several different multiplexing techniques in the following.
3.2.1. Time Division Multiplexing
To distinguish the transmitted signals at the receiver side, we can just activate only one transmitter
at one time instant. The next transmitter starts sending just after the one before has stopped, or more
precisely, after a delay of the maximum round trip time of the signal. This is called time division
multiplexing (TDM) or time division multiple access (TDMA). The sequence of the transmitters can
be chosen deterministic or randomly, the latter named randomized time division multiple access (R-
TDMA) [Xue et al., 2009]. TDM has low hardware requirements, because always only one antenna
transmits.
TDM allows to use conventional radar waveforms, e. g. chirps, which have already been optimized
for range and Doppler estimation. Since only one transmitter is active at one time instant, the max-
imum transmission power is limited by the maximum power of one antenna. One drawback of this
multiplexing technique arises, if the observed targets are non-stationary: Due to the motion of the
target, the received baseband signal is shifted by the corresponding Doppler frequency. This results
in a phase change between successive pulses which has to be estimated or compensated in the fol-
lowing signal processing, see Section 5.1 for more details. This is especially important in direction
of arrival (DOA) estimation, which relies on the phase relation of the signals. In other words, every
received signal originated from a slightly different environment. If all signals shall be processed
coherently, this has to be accounted for. We will discuss and investigate this in detail in this thesis.
3.2.2. Frequency Division Multiplexing
In frequency division multiplexing (FDM), different carrier frequencies are used for every transmitter
while the transmitted baseband signal stays the same for all transmitters. Each received signal is
mixed down with all used carrier frequencies, which results in NTx baseband signals. Obviously,
opposed to the TDM approach, all transmitters can send at the same time and the overall power
is not limited by the maximum power of one transmitter. In exchange, the hardware requirements
are higher. Moreover, for a fixed total bandwidth, the bandwidth for each Tx antenna is reduced.
Another disadvantage is the different phase of the received baseband signals: Due to the distinctive
carrier frequencies, the carrier wavelengths are not the same. Given a fixed distance to the target,
the ratio of the distance to the wavelength differs, resulting in a different phase of the received
baseband signal. Hence this phase has to be estimated or compensated in the following processing
steps, which is especially important in DOA estimation. The coupling of the range and angle can
also be visualized by the range-angle ambiguity function, cf. [Sun et al., 2014]. Doppler division
– 44 –
multiple access is another approach which can be considered as a special kind of FDM. In that case,
the center frequencies of the different Tx antennas are only slightly different, such that they can be
distinguished in the Doppler domain [Xue et al., 2009; Sun et al., 2014].
3.2.3. Code Division Multiplexing
Code Division Multiplexing (CDM) is another approach. The transmitted signals are modulated by
a unique code for every transmitter such that they can be distinguished on receiving. There are two
types of CDM, called CDM in fast time and CDM in slow time.
Consider a MIMO radar in which every transmitter sends several pulses with a unique shape of the
baseband signal. The number of pulses per transmitter is denoted by NPulse. If the baseband signals
of all pulses, sent at the same time instance, are orthogonal to each other, then we can distinguish
the signals at the receiver side in every pulse separately. This is called multiplexing in fast time [Li
and Stoica, 2009, Ch. 7], [Xue et al., 2009], since the modulation happens within one pulse. On the
other hand, modulations which extend over all NTx · NPulse pulses are called CDM in slow time.
3.2.3.1. CDM in Fast Time
Using orthogonal pulses in fast time has the advantage of distinguishing the transmitters in every
received pulse by using a matched filter [Haimovich et al., 2008], and the subsequent signal pro-
cessing is easier. For good separation of the signals, the pulses of the different transmitters have
to have a low cross correlation. On the other hand, a high autocorrelation with low range sidelobe
levels is required for a good range estimation and range compression. Hence a trade-off between
both properties has to be achieved [Xue et al., 2009]. High range sidelobes lead to a coupling of
the range and DOA. An example of a simulated range angle ambiguity function is given in [Sun
et al., 2014]. MIMO radars using modulation in fast time have been investigated e.g. in [Fletcher
and Robey, 2003; Robey et al., 2004]. In general, the hardware is much more demanding than in a
SIMO radar due to the generation of NTx orthogonal waveforms at the same time.
3.2.3.2. CDM in Slow Time
For each Tx antenna, the sequence of NPulse pulses is modulated by a unique code. Typically this is
achieved by a change of the phase of every pulse. Usually, these phase codes are orthogonal. After
receiving all NTx · NPulse pulses from all transmitters, the parts originating from a certain transmitter
can be reconstructed. Often the relative velocity of the target is estimated by the phase change of
successive pulses, e.g. by using a chirp sequence radar. Moreover, the DOA estimation also uses the
phase information of the baseband signals. Hence, when encoding of the signals, the Doppler as well
as the DOA estimation have to be done at once. This results in a high computational load. A typical
– 45 –
example is the application of a MIMO radar for ground moving target indication (GMTI), where
an angle Doppler map is constructed [Xue et al., 2009, 2010]. Code multiplexing can be combined
with transmit beamforming to focus the transmitted energy into a certain angular region. This is
investigated for the DOA estimation of multiple stationary targets in [Li et al., 2008]. Compared to
CDM in fast time, the signals of the pulses, which are transmitted at the same time instance, do not
have to have a low cross correlation. Therefore, they can achieve better autocorrelation properties.
In addition, the bandwidth requirements are lower and, opposed to TDM, the transmission power is
not limited by the transmission power of one antenna, since all antenna transmit at the same time.
3.2.4. Summary
Table 3.3.: Comparison of different multiplexing techniques. The symbolmeans ’yes’ and  means
’no’.
TDM FDM CDM CDM
fast time slow time
low hardware requirements     
total transmission power limited by     
transmission power of one antenna
coupling of Doppler frequency and DOA    
coupling of distance and DOA     
larger bandwidth requirements     
distinguishing of the Txs in every pulse     
good autocorrelation properties of Tx signals    
low computational load for encoding     
The advantages and drawbacks of the different multiplexing techniques are summarized in Tab. 3.3.
3.3. MIMO Radars for Automotive Applications and Relation to
our Work
Radar systems in automotive applications have to be produced at low cost. The hardware require-
ments for a TDM MIMO radar are less complex than for other multiplexing techniques. Therefore,
TDM is a promising approach. For radars in automobiles, the total transmission power is limited by
law. Usually this power limit is already achieved by only one Tx antenna. Hence, the disadvantage
of TDM that the total transmission power is limited by the transmission power of one antenna is not
crucial for automotive applications. Moreover, the transmission signals which are already optimized
for the distance and velocity estimation in SIMO radars can still be used in a TDM MIMO radar.
Due to these reasons, we investigate TDM MIMO radars in this thesis.
– 46 –
Proposed Implementations of MIMO Radars and Experimental Investigations The authors
of [Zwanetski and Rohling, 2012] suggest a MIMO radar using TDM with signals optimized for
automotive radars. The distance, relative velocity and DOA are estimated by evaluating the received
signals by a three dimensional discrete Fourier transform (DFT). In [Wintermantel, 2010], a radar
system is proposed in which the Tx antennas transmit chirp sequences with many chirps of short
duration, e.g. 512 chirps with a duration of 10 microseconds. The Tx antennas transmit sequentially
or all at the same time, but with a different phase modulation. The distance, velocity and DOA
are estimated by a three dimensional DFT. Due to special phase modulation, the transmitters can
be distinguished by using the DFT. The proposed system works using a chirp sequence of many
chirps. It cannot be used with a few linear frequency modulated continuous wave (FMCW) chirps.
Moreover, the hardware is more complex when using a phase modulation instead of TDM.
In [Feger et al., 2009; Jahn et al., 2012; Schmid et al., 2012; Guetlein et al., 2012, 2013a], experimen-
tal investigations of TDM MIMO radars are reported. In a TDM MIMO radar with non-stationary
targets, the phase shift of the baseband signal due to the Doppler effect has to be accounted for in the
DOA estimation as well, cf. Section 3.2.1. The authors of [Feger et al., 2009; Schmid et al., 2012]
propose to use a MIMO radar with receiving and transmitting antenna positions such that at least two
antenna elements of the virtual array have the same geometrical position (redundant position). Then,
for a non-stationary target, the unknown Doppler phase of the baseband signal can be estimated by
computing the phase difference of the antenna elements at the redundant positions. This Doppler
phase estimation is then used in the angle estimation algorithm. Since only the antenna elements at
the redundant positions are used to estimate the unknown Doppler phase, the noise at these antenna
elements has a larger influence on the DOA estimation than the noise at the other antenna elements,
which is a drawback of this approach. In [Guetlein et al., 2012] and [Guetlein et al., 2013a], two
different switching sequences are proposed in order to estimate the Doppler phase of the baseband
signal and experimental results are presented. In [Guetlein et al., 2013b], it is shown that one of the
switching sequences might be difficult to realize exactly in practice. In the mentioned experiments,
quantitative comparisons to different radar systems in the non-stationary case, especially in DOA
estimation, are still missing.
We fill this gap by investigating the DOA estimation using SIMO and TDM MIMO radars in the
following chapters. First we consider stationary targets and after that we extend the studies to non-
stationary targets. In doing so, we compare different radar types quantitatively.
– 47 –
4. DOA Estimation of Stationary Targets
We study the DOA performance of different radar systems. There are different performance mea-
sures of a radar system, e. g. the accuracy, the resolution and sidelobe levels, and the robustness, cf.
Section 2.2. A high accuracy of the DOA estimation is important in many applications. Consider an
automotive radar system: In order to decide if an object is on the right or the left lane of the street,
the DOA has to be more accurate, the larger the distance to the object. We investigate the maximum
achievable DOA accuracy in terms of the CRB. Since the CRB is a lower bound on the covariance
matrix of any unbiased estimator, the investigations based on the CRB are independent of the used
estimation algorithm. It enables to compare the maximum accuracy of different radar systems.
We give an overview of the considered radar system in Section 4.1. In Section 4.2, we introduce
the signal model for the SIMO radar and compute the CRB of the DOA. In Section 4.3, we consider
the TDM MIMO radar for a stationary target, i.e. the target is not moving relative to the radar. We
present the signal model, compute the CRB and finally compare it to the CRB of a SIMO radar
in Section 4.4. In Section 4.5, we compare the performance of a MIMO radar using code division
multiplexing (CDM) with orthogonal codes and TDM. We compare the CRBs of both systems and
show, using the concept of sufficient statistics, under which conditions the received signals of both
systems contain the same information. Section 4.5 is based on [Rambach and Yang, 2016b].1 The
results are summarized in Section 4.6.
4.1. Radar System
We consider a radar system consisting of a linear array with NRx receiving (Rx) and NTx transmitting
(Tx) colocated, isotropic antennas. An example of such a radar system is depicted in Fig. 4.1. The
positions of the NRx Rx antennas and NTx Tx antennas are given in dRx ∈ RNRx and dTx ∈ RNTx ,
respectively, in the unit of λ2pi where λ is the carrier wavelength. For example d
Rx
i = pi means that
the position of the i-th Rx antenna is pi · λ2pi = λ2 . The target is modeled as a point source and the
transmitted signal is narrowband. The target is in the far field. The DOA of the target, measured
perpendicular to the linear array, is denoted by ϑ, as depicted in Fig. 3.3 on page 37. The parameters
of the SIMO and MIMO radar under consideration are shown in Tab. 4.1. In the SIMO case, only one
Tx antenna transmits (Fig. 4.2a) or the Tx antennas transmit the same signal up to a complex scaling
1This chapter contains quotations of our own work [Rambach and Yang, 2016b] which are not marked additionally.
– 48 –
Tx2Rx1 Rx2 Rx3Tx1
position
[
λ
2 pi
]
0 pi 2.5 pi 3.5 pi 5 pi
Figure 4.1.: A radar system with NTx = 2 Tx and NRx = 3 Rx antennas at positions dTx = pi [0, 5]T ,
dRx = pi [1, 2.5, 3.5]T
Table 4.1.: Overview of the parameters of the SIMO and MIMO radar
SIMO TDM MIMO
Rx antennas NRx NRx
Tx antennas
NTx, transmitting the same NTxwaveform up to a phase shift
total Tx power of the whole radar system p p
time of one measurement cycle Tcycle Tcycle
number of FMCW chirps per cycle 1 NPulse
chirp duration Tcycle Tcycle/NPulse
number of samples per chirp Nsamples Nsamples/NPulse
number of measurement cycles L L
total measurement time Ttotal L Tcycle L Tcycle
factor, i. e. with a different amplitude and phase shift, in order to focus the transmitted energy to a
certain angular region (Fig. 4.2b). The total transmission power of the whole radar is p. Hence, if
only one Tx antenna transmits, its transmission power is p. If several Tx antennas transmit, the total
transmission power p has to be distributed among them. In the TDM MIMO system, the antennas
transmit one at a time, each with transmission power p (Fig. 4.2c). Thus, the SIMO and MIMO
radar use the same total transmission power. A measurement cycle consists of one or several chirps
and the signals during one cycle are processed coherently, i. e. one measurement cycle is a coherent
processing interval (CPI). The SIMO radar transmits one chirp per cycle and the MIMO radar NPulse
chirps of equal duration per cycle. We assume that the time gap between adjacent chirps is negligible.
The duration of one cycle is given by Tcycle, hence the duration of one chirp of the MIMO radar equals
Tcycle/NPulse. This ensures that both radar systems transmit the same total energy, which is crucial for
a fair comparison. One measurement cycle for the SIMO and MIMO radar is depicted in Fig. 4.3.
The radar systems transmits L measurement cycles. Hence the total transmission time Ttotal is
Ttotal = L Tcycle. (4.1)
We assume that the sampling rate is the same for both radar systems. The SIMO system takes Nsamples
of one chirp, and the MIMO radar Nsamples/NPulse samples per chirp for all NPulse chirps.
– 49 –
(a) SIMO radar with
one Tx antenna
(b) SIMO radar with several Tx anten-
nas
(c) TDM MIMO radar with succes-
sively transmitting antennas
Figure 4.2.: Different configurations of the SIMO and MIMO radars. The total Tx power p is the
same for all configurations.
t
fTx(t)
Tcycle
(a) SIMO radar
t
fTx(t)
Tcycle
(b) MIMO radar with NPulse = 4
Figure 4.3.: Example of the transmitted chirps of a SIMO and MIMO radar
4.2. SIMO Radar
We investigate the accuracy of the DOA estimation using a SIMO radar. We first present the signal
model and then derive the CRB.
4.2.1. Signal Model
The signal model of the SIMO radar is similar to the model (3.1) introduced in Section 3.1.2 in the
context of the virtual array. It is given by
x(l) = B(u) aRx(u) sSIMO(l) + n(l), l = 1, . . . L (4.2)
where x(l) is the signal after range processing. l denotes the cycle. Here
u = sin(ϑ) (4.3)
is the electrical angle with ϑ being the DOA of the target. B(u) is the transmit beampattern. For a
SIMO radar with one isotropic Tx antenna, depicted in Fig. 4.2a, B(u) = 1 ∀u. If several Tx antennas
– 50 –
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
1
2
u
|B(
u)
|
Figure 4.4.: Beampattern of an ULA of 5 isotropic Tx antennas with λ/2 spacing. All Tx antennas
transmit the same signal without a phase shift. The main beam is directed to u = 0.
transmit the same signal up to a phase shift, cf. Fig. 4.2b, B(u) changes depending on the position
and the phase shifts of the Tx antennas. The phase shifts can be chosen such that the transmitted
energy of the Tx antennas is directed to a certain DOA. An example of a transmit beampattern is
depicted in Fig. 4.4. Note that B(u) is normalized such that the total transmission power p remains
constant. aRx(u) is the steering vector with
aRx(u) = exp( j dRxu). (4.4)
sSIMO(l) is the complex signal strength and n(l) the additional noise. Defining
g(u) = B(u) aRx(u), (4.5)
the model reads
x(l) = g(u) sSIMO(l) + n(l). (4.6)
We make the following assumptions:
• n(l) is circular complex Gaussian with zero mean, spatially and temporally uncorrelated with
E
(
n(l) nH(m)
)
= δl,m σ
2 I. Here, I is the identity matrix and σ2 the variance of the noise. This
assumption can be justified in the following way: Usually, the baseband signal in the time
domain is processed by a fast Fourier transform (FFT). The FFT transformed signal is used
for the further signal processing steps. If the number of samples in the time domain baseband
signal is large enough, under some weak assumptions, the FFT transformed signal of the noise
has the following properties [Böhme, 1998]
– the real and imaginary part of the FFT signal are uncorrelated,
– both parts are Gaussian distributed with the same variance,
– the FFT signals at different frequencies are uncorrelated.
• Let τprop be the propagation time of the signal across the array and BW the bandwidth of the
– 51 –
signal. We assume that τprop  1BW , known as the narrowband array assumption [Dogandzic
and Nehorai, 2001].
• The target’s distance to the MIMO radar is much larger than the aperture of the radar such that
we have a far field situation. Hence the radar receives a plane wave and the radar cross section
as well as the DOA ϑ of the target and therefore also the electrical angle u are the same for all
antennas.
The unknown deterministic quantities to be estimated from x(l), 1 ≤ l ≤ L, are denoted by
Θ =
[
u, sSIMO(1), . . . , sSIMO(L), σ2
]T
. (4.7)
Note that if the SIMO radar transmits with a higher transmission power p, the signal strengths
|sSIMO(l)| are larger.
4.2.2. Cramer-Rao Bound
In the following, we derive the deterministic CRB for the electrical angle and discuss the result.
4.2.2.1. Derivation
In our model, the unknown parameter vector is given in (4.7). We want to derive that part of the
deterministic CRB of Θ which corresponds to the electrical angle u. Hence we have to compute the
Fisher information matrix (FIM) J of the whole parameter vector Θ and then its inverse J−1. After
that we take the entry of J−1 corresponding to u. This part is denoted by CRBu,SIMO. Following [Yau
and Bresler, 1992]
CRB−1u,SIMO = 2 L
σ2s,SIMO
σ2
Re(C) (4.8)
where
σ2s,SIMO =
1
L
L∑
l=1
|sSIMO(l)|2, (4.9)
C = DHP⊥g D, (4.10)
D =
∂g(u)
∂u
, (4.11)
P⊥g = I − g
(
gHg
)−1
gH. (4.12)
– 52 –
P⊥g is the projection matrix on the subspace which is orthogonal to the subspace spanned by g. After
some computations, this results in (cf. Appendix B)
CRBu,SIMO =
1
2 L
1
S SIMO
1
USIMO
(4.13)
with
S SIMO = SNRSIMO · NRx · |B(u)|2, (4.14)
SNRSIMO =
σ2s,SIMO
σ2
, (4.15)
USIMO = VarS(dRx). (4.16)
Here, VarS(dRx) is the sample variance of dRx. It is defined in (A.10) in Appendix A. There, some
important properties are derived as well. It can be written as
VarS(dRx) =
1
NRx
NRx∑
i=1
(
dRxi − ES(dRx)
)2
, (4.17)
with ES(dRx) =
1
NRx
1T dRx. (4.18)
Roughly speaking, VarS(dRx) quantifies how far the Rx antenna positions are distributed around the
mean value ES(dRx). SNRSIMO denotes the SNR at each Rx antenna. S SIMO is the total SNR of the
radar including the Rx antenna gain NRx and the Tx beam characteristics |B(u)|2. The term USIMO is
the sample variance of the Rx antenna positions.
The CRB CRBϑ,SIMO for the DOA ϑ can be easily derived from (4.13) by the rules of transformation
of parameters [Kay, 1993]. Since
∂u(ϑ)
∂ϑ
= cos(ϑ) (4.19)
CRBϑ,SIMO is given by
CRBϑ,SIMO =
1
cos(ϑ)2
CRBu,SIMO
=
1
2 L cos(ϑ)2
1
S SIMO
1
USIMO
. (4.20)
4.2.2.2. Discussion
The smaller the CRB, the more accurate the electrical angle can be estimated. According to (4.13)
and (4.14), the larger the SNR
σ2s,SIMO
σ2
, the smaller CRBu,SIMO.
Moreover, (4.14) shows that the transmit beampattern B is equivalent to an SNR gain. Only the
– 53 –
magnitude of B(u) at the target’s electrical angle u is important, not the shape or derivative of B(u).
In a SIMO radar, all Tx antennas transmit the same waveform up to a phase shift. Hence, the
maximal gain can be achieved, if the Tx beam is steered at the target’s true angle. This is known
as the beamforming gain. To be able to do that, the true angle has to be known approximately a
priori. The maximum value of |B| in the case of NTx isotropic Tx antennas and a constant total
transmit power is Bmax =
√
NTx, cf. Appendix C. Thus, the maximal achievable SNR gain in (4.14)
is B2max = NTx. If one Tx antenna is used, i. e. B(u) = 1, (4.13) simplifies to the expression derived in
[Stoica and Nehorai, 1989] and given in a more compact form in [Athley, 2005].
The influence of the position of the Rx antennas on the CRB is expressed by USIMO = VarS(dRx). The
larger VarS(dRx), the smaller CRBu,SIMO. This can be interpreted easily: Consider a radar with 2 Rx
antennas. The larger the distance between both antennas, the larger the phase change of the baseband
signal due to a change of the electrical angle. Hence, the radar is more sensitive to a change of the
electrical angle, which results in a higher accuracy. This is reflected in (4.13) and (4.16): The larger
the distance between both antennas, the larger VarS(dRx) and the smaller CRBu,SIMO.
4.3. MIMO Radar
Now we investigate the DOA estimation using a TDM MIMO radar. We present the signal model
and derive the CRB for the electrical angle.
4.3.1. Signal Model
The MIMO radar consists of a linear array of NTx Tx and NRx Rx isotropic, colocated antennas. The
positions of the Tx antennas in the sequence in which they transmit are given by dPulse ∈ RNPulse in
the unit of λ2pi . d
Pulse = [dTx1 , d
Tx
2 , d
Tx
1 ]
T means, for example, that the first antenna transmits, then the
second one, and again the first one, see Fig. 4.5. Thus, if an antenna transmits more than once, its
position occurs several times in dPulse. Similar to Section 3.1.2, the signal model for the MIMO radar
is given by
x(l) = aPulse(u) ⊗ aRx(u) sMIMO(l) + n(l), l = 1, . . . L, (4.21)
Time
Tx 1
Tx 2
Transmitting
antenna
Figure 4.5.: Example of a TDM sequence with dPulse = [dTx1 , d
Tx
2 , d
Tx
1 ]
T
– 54 –
where x(l) is the measured signal of cycle l after pulse compression.
aPulse(u) = exp( j dPulseu) (4.22)
is the steering vector of the transmitting antennas given by dPulse. aRx(u) is defined in (4.4). sMIMO(l)
is the complex signal of the target and n(l) the additional noise. Note that x(l), n(l) ∈ CNVirt , where
NVirt = NPulse · NRx. (4.23)
Thus, the measured signal x(l) has NVirt components due to the different transmitting antennas in
contrast to the SIMO case with NRx components. Because of the shorter chirp duration, the modulus
of sMIMO is in general smaller than the modulus of sSIMO, cf. Section 2.1.2. We define the steering
vector of the virtual array
aVirt(u) = aPulse(u) ⊗ aRx(u) ∈ CNVirt . (4.24)
aVirt(u) incorporates all transmitter-receiver combinations. It can be written as
aVirt(u) = exp( j dPulseu) ⊗ exp( j dRxu)
= exp( j dVirtu) (4.25)
with the antenna positions of the virtual TDM MIMO array
dVirt = 1NPulse ⊗ dRx + dPulse ⊗ 1NRx ∈ RNVirt . (4.26)
Note that dVirt already contains the transmitting sequence in dPulse. Hence the dimension of dVirt is
NRx NPulse and not NRx NTx. With these notations, we can write the signal model for the MIMO radar
as
x(l) = aVirt(u) sMIMO(l) + n(l), l = 1, . . . L. (4.27)
We make the same assumptions as in Section 4.2.1.
Similar to (4.7), the unknown deterministic parameters to be estimated from x(l), 1 ≤ l ≤ L, are
denoted by
Θ =
[
u, sMIMO(1), . . . , sMIMO(L), σ2
]T
. (4.28)
Note that in general |sMIMO(l)| < |sSIMO(l)|, since the MIMO radar transmits chirps with a shorter
chirp duration than the SIMO radar, but uses the same transmission power p as the SIMO radar.
Thus the transmitted energy per chirp is smaller, resulting in a smaller signal strength |sMIMO(l)|. We
investigate this in detail in Section 4.4.
– 55 –
4.3.2. Cramer-Rao Bound
The unknown parameter vector is given in (4.28). Again we are interested in that part of the CRB of
Θ which corresponds to the electrical angle u. We denote this part by CRBu,MIMO. The MIMO signal
model (4.27) has the same form as the SIMO model (4.6) if we set B(u) = 1. Hence we can use the
result for the SIMO radar (4.13) if we replace the variables accordingly. This gives
CRBu,MIMO =
1
2 L
1
S MIMO
1
UMIMO
(4.29)
with
S MIMO = SNRMIMO · NVirt, (4.30)
SNRMIMO =
σ2s,MIMO
σ2
, (4.31)
σ2s,MIMO =
1
L
L∑
l=1
|sMIMO(l)|2, (4.32)
UMIMO = VarS(dVirt). (4.33)
S MIMO is the total SNR of the MIMO radar system, similar to the SIMO radar. UMIMO depends on
the Rx and Tx antenna positions. Using the properties of VarS, cf. Appendix A.2.1 and A.2.4, the
sample variance of the virtual array positions can be expressed as
VarS(dVirt) = CovS(1NPulse ⊗ dRx + dPulse ⊗ 1NRx , 1NPulse ⊗ dRx + dPulse ⊗ 1NRx)
= CovS(1NPulse ⊗ dRx, 1NPulse ⊗ dRx) + CovS(1NPulse ⊗ dRx, dPulse ⊗ 1NRx)
+ CovS(dPulse ⊗ 1NRx , 1NPulse ⊗ dRx) + CovS(dPulse ⊗ 1NRx , dPulse ⊗ 1NRx)
= VarS(dRx) + VarS(dPulse). (4.34)
Thus
UMIMO = VarS(dRx) + VarS(dPulse). (4.35)
Hence the positions of the Rx as well as the positions of the Tx antennas influence the CRB of the
electrical angle.
4.4. Comparison of the SIMO and MIMO Radar
In order to see which radar system can achieve a higher accuracy, we compare CRBu,SIMO in (4.13)
and CRBu,MIMO in (4.29). Different comparisons of the CRBs for MIMO radars to other radars have
already been investigated, e. g. in [Li et al., 2008], but the comparisons depend on the precise type
– 56 –
of the MIMO radar and its application. Here we compare the TDM MIMO radar using an FMCW
or chirp sequence modulation to the SIMO radar. We discuss the result w. r. t. the application in
automobiles.
CRBu,SIMO and CRBu,MIMO have the same form but the formulas for the total SNR and the sample
variance of the antenna positions differ. Since the chirp durations of the SIMO and MIMO radar
differ the SNRs SNRSIMO and SNRMIMO are not equal. Hence, we have to compute the SNRs first.
This is done in the next section. Then, we compare the SNRs, the CRBs and discuss the results.
Numerical simulations are presented afterwards.
4.4.1. SNR Computation
In an FMCW and chirp sequence radar, the signal used for DOA estimation is gained by a discrete
Fourier transform (DFT) of the complex baseband signal of one chirp and taking the complex am-
plitude of the highest peak in the spectrum. We consider the complex baseband signal of one target,
i. e. a complex oscillation, and derive the SNR after doing the DFT. The measured, sampled signal
x(n) at one Rx antenna is
x(n) =
√
ρs s(n) + n(n), n ∈ Z (4.36)
where n is the time index and ρs is the transmitted energy per sample. s(n) is the complex signal
strength. It includes the RCS, damping effects due to the distance to the target, and the antenna
gain of the Rx and Tx antennas. n(n) denotes additional noise. Note that the phase shift due to the
position of the Rx antenna, which is described by aRx(u), and the transmit beampattern B(u) is not
incorporated here. The transmitted energy per sample ρs can be written as
ρs = p ∆t. (4.37)
where p is the transmitted power and ∆t the sampling time interval of the ADC for one sample.
Hence ∆t = 1/ fs with the sampling frequency fs. Let
s(n) = s0 e j (ωdiff n+ϕ) (4.38)
with the real and positive amplitude s0 ≥ 0. ωdiff is the angular difference frequency of the target and
ϕ the additional phase, cf. Section 2.1.2. Let the noise be stationary, zero mean, and white Gaussian
with variance σ2n and autocorrelation function
rn(n) = σ2n δ(n) =
σ
2
n if n = 0,
0 if n , 0.
(4.39)
– 57 –
The SNR before the Fourier transform, defined as the ratio of the received signal energy to the
variance of the noise, is
SNRbefore =
∣∣∣√ρs s(n)∣∣∣2
Var (n(n))
=
ρs s20
σ2n
. (4.40)
There is only a finite number of samples per chirp available, which we denote by N. We incorporate
this by multiplying x(n) with a real window function w(n) which is nonzero only for 0 ≤ n ≤
N − 1. Hence the discrete-time Fourier transform (DTFT) Xw(ω) of the windowed signal at angular
frequency ω is given by
Xw(ω) =
∞∑
n=−∞
w(n) x(n) e− jωn
=
N−1∑
n=0
w(n) x(n) e− jωn . (4.41)
Plugging in (4.36) yields
Xw(ω) =
√
ρs
N−1∑
n=0
w(n) s(n) e− jωn +
N−1∑
n=0
w(n) n(n) e− jωn
=
√
ρs S w(ω) + Nw(ω) (4.42)
with
S w(ω) =
N−1∑
n=0
w(n) s(n) e− jωn, (4.43)
Nw(ω) =
N−1∑
n=0
w(n) n(n) e− jωn . (4.44)
The complex value that is used for the DOA estimation is Xw(ωmax), where ωmax is the angular
frequency at which |Xw(ω)| is maximal, i. e.
ωmax = arg max
ω
|Xw(ω)| . (4.45)
For simplicity, we assume that the SNR is large enough such that the difference between ωmax and
the true angular frequency ωdiff is negligible, hence ωmax = ωdiff. Since we are doing a DFT, the
samples of Xw(ω) are only available at discrete frequencies ωk, k = 0, 1, . . . . We assume that, by
frequency interpolation or zero-padding, the exact value of ωmax is found. Then
Xw(ωmax) =
√
ρs S w(ωmax) + Nw(ωmax)
=
√
ρs S w(ωdiff) + Nw(ωdiff). (4.46)
– 58 –
The SNR after the DFT is defined as
SNRafter =
∣∣∣√ρs S w(ωmax)∣∣∣2
Var (Nw(ωmax))
. (4.47)
In Appendix D, it is shown that
SNRafter = SNRbefore · SNRgain (4.48)
with the SNR gain
SNRgain =
(∑N−1
n=0 w(n)
)2∑N−1
n=0 w2(n)
. (4.49)
SNRgain depends on the window used. For the rectangular window, w(n) = 1 for 0 ≤ n ≤ N − 1 and
thus SNRgain = N ∀N. For other windows, SNRgain ≤ N, which follows from the Cauchy-Schwartz
inequality. In Fig. 4.6, SNRgainN is plotted for some typical windows as a function of N. We see that
SNRgain
N
≈ const = wgain, for N > N0 = 100, (4.50)
where we have defined the window dependent gain wgain ≤ 1. Thus if the DFT length is larger than
N0, we can write
SNRgain ≈ wgain · N. (4.51)
101 102 103 104
0.4
0.6
0.8
1
window length N
SN
R
ga
in
N
rectangular
Chebyshev 60 dB
Hamming
Blackman-Harris
Figure 4.6.: SNR gain for different windows. Note, for the rectangular window SNRgainN = 1.
– 59 –
4.4.2. Comparison of the SNRs for SIMO and MIMO Radar
With the gained, results we determine
SNRSIMO =
1
L
L∑
l=1
|sSIMO(l)|2
σ2
(4.52)
in (4.15). We consider the l-th cycle and the k-th Rx antenna. Then σ2 = E(|nk(l)|2) = Var(nk(l)).
The signal and noise, without the Tx beampattern B(u) and aRxk (u), is given by (4.46):
sSIMO(l) + nk(l) =
√
ρs S w(ωmax) + Nw(ωmax). (4.53)
Hence
|sSIMO(l)|2
σ2
=
|sSIMO(l)|2
Var(nk(l))
=
∣∣∣√ρs S w(ωmax)∣∣∣2
Var (Nw(ωmax))
= SNRafter. (4.54)
This is valid for all l. Thus
SNRSIMO = SNRafter. (4.55)
The number of samples per chirp are Nsamples, cf. Tab. 4.1. Hence, for Nsamples > N0 and with (4.48)
to (4.51) follows
SNRSIMO = SNRbefore · wgain · Nsamples. (4.56)
This results in the total SNR in (4.14)
S SIMO = SNRbefore · wgain · Nsamples · NRx · |B(u)|2 . (4.57)
Analogously, we obtain for the MIMO radar
SNRMIMO = SNRbefore · wgain · NsamplesNPulse (4.58)
and with (4.30) and (4.23)
S MIMO = SNRbefore · wgain · NsamplesNPulse · NRx · NPulse
= SNRbefore · wgain · Nsamples · NRx. (4.59)
– 60 –
Comparing (4.57) and (4.59) yields
S SIMO = |B(u)|2 · S MIMO. (4.60)
Let us discuss this result. First we consider the case without a Tx beampattern, i. e. B(u) = 1. The
SNR of one chirp SNRMIMO is by the factor of NPulse smaller than SNRSIMO, since the chirp duration
is shorter. The total SNRs S MIMO and S SIMO are equal, since the MIMO radar uses NPulse chirps.
Now let us consider the influence of the Tx beampattern. B(u) increases S SIMO, if it is steered in the
direction of the target, or decreases S SIMO, if it is steered at another direction.
4.4.3. Comparison of the CRBs and Discussion
With the results gained in Section 4.4.2, we can compare the CRBs of the SIMO and MIMO radar
CRBu,SIMO (4.13) and CRBu,MIMO (4.29), respectively.
CRBu,SIMO
CRBu,MIMO
=
S MIMO UMIMO
S SIMO USIMO
=
VarS(dVirt)
|B(u)|2 VarS(dRx) (4.61)
where we have used (4.57) and (4.59). First we discuss the case without a Tx beampattern of the
SIMO radar, i. e. B(u) = 1, and afterwards with a Tx beampattern.
4.4.3.1. Without Tx Beampattern, B(u) = 1
If there is no Tx beampattern, i. e. the SIMO radar uses one omnidirectional Tx antenna, cf. Fig. 4.2a
on page 49, then B(u) = 1. Hence, the ratio of the CRBs is given by the ratio of the sample variance
of the virtual and Rx array. Using (4.34) we get
CRBu,SIMO
CRBu,MIMO
=
VarS(dVirt)
VarS(dRx)
= 1 +
VarS(dPulse)
VarS(dRx)
. (4.62)
Hence the virtual array of the MIMO radar has two effects. a) Due to the larger number of virtual
antennas, it compensates the SNR loss of one chirp: in the CRB only the total SNR S MIMO is relevant,
which is given by SNRMIMO ·NVirt. Due to the factor NVirt, it stays as large as SNRSIMO. b) In general,
the virtual aperture of the radar is larger then the one of the SIMO radar. This is reflected by the
sample variance VarS(dVirt). It contains not only the variance of the Rx antennas, VarS(dRx), but also
of the transmitting antennas, VarS(dPulse). Thus CRBu,MIMO becomes smaller. Illustrative spoken,
– 61 –
Rx1
Tx1
virtual array
Tx2
Rx2 Rx3
Figure 4.7.: Example of an ULA of Rx antennas. The Tx antennas are positioned, such that the
virtual array is an ULA as well. Antenna Rx 1 and Tx 1 are at the same position, they
are depicted slightly shifted just for illustrative purposes.
due to the larger virtual array, the digitally formed beam is sharper than that of the SIMO radar. As
an example, we consider a uniform linear array (ULA) of Rx antennas with a distance between the
antennas of ∆Rx. We place the Tx antennas of the MIMO radar such that the virtual array is also an
ULA, cf. Fig. 4.7. The sample variances are given by
VarS(dRx) =
1
12
(
N2Rx − 1
)
≈ 1
12
N2Rx for N
2
Rx  1 (4.63)
VarS(dVirt) =
1
12
(
N2Virt − 1
)
≈ 1
12
N2RxN
2
Pulse for N
2
Virt  1. (4.64)
Hence
VarS(dVirt)
VarS(dRx)
≈ N2Pulse. (4.65)
Thus the ratio of the CRBs increases as N2Pulse.
4.4.3.2. With Tx Beampattern
Up to now, we have considered the SIMO array without a Tx beampattern, i. e. one omnidirectional
Tx antenna. If the target’s DOA is known approximately a priori, the Tx antennas can transmit a
signal such that the DOA can be estimated more precisely. We consider the case, where the same
signal is transmitted from every Tx antenna, but phase shifted such that the transmitted energy is
focused on the a priori known DOA. This is called a sum beam, cf. Fig. 4.8 for an example. The
configuration of such a SIMO radar is depicted in Fig. 4.2b on page 49. To make a fair comparison,
the total transmit power p is fixed. It is shown in Appendix C that the maximal achievable beam-
forming gain at the angle u0, at which the beam is steered, subject to a constant transmit power p, is
given by
Bmax = |B(u0)| =
√
NTx. (4.66)
– 62 –
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
1
2
u
|B(
u)
|
Figure 4.8.: Example of a sum beam steered to u0 = 0.4 for an ULA of 5 isotropic Tx antennas with
λ/2 spacing. Here, |B(u0)| =
√
NTx =
√
5.
It is achieved when every Tx antenna transmits with the same power pNTx . Thus, from (4.61) follows
CRBu,SIMO
CRBu,MIMO
≥ Var
S(dVirt)
NTx VarS(dRx)
. (4.67)
Let us compare the SIMO radar with the maximal achievable beamforming gain to the TDM MIMO
radar. CRBu,SIMO at u0 is smaller than CRBu,MIMO if
|B(u0)|2 = NTx > Var
S(dVirt)
VarS(dRx)
. (4.68)
Consider an antenna array, where VarS(dPulse) = VarS(dRx). Then (4.68) is fulfilled for NTx > 3,
i. e. the beamforming gain is more important than the gain due to the virtual array. An example
of such an array is given in Fig. 4.9 a). On the other hand, the array depicted in Fig. 4.9 b) ful-
fills VarS(dPulse) > VarS(dRx), since the distance between the Tx antennas is as large as the maximal
distance between the Rx antennas. Here, the gain due to the virtual array has a larger impact than
the beamforming gain, and CRBu,MIMO < CRBu,SIMO. The disadvantage of the sum beam is the nec-
essary a priori information. Moreover, e. g. in automotive applications, different DOAs of several
targets are estimated in one measurement. Focusing the beam with the maximum beamforming gain
on all targets at once is not possible. On the contrary, the MIMO radar transmits an omni-directional
pattern, such that all DOAs can be estimated. This is an important advantage. A disadvantage of the
MIMO radar are the lower number of samples per chirp, due to the shorter chirp duration. There-
fore the accuracy and resolution of the frequency estimation becomes worse. The combination of
positionRx1 Rx2 Rx3 Rx4
Tx1 Tx2 Tx3 Tx4
(a) Here, VarS(dPulse) = VarS(dRx)
Rx1 Rx2 Rx3 Rx4
Tx1 Tx2
position
(b) Here, VarS(dPulse) > VarS(dRx)
Figure 4.9.: Comparison of VarS(dPulse) with VarS(dRx). In both examples (a) and (b) dPulse = dTx, i. e.
every Tx antenna transmits once.
– 63 –
several chirps, with the same frequency modulation, could overcome this problem. Due to the ad-
vantages, we focus on the MIMO radar in the following and neglect the possible beamforming gain
for convenience.
4.4.4. Notes on CRBu,SIMO
We show that CRBu,SIMO only depends on the overall transmission time Ttotal (4.1) and not on the
number of transmitted chirps, as long as every chirp has got a certain minimum duration.
The total SNR is given by (4.57)
S SIMO = SNRbefore wgain Nsamples NRx |B(u)|2 , for Nsamples ≥ N0. (4.69)
Plugging in (4.40) and using (4.37) yields
S SIMO = p ∆t
s20
σ2n
wgain Nsamples NRx |B(u)|2
∝ ∆t Nsamples (4.70)
Since the SIMO radar transmits one chirp per cycle, the duration of one cycle is
Tcycle = ∆t Nsamples (4.71)
and thus
S SIMO ∝ Tcycle. (4.72)
With (4.13) follows for CRBu,SIMO
CRBu,SIMO =
1
2L
1
USIMO
1
S SIMO
∝ 1
L Tcycle
=
1
Ttotal
. (4.73)
with Ttotal defined in (4.1). Since the radar transmits one chirp per cycle and L is the number of cycles,
L equals also the number of transmitted chirps. Hence, CRBu,SIMO stays the same, independent of
the number of transmitted chirps as long as Ttotal stays constant and the chirp duration is larger than
∆t N0.
– 64 –
4.4.5. Numerical Example
The CRB is a lower bound on the variance of any unbiased estimator. It is important to verify that
the computed CRB is a tight bound, i. e. it is actually achieved by an estimator. Otherwise the con-
clusions, drawn from the CRB, could be misleading. We do this by numerical simulations. We
compute the bias buˆ, standard deviation σuˆ, and root mean square error RMSEuˆ of the deterministic
Maximum Likelihood (ML) estimate uˆ of u by doing Monte-Carlo simulations. After that, we com-
pare the result to the CRB. We consider a radar system which consists of a linear array with 4 Rx
and Tx antennas. The Rx and Tx antennas are uniformly spaced with an antenna distance of λ/2,
i. e. dRx = dTx = pi · [0, 1, 2, 3]T . The DOA of the target is u = sin(10◦) and we consider L = 1 cycle.
We assume that the DOA of the target is not known approximately a priori and therefore the SIMO
radar uses 1 isotropic Tx antenna. Thus, the beamforming gain is B(u) = 1 and the signal model
(4.6) simplifies to
x(l) = aRx(u) sSIMO(l) + n(l). (4.74)
Note that the unknown parameter vector to be estimated is Θ in (4.7) and not u alone. According to
(E.23) in Appendix E, the ML estimate of u can be computed by
uˆ = arg max
u
L∑
l=1
∣∣∣∣(aRx(u))H x(l)∣∣∣∣2 , (4.75)
where we have dropped the normalization 1/‖aRx(u)‖2, since ‖aRx(u)‖ = const. uˆ is computed by
a grid search, followed by a local interpolation. The MIMO array under consideration transmits
NPulse = 4 pulses with transmission sequence dPulse = dTx. The signal model is given in (4.27). It
has exactly the same structure as (4.74) and the ML estimate uˆ can be computed analogously. We
consider different values of the total SNR S . For each value, 5000 Monte-Carlo simulations are
carried out to determine buˆ, σuˆ, and RMSEuˆ. Since B(u) = 1, according to (4.60), the total SNR S
of the SIMO and MIMO radar are equal. First we compare the bias buˆ to the standard deviation σuˆ.
The results for the SIMO radar are depicted in Fig. 4.10. It shows that |buˆ| is much smaller than σuˆ
for all SNR values. The same is true for the MIMO radar (not depicted here). Thus, the bias can be
neglected and σuˆ ≈ RMSEuˆ. Hence, we can compare RMSEuˆ to CRBu in the following. The results
are depicted in Fig. 4.11. It shows that both ML estimators reach the corresponding CRB above a
threshold of approximately 14 dB. If the SNR is too low, ambiguities arise in the estimation process.
This leads to a higher RMSE. The CRB is a bound which incorporates only the local properties of the
likelihood function and therefore does not include ambiguities. Because of this, the ML estimators
deviate from the CRBs below the threshold value. As expected, the MIMO radar achieves a lower
RMSE than the SIMO radar, due to its larger virtual aperture. Since the ML estimators indeed
achieve the CRBs, we can conclude that the CRB is a useful tool to study the DOA accuracy in our
applications. Here, we have investigated the behavior of the ML estimator for increasing SNR. For
– 65 –
0 5 10 15 20 25 30 35
10−4
10−3
10−2
10−1
100
Total SNR S in dB
|b uˆ
|,σ
uˆ
SIMO ML |buˆ|
SIMO ML σuˆ
Figure 4.10.: Absolute value of bias |buˆ| and standard deviation σuˆ of the estimated electrical angle
for the SIMO radar. |buˆ| is negligible compared to σuˆ.
0 5 10 15 20 25 30 35
10−3
10−2
10−1
100
Total SNR S in dB
R
M
SE
of
el
ec
tr
ic
al
an
gl
e
u
CRBu,SIMO
SIMO ML
CRBu,MIMO
MIMO ML
Figure 4.11.: RMSE of the electrical angle of a stationary target for a SIMO and MIMO radar
an increasing number of observations, asymptotic properties of the ML estimator can be shown, see
Appendix F for details. Nevertheless these asymptotic properties cannot be used in our automotive
radar application because the number of observations, which correspond here to the number of cycles
L, is small.
4.5. Comparison of Code and Time Division Multiplexing
In the following, we compare a MIMO radar using code division multiplexing (CDM) in slow time
with a MIMO radar using TDM. Both radars use the same Tx and Rx antenna array, only the multi-
plexing techniques are different. We consider several stationary targets. We investigate under which
circumstances both radar systems yield the same performance for DOA estimation. In order to do
that, we use the concept of sufficient statistics: The received signal X is transformed to a compressed
signal Y, which contains the same amount of information on the targets’ DOAs as the original re-
ceived signal X. Hence Y contains sufficient information for the DOA estimation. In general, the
– 66 –
XTDM
XCDM
YTDM
YCDM
identically distributed
compression
compression
Figure 4.12.: Relation between the received signals XTDM, XCDM and the compressed signals YTDM,
YCDM
received signal XCDM of the CDM MIMO radar is different from the received signal XTDM of the
TDM MIMO radar. We show under which conditions the compressed signal of the CDM radar
YCDM and the compressed signal of the TDM radar YTDM are identically distributed and therefore
the received signals XCDM and XTDM contain the same information on the DOAs, cf. Fig. 4.12. From
this follows that their performance in DOA estimation is the same.
We consider the signal model of [Li et al., 2008], adapted to our notation,
X =
NTargets∑
k=1
aRx(Uk) sk aTx
T
(Uk) Φ + N ∈ CNRx×NPulse (4.76)
with X being the received data and NPulse the number of chirps. NTargets is the number of tar-
gets. Uk and sk, 1 ≤ k ≤ NTargets are the DOAs and complex signals of the targets, respectively.
aRx(U) ∈ CNRx×1 and aTx(U) ∈ CNTx×1 are the steering vectors of the Rx and Tx array, respectively.
This model describes both the TDM and CDM MIMO radar. In the case of the CDM radar,
Φ ∈ CNTx×NPulse contains the known and deterministic transmitted codes, where row i of Φ is the
code of Tx i. For the TDM MIMO radar, Φ has to be chosen such that only one Tx transmits at a
time, i. e. only one entry of each column in Φ is unequal 0. Hence the TDM and CDM radar differ
only in Φ. N ∈ CNRx×NPulse is the noise. It is assumed that the columns of N are independent and
identically distributed circularly symmetric complex Gaussian random vectors with mean zero and
unknown covariance Cn.
N =
[
n1, . . . , nNPulse
]
, (4.77)
E
(
ni n
H
j
)
= δi j Cn ∈ CNRx×NRx . (4.78)
Note that in the case of one target and covariance matrix Cn = σ2I, for a TDM MIMO radar this
model is the same as the model presented in Section 4.3.1 for one cycle L = 1, using a different
notation. The model can be rewritten as
X = ARx(U) S ATxT (U)Φ + N (4.79)
with
ARx(U) = [aRx(U1), . . . , aRx(UNTargets)] ∈ CNRx×NTargets , (4.80)
ATx(U) = [aTx(U1), . . . , aTx(UNTargets)] ∈ CNTx×NTargets , (4.81)
– 67 –
U = [U1, . . . , UNTargets]
T , (4.82)
s = [s1, . . . , sNTargets]
T , (4.83)
S = diag(s) ∈ CNTargets×NTargets . (4.84)
The unknown parameters to be estimated are U, s,Cn. In [Li et al., 2008], the part of the deterministic
CRB corresponding to U and s is computed. It is shown that it only depends on MΦ
CRBU,s = CRBU,s(MΦ) (4.85)
with the correlation matrix of the transmitted codes
MΦ = ΦΦH ∈ CNTx×NTx . (4.86)
We consider transmitted codes Φ such that ΦΦH is invertible. A necessary condition for that is
NPulse ≥ NTx. Then, the received signal X can be compressed by multiplying it with ΦH(ΦΦH)−1/2,
where (.)−1/2 denotes the inverse of the Hermitian square root of a matrix. The compressed signal is
Y = XΦH (ΦΦH)−1/2
= ARx(U) S ATxT (U) (ΦΦH)1/2 + N˜ ∈ CNRx×NTx (4.87)
with
N˜ =
[
n˜1, . . . , n˜NTx
]
= NΦH (ΦΦH)−1/2 ∈ CNRx×NTx . (4.88)
It can be shown that choosing ΦH(ΦΦH)−1/2 to compress the signal guarantees that the noise term
stays temporally white [Li et al., 2008]. Moreover, N˜ and N have the same statistical properties [Li
et al., 2008], i. e. the columns of N˜ are independent and identically distributed circularly symmetric
complex Gaussian random vectors with mean zero and unknown covariance Cn
E
(
n˜i n˜
H
j
)
= δi j Cn ∈ CNRx×NRx , (4.89)
where Cn is the same matrix as in (4.78). Thus, the pdf of N˜ is independent of Φ. If the covariance
matrix Cn is known to be a scaled identity matrix
Cn = σ2I, (4.90)
then Y is a sufficient statistic for U, s [Li et al., 2008]. That means, Y contains the same amount of
information as X to estimate U, s. Note that Y in (4.87) only depends on MΦ = ΦΦH and not on Φ
alone.
In the following, we compute how MΦ looks like for CDM using orthogonal codes and show how to
choose a TDM sequence which yields the same correlation matrix MΦ. If the CDM and TDM radar
– 68 –
system have the same correlation matrix MΦ, CRBU,s is the same for both systems. Moreover, the
compressed signals of the systems YCDM and YTDM are identically distributed, cf. Fig. 4.12, since
YCDM and YTDM depend only on MΦ and not on Φ alone and the pdf of N˜ is independent of Φ.
4.5.1. Code Division Multiplexing using Orthogonal Codes
We compute how MΦ looks like when using CDM in slow time with orthogonal codes. The total
transmitting time is denoted by Ttotal. The maximum transmission power over all Tx antennas is
given by pmax. We divide Ttotal into NPulse periods of equal length
∆t =
Ttotal
NPulse
. (4.91)
Since we consider CDM in slow time, one period corresponds to one compressed radar pulse. We
write
Φ =

cT1
...
cTNTx
 (4.92)
where ci is the transmitted code of the ith Tx. The power constraint can be written as
NTx∑
n=1
|[Φ]nl|2 ≤ pmax∆t, 1 ≤ l ≤ NPulse. (4.93)
We consider the case where we transmit with the maximal available power, hence
NTx∑
n=1
|[Φ]nl|2 = pmax∆t. (4.94)
The orthogonality of the codes results in
cHi c j = δi j Ttotal p
mean
i (4.95)
where pmeani is the mean power of the i-th Tx over all pulses. Thus
MΦ = ΦΦH = Ttotal

pmean1 0
. . .
0 pmeanNTx
 . (4.96)
– 69 –
The transmitted energy WTx of all Tx antennas can be computed by
WTx = Tr
(
ΦΦH
)
=
NPulse∑
l=1
NTx∑
n=1
|[Φ]nl|2 =
NPulse∑
l=1
pmax∆t = pmax NPulse ∆t
= pmax Ttotal. (4.97)
4.5.2. Time Division Multiplexing
Given the same total transmitting time Ttotal and the same maximum transmission power pmax as in
the CDM case, we show how to construct a TDM scheme which yields the same correlation matrix
MΦ in (4.96).
Choose NTx pulses of length ∆ti such that
∆ti pmax = Ttotal pmeani , 1 ≤ i ≤ NTx. (4.98)
Hence
∆ti = Ttotal
pmeani
pmax
. (4.99)
We write Φ as in (4.92). Since now we transmit NTx pulses, Φ ∈ CNTx×NTx . We set
ci = [0, . . . ,
√
∆ti pmax, . . . , 0]T (4.100)
where all elements of ci are 0 except the i-th element. This ensures that only one Tx antenna is
transmitting at a time. MΦ can be computed with the help of
cHi c j = δi j p
max ∆ti
= δi j Ttotal pmeani (4.101)
which gives
MΦ = ΦΦH = Ttotal

pmean1 0
. . .
0 pmeanNTx
 . (4.102)
Hence such a TDM scheme results in the same correlation matrix MΦ as using CDM with orthogonal
codes (4.96). Moreover, the total transmitted energy WTx in (4.97) is the same.
– 70 –
4.5.3. Discussion
For a fixed total transmission time Ttotal and maximal transmission power pmax, we have derived
the correlation matrix MΦ for orthogonal CDM. We have shown that the same correlation matrix is
achieved for a TDM MIMO radar, if the TDM sequence is chosen appropriately. Since the CRB of
the DOAs (4.85) only depends on MΦ, the CRB is the same for both multiplexing schemes. Hence,
the CDM and TDM MIMO radar can achieve the same accuracy of the DOAs.
The compressed signal Y in (4.87) depends on MΦ. Moreover, the noise N˜ has the same statistical
properties as N with the same covariance matrix, cf. (4.89), independent ofΦ. Thus, N˜ has the same
pdf for the TDM and CDM MIMO radar. Therefore, for a CDM and TDM radar with the same MΦ,
the compressed signal of the CDM radar YCDM and of the TDM radar YTDM are identically distributed
and have the same likelihood function. If the noise covariance matrix is a scaled identity matrix
(4.90), the compressed signals are sufficient statistics for the DOA estimation [Li et al., 2008], i. e.
YCDM contains the same information on the DOAs as the received signal XCDM and YTDM contains the
same information on the DOAs as the received signal XTDM. Since YCDM and YTDM are identically
distributed, XCDM and XTDM contain the same information on the DOAs as well. In that case, the
CDM and TDM radars do not only have the same CRBs, but all other properties, like the ambiguity
and resolution, are equal since the likelihood functions of the compressed signals are the same. Thus,
by choosing the transmission matrixΦ of the TDM MIMO radar appropriately, it achieves the same
DOA performance as the CDM radar.
Obviously, if the power is not constrained by the overall power but by the transmission power per
Tx antenna, in general we cannot find a TDM scheme which results in the same CRB as the CDM,
since the CDM can transmit with a higher overall power. As discussed in Section 3.3, this is not a
concern for MIMO radars in automotive applications since the total transmission power is limited
by law and usually this power limit is already achieved by one Tx antenna.
4.6. Summary
We have investigated the DOA estimation of a stationary target. The CRBs of the DOA for a SIMO
and MIMO radar have been derived to compare the maximal achievable accuracy. Dependent on
the application, the SIMO radar can have a beamforming gain, if the target’s DOA is known ap-
proximately a priori. The MIMO radar has the advantage of a larger virtual aperture. Whether the
possible beamforming gain or the gain due to the virtual array dominates, depends on the positions
and number of the Rx and Tx antennas.
Moreover, a MIMO radar using CDM with orthogonal codes has been considered and compared to
a TDM MIMO radar. Under the constraint that the overall transmit power is limited, we have shown
that a TDM scheme can be constructed such that the CRB of the DOAs is the same for the TDM and
– 71 –
CDM radar. Moreover, if the noise covariance matrix has a certain structure, the received signals
of both radar systems contain the same information for DOA estimation. Then the CDM and TDM
radar do not only have the same CRBs of the DOAs, but all other properties which depend on the
likelihood function, e. g. the ambiguity, are the same.

– 73 –
5. DOA Estimation of Moving Targets
In a SIMO radar, the DOA can be estimated by analyzing the phase relation of the baseband signal
of the Rx channels. In a MIMO system, the phase relation between the transmitting channels is used
as well which makes it possible to construct a virtual array with a larger aperture than the receiving
or transmitting array alone. Therefore, it can achieve a more accurate DOA estimation. We consider
a TDM MIMO radar. However, a non-stationary target, i. e. a target which is moving relative to the
radar, results in an additional phase change of the baseband signal due to the Doppler effect. This
phase change distorts the phase change caused by the target’s DOA. For an accurate DOA estimation
it has to be considered as well, i. e. jointly estimated or canceled. In general, this leads to a decrease
in the DOA estimation accuracy which we want to avoid.
We analyze this situation in this chapter. We present the signal model in Section 5.1. We consider
both an one-dimensional antenna array for azimuth estimation and a two-dimensional antenna array
for azimuth and elevation estimation. In Section 5.2, the one target case is considered. In order
to analyze the maximal achievable DOA accuracy, we derive the CRB of DOA estimation for a
non-stationary target for a general TDM scheme. This allows a quantitative comparison of different
MIMO and SIMO radars. We use the results to find optimal TDM schemes which minimize the CRB.
The DOA estimation of two moving targets is investigated in Section 5.3. The CRB is derived and
the implications on the accuracy and resolution are investigated. In Section 5.4, the DOA estimation
of more than two targets is studied briefly. The results are summarized in Section 5.5.
This chapter is based on [Rambach and Yang, 2013, 2014; Rambach et al., 2014; Rambach and Yang,
2016a].1
5.1. Signal Model
We derive the signal model for DOA estimation for a TDM MIMO radar and a non-stationary target.
Non-stationary means that the target is moving relative to the radar. Since we are using a TDM
MIMO radar and the target is moving relative to the radar, the received signal of different transmitted
pulses changes. The signal used for DOA estimation is gained by a Fourier transform of the time
domain baseband signal, cf. Section 2.1.2. In order to see how the signal for the DOA estimation
1This chapter contains quotations of our own published works [Rambach and Yang, 2013, 2014; Rambach et al., 2014;
Rambach and Yang, 2016a] which are not marked additionally.
– 74 –
changes from pulse to pulse, we derive the baseband signal in the time domain for a moving target
and compute the phase change between different pulses in Section 5.1.1. After that, we present the
signal model for the DOA estimation using a TDM MIMO radar in Section 5.1.2.
5.1.1. Baseband Signal Model in the Time Domain
In a TDM MIMO radar, several Tx antennas transmit a signal, one after the other, cf. Fig. 5.1 for
an example. The data gained by the different Tx antennas is combined. To estimate the target’s
DOA, the phases of the complex baseband signals are processed. Since the target is moving relative
to the radar, the Doppler effect causes a phase shift of the baseband signal, which depends on the
transmission time. We compute this phase shift for different kinds of radars. The result is used
to establish an appropriate signal model for the DOA estimation of a moving target using a TDM
MIMO radar.
5.1.1.1. Baseband Signal of a Pulse Doppler Radar
For a pulse Doppler radar, the received baseband signal spulseDoppler(n) of pulse n has already been
computed in [Bühren, 2008]. It is given by
spulseDoppler(n) ∝ sin(ωDoppler n ∆t + ϕ0), (5.1)
where ωDoppler is the Doppler frequency, ∆t the difference between the transmission times of two
successive pulses and ϕ0 a constant, absolute phase. ωDoppler is given as
ωDoppler ≈ 2 vc ω (5.2)
for |v|  c, with the angular carrier frequency ω, the relative velocity of the target v and the speed
of light c. v is defined to be positive if the target moves away from the radar. This is the well known
Doppler frequency in radar applications. Hence, for a pulse Doppler radar, the phase shift ∆ϕ1k
Time
t1 t2
Transmitting
antenna
Tx 2
Tx 1
t3
Figure 5.1.: Example of two Tx antennas transmitting at different times. First Tx antenna 1 transmits,
then antenna 2 and after that antenna 1 again.
– 75 –
Radar
Radar
v · ∆tM,1k
Target
Target
Figure 5.2.: Illustrative explanation of the Doppler phase shift. [Openclipart, 2014, graphical parts
used and adapted]
between pulse 1 and k is
∆ϕ1k = ωDoppler ∆tM,1k, (5.3)
where ∆tM,1k is the difference between the transmission times of pulse 1 and k
∆tM,1k = tM,k − tM,1. (5.4)
Here, tM,i, i = 1, k, is the middle of the transmission time of the i-th pulse. Note that since all
transmitted pulses are equal, i. e. they have the same shape and duration, it does not matter if we
define tM,i to be the start time or the middle of the transmission time of the i-th pulse. The Doppler
phase shift ωDoppler ∆tM,1k can be interpreted in the following way, cf. Fig. 5.2: in the time ∆tM,1k the
target moves a distance of v ·∆tM,1k. With λ being the carrier wavelength, this corresponds to a phase
change of 2pi v ∆tM,1k
λ
= ω vc ∆tM,1k. The radar wave has to travel forth and back, which results in the
additional factor 2, i. e. the phase shift is ω 2 vc ∆tM,1k = ωDoppler ∆tM,1k.
(5.1) can easily be generalized to a complex baseband signal, which is obtained by using an IQ-
mixer.
5.1.1.2. Baseband Signal of an FMCW and Chirp Sequence Radar
We are interested in an FMCW and a chirp sequence radar. Both radar systems transmit linear
frequency modulated chirps. They differ in the chirp duration and the number of transmitted chirps.
A chirp sequence radar transmits many pulses of short durations, e. g. 1000 chirps with a chirp
duration of T = 10 µs. In an FMCW radar, few chirps of longer durations are transmitted, e. g. 4
chirps with durations of the order of ms. In these cases, the phase change of the received baseband
signal between two pulses is more complicated than (5.3) and will be investigated below. First we
consider one transmitted chirp and compute the received baseband signal. Having this result, we
compute the difference of the phases of the baseband signals of two chirps transmitted at different
transmission times. After that we discuss and summarize the result.
– 76 –
Baseband Signal of One Chirp Without loss of generality (w. l. o. g.) the Tx antenna starts to
transmit at t = 0. The transmitted signal at time t is given by
sTx(t) = ATx cos (ϕTx(t) + ϕ0) , t ∈ [0,T ], (5.5)
where ATx is the amplitude, ϕTx(t) the time dependent phase and ϕ0 the phase at t = 0. T is the
duration of the chirp. ϕTx(t) can be computed by
ϕTx(t) =
∫ t
0
ω(t˜) dt˜ (5.6)
with the linear changing angular frequency
ω(t) = ω0 + χ t, t ∈ [0,T ], (5.7)
where ω0 is the angular frequency at the beginning of the chirp and χ is the chirp rate, cf. Fig. 5.3.
Hence
ϕTx(t) = ω0 t + ϕ˜(t), (5.8)
ϕ˜(t) =
1
2
χ t2. (5.9)
The transmitted signal travels to the target, is reflected at the target and travels back to the radar. The
target is moving relative to the radar. Therefore, due to the Doppler effect, the signal is modified. To
compute the signal which is received by the radar, we express the transmitted signal by its inverse
Fourier transform
sTx(t) =
∫ ∞
−∞
S Tx(ω) e jωt
1
2pi
dω, (5.10)
where S Tx(ω) is the Fourier transform of sTx(t). ω is the angular frequency. Since the Maxwell
equations are linear, we can derive the change in frequency due to the Doppler effect for exp( jωt)
and transform back to the time domain afterwards. In the computations in Appendix G.1, it is shown
that the angular frequency of the received signal changes by ωDoppler defined in (5.2).
t
0
ω(t)
ω0
T
χ
Figure 5.3.: Linear frequency modulated chirp
– 77 –
The received signal sRx(t) can be computed from the transmitted one, modified by the Doppler effect
and time-delayed by the time which the electromagnetic wave needs to travel to the target and back
to the radar. sRx(t) is derived in detail in Appendix G.1 and and can be written as
sRx(t) = ARx cos (ϕRx(t)) , (5.11)
where ARx is the amplitude of the received signal and ϕRx(t) its time dependent phase given in (G.26).
ARx also includes the damping due to reflection at the target and the propagation of the signal.
The received signal sRx(t) is mixed in an IQ-mixer with the transmitted one, i. e. it is mixed with
ATx cos (ϕTx(t) + ϕ0) and ATx sin (ϕTx(t) + ϕ0). Afterwards a band pass filter is applied and the I- and
Q-signal are combined to a complex baseband signal. We assume the signal to be narrowband, i. e.
the bandwidth is much smaller than the carrier frequency. Thus the phase of the complex baseband
signal is given as the difference of the Tx phase ϕTx(t) + ϕ0 and the Rx phase ϕRx(t) in (G.26). In
Appendix G.2, the complex baseband signal s(t) is computed. It is given by
s(t) = A exp ( jϕbase(t) ) , (5.12)
with
ϕbase(t) = ϕbase,a + ϕbase,b(t) + ϕbase,c(t), (5.13)
ϕbase,a = ω0 τ0 (1 − β) − 12 χ τ
2
0, (5.14)
ϕbase,b(t) = t [ω0 2β + χ τ0 (1 − 3β)], (5.15)
ϕbase,c(t) = t22 χ β. (5.16)
Here, A is the complex amplitude of the baseband signal. It incorporates also the phase jump at the
reflection of the signal at the target. β is defined as
β =
v
c
. (5.17)
It is the ratio of the target’s velocity v and the speed of light c. τ0 is given by
τ0 = 2
d0
c
. (5.18)
It is the time the signal needs to travel to the target at distance d0 and back to the radar. d0 is the
distance of the target at t = 0. τ0 is also called the round trip time. For a moving target, the distance
between target and radar changes during the measurement.
Let us interpret the terms in (5.13). ϕbase,a is a time independent phase shift which depends on the
– 78 –
Table 5.1.: Maximal values of the parameters used to compute the order of magnitude of different
terms
Radar parameters
FMCW chirp sequence
carrier frequency f0 = 12pi ω0 76.5 GHz 76.5 GHz
chirp bandwidth F = 12pi Fω 1 GHz 1 GHz
chirp duration T 10 ms 100 µs
chirp rate d fdt =
1
2pi χ
1 GHz
5 ms = 2 · 1011 1s2 1 GHz100 µs = 1013 1s2
time difference ∆tM between last and first chirp 40 ms 20 ms
Target parameters
distance d 250 m
velocity v 80 ms
(
= 288 kmh
)
time delay τ0 of the target. ϕbase,b can be written as
ϕbase,b(t) = ωdiff t, (5.19)
ωdiff = ω0 2β + χ τ0 (1 − 3β) ≈ ω0 2β + χ τ0. (5.20)
ωdiff is the angular frequency difference between the transmitted and received signal due to the
Doppler effect ω0 2β and the propagation time τ0, i. e. the distance of the target. The difference
frequency is depicted in Fig. 2.5 on page 27. It is used to estimate the distance and relative velocity
of the target. For a chirp sequence radar with high chirp rate χ, the joint distance and velocity esti-
mation can be decoupled to a sequential distance and velocity estimation. For example, we consider
a chirp sequence radar with a chirp rate χ = 2pi · 1013 1s2 and carrier frequency f0 = 76.5 GHz. Then,
for a target with a large velocity v = 80 ms (288
km
h ) at a small distance d0 = 10 m, the second term
in (5.20) is still about 10 times larger than the first one and therefore ωdiff ≈ χ τ0. Thus only the
distance information is included in ωdiff. This is an advantage of the chirp sequence radar compared
to the FMCW radar, since in that case no matching algorithms are necessary.
Now we take a closer look at ϕbase,c(t). It can be seen as a distortion to the complex wave
A · exp { j (ϕbase,a + ϕbase,b(t) )} since it is a quadratic function in t. We estimate the maximum value
of ϕbase,c(t). The bandwidth Fω of the angular frequency of the chirp can be computed by
Fω = T χ. (5.21)
Then |ϕbase,c(t)| ≤ |ϕbase,c(T )| = 2 T Fω β. We use the values given in Tab. 5.1 as maximum val-
ues for the different parameters to compute the maximum value of |ϕbase,c(T )|. For a chirp se-
quence radar |ϕbase,c(T )| ≈ 0.3 follows. For the angle estimation, the phase ϕbase,a is used. It has
to be estimated modulo 2pi. For the estimation of ϕbase,a, in general ϕbase,c(t) is negligible, since
– 79 –
t
tstart,1 tM,1
ω(t)
tstart,k
ω0M
tM,k
Figure 5.4.: Several chirps transmitted at different start times tstart,i, with different chirp rates χi but
with the same middle frequencies ωi(tM,i) = const = ω0M ∀i
|ϕbase,c(t)| ≤ |ϕbase,c(T )|  2pi ∀ t ∈ [0,T ]. In the case of an FMCW radar, depending on the used
chirp parameters, ϕbase,c(T ) could be of the order of magnitude 10, i. e. it has to be considered. Oth-
erwise, the estimation of ϕbase,a could be wrong. In [Kulpa et al., 2000] a simple algorithm for the
estimation of nonlinear frequency distortions, like ϕbase,c(t), is investigated.
Phase Difference Between Two Chirps Now we compute the phase difference of the baseband
signal of the first transmitted chirp and the k-th chirp, k , 1. The chirps are transmitted at different
transmission times. In the following, the subscript i denotes that the corresponding variable belongs
to the i-th chirp. We study chirps which have the same carrier frequency ω0M in the middle of the
chirp 2
ω1(tM,1) = ωk(tM,k) = ω0M. (5.22)
tM,i is the time at the middle of chirp i
tM,i = tstart,i +
1
2
Ti, (5.23)
and tstart,i is the start time of the transmission of chirp i and Ti its duration, cf. Fig. G.1 on page 165.
An example of such chirps is depicted in Fig. 5.4. We compute the difference ∆ϕ1k of the phases of
the baseband signals at the time in the middle of the chirps. In Appendix G.4 it is shown that ∆ϕ1k
can be written as
∆ϕ1k = ϕbase,k(tM,k) − ϕbase,1(tM,1). (5.24)
Here
ϕbase,i(t) = ϕ
(i)
base(t − tstart,i), (5.25)
2Considering chirps which have the same frequency at the beginning of the chirps, ω0,1(tstart,1) = ω0,k(tstart,k) = ω0,
leads to analogous results as (5.31) and (5.35). In that case, ω0M has to be replaced by ω0, τM,1 by τ0,1, and ∆tM,1k
by ∆tstart,1k = tstart,k − tstart,1.
– 80 –
where ϕ(i)base(t) is the same as ϕbase(t) defined in (5.13), but the chirp dependent parameters ω0, τ0, χ
in (5.14) to (5.16) are replaced by ω0,i, τ0,i, χi, respectively. Computations in Appendix G.4 yield
∆ϕ1k = ωDoppler ∆tM,1k −
(
1
2
χk τ
2
M,k −
1
2
χ1 τ
2
M,1
)
(5.26)
with
ωDoppler = ω0M 2
v
c
, (5.27)
∆tM,1k = tM,k − tM,1. (5.28)
τM,i is the round trip time at the middle of chirp i, i. e. the distance of the target equals
dM,i = d0,i + v
1
2
Ti, (5.29)
with the chirp duration Ti and d0,i being the distance of the target at the start of the transmission of
chirp i. Hence
τM,i = 2
dM,i
c
. (5.30)
Let us interpret the result: (5.26) shows that the phase difference ∆ϕ1k contains a phase shift
ωDoppler ∆tM,1k due to the Doppler effect. This is the same phase shift as for the pulse Doppler radar
(5.3). But here ∆ϕ1k contains additional contributions due to the chirp rates χi and the distance of
the target, contained in τM,i. If χi = 0∀i, i. e. pulses with constant frequency are transmitted as in the
case of a pulse Doppler radar, (5.26) simplifies to the phase shift of a pulse Doppler radar (5.3).
We analyze and simplify (5.26) further. First we consider a chirp sequence radar, then an FMCW
radar.
Chirp Sequence Radar A chirp sequence radar transmits many equal chirps. Hence, all chirp
rates are equal with χ1 = χk = χ. In that case, (5.26) can be simplified further, cf. Appendix G.5,
and ∆ϕ1k has the simple form
∆ϕ1k = ωpseudoDoppler ∆tM,1k (5.31)
with the pseudo Doppler frequency
ωpseudoDoppler = 2
v
c
(
ω0M − χ τM,1) . (5.32)
We call it pseudo Doppler frequency, since it contains not only the Doppler frequency ωDoppler but
also a correction due to the round trip time τM,1. Let us interpret it: (ω0M−χ τM,1) is the Tx frequency
transmitted at time tM,1 − τM,1. If there was not a Doppler effect this would be the frequency received
– 81 –
t
tM,1
ω(t)
ω0M
τM,1
ω0M − χ τM,1
Tx
Rx
Figure 5.5.: Tx and Rx angular frequencies, neglecting the Doppler frequency, for the interpretation
of (5.32)
at the radar at time tM,1, cf. Fig. 5.5. ωpseudoDoppler is the Doppler frequency associated with this
frequency.
FMCW Radar In general, the chirp rates χi of the different chirps are not equal and we cannot
simplify ∆ϕ1k as in the case of a chirp sequence radar. In the typical signal processing chain of an
FMCW radar, the distance and velocity of the target are already estimated before the DOA estima-
tion. Using the estimated velocity, the distance of the target dM,i at the middle of chirp i can be
estimated as well for every chirp. Therefore we can use the estimated distances dˆM,1, dˆM,k to compute
a corrected phase difference ∆ϕcorrected,1k. We plug (5.30) in (5.26) to write
∆ϕ1k = ωDoppler ∆tM,1k − 2c2
(
χk d2M,k − χ1 d2M,1
)
. (5.33)
We define ∆ϕcorrected,1k as
∆ϕcorrected,1k = ∆ϕ1k +
2
c2
(
χk dˆ2M,k − χ1 dˆ2M,1
)
. (5.34)
Then
∆ϕcorrected,1k ≈ ωDoppler ∆tM,1k, (5.35)
where ωDoppler is given in (5.27). There is no exact equal sign in (5.35), since in general dˆM,1, dˆM,k
have an estimation error. Usually this error is negligible, as shown in Appendix G.6.
5.1.1.3. Summary and Discussion
We have presented the baseband signal in the time domain for a pulse Doppler, an FMCW and a
chirp sequence radar. We have computed the phase change between the first and the k-th pulse for
these different radar systems. For convenience, we repeat the results here. For a pulse Doppler radar
– 82 –
the phase change is given by (5.3)
∆ϕ1k = ωDoppler ∆tM,1k. (5.36)
For an FMCW radar, the phase change after distance and velocity estimation as well as a phase
correction is (5.35)
∆ϕcorrected,1k ≈ ωDoppler ∆tM,1k, (5.37)
and for the chirp sequence radar it is (5.31)
∆ϕ1k = ωpseudoDoppler ∆tM,1k, (5.38)
where ωpseudoDoppler is given in (5.32). Thus, for all of these radar types, the phase change has the
same structure. In the following, we abbreviate these three cases by
∆ϕ1k = ω∆t1k, (5.39)
where ω is defined appropriately, depending on the considered radar system, and ∆t1k is a shortcut
for ∆tM,1k. These phase shifts have to be incorporated in the signal model for the DOA estimation of
a moving target. W. l. o. g. we set the phase of the first chirp, due to the Doppler effect, to ωt1, where
t1 is the time in the middle of the first chirp. Then the phase shift ϕk of the k-th transmitted pulse is
ϕk = ω (t1 + ∆t1k)
= ω tk, (5.40)
with tk being the transmission time in the middle of the k-th chirp. In the TDM MIMO radar ϕk
are the phase shifts caused by the different pulses of different Tx antennas transmitted at different
times. We incorporate these phase shifts in the signal model for the virtual array of the TDM MIMO
radar in the next sections. Since the structure of the phase shifts are the same for the pulse Doppler,
FMCW, and chirp sequence radar, the DOA estimation model presented in the following section
works for all kinds of these radars.
5.1.2. Signal Model of a TDM MIMO Radar for DOA estimation
We present the signal model for the DOA estimation of a non-stationary target for a TDM MIMO
radar. We consider a radar system with a planar, i. e. two-dimensional antenna array in Sec-
tion 5.1.2.1. In Section 5.1.2.3, we present the signal model for a linear antenna array as a special
case of the planar one.
– 83 –
5.1.2.1. Planar Array
We investigate a TDM MIMO radar with a planar antenna array. We consider one non-stationary
target. Hence we extend the model given in Section 4.3.1 from a linear to a planar array and from a
stationary to a non-stationary target. Moreover, we include different transmission powers and chirp
lengths, which result in different energies transmitted by the Tx antennas.
The MIMO radar consists of NTx Tx and NRx Rx colocated, isotropic antennas. The transmitted
signal is narrowband. The moving target is modeled as a point target reflecting the transmitted
signals. The target is in the far field. The positions of the Rx antennas are given by
dRx,1, . . . , dRx,NRx ∈ R2 (5.41)
and the positions of the Tx antennas by
dTx,1, . . . , dTx,NTx ∈ R2. (5.42)
All positions are in the unit of λ2pi , where λ is the carrier wavelength corresponding to the middle
frequency of the chirps. The Tx antennas transmit a sequence of NPulse pulses at the time instances
given by t ∈ RNPulse . Here ti is the time in the middle of the i-th transmitted pulse as depicted in
Fig. 5.6. In the following, we call t just transmission times. The pulse durations are ∆t ∈ RNPulse and
the pulses are transmitted with transmission powers p ∈ RNPulse . The positions of the Tx antennas in
the sequence in which they transmit are denoted by
dPulse,1, . . . , dPulse,NPulse ∈ R2, (5.43)
also in the unit of λ2pi . We put the position vectors in matrices
DRx =
[
dRx,1, . . . , dRx,NRx
]T ∈ RNRx×2, (5.44)
DTx =
[
dTx,1, . . . , dTx,NTx
]T ∈ RNTx×2, (5.45)
DPulse =
[
dPulse,1, . . . , dPulse,NPulse
]T ∈ RNPulse×2. (5.46)
Timet1 t2 t3 t4
p1
p2
p4
p3
Transmitting
antenna
Tx 2
Tx 1
Figure 5.6.: Example of a TDM scheme: 2 transmitters transmitting at times t = [t1, t2, t3, t4]T with
power p = [p1, p2, p3, p4]T . Here, DPulse =
[
dTx,1, dTx,2, dTx,2, dTx,1
]T
.
– 84 –
y
x
z
target
ϕ
ϑ
antenna
array
uy
ux
Figure 5.7.: Azimuth ϑ and elevation ϕ of a target and its electrical angles ux, uy
If an antenna transmits more than once, its position occurs several times in DPulse, see Fig. 5.6 for
an example. Note that DTx contains only the possible Tx antenna positions, whereas DPulse contains
really the positions of the used Tx antennas. DPulse and t can be chosen independently. The planar
antenna array is located in the x, y-plane and the two unknown DOAs of the target are the azimuth ϑ
and elevation ϕ, cf. Fig. 5.7. The unit vector in the direction of the target in the Cartesian coordinate
system is
uunit =

ux
uy
uz
 =

cosϕ sinϑ
sinϕ
cosϕ cosϑ
 ∈ R3. (5.47)
For convenience, we did not choose the usual spherical coordinate system but we chose the coor-
dinate system such that ϑ and ϕ denote the azimuth and elevation, respectively. Since the array is
located in the x, y-plane, we need only the two electrical angles
u =
uxuy
 =
cosϕ sinϑsinϕ
 ∈ R2 (5.48)
in the following. The steering vector of the Rx array is
aRx(u) = exp( j DRxu) ∈ CNRx . (5.49)
The steering vector of the Tx array with the transmitting sequence DPulse is given by
aPulse(u) = exp( j DPulseu) ∈ CNPulse . (5.50)
Our model consists of L measurement cycles, where each cycle is made up of NPulse pulses as ex-
plained in Section 4.1. One cycle is one coherent processing interval (CPI). The transmission se-
– 85 –
quence of one cycle is given in DPulse and remains the same for all L measurement cycles. The
received baseband signal of the virtual array for cycle l is
x(l) =
{ √
ρ  exp( j tω)  aPulse(u)
}
⊗ aRx(u) s(l) + n(l) ∈ CNPulse·NRx , l = 1, . . . , L, (5.51)
where s(l) ∈ C is the unknown, deterministic complex target signal and n(l) the noise. ρ ∈ RNPulse
contains the transmission energies of the pulses, i. e. ρi is the energy of pulse i. The square root is to
be understood as an element by element operation, i. e. for any vector y ∈ RK , the expression z = √y
means that zi =
√
yi ∀i = 1, . . . ,K.  denotes the entrywise Hadamard product. The pulse energies
are given by the Hadamard product of the transmission powers and pulse durations
ρ = p  ∆t. (5.52)
ω is the target’s angular Doppler frequency. The term exp( j tω) describes the phase change in the
received signal x(l) due to the Doppler effect caused by the movement of the target, cf. (5.40).
This phase change has been derived in detail in Section 5.1.1. Note that in the model (5.51), the
transmission time ti is coupled to the position of the transmitting antenna dPulse,i, which is contained
in aPulse(u). Hence the time domain is coupled to the spacial domain and because of this, the model
has a different structure than many models in the literature which are space-time separable, e. g. the
SIMO radar model in [Dogandzic and Nehorai, 2001], and it does not satisfy the sufficient conditions
of space-time separability in [Pasupathy and Venetsanopoulos, 1974].
We define the new steering vector
b(u, ω) =
{ √
ρ  exp( j tω)  aPulse(u)
}
⊗ aRx(u) ∈ CNVirt , (5.53)
NVirt = NPulse · NRx, (5.54)
to write the signal as
x(l) = b(u, ω) s(l) + n(l) ∈ CNVirt , l = 1, . . . , L. (5.55)
For convenience, we write b(u, ω) in another way
b(u, ω) =
√
ρVirt  exp( j tVirtω)  aVirt(u) ∈ CNVirt , (5.56)
where we have defined
ρVirt = ρ ⊗ 1NRx ∈ RNVirt , (5.57)
tVirt = t ⊗ 1NRx ∈ RNVirt , (5.58)
aVirt(u) = aPulse(u) ⊗ aRx(u) ∈ CNVirt . (5.59)
– 86 –
aVirt(u) is the steering vector of the virtual array which includes all Rx-Tx combinations of all trans-
mitted pulses. It can also be written as
aVirt(u) = exp( j DVirtu) (5.60)
where the matrix with the positions of the virtual array is given by
DVirt = 1NPulse ⊗ DRx + DPulse ⊗ 1NRx ∈ RNVirt×2. (5.61)
Note that the transmission sequence is already incorporated in DPulse and thus in DVirt.
We make the following assumptions:
• n(l) is circular complex Gaussian with zero mean, spatially and temporally uncorrelated with
E
(
n(l) nH(m)
)
= δl,m σ
2 I. An explanation for this assumption is given in Section 4.2.1.
• Let τprop be the propagation time of the signal across the array and BW the bandwidth of the
signal. We assume that τprop  1BW , known as the narrowband array assumption [Dogandzic
and Nehorai, 2001].
• The target’s distance to the MIMO radar is much larger than the aperture of the radar such that
we have a far field situation. Hence the radar receives a plane wave and the radar cross section
as well as the target’s DOAs ϑ, ϕ, and therefore also the electrical angles u are the same for all
antennas.
• The DOAs do not change significantly during the L measurement cycles, i.e. the change is
much smaller than the DOA accuracy of the radar and can thus be ignored.
• The target moves with constant relative radial velocity during the L measurement cycles.
Hence the Doppler frequency ω is constant.
Thus the unknown quantities Θ to be estimated from x(l), 1 ≤ l ≤ L, are
Θ =
[
ux, uy, ω, s(1), . . . , s(L), σ2
]T
. (5.62)
Compared to a stationary target, we have to estimate the Doppler frequency ω additionally.
As an example, consider a radar with a carrier frequency of 76.5 GHz in an automotive application.
This corresponds to a carrier wavelength λ ≈ 4 mm. The antenna array is a rectangular array with
4 × 4 antennas spaced with half wavelength spacing. Thus the geometrical size of the antenna array
is approximately 6 mm×6 mm. Let the cycle time of the radar be 15 ms and consider a measurement
with L = 1 cycle. The target, another car, is located on the neighboring driving lane with a lateral
distance of 4 m and a radial distance of 30 m. It is moving ahead with a relative velocity of 15 ms .
The radar scattering center is located at the same height as the radar. This corresponds to the DOAs
ϕ = 0◦ and ϑ = 7.7◦. Obviously, the far field condition is fulfilled. The change of ϑ during the
– 87 –
measurement is 0.06◦, which is much smaller than the typical DOA accuracy of such a radar system.
On the other hand, the phase change during the measurement due to the Doppler effect is 113.8 · 2pi,
which is significant. The radial relative velocity changes by 10−2 %, therefore it can be neglected.
5.1.2.2. Notes and Discussion of the Model
In the model (5.55), the electrical angles u and the Doppler frequency ω have to be estimated jointly,
in addition to the other parameters in (5.62).
In a usual pulse Doppler and chirp sequence radar, the Doppler frequency ω, and therefore the
velocity v, is estimated from the phase shift from pulse to pulse, which is given by exp( j tω) in
(5.53). This is called the slow time domain. The successive estimation of ω and u, i. e. first ω is
estimated and using this result u is estimated, is worse or at most equally good as the joint estimation
of ω and u. Due to that, we investigate the estimation of both at once. In general, this introduces a
higher computational burden. As we will see in Section 5.2.3.2, by exploiting certain properties of
the signal model, the computational burden can be diminished.
In an FMCW radar, the velocity v of the target, and therefore its Doppler frequency, is estimated
before the DOA estimation. In general, the estimation error of v is too large to be used for the
DOA estimation. We show this in the following example: Consider a radar with carrier frequency
f0 = 76.5 GHz. An automotive radar can typically achieve an estimation accuracy of the velocity
σv = 0.1 ms . Using (5.39), the phase change between the first and second pulse is ∆ϕ12 = ω∆t12.
Plugging in the Doppler frequency (5.27) leads to ∆ϕ12 = 2 vc 2pi f0 ∆t12. A typical chirp duration is
about 5 ms. Hence we set ∆t12 = 5 ms. Then the error in the phase change is σ∆ϕ12 = 2
σv
c 2pi f0 ∆t12 ≈
1.6 ≈ pi/2. This is much too large to be used in the DOA estimation. The error in the phase shift
between the first and k-th pulse, k = 3, 4, . . . , is even larger. Thus the Doppler frequency and the
electrical angles have to be estimated jointly, together with the other parameters given in (5.62). The
result of the estimation of the Doppler frequency in the DOA estimation process could be used to
refine the first velocity estimation of the FMCW radar, which is done before the DOA estimation.
5.1.2.3. Linear Array
Now we consider a TDM MIMO radar where the Tx and Rx antennas are arranged in a linear array.
W. l. o. g. let the linear array be situated on the x-axes of the Cartesian coordinate system. The signal
model is a special case of the signal model introduced in Section 5.1.2.1. The positions of the Rx
and Tx antennas are just real numbers
dRx,1, . . . , dRx,NRx ∈ R, (5.63)
dTx,1, . . . , dTx,NTx ∈ R, (5.64)
– 88 –
respectively. The same is valid for the positions of the Tx antennas in the sequence in which they
transmit
dPulse,1, . . . , dPulse,NPulse ∈ R. (5.65)
The matrices DRx in (5.44) and DPulse in (5.46) simplify to vectors, which we denote by dRx and
dPulse, respectively.
dRx =
[
dRx,1 . . . dRx,NRx
]T
∈ RNRx , (5.66)
dPulse =
[
dPulse,1 . . . dPulse,NPulse
]T
∈ RNPulse . (5.67)
For convenience, we also define
dTx =
[
dTx,1 . . . dTx,NTx
]T
∈ RNTx . (5.68)
Since we consider a linear antenna array, we can only estimate one DOA. W. l. o. g. let the target
be located in the x, z-plane, i. e. ϕ = 0◦, cf. Fig. 5.7. Then the unknown DOA ϑ, the azimuth, is
measured perpendicular to the linear array as depicted in Fig. 5.8. For convenience, we denote the
x-component of the target’s electrical angle by u. It is given in (5.48) and here simplifies to
u = ux = sin(ϑ) ∈ R. (5.69)
We make the same assumptions as in Section 5.1.2.1. Hence we can write the signal model as a
special case of (5.55)
x(l) = b(u, ω) s(l) + n(l), l = 1, . . . , L, (5.70)
with
b(u, ω) =
{ √
ρ  exp( j tω)  aPulse(u)
}
⊗ aRx(u) ∈ CNVirt , (5.71)
aPulse(u) = exp( j dPulseu) ∈ CNPulse , (5.72)
aRx(u) = exp( j dRxu) ∈ CNRx . (5.73)
As in Section 5.1.2.1, we express b(u, ω) in another way:
b(u, ω) =
√
ρVirt  exp( j tVirtω)  aVirt(u) ∈ CNVirt , (5.74)
aVirt(u) = aPulse(u) ⊗ aRx(u) = exp( j dVirtu) ∈ CNVirt , (5.75)
dVirt = 1NPulse ⊗ dRx + dPulse ⊗ 1NRx ∈ RNVirt . (5.76)
– 89 –
The other variables remain the same as in Section 5.1.2.1, which we repeat here for convenience
s(l) ∈ C, (5.77)
ρVirt = ρ ⊗ 1NRx ∈ CNVirt , (5.78)
tVirt = t ⊗ 1NRx ∈ CNVirt . (5.79)
Given the observations x(l), l = 1, . . . , L, the unknown quantities Θ to be estimated are
Θ = [u, ω, s(1), . . . , s(L), σ2]T . (5.80)
z
x
target
ϑ
antenna array
Figure 5.8.: Linear antenna array. The DOA ϑ is measured perpendicular to the antenna array.
5.1.2.4. TDM Scheme
The TDM scheme which is used in the TDM MIMO radar is characterized by the parameters
ψTDM =
{
DPulse, t, ρ,NPulse
}
(5.81)
with
DPulse ∈ RNPulse×2, t, ρ ∈ RNPulse . (5.82)
For a linear array, DPulse simplifies to dPulse and
ψTDM =
{
dPulse, t, ρ,NPulse
}
. (5.83)
NPulse are the number of transmitted chirps, t the transmission times and ρ the energy of the transmit-
ted pulses. DPulse describes the sequence in which the Tx antennas transmit. DPulse contains also the
number of used Tx antennas, since different positions in DPulse correspond to different Tx antennas.
Note that ψTDM contains both discrete variables (NPulse) and continuous variables (t, ρ, positions in
DPulse).
– 90 –
In the following, we will see that the DOA performance of the TDM MIMO radar depends on the
chosen TDM scheme, i. e. on ψTDM. Thus, for a good DOA performance, ψTDM has to be optimized.
5.2. One Moving Target
We study the performance in DOA estimation of one moving target. In Section 5.2.1 we derive
the CRB for the DOA estimation using the model with a planar and linear array presented in Sec-
tion 5.1.2. We discuss the properties of the CRB and compare it to a SIMO radar and a MIMO radar
with a stationary target in Section 5.2.2. In Section 5.2.3, we present a theorem which gives suffi-
cient conditions such that the CRB of the DOAs for a moving target equals the CRB of the DOAs
for a stationary target, although the Doppler frequency of the target is not known. We investigate the
MIMO radar with a linear Rx an Tx array in more detail in Section 5.2.4. We derive optimal TDM
schemes in the sense that the CRB is minimized under different practical constraints.
5.2.1. Cramer Rao Bound
First we compute the CRB for a TDM MIMO radar with a planar array. After that we present the
CRB for a linear array.
5.2.1.1. CRB for a Planar Array
The unknown parameter vector to be estimated is given in (5.62). We are interested in that part of
the CRB of Θ which corresponds to the parameters u and ω. Similar to Section 4.2.2.1, we have
to obtain the FIM J of the whole parameter vector Θ and determine its inverse. From J−1, we take
the 3 × 3 block corresponding to [ux, uy, ω]T . This part is CRBu,ω. Using the results from [Yau and
Bresler, 1992],
CRB−1u,ω = 2 L
σ2s
σ2
Re(C) ∈ R3×3 (5.84)
where
C = DH P⊥b D ∈ C3×3, (5.85)
D =
[
∂b(u, ω)
∂ux
∂b(u, ω)
∂uy
∂b(u, ω)
∂ω
]
∈ CNVirt×3, (5.86)
P⊥b = I − Pb ∈ CNVirt×NVirt , (5.87)
Pb = b
(
bH b
)−1
bH ∈ CNVirt×NVirt , (5.88)
– 91 –
σ2s =
1
L
L∑
l=1
|s(l)|2 ∈ R. (5.89)
Note that Pb is the projection matrix on the subspace spanned by b. After some lengthy calculations,
cf. Appendix H, we can show that C is real and
CRB−1u,ω = 2 L
σ2s
σ2
C, (5.90)
C = NRx
(
1Tρ
) 

CovS(DRx)
0
0
0 0 0
 +
 CovWS(DPulse, ρ) CovWS(t,DPulse, ρ)CovWS(DPulse, t, ρ) VarWS(t, ρ)

 . (5.91)
CovS is the sample covariance or cross-covariance and CovWS the weighted sample covariance or
cross-covariance. VarWS is the weighted sample variance. The definitions and properties are given
in Appendix A. As an example for the weighted sample variance, we consider VarWS(t, ρ). It can be
written as
VarWS(t, ρ) =
1
1Tρ
NPulse∑
i=1
(
ti − EWS(t, ρ)
)2
ρi, (5.92)
EWS(t, ρ) =
1
1Tρ
NPulse∑
i=1
ti ρi. (5.93)
It has a similar form as the usual sample variance, but each value is weighted by the transmission
energy ρi of the i-th pulse.
5.2.1.2. CRB for a Linear Array
In the important case of a linear array of Tx and Rx antennas, the CRB can be computed completely
analog to the planar case, or it can be derived as a special case of the CRB for the planar array. We
use the latter approach which is much simpler. Setting
DRx =
[
dRx 0
]
, (5.94)
DPulse =
[
dPulse 0
]
, (5.95)
(5.91) reads
C = NRx
(
1Tρ
) 

VarS(dRx) 0 0
0 0 0
0 0 0
 +

CovWS(dPulse, ρ) 0 CovWS(t, dPulse, ρ)
0 0 0
CovWS(dPulse, t, ρ) 0 VarWS(t, ρ)

 . (5.96)
– 92 –
Logically, there is no information about uy, since the antennas are arranged on a linear array on the
x-axes. The inverse of the CRB for the electrical angle u = ux and ω is
CRB−1u,ω = 2 L
σ2s
σ2
Re(C) ∈ R2×2, (5.97)
with
C = NRx
(
1Tρ
) 
VarS(dRx) 00 0
 +
 CovWS(dPulse, ρ) CovWS(t, dPulse, ρ)CovWS(dPulse, t, ρ) VarWS(t, ρ)

 . (5.98)
We are interested in the CRB CRBu of the electrical angle u. Assuming
det
(
CRB−1u,ω
)
, 0,
it is given by
CRBu =
[
CRBu,ω
]
1,1
=
1
2 L
1
S
1
U
∈ R (5.99)
with
S =
σ2s
σ2
NRx
(
1Tρ
)
∈ R, (5.100)
U = VarS(dRx) + VarWS(dPulse, ρ) −
(
CovWS(dPulse, t, ρ)
)2
VarWS(t, ρ)
∈ R. (5.101)
Note that since ρi > 0, i = 1, . . . ,NPulse, and all values in t have to be different for a meaningful
physical system, VarWS(t, ρ) , 0.
5.2.2. Properties of the CRB
In the following, we discuss and interpret the derived CRB. Moreover, we proof important properties
of the CRB.
– 93 –
5.2.2.1. Structure
CRB−1u,ω in (5.90) has the structure
CRB−1u,ω = 2L
σ2s
σ2
NRx
(
1Tρ
)
·

Contribution of SIMO
0
0
0 0 0
 +

Contribution of MIMO moving
moving target

 (5.102)
The first part is determined by DRx, i. e. it is the contribution of the Rx array to the DOA estimation.
This 2 × 2-block can be expressed as
CovS(DRx) =

ux uy
ux VarS(dRx,x) CovS(dRx,x, dRx,y)
uy CovS(dRx,x, dRx,y) VarS(dRx,y)
 (5.103)
where dRx,x and dRx,y are the first and second column of DRx, respectively. They contain the x- and
y-positions of the Rx antennas, respectively. The small variables ux, uy on the outside of the matrix
in (5.103) depict the parameters to which the matrix entries correspond. VarS(dRx,x) describes the
information on ux gained by the different x-positions of the Rx antennas. It is the analog expression to
(4.16), which appears in the CRB for DOA estimation of one stationary target using a SIMO radar.
Analogously, VarS(dRx,y) denotes the information on uy. CovS(dRx,x, dRx,y) describes the coupling
between ux and uy due to the Rx antenna positions.
The matrix CovWS(DPulse, ρ) in (5.91) contains the contributions of the different Tx antenna po-
sitions to the DOA estimation. It has the same structure as CovS(DRx), but the sample co-
variance is weighted with the transmission energies of the different antennas. The expressions
CovWS(DPulse, t, ρ), CovWS(t,DPulse, ρ), and VarWS(t, ρ) in the second matrix in (5.91) are related to
the movement of the target. For convenience, we define dPulse,x and dPulse,y to be the first and second
column of DPulse, respectively. Then the off-diagonal matrix block in (5.91) reads
CovWS(t,DPulse, ρ) =

ω
ux CovWS(t, dPulse,x, ρ)
uy CovWS(t, dPulse,y, ρ)
 ∈ R2×1. (5.104)
It describes the coupling between ω and ux and the coupling between ω and uy. The term VarWS(t, ρ)
in (5.91) is the information on ω, gained by the different transmission times.
– 94 –
5.2.2.2. Comparison to CRB for a Stationary Target and to CRB for a SIMO Radar
We compute the CRB for DOA estimation of a stationary target and the CRB for a SIMO radar in
order to compare them to the general CRB we have derived.
CRB for the TDM MIMO Radar and a Stationary Target If the target is stationary, i. e. it does
not move relative to the radar, the Doppler frequency vanishes, i. e. ω = 0. If this is known a priori,
there is no need to estimate it. In this case, the CRB of the electrical angles CRBu,stat can be derived
by using the model (5.55) with ω = 0 and computing the CRB analogously to Section 5.2.1.1. In
this case, the unknown parameter vector Θ is
Θ =
[
ux, uy, s(1), . . . , s(L), σ2
]T
(5.105)
and the part of the CRB corresponding to u for the TDM MIMO radar for a stationary target is
CRB−1u,stat = 2 L
σ2s
σ2
Re(Cstat) ∈ R2×2 (5.106)
with
Cstat = DHstat P
⊥
b Dstat ∈ C2×2, (5.107)
Dstat =
[
∂b(u)
∂ux
,
∂b(u)
∂uy
]
∈ CNVirt×2. (5.108)
The other variables remain the same and are given in (5.87) to (5.89). Completely analogous com-
putations to Appendix H yield
CRB−1u,stat = 2 L
σ2s
σ2
Cstat ∈ R2×2, (5.109)
Cstat = NRx
(
1Tρ
) {
CovS(DRx) + CovWS(DPulse, ρ)
}
∈ R2×2. (5.110)
Here we see the information gain CovS(DRx) and CovWS(DPulse, ρ) due to the different Rx and Tx
antenna positions, respectively. CovWS(DPulse, ρ) depends on the positions of the Tx antennas in
DPulse, but not on the sequence in which they transmit, since CovWS(DPulse, ρ) remains the same if the
elements in DPulse and ρ are permuted in parallel.
In the case of a linear array, we substitute DRx and DPulse by dRx and dPulse, respectively, as in Sec-
tion 5.2.1.2. This results in
CRBu,stat =
1
2L
1
S
1
Ustat
∈ R, (5.111)
S =
σ2s
σ2
NRx
(
1Tρ
)
∈ R, (5.112)
Ustat = VarS(dRx) + VarWS(dPulse, ρ) ∈ R. (5.113)
– 95 –
Note that for pulses with equal transmission energy ρ ∝ 1NTx and using the notation σ2s,MIMO instead
of σ2s , this is the same expression as (4.29) in Section 4.3.2.
CRB for a SIMO Radar and a Moving Target The CRB for a SIMO radar and a moving target
can be deduced from the CRB in (5.90) and (5.91) by using only one Tx antenna, i. e. setting
DPulse =
[
dTx,1, . . . , dTx,1
]T ∈ RNPulse×2. (5.114)
Note that by doing so, we do not include the beamforming gain of a phased array with NTx Tx
antennas, which can be achieved by sending the same waveform phase shifted with all Tx anten-
nas. This is already discussed in Section 4.4.3.2. Due to (5.114), CovWS(DPulse, ρ) = 02×2 and
CovWS(t,DPulse, ρ) = 0 in (5.91), i. e. the DOAs always decouple from the Doppler frequency. This
is logical since there is no information gain on the DOAs by different Tx positions. Thus the move-
ment of the target has no influence on the CRB of the electrical angles. Due to (5.90) and (5.91), the
part of the CRB corresponding to the electrical angles using a planar SIMO radar is given by
CRB−1u,SIMO = 2 L
σ2s
σ2
CSIMO ∈ R2×2, (5.115)
CSIMO = NRx
(
1Tρ
)
CovS(DRx) ∈ R2×2, (5.116)
regardless of the movement of the target.
For the special case of a linear Rx array, the CRB for the electrical angle u can be derived from
(5.99) to (5.101) analogously. This results in
CRBu,SIMO =
1
2 L
1
S
1
USIMO
∈ R, (5.117)
S =
σ2s
σ2
NRx
(
1Tρ
)
∈ R, (5.118)
USIMO = VarS(dRx) ∈ R. (5.119)
This is the same expressions as the CRB for a stationary target computed in Section 4.2.2.1, (4.13)
if we use only one pulse with ρ = 1 and neglect the beampattern, i. e. B(u) = 1. Note that we used
the same number of pulses and the same pulse energies ρ for the SIMO and MIMO radar. Due to
this, the signal strength σ2s is the same for the SIMO and MIMO radar in contrast to Section 4.
Comparison Let us compare the CRBs for the MIMO radar with a moving target, for the MIMO
radar with a stationary target and for the SIMO radar given in (5.90), (5.109) and (5.115), respec-
tively. They differ only in the matrix C. For the sake of clarity, we repeat the matrices here without
the factor NRx
(
1Tρ
)
CSIMO ∝ CovS(DRx) ∈ R2×2, (5.120)
– 96 –
Cstat ∝
{
CovS(DRx) + CovWS(DPulse, ρ)
}
∈ R2×2, (5.121)
C ∝


CovS(DRx)
0
0
0 0 0
 +
 CovWS(DPulse, ρ) CovWS(t,DPulse, ρ)CovWS(DPulse, t, ρ) VarWS(t, ρ)

 ∈ R
3×3. (5.122)
Obviously, in the SIMO radar only the different Rx antenna positions DRx are relevant. In the sta-
tionary MIMO case, there is an additional gain CovWS(DPulse, ρ) due to the different Tx positions.
The MIMO radar with a moving target has the same gain as well, but also receives an additional
coupling term CovWS(t,DPulse, ρ). In general, this term leads to a larger CRB of the electrical angles.
The upper 2 × 2 block of CRBu,ω corresponds to u. We denote this block by CRBu. If the coupling
term CovWS(t,DPulse, ρ) vanishes, CRBu is equal to CRBu,stat, which is the following lemma. We use
the following notation: ”A =⇒ B“ means ”A is sufficient for B“.
Lemma 5.2.1. CovWS(t,DPulse, ρ) = 02×2 =⇒ CRBu = CRBu,stat.
Hence, the MIMO radar with the moving target can have the same DOA CRB as for a stationary
target, despite the unknown Doppler frequency. We show in Section 5.2.3 under which conditions
the coupling term vanishes and how the vanishing of the coupling can be interpreted.
Comparison: Linear Array We compare the CRBs in the special case of a radar system with a
linear array. The CRBs are given in (5.99), (5.111), (5.117) and differ by the term U:
USIMO = VarS(dRx), (5.123)
Ustat = VarS(dRx) + VarWS(dPulse, ρ), (5.124)
U = VarS(dRx) + VarWS(dPulse, ρ) − Upenalty, (5.125)
with Upenalty =
(
CovWS(dPulse, t, ρ)
)2
VarWS(t, ρ)
. (5.126)
In the moving target case, the performance degradation due to the coupling given by Upenalty can be
seen explicitly, since we could easily invert the CRB. Moreover, the following theorem holds, where
ψTDM is defined in (5.83).
Theorem 1. For a radar system with a linear array and for a fixed ψTDM, the following inequalities
hold for CRBu
CRBu,SIMO ≥ CRBu ≥ CRBu,stat, (5.127)
where CRBu,SIMO,CRBu,CRBu,stat are given in (5.117),(5.99) and (5.111), respectively. Moreover,
CRBu = CRBu,stat ⇐⇒ CovWS(dPulse, t, ρ) = 0. (5.128)
– 97 –
Here, the notation ”A ⇐⇒ B“ stands for ”A is necessary and sufficient for B“. The proof is given
in Appendix I.
Theorem 1 states that for a given TDM scheme, the CRB for a moving target is always bounded by
the CRB of the corresponding SIMO radar and the CRB of the same MIMO radar with the same
TDM scheme, but with a stationary target. Hence we can easily compute the MIMO gain compared
to the SIMO radar and the loss in accuracy due to the movement of the target. By using the same
MIMO radar hardware (NTx Tx antennas and NRx Rx antennas), the choice of the TDM scheme ψTDM
has a significant impact on the DOA accuracy. In the worst case CRBu = CRBu,SIMO, and in the best
case CRBu = CRBu,stat. The latter is achieved if the coupling term CovWS(dPulse, t, ρ) vanishes. This
is only one condition on the TDM scheme. ψTDM has further degrees of freedom, i. e.
• the position of the Tx antennas and the sequence in which they transmit, given in dPulse,
• the transmitting times t,
• the energy of the transmitted pulses ρ, and
• the number of transmitted pulses NPulse.
In order to find TDM schemes which minimize CRBu (5.99), ψTDM has to be optimized further. This
is studied in detail in Section 5.2.4.
We present 2 examples of the TDM scheme which actually reach the bounds of Theorem 1. We
consider an uniform linear array (ULA) of 4 Tx antennas with NPulse = 4, ρ ∝ 1 and t ∝ [0, 1, 2, 3]T .
Example 5.2.1 (Bad TDM Schemes). We set dPulse = c1 t + c2 with constants c1, c2. In this case,
Upenalty = VarS(dPulse) and CRBu = CRBu,SIMO. This is the case if the Tx antennas transmit in the
order of their geometric arrangement at equally spaced time instants, cf. Fig. 5.9.
Example 5.2.2 (Good TDM Scheme). The lower bound in (5.127) is reached when using the TDM
scheme dPulse = [dTx1 , d
Tx
4 , d
Tx
4 , d
Tx
1 ]
T , since CovWS(dPulse, t, ρ) = 0 is satisfied. This TDM scheme is
also depicted in Fig. 5.9.
Time
Tx 1
Tx 2
t1 t2 t3 t4
Tx 3
Tx 4
good TDM
bad TDM
Transmitting
antenna
Figure 5.9.: Good and bad TDM scheme for an ULA of 4 Tx antennas and t ∝ [0, 1, 2, 3]T
– 98 –
Simulations We present numerical simulations to verify and foster the understanding of the theo-
retical results. We examine a MIMO radar with a linear antenna array of Rx and Tx antennas and
consider the maximum likelihood (ML) estimator of the unknown parameter vector Θ in (5.80). We
determine the root mean square error (RMSE) of the ML estimate uˆ by doing Monte-Carlo simu-
lations. We do this for different TDM schemes. According to (E.23), given the signal x, the ML
estimates uˆ, ωˆ can be computed by  uˆ
ωˆ
 = arg maxu,ω | b(u, ω)H x |2. (5.129)
The MIMO radar under consideration consists of 4 Rx and 4 Tx antennas, uniformly spaced with
an antenna distance of λ/2, i.e. dRx = dTx = pi · [0, 1, 2, 3]T . We choose L = 1 cycle with NPulse = 4
pulses and set the transmitting time instants to t = [0, 1, 2, 3]T , where t is normalized such that it is
dimensionless. Moreover, ρ = 1NPulse 1, i. e. all pulses are transmitted with the same energy. The tar-
get’s electrical angle u is set to u = sin(10◦) and, for the non-stationary case, the Doppler frequency
ω is chosen to be ω = 1.3.
First we consider a TDM scheme with dPulse = dTx, i. e. it is has the form of the bad TDM scheme
described in Example 5.2.1 and is depicted in Fig. 5.9. The CRB CRBu,SIMO of a SIMO radar with
the same Rx antennas, using only one Tx antenna with NPulse = 4, is computed as the upper bound for
CRBu. The CRB CRBu,stat of a MIMO radar and a stationary target with the same dPulse is computed
as the lower bound for CRBu. The simulations are done for a total SNR S from 0 to 35 dB. 3000
Monte-Carlo simulations are carried out for each value of S in order to determine the RMSE of the
ML estimator. The ML estimates are computed by doing a 2-dimensional search on a grid, with a
resolution of 3.5 · 10−4 for the electrical angle and 6 · 10−4 pi for the normalized Doppler frequency,
followed by a quadratic interpolation. The simulations show that the modulus of the bias |buˆ| is much
smaller than the standard deviation σuˆ and thus negligible. Fig. 5.10 shows that the ML estimator
of the TDM MIMO radar with the bad TDM scheme achieves only the upper bound CRBu,SIMO as
expected. Hence, in this case, there is no MIMO gain at all compared to the SIMO radar.
Next we consider a TDM scheme which uses only the two outermost Tx antennas dTx1 , d
Tx
4 and choose
dPulse = [dTx1 , d
Tx
4 , d
Tx
4 , d
Tx
1 ]
T , i. e. the good TDM scheme of Example 5.2.2. The other parameters
of the simulation are the same as in the former simulation. Again, we compute the upper bound
CRBu,SIMO, which is the same as in the former simulation, and the lower bound CRBu,stat using
the same TDM scheme dPulse = [dTx1 , d
Tx
4 , d
Tx
4 , d
Tx
1 ]
T . For comparison, we carry out an additional
simulation to compute the RMSE of uˆ for the MIMO radar with a stationary target. Both simulations
show that the bias is negligible compared to the standard deviation. The CRBs and the RMSE of
the ML estimators are depicted in Fig. 5.11. It shows that the ML estimator with the good TDM
scheme reaches indeed the lower bound CRBu,stat, as does the ML estimator in the stationary case.
The threshold values of S at which the ML estimators reach the CRB is approximately the same for
the stationary and non-stationary case. Note that the ML estimator using the bad TDM scheme in
– 99 –
0 5 10 15 20 25 30 35
10−3
10−2
10−1
100
Total SNR S in dB
R
M
SE
of
el
ec
tr
ic
al
an
gl
e
u SIMO CRBu,SIMO
MIMO stat. CRBu,stat
MIMO non-stat. ML bad TDM
Figure 5.10.: MIMO radar DOA estimation with a bad TDM scheme. The ML estimator achieves
only CRBu,SIMO
0 5 10 15 20 25 30 35
10−3
10−2
10−1
100
Total SNR S in dB
R
M
SE
of
el
ec
tr
ic
al
an
gl
e
u SIMO CRBu,SIMO
MIMO stat. CRBu,stat
MIMO stat. ML
MIMO non-stat. ML good TDM
Figure 5.11.: MIMO radar DOA estimation with a good TDM scheme. The ML estimator achieves
CRBu,stat
Fig. 5.10 achieves only CRBu,SIMO, which is the same in Fig. 5.10 and 5.11. Hence, with the same
hardware platform, we can achieve a gain of 4.47 dB for the same DOA accuracy by just changing
the transmission sequence dPulse of the Tx antennas.
5.2.3. Decoupling of Electrical Angles from the Doppler Frequency
In the last section we have shown that if the coupling term vanishes, i. e. CovWS(DPulse, t, ρ) = 0,
the electrical angles decouple from the Doppler frequency in the CRB. Moreover, the CRB of the
electrical angles for the non-stationary and stationary target are equal. In the following, we present
a theorem which gives conditions for a TDM scheme such that the coupling term CovWS(DPulse, t, ρ)
vanishes. Moreover, we interpret the vanishing of the coupling term to improve the understanding.
– 100 –
We present several examples and some numerical simulations to verify the theoretical findings.
5.2.3.1. Theorem for Decoupling of the Electrical Angles from the Doppler Frequency
We denote by t(k) the time instances when the k-th Tx antenna transmits and by ρ(k) the corresponding
energies of the transmitted pulses, cf. Fig. 5.12 for an example.
Theorem 2.
• It holds
EWS
(
t(k), ρ(k)
)
= EWS
(
t(l), ρ(l)
)
∀ k, l ∈ {1, . . . ,NTx}
=⇒ CovWS
(
DPulse, t, ρ
)
= 0. (5.130)
• In the case of two transmitting Tx antennas, i. e.
NTx = 2, dTx,1 , dTx,2, 1Tρ(1), 1Tρ(2) > 0,
EWS
(
t(1), ρ(1)
)
= EWS
(
t(2), ρ(2)
)
⇐⇒ CovWS
(
DPulse, t, ρ
)
= 0. (5.131)
Note that for a linear array, CovWS
(
DPulse, t, ρ
)
= 0 in (5.130) and (5.131) simplifies to
CovWS
(
dPulse, t, ρ
)
= 0 and dTx,1 , dTx,2 becomes dTx1 , d
Tx
2 .
Proof. Here we present just the main idea of the proof. For the detailed proof see Appendix J.
Let us consider the case of a linear array. dPulse contains only NTx different values, the positions of
the Tx antennas. It can be shown that CovWS(dPulse, t, ρ) = CovWS(dTx, t˜, ρ˜), where t˜, ρ˜ ∈ RNTx can be
computed from t, ρ. Thus, the weighted covariance of NPulse-dimensional vectors can be reformulated
as weighted covariance of NTx-dimensional vectors. Therefore, the number of different elements in
dPulse is decisive. The elements of t˜ are t˜k = EWS(t(k), ρ(k)). If all elements in t˜ are equal, which is
the condition in the theorem, t˜ ∝ 1 and therefore CovWS(dTx, t˜, ρ˜) = CovWS(dPulse, t, ρ) = 0. This is
stated in (5.130).
Time
Tx 1
Tx 2
Transmitting
antenna
t1
ρ2 ρ3 ρ4
ρ1 ρ5
t2 t3 t4 t5
Figure 5.12.: Example of two transmitting antennas with t(1) = [t1, t5]T , ρ(1) = [ρ1, ρ5]T and
t(2) = [t2, t3, t4]T , ρ(2) = [ρ2, ρ3, ρ4]T
– 101 –
5.2.3.2. Discussion
In the following, we discuss some consequences of the theorem. First let us consider the case of
equal transmission energy, i. e. ρ ∝ 1. Thus the weighted sample mean EWS simplifies to simple
sample mean ES. From Theorem 2 it follows: If the mean transmission time of every Tx antenna
is the same, then the electrical angles decouple from the Doppler frequency in the CRB. This can
be roughly interpreted in the following way: The mean transmission time ES(t(k)) is the effective
measurement time instance of the data gained from Tx antenna k. Equal mean transmission times
means that the effective measurement time instances of all Tx antennas are the same. In average,
this results in the same Doppler phase for all Tx antennas. Therefore, the movement of the target
does not influence the DOA estimation. This is what Theorem 2 tells us. If ρ 6∝ 1, the transmission
times of the k-th Tx antenna are weighted by the corresponding weights in ρ in order to compute the
effective measurement time of the k-th antenna EWS(t(k), ρ(k)).
We illustrate this interpretation by the following examples. Consider a linear antenna array with 2
Rx and 4 Tx antennas. The antenna positions are dRx = ∆Rx · [0, 1]T and dTx = ∆Tx · [0, 1, 2, 3]T ,
where ∆Rx and ∆Tx denotes the inter antenna spacing, cf. Fig. 5.13. We consider one cycle L = 1
and set NPulse = 4, ρ ∝ 1 and t = ∆t · [0, 1, 2, 3]T , with ∆t being the time difference between two
successive pulses. The steering vector b(u, ω) in (5.74) is written as
b(u, ω) =
√
ρVirt  exp( j φVirt) (5.132)
with the definition of the phase of the virtual array
φVirt = dVirtu + tVirtω. (5.133)
We consider different TDM schemes. First we present the bad TDM scheme
dPulse = [dTx1 , d
Tx
2 , d
Tx
3 , d
Tx
4 ]
T depicted in Fig. 5.9. In Fig. 5.14a and 5.14b, φVirti is shown for the
different antenna positions of the virtual array for a stationary and a moving target, respectively. For
the stationary target, ω = 0, φVirt changes proportional to the antenna positions of the virtual array. If
ω , 0, the movement of the target introduces an additional stepwise increment of φVirt for every Tx
antenna which transmits. Hence the phase change due to the different Tx antenna positions contains
a phase change due to the DOA and due to Doppler effect. In this example, there is no information
Rx array
dRx1 d
Rx
2
∆Rx
Tx array
dTx1 d
Tx
2
∆Tx
dTx3 d
Tx
4
dTxid
Rx
i
Figure 5.13.: Rx and Tx antenna positions of the considered example
– 102 –
dVirti
φVirti
(a) ω = 0
dVirti
φVirti
ω · ∆t
(b) ω , 0
Figure 5.14.: Change of the virtual phase φVirti , bad TDM
dVirti
φVirti
(a) ω = 0
dVirti
φVirti
ω · ∆t
(b) ω , 0
Figure 5.15.: Change of the virtual phase φVirti , bad TDM, with an Rx array containing only 1 Rx
antenna
gain due to the different Tx antennas. This can be understood as follows: Let us assume that the Rx
array contains only one Rx antenna. Then, the virtual array is the same as dPulse. For this scenario,
the changes of φVirti are depicted in Fig. 5.15. For a moving target, it cannot be distinguished if the
phase change origins from an electrical angle u , 0, from the Doppler effect or from a combination
of both. Hence the different Tx antennas do not yield additional information on the DOA estimation.
Next consider the good TDM scheme dPulse = [dTx1 , d
Tx
4 , d
Tx
4 , d
Tx
1 ]
T which is also shown in Fig. 5.9.
The change of φVirti is plotted in Fig. 5.16. For ω = 0, the virtual phases are located on a straight
line. Some of the virtual antenna positions coincide, but are depicted just next to each other for
illustrative purposes. The corresponding virtual phases are depicted in the same manner. For ω , 0,
there is again an additional Doppler phase change. The average phase change due to the Doppler
effect for each Tx antenna is symbolized by the red and blue dot. In this example, the average
transmission time for the first and fourth Tx antenna is EWS(t(k), ρ(k)) = ES(t(k)) = 1.5 ∆t, k = 1, 4.
Hence the average Doppler phase change is the same for both Tx antennas, as depicted in Fig. 5.16b,
and equals ω · 1.5 ∆t. Then, due to Theorem 2, CovWS(dPulse, t, ρ) = 0 and therefore, because of
Theorem 1, CRBu = CRBu,stat. For comparison, we also plot the change φVirti for a SIMO radar in
Fig. 5.17. Here, dPulse = [dTx1 , d
Tx
1 , d
Tx
1 , d
Tx
1 ]
T , i. e. only one Tx antenna transmits.
Theorem 2 gives us an important design criterion for TDM schemes: If the TDM scheme satisfies
the conditions in (5.130), then the CRB of the electrical angles for a moving target is as large as for
a stationary target, despite the unknown Doppler frequency. Thus the whole virtual aperture of the
MIMO radar can be used for the DOA estimation.
– 103 –
dVirti
φVirti
(a) ω = 0
dVirti
φVirti
ω · 1.5 ∆t
(b) ω , 0
Figure 5.16.: Change of the virtual phase φVirti , good TDM
dVirti
φVirti
4-
tim
es
4-
tim
es
(a) ω = 0
dVirti
φVirti
ω · ∆t
(b) ω , 0
Figure 5.17.: Change of the virtual phase φVirti , SIMO array
Note that if the electrical angles are decoupled from the Doppler frequency in the CRB, then the
DOAs ϑ, ϕ are decoupled from the Doppler frequency in the corresponding CRB as well. The
Jacobian matrix of the transform ψ = ψ(Θ) with ψ = [ϑ, ϕ, ω, s(1), . . . , s(L), σ2]T and Θ given in
(5.62) is used to compute the CRB of ψ from the CRB of Θ. The transform between ux, uy and ϑ, ϕ
is given in (5.48). Since the Jacobian matrix is block diagonal, the statement ”if ux, uy are decoupled
from ω in the CRB, then ϑ, ϕ are decoupled from ω in the CRB as well“ is obvious.
To foster the understanding, we present several examples.
Example: Planar Antenna Array. We investigate a radar system with an Rx array
DRx = pi
0 0 1 10 1 0 1

T
and a Tx array DTx = pi
0 00 2

T
in the unit of λ2pi . The radar transmits at uni-
−0.5 0 0.5 1
0
1
2
x in λ
y
in
λ
(a) Rx array
−0.5 0 0.5 1
0
1
2
x in λ
(b) Tx array
−0.5 0 0.5 1
0
1
2
x in λ
(c) Virtual array
Figure 5.18.: Positions of the antenna arrays
– 104 –
t1 t2 t3
Tx 2
Tx 1
Timet4
Transmitting
antenna
Figure 5.19.: Transmission sequence which satisfies the condition of Theorem 2
formly spaced times instances t ∝ [0, . . . ,NPulse − 1]T with equal pulse energies ρ ∝ 1. The antenna
positions are depicted in Fig. 5.18. First consider the transmission sequence DPulse =
[
dTx,1, dTx,2
]T
,
i. e. both Tx antennas transmit once, one after the other. This TDM sequence does not satisfy the
condition of Theorem 2. By computing the CRB, we can show that the CRB of the electrical angles
CRBu is the same as that of a SIMO radar with DPulse =
[
dTx,1, dTx,1
]T
, i. e. only one Tx antenna
transmits twice. Thus the MIMO radar does not yield any gain despite of the different Tx antennas.
The reason is that it cannot be distinguished if the phase difference of the received signals from the
two different Tx antennas is due to the movement of the target or due to its DOAs. Now we consider
the transmission sequence DPulse =
[
dTx,1, dTx,2, dTx,2, dTx,1
]T
, which is depicted in Fig. 5.19. Due
to its symmetry, it satisfies the condition of the theorem and thus achieves the same CRBu as that
of a MIMO radar with a stationary target. In other words, the choice of a suitable TDM sequence
DPulse satisfying (5.130) can significantly increase the DOA accuracy without changing the Rx and
Tx arrays.
Example: Local Decoupling of a Linear Array. We consider a linear Rx and Tx array. The Rx ar-
ray is a 4 element ULA with half wavelength spacing, i. e. dRx = pi · [0, 1, 2, 3]T , and the positions of
the Tx antennas are dTx = pi · [0, 3]T . We choose L = 1 cycle with NPulse = 4 pulses and set the trans-
mitting time instants to t = [0, 1, 2, 3]T . Here t is normalized such that it is dimensionless. The trans-
Figure 5.20.: Beampattern for dPulse,a. u and ω do not decouple.
– 105 –
Figure 5.21.: Beampattern for dPulse,b. u and ω are decoupled in the vicinity of the global maximum.
mitted pulses have the same energy, ρ = 1. The target’s parameters are u0 = 0.2 and ω0 = 0.3 · 2 pi.
We compare two TDM schemes dPulse,a =
[
dTx1 , d
Tx
1 , d
Tx
2 , d
Tx
2
]T
and dPulse,b =
[
dTx1 , d
Tx
2 , d
Tx
2 , d
Tx
1
]T
. We
consider the beampattern
B(Θ) =
∣∣∣aH(Θ0) a(Θ)∣∣∣2 1‖a(Θ)‖2 (5.134)
with the parameter vector
Θ = [u, ω]T (5.135)
and the true parameter vector
Θ0 = [u0, ω0]
T . (5.136)
We normalize B(Θ) such that the maximum value equals 1. We denote this by Bnorm(Θ). B(Θ) is
the expression which is maximized to obtain the ML estimator Θˆ (E.23), if we replace the measured
signal x(1) by a(Θ0) s(1), i. e. we ignore the noise. In Fig. 5.20, Bnorm is depicted for the TDM scheme
dPulse,a. Note that this TDM scheme does not satisfy the decoupling condition (5.131). Fig. 5.20
shows that the contour lines of the main peak are ellipses and the main axes of the ellipses are
rotated with respect to the coordinate axes, i. e. the electrical angle and the Doppler frequency are
coupled. The TDM scheme dPulse,b satisfies the decoupling condition. The normalized beampattern
for this case is depicted in Fig. 5.21. Here the main axes of the elliptic contour lines are parallel to
the coordinate axes, i. e. u and ω decouple. Since the decoupling condition is based on the CRB, it is
only valid locally, i. e. in the vicinity of the main peak. This can also be seen in Fig. 5.21: There is no
global decoupling. Hence, for a fixedω , ω0 the maximum w. r. t. u can be at a different value u , u0.
As long as the main peak is concerned, the maximum w. r. t. u is at the same value u0 independent
of ω. This property can be used to minimize the computational cost of estimation algorithms. For
– 106 –
example, only the data gained by one Tx antenna can be used to estimate ω roughly. Then, by an
one-dimensional search in u, the main peak can be easily identified. With the knowledge of the local
decoupling of u and ω, local optimization algorithms can be used to find the exact maximum.
5.2.3.3. Numerical Simulations
To verify the theoretical results of Theorem 2, we present numerical simulations. We study a planar
antenna array with a single moving target in one experiment and a stationary target in a second
experiment. For the moving target, the signal model (5.55) has the same structure as the model (E.1)
in Appendix E. Therefore, according to (E.23), the deterministic ML estimates of ux, uy, ω can be
computed by 
uˆx
uˆy
ωˆ
 = arg maxux,uy,ω
L∑
l=1
|b(ux, uy, ω)H x(l)|2. (5.137)
Note that the unknown parameter vector of the ML estimator is Θ in (5.62) and not only ux, uy and
ω.
In the case of the stationary target, the signal model is given by (5.55) with ω = 0. The ML estimates
of ux, uy can be computed analogouslyuˆxuˆy
 = arg max
ux,uy
L∑
l=1
|b(ux, uy, ω = 0)H x(l)|2. (5.138)
By doing Monte-Carlo simulations, we estimate the covariance matrix Cuˆ of uˆ. The error ellipse of uˆ
is defined by (uˆ− u)T Cuˆ(uˆ− u) = r2, where r is a constant.
√
det Cuˆ is proportional to the area of the
error ellipse [McWhorter and Scharf, 1993]. Hence
√
det Cuˆ is a measure of the area of the region in
which the target is supposed to be. It contains the variance of uˆx, of uˆy as well as the correlation of
both. Under some regularity conditions, CRBu is a lower bound for Cuˆ for any unbiased estimator.
The simulations show that the modulus of each element in buˆb
T
uˆ is much smaller then the modulus
0 1 2 3
0
1
2
x in λ
y
in
λ
(a) Rx array
0 1 2 3
0
1
2
x in λ
(b) Tx array
0 1 2 3
0
1
2
x in λ
(c) Virtual array
Figure 5.22.: Positions of the antenna arrays used in the simulation
– 107 –
0 5 10 15 20 25 30 35 40
10−6
10−5
10−4
10−3
10−2
10−1
100
Total SNR S in dB
√ det
C
R
B
u,
√ det
C
uˆ
CRBu,stat
Cuˆ, MLstat
Cuˆ, MLmov
CRBu,SIMO
Figure 5.23.: Comparison: Estimation of DOAs of a moving target and stationary target. The TDM
scheme satisfies the decoupling condition of Theorem 2.
of the corresponding element in the covariance matrix Cuˆ, where buˆ is the bias of uˆ. Therefore, buˆ is
negligible. Due to this, we compute
√
det CRBu and compare it to
√
det Cuˆ. The radar system has
the following parameters: The number of measurement cycles is L = 1 and all pulses are transmitted
with equal energy (p = 1) and transmission times t = [0, . . . ,NPulse−1]T . ρ and t are normalized such
that they are dimensionless. The Rx and Tx antenna arrays are given by DRx = pi
0 0 1 10 1 0 1

T
and
DTx = pi
0 3 30 0 2

T
, respectively. The positions of the antenna arrays are depicted in Fig. 5.22. The
transmission sequence is DPulse =
[
dTx,1, dTx,2, dTx,3, dTx,3, dTx,2, dTx,1
]T
. Since every Tx antenna
transmits two times, all positions in DVirt appear twice. Due to the symmetry of the transmission
sequence DPulse, the condition (5.130) of Theorem 2 is satisfied. Consequently, the CRB of the
electrical angles for the moving and stationary target are equal. The target’s DOAs are ϑ = 10.1◦,
ϕ = 4.3◦ and the normalized angular Doppler frequency is zero for the stationary target andω = 0.4 pi
for the moving target. For convenience, we define the total SNR
S = NRx
(
1Tρ
) σ2s
σ2
. (5.139)
The simulations are run for different values of S , where 3000 trials are done for each value of S . The
ML estimates (5.137) and (5.138) are computed by a grid search followed by a local optimization.
The grid resolution is 2 · 10−2 for ux and uy, and 2 pi · 10−2 for ω. We use a simplex search method for
the local optimization, which is implemented in the programming language Matlab in the function
fminsearch. The results are depicted in Fig. 5.23. The ML estimators with the stationary (MLstat) and
moving target (MLmov) both achieve the CRB beyond the threshold point of approximately 15 dB.
– 108 –
0 5 10 15 20 25 30 35 40
10−6
10−5
10−4
10−3
10−2
10−1
100
Total SNR S in dB
√ det
C
R
B
u,
√ det
C
uˆ
CRBu,stat
CRBu
Cuˆ, MLmov
CRBu,SIMO
Figure 5.24.: Comparison: Estimation of DOAs of a moving target and stationary target. The TDM
scheme does not satisfy the decoupling condition of Theorem 2.
Beyond the threshold point, MLmov achieves the same value of
√
det Cuˆ as MLstat, despite of the
unknown Doppler frequency.
Now we do the experiment for a moving target with the same radar system again, but this time we
consider the TDM sequence DPulse =
[
dTx,1, dTx,1, dTx,2, dTx,2, dTx,3, dTx,3
]T
. It does not satisfy the
condition of Theorem 2. The bias is negligible as in the former simulation. For the stationary target,
CRBu,stat and the performance of MLstat are the same as with the former sequence DPulse, since every
Tx antenna transmits twice and the sequence in which they transmit is irrelevant. Fig. 5.24 shows
that MLmov reaches the CRB at the threshold of approximately 15 dB. But this time
√
det CRBu,stat <√
det CRBu <
√
det CRBu,SIMO, i. e. the radar system with a moving target does not reach the same
CRB as with a stationary target.
Thus, using the same radar hardware, i. e. the same Rx and Tx antennas, the DOA accuracy can be
significantly increased by choosing an appropriate TDM scheme.
5.2.4. Optimal TDM Schemes for Linear Arrays
We are interested in TDM schemes which minimize CRBu in (5.99) for a linear antenna array. In the
following, we call these schemes optimal TDM schemes. A TDM scheme is characterized by the
parameters ψTDM =
{
dPulse, t, ρ,NPulse
}
in (5.83). Theorem 2 states a sufficient condition under which
CovWS(dPulse, t, ρ) = 0 and therefore, due to Theorem 1, CRBu = CRBu,stat. ψTDM contains further
degrees of freedom which can be optimized to minimize CRBu.
Typically, applications introduce constraints on some of the parameters ψTDM. Important variables,
which can be used to express constraints are
• the transmission energy of the pulses ρ,
– 109 –
• the position of the Tx antennas in dPulse,
• the transmission power of the pulses p,
• the duration of the pulses ∆t,
• the minimal time gap between two successive pulses ∆gap, and
• the overall transmission time of one cycle Tcycle.
Dependent on the application these parameters can be already fixed, constrained to a certain region,
or they can be freely chosen. An example of a constrain is a limited transmission power or a limited
geometrical size of the antenna array. As another example, a constraint on the transmission times t
follows from a given ∆t and a given minimal ∆gap.
In general, different constraints result in different minimal values of CRBu. In the next sections, we
investigate optimal TDM schemes for two different cases. These sections are based on [Rambach
et al., 2014] and on the master thesis [Vogel, 2013] which I supervised.3
5.2.4.1. Optimal TDM Schemes Under Limited Transmission Energy and Aperture Size
We determine the minimum value of CRBu with respect to ψTDM (5.83) and derive conditions on
ψTDM under which this minimum is reached. We consider TDM schemes which satisfy the energy
constraint
1Tρ ≤ ρmax, (5.140)
i. e. the transmitted energy per cycle is limited by ρmax. Note that CRBu in (5.99) is scaled by
1/
(
1Tρ
)
. Obviously, (5.140) is a physically meaningful constraint. Moreover, the Tx antennas have
to be placed within the real aperture range
[
dTxmin, d
Tx
max
]
, i. e.
dTxmin ≤ dPulsei ≤ dTxmax, i = 1, . . . , NPulse. (5.141)
Theorem 3.
1. CRBu is minimal w. r. t. ψTDM if and only if
a) only the 2 outermost Tx antennas with dTx1 = d
Tx
min and d
Tx
2 = d
Tx
max are used,
b) 1Tρ(1) = 1Tρ(2),
c) 1Tρ = ρmax,
d) EWS(t(1), ρ(1)) = EWS(t(2), ρ(2)),
3These sections contain quotations of our own published work [Rambach et al., 2014] which are not marked addition-
ally.
– 110 –
where t(1) and t(2) are the time instances when the 1. and 2. Tx antenna transmit and ρ(1) and
ρ(2) are the energies of their transmitted pulses, respectively.
2. The minimal value of CRBu w. r. t. ψTDM is given by
CRBu,min =
1
2L
1
S max
1
Umax
, (5.142)
S max =
σ2s
σ2
NRx ρmax, (5.143)
Umax = VarS(dRx) +
1
4
(
dTxmax − dTxmin
)2
. (5.144)
Proof. 1. To minimize CRBu in (5.99), S in (5.100) and U in (5.101) have to be maximized.
S is maximal iff 1Tρ = ρmax, since the other variables are fixed. This is condition 1c. U is
independent of a scaling of ρ. To maximize U, we have to maximize
Ω = VarWS(dPulse, ρ) −
(
CovWS(dPulse, t, ρ)
)2
VarWS(t, ρ)
(5.145)
since dRx is given. VarWS(dPulse, ρ) is maximal iff only the two outermost Tx antennas are
used and 1Tρ(1) = 1Tρ(2). This is proven in Appendix K. Using only two Tx antennas,
due to Theorem 2 (5.131), CovWS(dPulse, t, ρ) = 0 iff EWS(t(1), ρ(1)) = EWS(t(2), ρ(2)). The two
conditions 1Tρ(1) = 1Tρ(2) and EWS(t(1), ρ(1)) = EWS(t(2), ρ(2)) do not contradict. An example
which fulfills both conditions is presented below. Thus, max
dPulse,ρ
VarWS(dPulse, ρ) is independent
whether CovWS(dPulse, t, ρ) = 0 is fulfilled or not. Hence, VarWS(dPulse, ρ) can be maximized
and CovWS(dPulse, t, ρ) = 0 at the same time. Consequently, Ω is maximal iff VarWS(dPulse, ρ) is
maximal and at the same time CovWS(dPulse, t, ρ) = 0. Therefore, Ω is maximal iff conditions
1a, 1b and 1d are fulfilled.
2. The minimal value of CRBu can be determined by choosing ψTDM such that the conditions of
the 1. part of the Theorem are fulfilled.

The conditions of the theorem can be interpreted in the following way: Using the two outermost Tx
antennas makes the virtual aperture as large as possible. 1Tρ(1) = 1Tρ(2) means that the same amount
of energy is transmitted from each Tx antenna. Roughly spoken, this yields the same amount of
spatial information for both Tx antennas. 1Tρ = ρmax states to transmit as much energy as possible,
which obviously leads to the highest possible SNR. The condition EWS(t(1), ρ(1)) = EWS(t(2), ρ(2)) has
already been interpreted in the context of Theorem 2. Due to this condition and with the help of
Theorem 2 it follows that the minimal value of CRBu w. r. t. ψTDM is as large as CRBu,stat, the CRB of
the electrical angle of a stationary target using the same radar and TDM scheme. Thus the achievable
– 111 –
Time
TX 1
TX 2
t1 t2 t3 t4
Transmitting
antenna
(a) TDM scheme with dPulse = [dTxmin, d
Tx
max, d
Tx
max, d
Tx
min]
T , t ∝ [0, 1, 2, 3]T and ρ ∝ 1
Time
Tx 1
Tx 2
t1 t2 t3
ρ1
ρ2
ρ3
Transmitting
antenna
(b) TDM scheme with dPulse = [dTxmin, d
Tx
max, d
Tx
min]
T , t ∝ [0, 1, 2]T and ρ ∝ [1, 2, 1]T
Figure 5.25.: Examples of TDM schemes satisfying the conditions of Theorem 3
DOA accuracy is as large as if the target is not moving relative to the radar and the complete virtual
aperture of the MIMO radar can be used as indicated in (5.144).
Theorem 3 helps us to find TDM schemes which achieve the smallest possible CRBu. If we optimize
ψTDM, this theorem tells us which conditions have to be fulfilled in order to yield an optimal TDM
scheme.
We give two examples of optimal TDM schemes.
Example 5.2.3. The TDM scheme uses NPulse = 4 pulses and the same energy per pulse, ρ = ρ
max
4 1.
Moreover, dPulse = [dTxmin, d
Tx
max, d
Tx
max, d
Tx
min]
T , t ∝ [0, 1, 2, 3]T . The transmission sequence is depicted in
Fig. 5.25a. Since the energy per pulse is constant, we can easily see that 1d is satisfied due to the
symmetry in Fig. 5.25a.
Example 5.2.4. The radar transmits NPulse = 3 pulses with different pulse energies, ρ = ρ
max
4 [1, 2, 1]
T ,
dPulse = [dTxmin, d
Tx
max, d
Tx
min]
T and t ∝ [0, 1, 2]T . This is depicted in Fig. 5.25b.
5.2.4.2. Optimal TDM Schemes with Constant Transmission Energy and Equidistant
Transmission Times
Theorem 3 states conditions for optimal TDM schemes under the two physically meaningful con-
straints of limited transmission energy (5.140) and limited geometrical size (5.141). Typical appli-
cations introduce additional constraints on the TDM scheme. In general, the additional constraints
result in other optimal TDM schemes. In the following, we search for optimal TDM schemes un-
der the constraints (5.140), (5.141) and additional practical constraints: We consider TDM schemes
– 112 –
which contain pulses transmitted at equidistant time instances with the same transmission energy
per pulse, i. e. ρ ∝ 1. This is e. g. the case in automobile radar systems which transmit a sequence of
identical chirps with the maximum available power pmax. We consider two cases: a) the number of
pulses NPulse in the TDM scheme is given and b) can be chosen freely.
For a given number of pulses NPulse, we derive the conditions a TDM scheme has to fulfill in order
to minimize CRBu. Dependent on NPulse the TDM schemes achieve different values of CRBu. We
show for which values of NPulse the smallest CRBu is achieved, which is case b).
First we investigate case a). W. l. o. g. we set dTxmin = −1 and dTxmax = +1 and
t = [1, . . . ,NPulse]T − ES
(
[1, . . . ,NPulse]T
)
1, (5.146)
i. e. ES(t) = 0. Here, t is normalized such that it is dimensionless. Analog to the proof of Theorem 3,
to minimize CRBu in (5.99), S in (5.100) and U in (5.101) have to be maximized. S is maximal iff
1Tρ = ρmax. Thus
ρ =
ρmax
NPulse
1. (5.147)
To maximize U, we have to maximize Ω in (5.145). Since ρ ∝ 1, the functions EWS, VarWS and
CovWS simplify to ES, VarS and CovS, respectively. Thus we have to maximize
Ω(dPulse) = VarS(dPulse) −
(
CovS(dPulse, t)
)2
VarS(t)
. (5.148)
The optimal solutions of dPulse are given by
dPulse,opt = arg max
dPulse
Ω(dPulse). (5.149)
Note that this optimization problem contains continuous variables (antenna positions in dPulse) and
discrete variables (NPulse).
Analog to Theorem 3 an optimum is achieved by using only the two outermost antennas [Vogel,
2013], i. e. dPulsei ∈
{
dTxmin, d
Tx
max
}
∀ i. This simplifies the optimization problem significantly, since the
number and positions of the Tx antennas are already fixed. In contrast to Theorem 3, in general,
for a fixed NPulse, we cannot maximize VarS(dPulse) and achieve CovS(dPulse, t) = 0 at the same time.
For different NPulse, the maximal value of Ω and the criteria for an optimal TDM scheme differ.
Therefore, we have to consider different cases of NPulse. This leads to the following Theorem. In the
following, we use the short notation NPulse ∈ 4N instead of NPulse ∈ 4n, n ∈ N. For the other cases,
we use an analog notation. Moreover, we call a TDM scheme balanced, if only 2 Tx antennas are
used and both antennas transmit equally often.
– 113 –
Theorem 4. Under the constraints of limited transmission energy (5.140), limited geometrical size
(5.141), same transmission energy per pulse, ρ ∝ 1, and equidistant transmission times, an optimal
TDM scheme has to satisfy the conditions
a) only the 2 outermost Tx antennas with dTx1 = d
Tx
min and d
Tx
2 = d
Tx
max are used,
b) ρ = ρ
max
NPulse
1.
Moreover, dependent on NPulse, the following holds:
NPulse ∈ 4N An optimal TDM scheme has to be balanced and fulfill CovS(dPulse, t) = 0 which is
equivalent to ES(t(1)) = ES(t(2)). The optimal value for Ω is
Ω(dPulse,opt)
∣∣∣∣
NPulse∈4N
= 1. (5.150)
NPulse ∈ 4N − 2 An optimal TDM scheme has to be balanced and fulfill CovS(dPulse, t) = ± 1NPulse . The
maximum Ω for the minimal CRBu is
Ω(dPulse,opt)
∣∣∣∣
NPulse∈(4N−2)
= 1 − 1
N2Pulse
· 12
N2Pulse − 1
. (5.151)
NPulse ∈ 2N + 1 In an optimal TDM scheme, one Tx antenna has to transmit one pulse more than
the other Tx antenna and the TDM scheme has to fulfill ES(t(1)) = ES(t(2)). The maximum Ω for the
minimal CRBu is
Ω(dPulse,opt)
∣∣∣∣
NPulse∈(2N+1)
= 1 − 1
N2Pulse
. (5.152)
Proof. In the following we present the reason for the case differentiation. The proof of the conditions
on the TDM scheme for each case is shown in Appendix L. To maximize Ω, we have to maximize
VarS(dPulse) and to minimize |CovS(dPulse, t)| at the same time. First we consider VarS(dPulse). For even
NPulse, both Tx antennas can be used equally often such that VarS(dPulse) is maximal, VarS(dPulse) = 1.
For odd NPulse, one antenna is used more often than the other and a sample variance of 1 cannot
be achieved. Hence optimal TDM schemes have to be determined for even and odd NPulse sepa-
rately. A further case differentiation is required to minimize CovS(dPulse, t). For odd NPulse, it can
be shown that min |CovS(dPulse, t)| = 0. For even NPulse with NPulse ∈ 4N, the minimum value is also
min |CovS(dPulse, t)| = 0. If NPulse is even with NPulse ∈ 4N − 2, then min |CovS(dPulse, t)| = 0.5. Thus
we have to make a second case differentiation for even NPulse. Therefore, we have to consider the
cases NPulse ∈ 4N, NPulse ∈ 4N − 2 and NPulse ∈ 2N + 1. See [Vogel, 2013] for more details.
For a given number of pulses NPulse, Theorem 4 states the conditions a TDM has to satisfy in or-
der to minimize CRBu. Moreover, the maximum value of Ω and therefore the minimal value of
CRBu is specified. An overview of the optimal TDM schemes is shown in Tab. 5.2. It shows
that the largest value of Ω is achieved for NPulse ∈ 4N. Note that the difference between Ω using
NPulse ∈ 4N pulses and using NPulse < 4N pulses vanishes at least with 1/N2Pulse. Thus, the larger
– 114 –
Table 5.2.: Overview of the optimal TDM schemes
NPulse ∈ 4N NPulse ∈ (4N − 2) NPulse ∈ (2N + 1)
balanced    
VarS(dPulse) 1 1 1 − 1N2Pulse
decoupling of u and ω   
(CovWS(dPulse,opt,t))2
VarS(t)
0 1N2Pulse
· 12N2Pulse−1 0
Ω(dPulse,opt) 1 1 − 1N2Pulse ·
12
N2Pulse−1
1 − 1N2Pulse
NPulse, the smaller the difference. As an example, NPulse = 6 yields Ω(dPulse,opt) ≈ 0.99 and NPulse = 7
yields Ω(dPulse,opt) ≈ 0.98. Hence Ω(dPulse,opt) differs only by a few percent for different NPulse.
From the above findings follows: If NPulse can be chosen freely, which we denoted by case b), then
an optimal TDM scheme has to use NPulse ∈ 4N pulses, be balanced and satisfy ES(t(1)) = ES(t(2)),
since due to Theorem 4, it achieves the smallest CRBu.
Thus for given or freely selectable NPulse, we have derived the conditions optimal TDM schemes
have to fulfill. We present several examples of optimal TDM schemes for the different cases.
Example 5.2.5. NPulse ∈ 4N. Such optimal TDM schemes can be constructed by building schemes
which are balanced and symmetric to the center index, i.e. dPulsei = d
Pulse
NPulse+1−i. The symmetry
ensures that ES(t(1)) = ES(t(2)). An example is dPulse,opt1 = [1, 1,−1,−1,−1,−1, 1, 1]T . dPulse,opt2 =
[1,−1,−1, 1,−1, 1, 1,−1]T is also optimal, but not of this structure. Both schemes are depicted in
Fig. 5.26.
Example 5.2.6. NPulse ∈ 4N − 2. Such optimal TDM schemes can be constructed by building
schemes which are balanced and symmetric to the center index, except for the elements which
are closest to the center of the sequence. An example of such an optimal TDM scheme is
dPulse,opt1 = [−1, 1, 1,−1, 1,−1]T . dPulse,opt2 = [−1, 1, 1,−1,−1, 1]T is also optimal, but of a different
structure. These schemes are depicted in Fig. 5.27.
Example 5.2.7. NPulse ∈ 2N + 1. Such optimal TDM schemes can be constructed by building
schemes where one antenna transmits one pulse more than the other and which are symmetric to the
center index. An example is dPulse,opt1 = [−1,−1, 1, 1, 1,−1,−1]T . dPulse,opt2 = [−1, 1, 1,−1,−1,−1, 1]T
is also optimal, but not of this structure. Both examples are illustrated in Fig. 5.28.
– 115 –
Time [arb. units]1 3 5
Tx 1
7
Tx 2
2 4 6 8
Transmitting
antenna
(a) Symmetric optimal TDM scheme
Time [arb. units]1 3 5
Tx 1
7
Tx 2
2 4 6 8
Transmitting
antenna
(b) Asymmetric optimal TDM scheme
Figure 5.26.: Example of optimal TDM schemes with NPulse = 8
Time [arb. units]1 3 5
Tx 1
Tx 2
2 4 6
Transmitting
antenna
(a) Optimal symmetric TDM scheme except for the elements closest to the center of the sequence
Time [arb. units]1 3 5
Tx 1
Tx 2
2 4 6
Transmitting
antenna
(b) Optimal TDM scheme of a different structure
Figure 5.27.: Example of optimal TDM schemes with NPulse = 6
In the examples, we have presented how to construct TDM schemes of a certain structure, which
fulfill the conditions of Theorem 4. An algorithm which constructs all optimal TDM schemes is
presented in [Vogel, 2013].
Simulations To verify and foster the understanding of the theoretical results, we present some
numerical simulations.
We consider a MIMO radar which consists of 4 Rx and 4 Tx antennas. The antennas are uni-
formly spaced with an antenna distance of λ/2, i.e. dRx = dTx = pi · [0, 1, 2, 3]T . We choose
L = 1 cycle and consider optimal TDM schemes for 3 different values of NPulse, which corre-
– 116 –
Time [arb. units]1 3 5
Tx 1
7
Tx 2
2 4 6
Transmitting
antenna
(a) Symmetric optimal TDM scheme
Time [arb. units]1 3 5
Tx 1
7
Tx 2
2 4 6
Transmitting
antenna
(b) Asymmetric optimal TDM scheme
Figure 5.28.: Example of optimal TDM-schemes with NPulse = 7
spond to the 3 different cases NPulse ∈ 4N, NPulse ∈ (4N − 2) and NPulse ∈ (2N + 1). The transmit-
ting time instants are t = [0, 1, . . . ,NPulse − 1]T , where we have normalized t such that it is di-
mensionless. The optimal TDM sequences, which we use for the simulations, are taken from the
examples above: dPulse,opt = [1,−1,−1, 1,−1, 1, 1,−1]T ∈ R8, dPulse,opt = [−1, 1, 1,−1,−1, 1]T ∈ R6
and dPulse,opt = [−1, 1, 1,−1,−1,−1, 1]T ∈ R7. The target’s electrical angle is u = sin(10◦) and the
Doppler frequency is ω = 1.3. For each TDM scheme, we compute the RMSE of the deterministic
ML estimate of u by doing Monte-Carlo simulations. Note that the unknown parameter vector of
the ML estimator is Θ in (5.80). We do this for different values of the total SNR S and compare the
RMSE with CRBu. The ML estimates are computed as in the former simulations, cf. (5.129).
The result is depicted in Fig. 5.29. The difference between the CRBs for the optimal TDM schemes
is barely notable, since they differ only slightly, as computed above. The plot shows that the ML
estimators achieve the corresponding CRBs above a threshold of approximately 16 dB. Hence, as
long as an optimal TDM scheme is chosen, the number of chirps NPulse has only a small influence
on the achievable accuracy. For comparison, CRBu,SIMO, the CRB of the SIMO radar which uses the
same antenna array but only one Tx antenna, is also depicted. CRBu,SIMO is achieved by a MIMO
radar using a bad TDM scheme, cf. Theorem 1.
– 117 –
0 5 10 15 20 25 30 35
10−3
10−2
10−1
100
Total SNR S in dB
R
M
SE
of
el
ec
tr
ic
al
an
gl
e
u
NPulse = 6,CRBu
NPulse = 6, ML
NPulse = 7,CRBu
NPulse = 7, ML
NPulse = 8,CRBu
NPulse = 8, ML
CRBu,SIMO
Figure 5.29.: RMSE of ML and CRB of optimal TDM schemes for different NPulse
– 118 –
5.3. Two Moving Targets
Consider a radar system which measures the signal of several targets, which have the same distance
and relative radial velocity. The radar system cannot distinguish the targets in their distance and
velocity, hence it has to estimate the DOAs of several targets. Since it is unlikely that more than
two targets have nearly the same distance and velocity, we focus on the two target case. A typical
automotive example, in which this situation arises, is depicted in Fig. 5.30: a car is driving in the
middle of an alley, where other cars are parking on the left and right side. Due to the symmetry of
the figure, the two cars which are located at the same distance have the same relative radial velocity.
Figure 5.30.: Example of a scenario, in which two parking targets have the same distance. Due to the
symmetry of the figure, they also have the same relative radial velocity v. [Openclipart,
2014, graphical parts used and adapted]
We are interested in the accuracy and resolution of a TDM MIMO radar system for the DOA esti-
mation of two targets, which are moving relative to the radar. We want to compare different TDM
MIMO radars, and compare the MIMO to a SIMO radar. We aim to find TDM schemes, which
achieve a high accuracy and resolution. In [Li et al., 2008] and [Li and Stoica, 2009, Chapter 2],
the CRB for the DOA of several stationary targets for a MIMO radar is computed. The authors use
it to optimize the transmit waveform of the MIMO radar. Here, we derive and investigate the CRB
for two non-stationary targets, i. e. targets moving relative to the radar, for a TDM MIMO radar.
We derive conditions for TDM schemes such that the DOAs and Doppler frequencies decouple in
the CRB. Hence the DOA estimation can be as accurate as if the Doppler frequencies are known a
priori. Using the CRB, we define a statistical resolution limit to separate both targets. We show that
the TDM MIMO radar can achieve a better resolution than the SIMO radar, despite the unknown
Doppler frequencies. This section is based on [Rambach and Yang, 2014]4.
We introduce the signal model in Section 5.3.1. In Section 5.3.2 the CRB is computed and inter-
preted. We derive the conditions for the decoupling of the DOAs and Doppler frequencies in Sec-
4This section contains quotations of our own published work [Rambach and Yang, 2014] which are not marked addi-
tionally.
– 119 –
tion 5.3.3. To confirm the analytical results, numerical simulations are presented in Section 5.3.4.
We investigate the angular resolution in Section 5.3.5 and summarize the results in Section 5.3.6.
5.3.1. Signal Model
Before we present the signal model, consider two targets which the radar system cannot separate in
their distances or relative radial velocities. This does not necessarily mean that the relative velocities
are equal, but the difference of the velocities is smaller than the velocity resolution of the radar
system. A typical automotive radar system with a carrier frequency of 76.5 GHz achieves a resolution
in the velocity of approximately δv = 1 m/s [Reiher, 2012]. The carrier wavelength is λ ≈ 3.9 mm.
Let us assume a measurement time of one cycle Tcycle = 5 ms. The difference of the distance, the
targets are moving during Tcycle, is up to Tcycle·δv = 5 mm, which is larger than the carrier wavelength.
Hence it is important to incorporate this in the signal model for the DOA estimation. Therefore, we
have to introduce in the signal model an own Doppler frequency for every moving target.
We present the signal model for the DOA estimation of NTargets moving targets for a TDM MIMO
radar. The two target model is a special case which is obtained by setting NTargets = 2. The radar
consists of a linear array with NTx Tx and NRx Rx colocated, isotropic antennas. As in Section 5.1.2
the targets are in the far field, are modeled as point targets and reflect the narrowband transmitted
signal. To distinguish between the two dimensional electrical angle u of one target in Section 5.1.2.1,
we denote the electrical angles of the targets by
U =
[
U1, . . . ,UNTargets
]T ∈ RNTargets , (5.153)
where Ui is the electrical angle of the i-th target. The Doppler frequencies and complex signal
strengths of the targets are denoted by
ω =
[
ω1, . . . , ωNTargets
]T ∈ RNTargets , (5.154)
s(l) =
[
s1(l), . . . , sNTargets(l)
]T ∈ CNTargets , l = 1, . . . , L, (5.155)
respectively. The signal is made up of the sum of the signals of each target. Hence, using the one
target signal model for a linear array (5.70), we get
x(l) =
NTargets∑
k=1
b(Uk, ωk) sk(l)
 + n(l)
= B(U, ω) s(l) + n(l), l = 1, . . . , L, (5.156)
with
b(U, ω) =
√
ρVirt  exp( j · tVirtω)  aVirt(U) ∈ CNVirt , (5.157)
– 120 –
B(U, ω) =
[
b(U1, ω1) . . . b(UNTargets , ωNTargets)
]
∈ CNVirt×NTargets . (5.158)
All other variables are given in (5.75) to (5.79). We make the same assumptions as stated in Sec-
tion 5.1.2.1, where the assumptions concerning the target apply to all targets.
The unknown parameters Θ to be estimated from x(l), 1 ≤ l ≤ L, are
Θ =
[
UT , ωT , sT (1), . . . , sT (L), σ2
]T
. (5.159)
5.3.2. Cramer Rao Bound
We present the deterministic CRB for the DOAs and Doppler frequencies for the multiple target
model (5.156) in Section 5.3.2.1 and simplify it further for the two target case, NTargets = 2. In
5.3.2.2 we interpret the resulting formula of the CRB.
5.3.2.1. Derivation of the CRB
The unknown parameter vector is given in (5.159). We denote the electrical angle and Doppler
frequency of each target by
Θ(i) = [Ui, ωi]T , i = 1, . . . ,NTargets. (5.160)
We want to obtain the part of the CRB corresponding to the parameters Θ(1), . . . ,Θ(NTargets). This
part is denoted by CRB(
Θ(1),...,Θ
(NTargets)
). Thus we have to compute the FIM J of the whole parameter
vector Θ and then its inverse J−1. We take that 2NTargets × 2NTargets block of J−1 corresponding to
Θ(1), . . . ,Θ(NTargets). This part is CRB(
Θ(1),...,Θ
(NTargets)
). Following [Yau and Bresler, 1992],
CRB−1(
Θ(1),...,Θ
(NTargets)
) = 2 L
σ2
Re
[
C 
(
ST ⊗ 12×2
)]
∈ R2NTargets×2NTargets (5.161)
with
C = DH P⊥B D ∈ C2NTargets×2NTargets , (5.162)
D =
[
D1 . . . DNTargets
]
∈ CNVirt×2NTargets , (5.163)
Di =
[
∂b(Ui, ωi)
∂Ui
∂b(Ui, ωi)
∂ωi
]
∈ CNVirt×2, i = 1, . . . ,NTargets, (5.164)
P⊥B = I − PB ∈ CNVirt×NVirt , (5.165)
PB = B
(
BH B
)−1
BH ∈ CNVirt×NVirt , (5.166)
S =
1
L
L∑
l=1
s(l) sH(l) ∈ CNTargets×NTargets . (5.167)
– 121 –
For the sake of clarity, we show which parameters the different elements of CRB−1(Θ(1),Θ(2)) correspond
to:
CRB−1(
Θ(1),...,Θ
(NTargets)
) =

U1 ω1 ... UNTargets ωNTargets
U1 × × . . . × ×
ω1 × × . . . × ×
...
...
...
. . .
...
...
UNTargets × × . . . × ×
ωNTargets × × . . . × ×

Further Analytical Computations for Two Targets To gain a deeper insight, we compute a more
compact expression for the case of two targets, i. e. NTargets = 2. In that case CRB(Θ(1),Θ(2)) ∈ C4×4.
We abbreviate the matrix elements in S by
S =
σ2s,1 csc∗s σ2s,2
 ∈ C2×2. (5.168)
Hence
σ2s,i =
1
L
L∑
l=1
|si(l)|2 (5.169)
is the mean of the i-th target’s signal strength over all cycles and
cs =
1
L
L∑
l=1
s1(l)s∗2(l) (5.170)
is the correlation coefficient between the two target signals. After some lengthy calculations, which
are given in Appendix M, we can express the matrix C ∈ C4×4 by
C = C1 + C2 − C3, (5.171)
C1 =
 ρVirtsum CovS(V1) b∗∆sum CovWS(V1, b∗∆Rx)b∆sum CovWS(V1, b∆Rx) ρVirtsum CovS(V1)
 , (5.172)
C2 =
 ρVirtsum CovWS(V2, ρ) b∗∆sum CovWS(V2, b∗∆Tx)b∆sum CovWS(V2, b∆Tx) ρVirtsum CovWS(V2, ρ)
 , (5.173)
C3 =
ρVirtsum
ρVirtsum
2 − |b∆sum|2
·
 |b∆sum|2 m∗mT −ρVirtsum b∗∆sum m∗mH−ρVirtsum b∆sum m mT |b∆sum|2 m mH
 (5.174)
– 122 –
with
ρVirtsum = 1
TρVirt = NRx1Tρ ∈ R, (5.175)
∆U = U1 − U2 ∈ R, (5.176)
∆ω = ω1 − ω2 ∈ R, (5.177)
b
∆Tx = ρ  exp( j · t ∆ω)  exp( j · dPulse∆U) ∈ CNPulse , (5.178)
b
∆Rx = exp( j · dRx∆U) ∈ CNRx , (5.179)
b
∆
= b
∆Tx ⊗ b∆Rx ∈ CNVirt , (5.180)
b∆sum = 1T b∆ =
(
1T b
∆Tx
) (
1T b
∆Rx
)
∈ C, (5.181)
m =
[
ES(V1) − EWS(V1, b∆Rx) + EWS(V2, ρ) − EWS(V2, b∆Tx)
]T ∈ C2, (5.182)
V1 =
[
dRx 0
]
∈ CNRx×2, (5.183)
V2 =
[
dPulse t
]
∈ CNPulse×2. (5.184)
The definitions and properties of EWS,ES,CovWS,CovS are given in Appendix A. Here we have
assumed that all expressions are finite and no division by 0 occurs. Since EWS(.,w), CovWS(.,w)
with complex weight vector w contains the division by 1T w, this means that 1T w , 0 for all weight
vectors.
5.3.2.2. Notes on the CRB
Interpretation of C The matrix C consists of 3 terms, C1,C2 and C3, which we interpret in the
following.
C1 is especially determined by the Rx-array dRx. The 2 × 2-blocks on the diagonal contain
CovS(V1) =

Ui ωi
Ui VarS(dRx) 0
ωi 0 0
. (5.185)
It describes the influence of dRx on the estimation of U1 and U2. The 0 entries show that CovS(V1)
does not contain any information about ω. VarS(dRx) is the same expression as the one which appears
in the CRB for a SIMO radar for the one target DOA estimation (4.16). The off-diagonal 2×2-blocks
contain
CovWS(V1, b∆Rx) =

Ui ωi
U j VarWS(dRx, b∆Rx) 0
ω j 0 0
, i , j, (5.186)
– 123 –
or the complex conjugate of it. There is again no information aboutω. VarWS(dRx, b
∆Rx) expresses the
coupling between the two electrical angles. The coupling depends on dRx and the angle separation
∆U of the two targets.
C2 is particularly determined by the TDM sequence, namely dPulse and t.
CovWS(V2, ρ) =

Ui ωi
Ui VarWS(dPulse, ρ) CovWS(dPulse, t, ρ)
ωi CovWS(dPulse, t, ρ) VarWS(t, ρ)
 (5.187)
is analog to the one-target DOA estimation, Section 5.2.2: VarWS(dPulse, ρ) describes the amount of
information on U1, U2 due to the Tx antennas. VarWS(t, ρ) contains the information for estimating
ω1 and ω2 due to the different Tx times. CovWS(dPulse, t, ρ) is the coupling between U and ω of the
same target. If dPulse, t and ρ are chosen in a suitable way, this coupling can be reduced to 0, cf.
Section 5.2.3. The 2 × 2-block b∆sum CovWS(V2, b∆Tx) expresses the coupling between both targets.
Since
CovWS(V2, b∆Tx) =

Ui ωi
U j VarWS(dPulse, b∆Tx) Cov
WS(dPulse, t, b
∆Tx)
ω j CovWS(dPulse, t, b∆Tx) Var
WS(t, b
∆Tx)
, i , j, (5.188)
∆U as well as ∆ω affect the amount of coupling between the two targets’ electrical angles and
Doppler frequencies.
The matrix C3 can be interpreted in the following way: C3 origins from DHPBD. PB is the pro-
jector onto the subspace W spanned by the column vectors of B, namely the steering vectors
b(U1, ω1), b(U2, ω2). D contains the derivatives of the steering vectors. Those vectors are perpen-
dicular to the subspace W iff DHPBD = 0. It can be shown, without loss of generality, that this is
equivalent to C3 = 0. Hence, roughly speaking, C3 denotes the part of the derivatives of the steering
vectors which is in W. The smaller the diagonal elements of C3, the more information is contained in
the diagonal elements of C. This is in accordance to the geometrical interpretation: the radar system
is sensitive if the change of the steering vectors, due to a change of parameters, is perpendicular
to the steering vectors. Note that C3 has mixed contributions from both the Rx and Tx array and
contains couplings of both targets’ DOAs and Doppler frequencies.
All matrices C1,C2 and C3 contain b∆sum =
(
1T b
∆Tx
) (
1T b
∆Rx
)
. The factor
1T b
∆Rx =
NRx∑
i=1
exp( j · dRxi ∆U) (5.189)
– 124 –
can be considered as the complex Rx beampattern at ∆U. Similarly,
1T b
∆Tx =
NPulse∑
i=1
ρi · exp( j · ti ∆ω) · exp( j · dPulsei ∆U) (5.190)
is the complex Tx beampattern at ∆U,∆ω weighted with ρ. Hence, b∆sum is the product of these
beampatterns.
Other notes on the CRB CRB(Θ(1),Θ(2)) depends not only on the position of the Tx antennas, but
also on the TDM scheme described by the transmitting order dPulse and transmitting times t. Hence,
the choice of a good TDM scheme is crucial in order to achieve a high accuracy.
Moreover, CRB(Θ(1),Θ(2)) is a function of ∆U, ∆ω, σ
2
s,1, σ
2
s,2, cs
CRB(Θ(1),Θ(2)) = CRB(Θ(1),Θ(2))(∆U, ∆ω, σ
2
s,1, σ
2
s,2, cs). (5.191)
Hence it does not depend on the absolute values of Ui, ωi, but only on their differences. Moreover,
it does not depend on s1(l), s2(l), l = 1, . . . , L, but on the mean signal strengths σ2s,1, σ
2
s,2 and the
correlation coefficient cs defined in (5.169) and (5.170), respectively. Note that for the important
case of one cycle, i. e. L = 1, σ2s,i = |si|2 and cs = s1 s∗2.
5.3.3. Decoupling of Electrical Angles and Doppler Frequencies
In Section 5.2, the DOA estimation of one moving target was investigated. It was shown that TDM
schemes can be found such that the CRB of the DOA has the same value as for a stationary target.
Similarly, we are interested here in TDM schemes which lead to a decoupling of the electrical angles
and Doppler frequencies in the CRB. CRB(Θ(1),Θ(2)) depends on several parameters of the targets cf.
(5.191). Because of this, it is difficult to find good TDM schemes for a general setting. Therefore,
we focus first on the case with L = 1 cycle and consider two targets, which have the same signal
strength, i. e. |s1| = |s2|. Here we have abbreviated si(1) by si, i = 1, 2. In Fig. 5.31, the element
of the CRB which corresponds to the first target’s electrical angle CRBU1 =
[
CRB(Θ(1),Θ(2))
]
1,1
is
plotted for varying ∆ω and ∆U. We have used the following parameters: NPulse = 4, ρ = 1NPulse 1,
the SNR defined as SNR =
σ2s,1
σ2
is 30 dB, the Rx array is a 4-element ULA with half-wavelength
spacing dRx = pi · [−1.5, −0.5, 0.5, 1.5]T . Moreover, dTx = pi · [−2, 2]T , dPulse = [dTx1 , dTx2 , dTx2 , dTx1 ]T ,
t = [0, 1, 2, 3]T , where t is normalized such that it is unitless, and cs = |s1| · |s2| e j·0.3 pi. Fig. 5.31
shows that for a fixed ∆U, CRBU1 is maximal in the vicinity of ∆ω = 0. We have studied other TDM
schemes and different values of cs, which yield a qualitative similar result, i. e. for a fixed ∆U, the
maximum of CRBU1 is located in the vicinity of ∆ω = 0.
Due to this, we investigate in the following the case ∆ω = 0, i. e. the two targets have exactly the
same Doppler frequency. Hence the two targets can be distinguished only by their DOAs. Note,
– 125 –
∆ U
∆ 
ω
 
[2pi
]
 
 
−1 −0.5 0 0.5 1
−0.4
−0.2
0
0.2
0.4
−3
−2.5
−2
−1.5
−1
−0.5
Figure 5.31.: log
( √
CRBU1
)
for varying ∆ω and ∆U
since the estimator does not know a priori ∆ω = 0, both Doppler frequencies ω1 and ω2 have to be
estimated.
Theorem 5. Let ∆ω = 0. The elements of CRB(Θ(1),Θ(2)) corresponding to U1,U2 decouple from the
elements corresponding to ω1, ω2 if the TDM scheme satisfies
EWS
(
t(k), ρ(k)
)
= EWS
(
t(l), ρ(l)
)
∀k, l ∈ {1, . . . ,NTx}. (5.192)
Here, t(k) denotes the transmit times of the k-th Tx antenna and ρ(k) the corresponding energies of the
transmitted pulses.
Proof. This theorem is a special case of the more general Theorem 6 which we present and prove in
Section 5.4.
The theorem gives a sufficient condition for the decoupling of the electrical angles and Doppler
frequencies. The decoupling means that the CRB has the following form
CRB(Θ(1),Θ(2)) =

U1 ω1 U2 ω2
U1 . 0 . 0
ω1 0 . 0 .
U2 . 0 . 0
ω2 0 . 0 .

. (5.193)
Thus, after a permutation of the parameters U1,U2, ω1, ω2, the CRB has a block diagonal structure.
We denote the first 2 × 2 block for U = [U1,U2]T by CRBU . Moreover, due to the decoupling,
– 126 –
CRBU is equal to the CRB when the Doppler frequencies are known and do not have to be estimated
CRBU = CRBU, known ω. (5.194)
Hence, if a TDM scheme satisfying (5.192) is chosen, a DOA accuracy can be achieved which is as
good as if the Doppler frequencies are known a priori.
Let us consider the case of two stationary targets, i. e. the targets do not move relative to the radar
and that is known a priori. In that case, ω1 = ω2 = 0 and therefore ∆ω = 0. Since CRB(Θ(1),Θ(2)) only
depends on ∆ω and not on the absolute values ω1, ω2, cf. (5.191), CRBU and therefore CRBU, known ω
also depend on ∆ω. Hence the CRB for the two electrical angles of two stationary targets CRBU, stat
is the same as CRBU, known ω with ∆ω = 0. Thus
CRBU = CRBU, stat, (5.195)
i. e. if the TDM scheme satisfies (5.192), the achievable DOA accuracy is as good as if the targets are
not moving relative to the radar. Hence the complete virtual aperture can be used for the estimation
of the electrical angles despite the unknown Doppler frequencies.
Note that (5.192) is the same condition as for the decoupling of electrical angle and Doppler fre-
quency in the one target case, cf. Section 5.2.3.1 Theorem 2. Moreover, due to Theorem 2, from
(5.192)
CovWS
(
dPulse, t, ρ
)
= 0 (5.196)
follows, and for two Tx antennas (5.192) and (5.196) are equivalent.
To illustrate the findings, we consider a radar system with the following parameters: The number
of measurement cycles is L = 1 and the Rx array is dRx = pi · [−1.5,−0.5, 0.5, 1.5]T . The Tx
antenna positions are dTx = pi · [−2, 2]T . Both targets have the same signal strength s1 = s2 = 1
and we set ∆ω = 0. The SNR is SNR = |s1|2/σ2 = 30 dB. The transmitted energy per pulse is
ρ = 1NPulse 1 ensuring a constant total transmitted energy independent of the number of pulses. The
transmission times are t = [0, . . . ,NPulse − 1]T . Since both targets have the same signal strength and
∆ω = 0, the CRBs of their electrical angles are equal, which we denote by CRBU in the following:
CRBU =
[
CRBU
]
1,1
=
[
CRBU
]
2,2
. In Fig. 5.32,
√
CRBU is depicted for varying ∆U. Three different
TDM schemes are considered: TDM scheme 1, for which the electrical angles decouple from the
Doppler frequencies, and TDM scheme 2 and 3 which do not satisfy condition (5.192). Moreover,
the CRBs for one moving target, which are computed in Section 5.2.1, are depicted. Fig. 5.32 shows
that for ∆ω = 0 in general CRBU,TDM 1 ≤ CRBU,TDM 2. For large ∆U, CRBU for the two-target case
reaches CRBu for one moving target. TDM scheme 3 achieves the same CRBU as using only one
transmitter, i.e. the SIMO case, because the Doppler frequencies have to be estimated as well. Since
both Tx transmit only once, it cannot be distinguished by the different Tx positions if the phase
– 127 –
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10−3
10−2
10−1
100
∆U
√ C
R
B
U
SIMO dPulse = [dTx1 , d
Tx
1 , d
Tx
1 , d
Tx
1 ]
T
MIMO TDM 1: dPulse = [dTx1 , d
Tx
2 , d
Tx
2 , d
Tx
1 ]
T
MIMO TDM 2: dPulse = [dTx1 , d
Tx
1 , d
Tx
2 , d
Tx
2 ]
T
MIMO TDM 3: dPulse = [dTx1 , d
Tx
2 ]
T
1 target, SIMO
1 target, MIMO TDM 1
Figure 5.32.:
√
CRBU for DOA estimation of two and one moving target
difference is due to the Doppler or due to the DOA. This is the same observation as in the one-target
case.
We do the same investigation again but with a different correlation coefficient cs, by setting s1 = 1
and s2 = e j·0.3 pi. This yields qualitatively the same result, i. e. in general CRBU,TDM 1 ≤ CRBU,TDM 2,
but the absolute values of the CRBs are different compared to the case with s1 = s2 = 1.
In order to see if TDM schemes which satisfy the decoupling condition (5.192) yield also a small
CRB for ∆ω , 0, we compute CRBU1 for varying ∆ω and fixed ∆U = 0.2. The other parameters
are the same as before, with s1 = s2 = 1. The CRB of the electrical angle of the first target CRBU1
is depicted in Fig. 5.33 for three different TDM schemes. TDM scheme 1 satisfies the decoupling
condition (5.192), TDM scheme 2 and 3 do not. The figure shows that CRBU1,TDM 1 is smaller than
CRBU1,TDM 2 and CRBU1,TDM 3 for a large part of the values of ∆ω, but not for all ∆ω. Moreover, for
large |∆ω| TDM scheme 1 nearly achieves the CRB for one stationary target. As before, doing the
same investigation with s1 = 1 and s2 = e j·0.3 pi yields a qualitative similar result.
Hence, in general, a TDM scheme satisfying the decoupling condition (5.192) does not yield a
smaller CRB than other TDM schemes for all possible target parameters. But for the important case
∆ω = 0, due to (5.194), such a TDM scheme yields a CRBU which is as small as if the Doppler
frequencies are known a priori.
5.3.4. Numerical Simulation
In the following, we confirm the analytical findings by numerical simulations. We do a Monte-Carlo
simulation to determine the RMSE of the ML estimator for the electrical angles U1 and U2 of the 1.
and 2. target, respectively. We compare the RMSE to the computed CRB. The signal model (5.156)
– 128 –
Figure 5.33.:
√
CRBU1 for varying ∆ω and different TDM schemes
has the same structure as the one presented in Appendix E. Hence we can use the result (E.22) and
the ML estimator [Uˆ, ωˆ]T of [U, ω]T is given by
[Uˆ, ωˆ]T = arg max
U,ω
Tr
(
PB(U,ω) Rˆ
)
(5.197)
with
PB(U,ω) = B(U, ω) B+(U, ω), (5.198)
Rˆ =
1
L
L∑
l=1
x(l)xH(l). (5.199)
B+ is the matrix pseudo inverse of B, which can be computed by
B+ =
(
BHB
)−1
BH, (5.200)
under the assumption that BHB is invertible.
The parameters are the same as for the simulation presented in Fig. 5.32, but with SNR = 25 dB.
We consider TDM scheme 1 with dPulse = [dTx1 , d
Tx
2 , d
Tx
2 , d
Tx
1 ]
T . 3000 Monte-Carlo simulations are
done for every value of ∆U. Since the ML estimator (5.197) is nonlinear in U1,U2, ω1, ω2, the ML
estimates are computed by a 4-dimensional grid search, in order to find the global maximum of the
likelihood, followed by a local optimization. Since the grid search is computational expensive, for
∆U ≥ 0.2, the ML estimates are computed by a local optimization starting at the true value. This is
possible, since the ML estimator already achieves the CRB, as shown in Fig. 5.34, and hence global
errors are negligible. In this case, the number of Monte-Carlo simulations is 10000.
– 129 –
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10−3
10−2
10−1
100
∆U
R
M
SE
,
√ C
R
B
U
MIMO, RMSE ML U1
MIMO, RMSE ML U2
MIMO, CRB
1 target, MIMO CRB
Figure 5.34.: DOA estimation of two moving targets with TDM scheme 1,
dPulse = [dTx1 , d
Tx
2 , d
Tx
2 , d
Tx
1 ]
T . Comparison of
√
CRBU and the RMSE of the max-
imum likelihood estimator for U1 and U2
Fig. 5.34 shows that for ∆U ≥ 0.13, the ML estimator reaches the CRB. For large ∆U, the one-target
CRB is achieved. For ∆U < 0.13, the ML deviates from the CRB due to global errors. If ∆U is very
small, the RMSE of the U1- and U2-estimator differ. The reason is that for the computation of the
RMSE, the smaller estimated electrical angle is always assigned to the 1. target. Note that for small
∆U the RMSE could be even smaller than the CRB, since the CRB does not incorporate the fact that
the DOAs are always bounded by −90◦ and 90◦ and therefore −1 ≤ U1,U2 ≤ 1.
With this, we have confirmed the derived CRB by simulations.
5.3.5. Angular Resolution
An important performance criterion of a radar system is its angular resolution, i.e. the angular sepa-
ration at which two targets can be distinguished. In literature, there are several different definitions
of the resolution, cf. Section 2.2.3. We use the definition of the resolution given in [El Korso et al.,
2011], which works for multiple targets and multiple parameters of interest. Note that it is not clear
yet, which estimators fulfill the conditions used in the definition in [El Korso et al., 2011] and which
actually achieve the resolution. This is an ongoing research topic. The definition enables us to use
the two-target CRB in order to define the angular resolution analytically. The unknown parameters
of the signal model are given in (5.159). Instead of estimating U1,U2, ω1, ω2, we can also rewrite
the model and estimate U1,∆U, ω1,∆ω. Hence the unknown parameters are
Θ = [U1, ω1,∆U,∆ω, s(1), . . . , s(L), σ2]T . (5.201)
– 130 –
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10−3
10−2
10−1
100
δ
√ C
R
B
δ
SIMO dPulse = [dTx1 , d
Tx
1 , d
Tx
1 , d
Tx
1 ]
T
MIMO TDM 1: dPulse = [dTx1 , d
Tx
2 , d
Tx
2 , d
Tx
1 ]
T
MIMO TDM 2: dPulse = [dTx1 , d
Tx
1 , d
Tx
2 , d
Tx
2 ]
T
MIMO TDM 3: dPulse = [dTx1 , d
Tx
2 ]
T
δ/η
Figure 5.35.: Angular resolution for DOA estimation of two moving targets: it is obtained by the
intersection of δ/η and
√
CRBδ(δ).
For the computation of the resolution, the parameters of interest are ∆U,∆ω and the nuisance para-
meters are [U1, ω1, s(1), . . . , s(L), σ2]T . We investigate the case with ∆ω = 0 and equal signal
strengths s1 = s2. Then, using (5.191) and due to symmetry, CRB(Θ(1),Θ(2)) depends on |∆U |. The
angular resolution δARL is defined implicitly as the solution of [El Korso et al., 2011]
δARL = η
√
CRBδ(δARL). (5.202)
Here, η is a factor which depends on the probability of false alarm PFA and probability of detection
PD
η = η(PFA, PD) (5.203)
and CRBδ is the CRB of the parameter δ = |∆U |. Note that CRBδ is a function of δ, i. e. CRBδ =
CRBδ(δ). It can be computed by using the rule of transformation of parameters [Kay, 1993] and is
given by [El Korso et al., 2011]
CRBδ =
[
CRBU
]
1,1
+
[
CRBU
]
2,2
− 2
[
CRBU
]
1,2
. (5.204)
Using (5.202) the resolution can be obtained by the intersection of δ/η and
√
CRBδ(δ). We set
PFA = 0.01 and PD = 0.9 which results in η ≈ 14.9. For an interpretation of this resolution definition
see Section 2.2.3.
δ/η and
√
CRBδ are depicted in Fig. 5.35 for different TDM schemes. The parameters of the radar
system are the same as in Fig. 5.34. The TDM scheme 1 with dPulse = [dTx1 , d
Tx
2 , d
Tx
2 , d
Tx
1 ]
T achieves
– 131 –
the best resolution δARL ≈ 0.09. In this case, dPulse satisfies the condition of Theorem 5 which yields
a decoupling of the electrical angles and Doppler frequencies. TDM scheme 2 does not satisfy the
condition of Theorem 5. It achieves a worse resolution. TDM scheme 3 achieves the same resolution
as the SIMO radar, i.e. using only one transmitter. In that case δARL ≈ 0.2. This behavior is similar
to the CRBs depicted in Fig. 5.32 and explained there. Hence, the resolution of the MIMO radar
using TDM scheme 1 is approximately half as large as the resolution of the SIMO radar or the MIMO
radar using TDM scheme 3.
The resolution of a TDM MIMO radar depends crucially on the chosen TDM scheme and can be
significantly improved compared to a SIMO radar by optimizing the TDM scheme.
5.3.6. Summary
We have investigated the DOA estimation of two moving targets for a TDM MIMO radar. After
introducing the signal model, the CRB for the parameters of interest, the DOAs and Doppler fre-
quencies, was derived and we interpreted the resulting equation. Since the CRB depends on many
target parameters and due to the complexity of CRB, it is difficult to find general statements which
are valid for all target parameters. Because of this, we investigated the important case where both
targets have the same Doppler frequency, ∆ω = 0. For this case, conditions for TDM schemes were
derived such that the CRB of the DOAs is decoupled from the CRB of the Doppler frequencies.
These conditions can be used to find good TDM schemes. These TDM schemes achieve a DOA
accuracy which is as good as if the Doppler frequencies are known a priori. We compared the CRB
of MIMO radars using different TDM schemes. Moreover, we have confirmed the analytical compu-
tations by numerical simulations. These simulations show that the ML estimator indeed achieves the
CRB. Moreover, the CRB was used to define the angular resolution. We have shown that the angular
resolution depends on the chosen TDM scheme of the MIMO radar and can be better than that of the
SIMO radar, despite the unknown Doppler frequencies. Moreover, in the investigated scenario, the
TDM scheme which fulfills the decoupling condition achieves a high angular resolution.
5.4. Arbitrary Number of Moving Targets
In the previous sections, we have investigated the DOA estimation of one and two moving targets. In
doing so, we have derived the CRB for the electrical angles and Doppler frequencies. One important
result are the conditions which are sufficient for a decoupling of the electrical angles and Doppler
frequencies in the CRB (Theorem 2 and 5). If these conditions are fulfilled, a DOA accuracy can
be achieved as good as if the Doppler frequencies are already known a priori. The conditions are
the same for the one target and two target case. Hence the question arises, if the same decoupling
conditions are also true in the case of an arbitrary number of moving targets. The part of CRB which
– 132 –
corresponds to the electrical angles and Doppler frequencies CRB(
Θ(1),...,Θ
(NTargets)
) is given in (5.161).
A further simplified analytical expression as in the case of two targets is not feasible or only feasible
with a high effort. We present and proof the conditions which lead to a decoupling by using the
result (5.161). As we will see in the following, we do not need a more compact expression in order
to state and prove the decoupling conditions.
Decoupling of Electrical Angles and Doppler Frequencies We denote the differences of the
Doppler frequencies by
∆ωqr = ωq − ωr, q, r ∈ {1, . . . ,NTargets}. (5.205)
As in the two target case, we focus on the case in which the targets cannot be distinguished by their
Doppler frequency, i. e. ∆ωqr = 0 ∀ q, r ∈ {1, . . . ,NTargets}. The following theorem gives sufficient
conditions which lead to a decoupling in that case.
Theorem 6. Let
∆ωqr = 0 ∀ q, r ∈ {1, . . . ,NTargets}. (5.206)
If
EWS
(
t(k), ρ(k)
)
= EWS
(
t(l), ρ(l)
)
∀k, l ∈ {1, . . . ,NTx}, (5.207)
then the elements in CRB(
Θ(1),...,Θ
(NTargets)
) corresponding to the electrical angles U decouple from the
elements corresponding to the Doppler frequencies ω.
Here, t(k) denotes the transmit times of the k-th Tx antenna and ρ(k) the corresponding energies of the
transmitted pulses.
Proof. The theorem states that if (5.206) and (5.207) are fulfilled, the elements marked with × are 0
CRB(
Θ(1),...,Θ
(NTargets)
) =

U1 ω1 U2 ω2 ... UNTargets ωNTargets
U1 . × . × . . . . ×
ω1 × . × . . . . × .
U2 . × . × . . . . ×
ω2 × . × . . . . × .
...
...
...
...
...
. . .
...
...
UNTargets . × . × . . . . ×
ωNTargets × . × . . . . × .

. (5.208)
– 133 –
Thus after a reordering of the parameters U1, . . . ,UNTargets , ω1, . . . , ωNTargets , the matrix
CRB(
Θ(1),...,Θ
(NTargets)
) is block diagonal. Therefore, it is sufficient to prove that the correspond-
ing elements in CRB−1(
Θ(1),...,Θ
(NTargets)
) are 0, since the inversion keeps the block diagonal structure. In
order to prove this, we express C in (5.161) by
C = DH P⊥B D
= C(1) − C(2) (5.209)
with the definitions
C(1) = DH D ∈ C2NTargets×2NTargets , (5.210)
C(2) = DH B
(
BH B
)−1
BH D ∈ C2NTargets×2NTargets . (5.211)
We show that the relevant elements in C(1) are 0. The computation and inversion of the matrix BH B
is not feasible in general. Therefore, we show which elements in BH D are 0 in order to conclude
that the relevant elements in C(2) are 0 without the need to compute
(
BH B
)−1
. Hence, we can proof
the theorem without computing the CRB in detail. See Appendix N for the detailed proof.
Interpretation and Discussion In the worst case, if all targets have the same Doppler frequency,
Theorem 6 states that by choosing an appropriate TDM scheme, the CRB for the electrical angles
is as large as if the Doppler frequencies are already known. Note that we do not assume that it
is a priori known that the Doppler frequencies are equal, i. e. all Doppler frequencies have to be
estimated. Hence the decoupling condition does not only work for one or two moving targets, but
also for an arbitrary number of moving targets. A DOA accuracy can be achieved which is as large
as if the Doppler frequencies are known a priori.
Note that it is not sufficient if only some of the Doppler frequencies are the same. For example,
consider 3 moving targets with ω1 = ω2 , ω3. The theorem does not tell you anything about the
decoupling in this case. In fact, the Doppler frequencies and electrical angles of the first and second
target do not decouple in general. This can be easily verified by computing CRB(
Θ(1),...,Θ
(NTargets)
) (5.161)
numerically for a specific counterexample with NTargets = 3. Thus the influence of the third target
prevents the decoupling of the first and second target.
5.5. Summary and Conclusions
In this chapter we have studied the DOA estimation of moving targets using a TDM MIMO radar. We
derived the signal model and saw that the relative movement of the targets causes an additional phase
shift due to the Doppler effect. Thus, the Doppler frequencies of the targets have to be estimated in
addition to the DOAs. In general, this leads to a decrease in the DOA accuracy.
– 134 –
We analyzed the DOA estimation of one, two and several moving targets. In order to do that, we
derived the CRB for these cases.
For one moving target, we compared the CRB of a SIMO radar to that of a MIMO radar with a
stationary target. The results of the comparison for a planar array are given in Lemma 5.2.1 and
for a linear array in Theorem 1. After that sufficient conditions were presented in Theorem 2 which
lead to the vanishing of the coupling term between the electrical angles and the Doppler frequency
in the CRB. If these conditions are satisfied, the CRB of the electrical angles for the moving target
and the stationary target are equal. This theorem is the main result of this chapter. We demonstrated
by numerical simulations that a DOA accuracy can be achieved, which is as good as if the Doppler
frequencies are known a priori. Moreover we derived optimal TDM schemes which minimize the
CRB of a linear MIMO array under different constraints. Theorem 3 states optimal TDM schemes
under a limited transmission power and geometrical size of the antenna array. In Theorem 4 optimal
TDM schemes are presented under the additional constraints of equidistant transmission times and
equal transmission energy of the pulses.
For two moving targets, we investigated the decoupling condition for the special case, in which both
targets have the same Doppler frequency (Theorem 5). We also investigated the resolution, which
was defined using the two target CRB. Finally we extended the considerations to an arbitrary number
of moving targets which is presented in Theorem 6.
– 135 –
6. Conclusions and Future Work
We summarize the presented work, draw conclusions and give an outlook on possible future work.
6.1. Conclusions
MIMO radars with colocated Tx and Rx antennas are a promising radar type to achieve a high
performance in DOA estimation. They can be useful in automotive applications, which require good
DOA estimates although the antenna arrays are limited to a small geometrical size and only a small
number of antennas is used. We have investigated TDM MIMO radars and studied the DOA accuracy
as well as the resolution.
The background and basics of radar systems have been presented in Chapter 2. We explained the
working principle of a pulse Doppler, an FMCW, and a chirp sequence radar. After that we in-
troduced the concept of parameter estimation and the statistical properties of an estimator: Bias,
covariance and mean square error. We defined the accuracy and introduced the CRB as a lower
bound on the covariance matrix. Finally, a definition of the resolution has been presented which is
based on the CRB from estimation theory and the general likelihood ratio test (GLRT) from detection
theory.
Chapter 3 gives an overview on MIMO radars and their current state of the art. First we introduced
the two different types of MIMO radars: the MIMO radar with widely separated antennas and the
MIMO radar with colocated antennas. We focused on the colocated MIMO radar and explained the
concept of the virtual array. We gave an overview on the research which was already done on MIMO
radars. After that different multiplexing techniques, frequency, code, and time division multiplexing,
have been discussed. We explained their advantages and disadvantages. A TDM MIMO radar has
low hardware requirements and can use the already optimized transmit waveforms used in SIMO
radars. These are two important advantages for the use as an automobile radar system. Because of
this, we focused on the investigation of TDM MIMO radars in this thesis. We described experimental
investigations done with TDM MIMO radars and noted that quantitative and theoretical comparisons
to different MIMO radars were still missing.
In Chapter 4, we investigated the DOA estimation of stationary targets. We introduced the signal
models for the DOA estimation using a SIMO and MIMO radar, respectively. To study the maximal
– 136 –
achievable DOA accuracy, the CRB for the DOA has been derived for the SIMO and MIMO radar.
In order to compare both systems, we computed the SNR after the baseband signal has been pulse
compressed. This allows a quantitative comparison. The SIMO has the advantages of a Tx beam-
forming gain, if the beam is steered in the target’s direction. To do this, the target’s DOA has to be
known approximately a priori. The MIMO radar offers a larger virtual aperture than the SIMO radar.
Hence, dependent on the application and the positions of the Rx and Tx antennas, the MIMO or
the SIMO radar can achieve a higher accuracy. In automotive applications, the advantage of a large
aperture is important and due to this we focused our investigations on MIMO radars. Moreover, we
compared the TDM to a CDM MIMO radar. We studied the case that one Tx antenna can transmit
the maximal allowed total transmission power. In that case, the CRBs of the DOAs for both systems
are equal. Moreover, we showed with the help of sufficient statistics that, for a certain structure
of the noise covariance matrix, the received signals of both radar types contain the same amount
of information on the DOA. Then, all properties which depend on the likelihood function, e. g. the
sidelobe levels and resolution, are the same for the CDM and TDM MIMO radar.
The main part of our research is devoted to the study of DOA estimation of moving targets in Chap-
ter 5. A target which is moving relative to the radar causes a phase shift in the baseband signal due to
the Doppler effect in addition to the phase shift due to the DOA. In general, this leads to a decrease
in the DOA accuracy, since the Doppler frequency has to be estimated as well. First we derived the
signal model of the baseband signal of a transmitted chirp in order to compute the phase relation be-
tween the pulse compressed signal of successive chirps. Using this result, we introduced the signal
model for DOA estimation of a moving target using a TDM MIMO radar. We computed the CRB
for the DOA and Doppler frequency for planar Tx and Rx arrays in order to analyze the maximal
achievable DOA accuracy. A theorem was deduced which characterized TDM schemes such that
the DOA decouples from the Doppler frequency in the CRB. Hence, the CRB of the DOAs is as
large as the CRB for the DOAs of a stationary target. Therefore, the whole virtual aperture of the
MIMO radar can be used for the DOA estimation. Moreover, for linear arrays, we derived optimal
TDM schemes which minimize the CRB of the DOA under different practical boundary conditions.
Numerical simulations were presented, which confirmed the analytical findings.
We also investigated the DOA performance of a TDM MIMO radar in the estimation of the DOAs of
several targets. We considered the two target case and computed the CRB of the DOAs and Doppler
frequencies. For the case that both targets have the same Doppler frequency, TDM schemes were
derived which result in a decoupling of the DOAs of the Doppler frequencies. In that case, similar to
the one target DOA estimation, the CRB of the DOAs is the same as for stationary targets. Numerical
simulations were carried out to verify the analytical computations. With the help of the two target
CRB, we computed the DOA resolution of the TDM MIMO radar. Finally, we investigated the DOA
estimation of more than two targets. Similar to the two target case, we derived a theorem which
states conditions on the TDM schemes such that the DOAs decouple from the Doppler frequencies.
The results show that a TDM MIMO radar is a suitable system to be used in an automobile. It is
– 137 –
important to choose an appropriate TDM scheme. With a good TDM scheme, the TDM MIMO radar
can achieve a high DOA performance despite a small geometrical size.
6.2. Future Work
We have investigated the DOA accuracy of a TDM MIMO radar. Another important property are
the sidelobe levels in DOA and Doppler estimation. In a future work, the sidelobe levels of different
TDM schemes can be studied and compared. Moreover, the Rx and Tx antenna positions, including
the TDM scheme, can be optimized such that a high DOA accuracy is obtained under the constraint
of a maximum sidelobe level. This ensures that the probability of outliers, i. e. completely wrong
estimates, is small.
For stationary targets we have compared the TDM and CDM MIMO radars. The investigation can
be generalized to moving targets, although an analytical comparison could be difficult to obtain.
Another issue is the robustness of a radar against different perturbation which are not considered
in the model. For example, the positions of the Rx and Tx antennas may not coincide with the
values used in the model due to fabrication uncertainties. It is interesting and important to know,
how sensitive the radar reacts on perturbations. A first way to study the robustness can be achieved
by numerical simulations: the estimator is evaluated with a signal which does not coincide with the
modeled signal, but is a perturbed version of it.
Another topic is multipath propagation. The transmitted signal does not travel directly to the tar-
get and back, but is reflected one or several times in between. In an automotive environment, an
additional reflection can occur at the traffic barrier for example. Then, in general the direction of
departure (DOD) and the direction of arrival is not the same anymore. Therefore, the virtual array
concept has to be modified to include this fact. This results in a new signal model for the DOA and
the additional DOD estimation with a MIMO radar.
Moreover, in a future work the anisotropy of the Rx and Tx antennas can be incorporated in the
signal model. This is important when evaluating measured signals, since real antennas do not have
an isotropic characteristic. After measuring the antenna characteristics, the CRB of the DOA can be
calculated by determining the derivatives of the measured steering vectors numerically. Then, the
CRB can be compared to the experimentally determined MSE of the estimated DOA.

– 139 –
A. Sample Statistics
We define the sample statistics of matrices and vectors in Section A.1 and derive some properties in
Section A.2.
A.1. Definitions
We define different sample statistics for two matrices X ∈ CK×N1 ,Y ∈ CK×N2 and one weight vector
w ∈ CK with 1T w , 0. K can be regarded as the number of samples, N1 and N2 as the number of
variables. We define the
• weighted sample mean
µw
X
= EWS(X,w) =
1
1T w
wT X ∈ C1×N1 , (A.1)
• weighted sample cross-correlation
CorrWS(X,Y,w) =
1
1T w
YH diag(w) X ∈ CN2×N1 , (A.2)
• weighted sample cross-covariance
CovWS(X,Y,w) = CorrWS
(
X − 1 EWS(X,w) ,Y − 1 EWS(Y,w∗) ,w
)
∈ CN2×N1 , (A.3)
• weighted sample covariance
CovWS(X,w) = CovWS(X,X,w) ∈ CN1×N1 . (A.4)
If w ∝ 1K , all weights in w are equal and the above defined weighted sample mean, cross-correlation,
cross-covariance and covariance simplify to the simple sample mean ES(X), sample cross-correlation
– 140 –
CorrS(X,Y), sample cross-covariance CovS(X,Y) and sample covariance CovS(X), respectively,
ES(X) =
1
K
1TKX ∈ C1×N1 , (A.5)
CorrS(X,Y) =
1
K
YHX ∈ CN2×N1 , (A.6)
CovS(X,Y) = CorrS
(
X − 1 ES(X) ,Y − 1 ES(Y)
)
∈ CN2×N1 , (A.7)
CovS(X) = CovS(X,X) ∈ CN1×N1 . (A.8)
In the special case of a vector x ∈ CK , we write the weighted sample variance VarWS(x,w) and the
sample variance VarS(x) instead of CovWS(x,w) and CovS(x), respectively, i. e.
VarWS(x,w) = CovWS(x,w) ∈ C, (A.9)
VarS(x) = CovS(x) ∈ C. (A.10)
A.2. Lemmas of the Sample Statistics
We state different lemmas of the sample statistics defined above. The proofs are given in Sec-
tion A.2.5.
In the following, we assume
• s, s1, s2 ∈ C.
• w, z ∈ CK and c ∈ CM, with 1T w, 1T z, 1T c , 0 and 1T (z ⊗ c) , 0.
• ⊗ is the Kronecker tensor product. For two matrices C ∈ CS×T ,D ∈ CU×V it is defined as
C ⊗ D =

C11D . . . C1T D
...
. . .
...
CS 1D . . . CS T D
 ∈ CS ·U×T ·V . The Kronecker product of two vectors is a special
case of this definition.
• U,X ∈ CK×N1 ,V,Y ∈ CK×N2 and A ∈ CM×P1 ,B ∈ CM×P2 (K,M can be regarded as the number
of samples, N1,N2, P1, P2 as the number of variables).
• 1K ∈ RK is a vector with all elements equal 1. 1 is a shortcut with the dimension following
from the context.
• 1K×M ∈ RK×M is a matrix with all elements equal 1. 1 is a shortcut with the dimensions
following from the context.
• 0K×M ∈ RK×M is a matrix with all elements equal 0. 0 is a shortcut with the dimensions
following from the context.
– 141 –
A.2.1. Lemmas of ES,CorrS,CovS
Lemma A.2.1. ES,CorrS,CovS are special cases of EWS,CorrWS,CovWS respectively. Hence all
properties of EWS,CorrWS,CovWS apply to ES,CorrS,CovS, respectively, as well.
A.2.2. Lemmas of EWS
Lemma A.2.2. From the definitions
EWS(X,w) = CorrWS(X, 1K ,w),
EWS(Y,w)H = CorrWS(1K ,Y,w
∗)
follow.
Due to Lemma A.2.2 and the properties of CorrWS given in Section A.2.3, the subsequent lemmas
follow:
Lemma A.2.3. EWS is linear in its first argument
EWS(s X,w) = s EWS(X,w)
EWS(U + X,w) = EWS(U,w) + EWS(X,w)
Lemma A.2.4.
EWS(X ⊗ A, z ⊗ c) = EWS(X, z) ⊗ EWS(A, c) ∈ C1×N1P1
Lemma A.2.5.
EWS(X ⊗ 1M×P1 , z ⊗ c) = EWS(X, z) ⊗ 1TP1 ∈ C1×N1P1
EWS(1K×N1 ⊗ A, z ⊗ c) = 1TN1 ⊗ EWS(A, c) ∈ C1×N1P1
Lemma A.2.6.
EWS(X ⊗ 1M×P1 , 1K ⊗ c) = ES(X) ⊗ 1TP1
EWS(1K×N1 ⊗ A, z ⊗ 1M) = 1TN1 ⊗ ES(A)
A.2.3. Lemmas of CorrWS
Lemma A.2.7. From the definition follows
CorrWS(X,Y,w) = CorrWS(Y,X,w∗)H
– 142 –
Lemma A.2.8. From the definition follows that CorrWS is sesquilinear in its first two arguments
CorrWS(s1X, s2Y,w) = s1s∗2 Corr
WS(X,Y,w)
CorrWS(U + X,V + Y,w)
= CorrWS(U,V,w) + CorrWS(U,Y,w) + CorrWS(X,V,w) + CorrWS(X,Y,w)
Lemma A.2.9.
CorrWS(X ⊗ A,Y ⊗ B, z ⊗ c) = CorrWS(X,Y, z) ⊗ CorrWS(A,B, c) ∈ CN2P2×N1P1
Lemma A.2.10.
CorrWS(X ⊗ 1M×P1 ,Y ⊗ 1M×P2 , z ⊗ c) = CorrWS(X,Y, z) ⊗ 1P2×P1 ∈ CN2P2×N1P1
CorrWS(1K×N1 ⊗ A, 1K×N2 ⊗ B, z ⊗ c) = 1N2×N1 ⊗ CorrWS(A,B, c) ∈ CN2P2×N1P1
Lemma A.2.11.
CorrWS(X ⊗ 1M×P1 ,Y ⊗ 1M×P2 , 1K ⊗ c) = CorrS(X,Y) ⊗ 1P2×P1 ∈ CN2P2×N1P1
CorrWS(1K×N1 ⊗ A, 1K×N2B, z ⊗ 1M) = 1N2×N1 ⊗ CorrS(A,B) ∈ CN2P2×N1P1
A.2.4. Lemmas of CovWS
Lemma A.2.12.
CovWS(X,Y,w) = CorrWS(X,Y,w) − EWS(Y,w∗)H EWS(X,w)
= CorrWS(X,Y,w) − EWS(Y∗,w)T EWS(X,w)
Lemma A.2.13. Due to Lemma A.2.12 and the properties of EWS and CorrWS, it follows that CovWS
is sesquilinear in its first two arguments
CovWS(s1X, s2Y,w) = s1s∗2 Cov
WS(X,Y,w)
CovWS(U + X,V + Y,w)
= CovWS(U,V,w) + CovWS(U,Y,w) + CovWS(X,V,w) + CovWS(X,Y,w)
Lemma A.2.14. Due to Lemma A.2.12 and the properties of EWS and CorrWS
CovWS(X ⊗ A,Y ⊗ B, z ⊗ c)
= CorrWS(X,Y, z) ⊗ CorrWS(A,B, c) −
(
EWS(Y, z∗)H EWS(X, z)
)
⊗
(
EWS(B, c∗)H EWS(A, c)
)
. CovWS(X,Y, z) ⊗ CovWS(A,B, c)
– 143 –
follows.
Lemma A.2.15.
CovWS(1K×N1 ⊗ A,Y ⊗ 1M×P2 , z ⊗ c) = 0N2P2×N1P1
Lemma A.2.16. Due to Lemma A.2.12 and the properties of EWS and CorrWS,
CovWS(X ⊗ 1M×P1 ,Y ⊗ 1M×P2 , z ⊗ c) = CovWS(X,Y, z) ⊗ 1P2×P1 ∈ CN2P2×N1P1
CovWS(1K×N1 ⊗ A, 1K×N2 ⊗ B, z ⊗ c) = 1N2×N1 ⊗ CovWS(A,B, c) ∈ CN2P2×N1P1
follow.
Lemma A.2.17. From Lemma A.2.16,
CovWS(X ⊗ 1M×P1 ,Y ⊗ 1M×P2 , 1K ⊗ c) = CovS(X,Y) ⊗ 1P2×P1 ∈ CN2P2×N1P1
CovWS(1K×N1 ⊗ A, 1K×N2B, z ⊗ 1M) = 1N2×N1 ⊗ CovS(A,B) ∈ CN2P2×N1P1
follow.
A.2.5. Proofs
A.2.5.1. Proofs of Lemmas of CorrWS
Proof of A.2.9.
CorrWS(X ⊗ A,Y ⊗ B, z ⊗ c) = 1
(1 ⊗ 1)T (z ⊗ c) (Y ⊗ B)
H
(
diag(z ⊗ c)
)
(X ⊗ A)
=
1
1T z · 1T c
(
YH ⊗ BH
) (
diag(z) ⊗ diag(c)
)
(X ⊗ A)
=
 11T zYH diag(z)X
 ⊗ ( 11T cBH diag(c)A
)
= CorrWS(X,Y, z) ⊗ CorrWS(A,B, c)

Proof of A.2.10. Using Lemma A.2.9 results in
CorrWS(X ⊗ 1M×P1 ,Y ⊗ 1M×P2 , z ⊗ c) = CorrWS(X,Y, z) ⊗ CorrWS(1M×P1 , 1M×P2 , c)
= CorrWS(X,Y, z) ⊗ 1P2×P1
The other property follows analogously. 
– 144 –
Proof of A.2.11. Using Lemma A.2.9 we get
CorrWS(X ⊗ 1M×P1 ,Y ⊗ 1M×P2 , 1K ⊗ c) = CorrWS(X,Y, 1) ⊗ CorrWS(1M×P1 , 1M×P2 , c)
= CorrS(X,Y) ⊗ 1P2×P1
The other property follows analogously. 
A.2.5.2. Proofs of Lemmas of CovWS
Proof of A.2.12.
CovWS(X,Y,w)
=
1
1T w
(
Y − 1 EWS(Y,w∗)
)H
diag(w)
(
X − 1 EWS(X,w)
)
=
1
1T w
(
Y∗ − 1 EWS(Y∗,w)
)T
diag(w)
(
X − 1 EWS(X,w)
)
=
1
1T w
(
YH diag(w)X − YH diag(w)1µw
X
− µw
Y∗
T 1T diag(w)X + µw
Y∗
T 1T diag(w)1µw
X
)
= CorrWS(X,Y,w) − µw
Y∗
Tµw
X
− µw
Y∗
Tµw
X
+ µw
Y∗
Tµw
X
= CorrWS(X,Y,w) − EWS(Y∗,w)T EWS(X,w)
= CorrWS(X,Y,w) − EWS(Y,w∗)H EWS(X,w)

Proof of A.2.15. Due to Lemma A.2.14 and A.2.2
CovWS(1K×N1 ⊗ A,Y ⊗ 1M×P2 , z ⊗ c)
= CorrWS(1K×N1 ,Y, z) ⊗ CorrWS(A, 1M×P2 , c)
−
(
EWS(Y, z∗)H EWS(1K×N1 , z)
)
⊗
(
EWS(1M×P2 , c
∗)H EWS(A, c)
)
=
(
EWS(Y, z∗)H1TN1
)
⊗
(
1P2 E
WS(A, c)
)
−
(
EWS(Y, z∗)H1TN1
)
⊗
(
1P2 E
WS(A, c)
)
= 0

– 145 –
B. Derivation of the Cramer-Rao Bound for
a SIMO Radar
We derive the Cramer-Rao bound CRBu,SIMO in (4.13) for the electrical angle u using a SIMO radar,
cf. Section 4.2.2.1.
We set
C1 = DHD (B.1)
C2 = DHg(gHg)−1gHD. (B.2)
Hence
C = C1 −C2. (B.3)
We use the notation
a = aRx = exp( j dRxu), (B.4)
B(u) = f (u) · e jβ(u) (B.5)
with f (u) ∈ R, f (u) ≥ 0, β(u) ∈ R. Hence,
g(u) = B(u) a(u). (B.6)
In the following, we use an upper dot to denote the derivative with respect to u, e. g. f˙ = ∂ f
∂u . With
(4.11) and (4.5), D can be computed by
D = g˙ = B˙a + Ba˙. (B.7)
In the following,  denotes the entrywise Hadamard product, i. e. x = y  z means xi = yi · zi∀i. We
need the following identities for further computations:
aHa = NRx, (B.8)
a˙ = jdRx  a, (B.9)
– 146 –
aHa˙ = jNRx ES(dRx), (B.10)
a˙Ha˙ = NRx CorrS(dRx, dRx), (B.11)
B˙ = f˙ e jβ + j f β˙ e jβ = e jβ( f˙ + j f β˙), (B.12)∣∣∣B˙∣∣∣2 = f˙ 2 + f 2β˙2, (B.13)
B∗B˙ = f f˙ + j f 2β˙, (B.14)
where ES and CorrS are defined in (A.5) and (A.6), respectively. Then
C1 = g˙Hg˙
= (B˙a + Ba˙)H(B˙a + Ba˙)
=
∣∣∣B˙∣∣∣2 aHa + B˙∗BaHa˙ + B∗B˙a˙Ha + |B|2a˙Ha˙
=
∣∣∣B˙∣∣∣2 NRx + jB˙∗BNRx ES(dRx) − jB∗B˙NRx ES(dRx) + |B|2NRx CorrS(dRx, dRx)
=
∣∣∣B˙∣∣∣2 NRx + j ( f f˙ − j f 2β˙ − ( f f˙ + j f 2β˙)) NRx ES(dRx) + |B|2NRx CorrS(dRx, dRx)
= ( f˙ 2 + f 2β˙2)NRx + 2 f 2β˙NRx ES(dRx) + f 2NRx CorrS(dRx, dRx). (B.15)
C2 can be computed by
C2 = DHg
(
gHg
)−1
gHD
= g˙Hg
(
gHg
)−1
gHg˙
=
(
gHg˙
)∗ (
gHg
)−1
gHg˙
=
1
‖g‖2
∣∣∣∣gHg˙∣∣∣∣2 , (B.16)
gHg˙ = (Ba)H(B˙a + Ba˙)
= B∗B˙NRx + j|B|2NRx ES(dRx)
= f f˙ NRx + jNRx
(
f 2β˙ + f 2 ES(dRx)
)
, (B.17)
‖g‖2 = f 2NRx, (B.18)
C2 =
NRx2
f 2NRx
[(
f f˙
)2
+
(
f 2β˙ + f 2 ES(dRx)
)2]
= f˙ 2NRx + f 2NRx
(
β˙ + ES(dRx)
)2
= f˙ 2NRx + f 2NRx
(
β˙2 + 2β˙ES(dRx) + ES(dRx)2
)
. (B.19)
Putting the expressions (B.15) and (B.19) in (B.3) yields
C = f 2NRx
(
CorrS(dRx, dRx) − ES(dRx)2
)
= f 2NRx VarS(dRx), (B.20)
– 147 –
where we have used the property of VarS given in Lemma A.2.1 and A.2.12. Since all variables in
(B.20) are real, C ∈ R. Using (4.8) and substituting f by |B| results in
CRB−1u,SIMO = 2L
σ2s,SIMO
σ2
|B(u)|2NRx VarS(dRx), (B.21)
from which (4.13) follows.

– 149 –
C. Maximal Beamforming Gain
We derive the maximal achievable beamforming gain for a SIMO radar with NTx Tx antennas which
transmit the same waveform s(t) up to a complex scaling factor. The transmitted signal of the i-th
Tx antenna can be written as
sTxi (t) = cis(t) (C.1)
with the complex scaling factor ci. We put the scaling factors ci in a vector c. Then, the overall
transmitted signal at the target’s electrical angle u is
sTx(t) =
(
aTx(u)
)T
cs(t) (C.2)
with the Tx steering vector aTx(u).
W. l. o. g. , we set the target’s electrical angle u = 0. Then
sTx(t) = s(t)1T c. (C.3)
Thus the beampattern is
B(0) = 1T c. (C.4)
To find the maximal value Bmax of |B(0)|
Bmax = max
c
|B(0)|, (C.5)
1T c has to be maximal or minimal. Obviously, B(0) has an extremum, if the phase of ci is the same
for all i. W. l. o. g. we choose ci ∈ R, i = 1, . . .NTx.
The total transmission power has to be kept constant. In the case of one transmitter (NTx = 1),
|c1|2 = 1. Hence the total power constraint reads
NTx∑
i=1
|ci|2 = 1 (C.6)
– 150 –
and since ci ∈ R
NTx∑
i=1
c2i = 1. (C.7)
Thus, we have to find the extremum of the function f (c)
f (c) = 1T c (C.8)
s.t. g(c) =
NTx∑
i=1
c2i = 1. (C.9)
We use the method of Lagrange Multipliers [Bronstein et al., 2008] with the Lagrangian multiplier
λ and set
Λ(c) = f (c) + λ
(
g(c) − 1
)
. (C.10)
Taking the derivate of Λ(c) with respect to c and setting it to 0 yields
∂Λ
∂c
= 1 + λ2c != 0. (C.11)
Hence
λ = − 1
2ci
, i = 1, . . .NTx. (C.12)
Therefore
ci = c j, i, j = 1, . . .NTx. (C.13)
Thus, with (C.9) follows
NTx∑
i=1
c2i = NTxc
2
i = 1, i = 1, . . .NTx, (C.14)
and ci = ±
√
1
NTx
, i = 1, . . .NTx. (C.15)
This yields for Bmax in (C.5)
Bmax = |1T c|
= NTx
√
1
NTx
=
√
NTx. (C.16)
Thus, the maximal achievable beamforming gain for a SIMO radar using NTx antennas is
√
NTx.
– 151 –
D. Computation of the SNR after a Discrete
Fourier Transform
We consider a complex oscillation exp ( j (ωdiff n + ϕ)) and compute the DFT of it. We derive the
SNR of the complex amplitude of the highest peak in the spectrum as explained in Section 4.4.1.
Then, we compute the SNR gain of the DFT.
The SNR after the DFT is (4.47)
SNRafter =
ρs|S w(ωmax)|2
Var(Nw(ωmax))
(D.1)
and since ωmax = ωdiff
SNRafter =
ρs|S w(ωdiff)|2
Var(Nw(ωdiff))
. (D.2)
We first compute |S w(ωdiff)|2. Since w(n) is nonzero only for 0 ≤ n ≤ N−1 and using the convolution
theorem [Kammeyer and Kroschel, 2012], S w(ω) in (4.43) can be written as
S w(ω) =
N−1∑
n=0
w(n)s(n) e− jωn
=
∞∑
n=−∞
w(n)s(n) e− jωn
= (W ∗ S )(ω) (D.3)
where the DTFT of w(n) is
W(ω) =
∞∑
n=−∞
w(n) e− jωn =
N−1∑
n=0
w(n) e− jωn (D.4)
and
S (ω) =
∞∑
n=−∞
s(n) e− jωn
= s0 2pi e jϕ δ(ω − ωdiff). (D.5)
– 152 –
Hence
S w(ω) =
1
2pi
∫ pi
−pi
W(ω˜)S (ω − ω˜) dω˜
=
1
2pi
∫ pi
−pi
W(ω˜) s0 2pi e jϕ δ(ω − ω˜ − ωdiff) dω˜
= s0W(ω − ωdiff) e jϕ (D.6)
and thus
S w(ωdiff) = s0W(0) e jϕ . (D.7)
From (D.4) follows
W(0) =
N−1∑
n=0
w(n) (D.8)
and therefore, since w(n) ∈ R,
|S w(ωdiff)|2 = s20
N−1∑
n=0
w(n)
2 . (D.9)
Next, we compute the variance of the noise. With Nw(ω) in (4.44) and using (4.39) we get
Var(Nw(ω)) = E
(
|Nw(ω)|2
)
=
N−1∑
m=0
N−1∑
n=0
w(m)w(n) e− jω(m−n) E [n(m)n∗(n)]
=
N−1∑
m=0
N−1∑
n=0
w(m)w(n) e− jω(m−n) σ2nδ(m − n)
= σ2n
N−1∑
n=0
w(n)2. (D.10)
Plugging the expressions in (D.2) and using (4.40) yields
SNRafter =
ρss20
σ2n
(∑N−1
n=0 w(n)
)2∑N−1
n=0 w(n)2
= SNRbefore · SNRgain (D.11)
with the SNR gain
SNRgain =
(∑N−1
n=0 w(n)
)2∑N−1
n=0 w(n)2
. (D.12)
– 153 –
E. Maximum Likelihood Estimator for an
NTargets Target Model
We present the Maximum Likelihood (ML) estimator for the estimation of the unknown parameters
of NTargets targets with the following signal model
x(l) =
NTargets∑
k=1
a
(
Θk
)
sk(l) + n(l), l = 1, . . . , L (E.1)
where L ∈ R is the number of measurement cycles, x(l) ∈ CNant is the complex signal of the Nant
antennas. sk(l) ∈ C is the unknown deterministic complex signal strength of the k-th target of
measurement cycle l. n(l) ∈ CNant is the additional complex noise. a
(
Θk
)
is a function which depends
on the k-th target’s unknown parameter vector Θk ∈ RM. Hence, M is the number of unknowns per
target in addition to the unknown signal strengths sk(l). a
(
Θk
)
can be e. g. the usual array steering
vector which depends on the DOA of a target, but it can also be a more general function which
depends on other parameters like the target’s Doppler frequency.
In literature, such an ML estimator is called deterministic maximum likelihood (DML) estimator
[Krim and Viberg, 1996], since the unknown signal strengths do not have a probability density
function but are assumed to be deterministic.
The noise n(l) is assumed to be circular complex Gaussian with zero mean, spatially and temporally
uncorrelated with
E
(
n(l) nH(m)
)
= δl,m σ
2 INant×Nant . (E.2)
Here, σ2 is the unknown variance.
We stack the signal strengths in a vector
s(l) =
[
s1(l), . . . , sNTargets(l)
]T ∈ CNTargets , l = 1, . . . , L. (E.3)
Hence, the unknown quantities Θ to be estimated from x(l), l = 1, . . . , L, are given by
Θ =
[
ΘT1 , . . . ,Θ
T
NTargets
, sT (1), . . . , sT (L), σ2
]T
. (E.4)
– 154 –
To compute the ML estimator of Θ, we write the signal model in a more compact form. We define
the unknown parameter matrix
Θ =
[
Θ1 . . . ΘNTargets
]
∈ RM×NTargets (E.5)
and the matrix
A (Θ) =
[
a
(
Θ1
)
. . . a
(
ΘNTargets
)] ∈ CNant×NTargets . (E.6)
Then
x(l) = A (Θ) s(l) + n(l). (E.7)
We give an overview how to derive the ML estimator of Θ similar to [Krim and Viberg, 1996;
Böhme, 1985]. The likelihood function fx(l)
(
x(l);Θ, s(l), σ2
)
of x(l) is complex normal distributed
fx(l)
(
x(l);Θ, s(l), σ2
)
∼ CN
(
A(Θ)s(l), σ2 INant×Nant
)
. (E.8)
Thus the pdf is given by [Kay, 1993]
fx(l)
(
x(l); Θ, s(l), σ2
)
=
1
piNant det(σ2 INant×Nant)
·
exp
{
−
(
x(l) − A(Θ)s(l)
)H (
σ2 I
)−1 (
x(l) − A(Θ)s(l)
)}
=
1
piNantσ2Nant
exp
−‖x(l) − A(Θ)s(l)‖2
σ2
 . (E.9)
Since the observations x(l) are independent, the likelihood of all observations
X =
[
x(1) . . . x(L)
]
∈ CNant×L (E.10)
is
f
(
X; Θ
)
=
L∏
l=1
fx(l)
(
x(l);Θ, s(l), σ2
)
=
1
piL Nantσ2 L Nant
exp
− L∑
l=1
‖x(l) − A(Θ)s(l)‖2
σ2
 . (E.11)
The ML estimate Θˆ of Θ are the arguments which maximize f
(
X; Θ
)
, given the observations X
Θˆ = arg max
Θ
f
(
X; Θ
)
. (E.12)
Since the logarithm is strictly monotonically increasing, Θˆ can also be computed by minimizing the
– 155 –
negative logarithm of the likelihood, called log-likelihood lML.
Θˆ = arg min
Θ
lML
(
X; Θ
)
(E.13)
with
lML
(
X; Θ
)
= − ln f
(
X; Θ
)
= −L Nant ln
(
piσ2
)
+
1
σ2
L∑
l=1
‖x(l) − A(Θ)s(l)‖2. (E.14)
The minima with respect to s(l) and σ2 can be computed explicitly [Krim and Viberg, 1996]. A
convenient way to derive the minima with respect to the complex vectors s(l) is the use of Wirtinger
derivatives introduced by Wirtinger [Wirtinger, 1927]. The ML estimates are
σˆ2 =
1
Nant
Tr
(
P⊥A(Θ) Rˆ
)
, (E.15)
sˆ(l) = A+(Θ) x(l), l = 1, . . . , L, (E.16)
with the sample correlation matrix Rˆ
Rˆ =
1
L
L∑
l=1
x(l)xH(l) =
1
L
X XH ∈ CNant×Nant . (E.17)
A+ is the Moore-Penrose pseudo-inverse of A,
A+ =
(
AHA
)−1
AH ∈ CNTargets×Nant , (E.18)
and P⊥A is the orthogonal projector onto the nullspace of A
H,
PA = A A+ ∈ CNant×Nant , (E.19)
P⊥A = I − PA ∈ CNant×Nant . (E.20)
Plugging these ML estimates in (E.14) and (E.13), we can compute the ML estimates Θˆ of the
parameters Θ. This results in
Θˆ = arg min
Θ
Tr
(
P⊥A(Θ)Rˆ
)
(E.21)
or equivalently
Θˆ = arg max
Θ
Tr
(
PA(Θ)Rˆ
)
. (E.22)
In general, this is a nonlinear function of Θ which cannot be simplified further.
– 156 –
For the special case of one target, NTargets = 1, Θ = Θ1, (E.22) can be simplified further and the ML
estimator Θˆ1 of Θ1 reads
Θˆ1 = arg maxΘ1
Tr
(
a
(
aHa
)−1
aH X XH
)
= arg max
Θ1
L∑
l=1
∣∣∣aH(Θ1) x(l)∣∣∣2 1‖a(Θ1)‖2 . (E.23)
– 157 –
F. Asymptotic Properties of Maximum
Likelihood Estimators
The deterministic CRB is a lower bound for unbiased estimators [Kay, 1993]. In general, the de-
terministic maximum likelihood (ML) estimator is not unbiased and we cannot conclude that the
CRB is a lower bound for the ML estimator. Because of this, we carry out numerical simulations
to see if the ML estimator achieves the CRB. In the considered radar applications, we investigate
this for increasing SNR. Analytical results can be obtained for an increasing number of observations
[Böhme, 1998; Yang, 2011]. We briefly present them here.
We consider a sequence of ML estimators, where the number of observations increases. Let
X1, . . . , XN be N independent identically distributed random observations. The pdf of each observa-
tion is fn(xn; Θ) with the unknown deterministic parameter Θ. Let J be the Fisher information matrix
(FIM) for one observation. Let ΘˆN be the ML estimator of Θ given N observations. Under some
regularity conditions, the following properties hold:
• ΘˆN is consistent: ΘˆN converges for N → ∞ to Θ in probability.
• ΘˆN is asymptotically unbiased: limN→∞E(ΘˆN) = Θ.
• ΘˆN is asymptotically normal distributed: the cumulative distribution function of
√
N(ΘˆN −Θ)
converges for N → ∞ to the cumulative distribution function of a normal distributed, zero
mean random vector with covariance matrix C.
• ΘˆN is asymptotically efficient: C = J−1, where C is the covariance matrix of the asymptotic
distribution.

– 159 –
G. Baseband Signal Model for a Moving
Target in the Time Domain
Here we present the calculations used to derive the baseband signal s(t) and the phase difference ∆ϕ
presented in Section 5.1.1.
G.1. Computation of the Received Signal
To compute the signal sRx(t) which is received by the radar, we first compute the Doppler frequency
change of the wave exp( jωt) as explained in Section 5.1.1.2. After that we use the result to calculate
sRx(t).
Consider the transmitted wave exp( jωt). We compute the frequency change of the signal which
reaches the target. Note that in classical physics, the change in frequency of a sound signal is
different for the following two cases: a) the Rx is at rest relative to the transmission medium and
the Tx is moving and b) the Rx is moving relative to the transmission medium and the Tx is at rest.
Here, we consider electromagnetic waves which do not need a transmission medium. For an exact
derivation, we have to use theory of relativity. From that it follows: In the frame of reference of the
target, the target receives a signal with the angular frequency [Rebhan, 2001]
ω′ =
√
ζ ω, (G.1)
ζ =
1 − β
1 + β
, (G.2)
β =
v
c
, (G.3)
where the relative velocity v is defined to be positive if the target moves away from the radar. c is
the speed of light. The signal is reflected at the target. We treat this as a signal which is transmitted
from the target to the radar with transmission frequency ω′. Since the target moves relative to the
radar, the frequency changes once again by the same factor. Thus the radar system receives a signal
with the angular frequency
ω′′ =
√
ζ ω′ = ζ ω. (G.4)
– 160 –
Computing the Taylor series of ζ w. r. t. β at β = 0 yields
ζ(β) = 1 − 2β + O(β2). (G.5)
Thus the angular frequency changes by
ωDoppler = ω − ω′′ ≈ 2 vc ω (G.6)
for |v|  c. Now we compute the received signal. Let us ignore the phase jump at the reflection at
the target and let us ignore the time delay between the transmitted and received signal for a moment.
Then, with (5.10), the radar receives
sRx0(t) ∝
∫ ∞
−∞
S Tx(ω) e jω
′′ t 1
2pi
dω (G.7)
=
∫ ∞
−∞
S Tx(ω) e jζω t
1
2pi
dω (G.8)
= sTx(ζt). (G.9)
Now we compute the time delay. In the frame of reference of the radar, the target’s position xtarget(t)
is given by
xtarget(t) = d0 + vt, (G.10)
where d0 is the distance of the target at the start of the transmission of the chirp, i. e. t = 0. Consider
a short electromagnetic pulse which is transmitted from the radar at t = 0. It’s position xem(t) is
xem(t) = ct. (G.11)
The time t0 at which the pulse reaches the target is the solution of
xem(t0) = xtarget(t0). (G.12)
It follows
t0 =
d0
c − v . (G.13)
The pulse is reflected at the target and needs to travel back exactly the same distance xtarget(t0). Thus
the radar receives the reflected pulse at
τ = 2 t0 = 2
d0
c − v = κ τ0, (G.14)
τ0 = 2
d0
c
, (G.15)
– 161 –
κ =
1
1 − β, (G.16)
where τ0 is the time delay of a stationary target at distance d0. Thus the received signal, including
the time delay, is given by
sRx(t) = sRx0(t − τ) ∝ sTx (ζ(t − τ)) . (G.17)
The transmitted signal sTx(t) is defined in (5.5). The amplitude of the received signal is smaller than
the transmitted one due to the damping and the radar cross section of the target. Furthermore, the
duration of the signal changes due to relativistic reasons, and since the energy has to be conserved,
the amplitude changes as well. We include all these effects in the amplitude ARx. Moreover, in
general a phase jump occurs at the reflection of the signal which we denote by ϕtarget. Thus the
received signal is
sRx(t) = ARx cos (ϕRx(t)) (G.18)
with
ϕRx(t) = ϕTx (ζ(t − τ)) + ϕ0 + ϕtarget (G.19)
and ϕTx(t) in (5.8) is the time dependent phase of the transmitted signal and ϕ0 the phase of the
transmitted signal at t = 0. Now we compute
ϕTx (ζ(t − τ)) = ϕTx (ζ(t − κτ0)) (G.20)
in more detail. Plugging in (5.8) yields
ϕTx (ζ(t − κτ0)) = ω0 (ζ(t − κτ0)) + 12 χ (ζ(t − κτ0))
2 . (G.21)
We compute the first order truncated Taylor series of (G.21) w. r. t. β at β = 0. With
ζ(β) = 1 − 2 β + O(β2), (G.22)
κ(β) = 1 + β + O(β2), (G.23)
this results in
ϕTx (ζ(t − κτ0)) = ϕTx(t) − ϕbase(t) + O(β2), (G.24)
with
ϕbase(t) = ω0 τ0 (1 − β) − 12 χ τ
2
0 (1 − 2β) + t [ω0 2β + χ τ0 (1 − 3β)] + t22 χ β. (G.25)
– 162 –
Thus the phase of the received signal is
ϕRx(t) = ϕTx(t) − ϕbase(t) + ϕ0 + ϕtarget. (G.26)
G.2. Baseband Signal After the IQ-Mixer
We consider an IQ-mixer which mixes the received signal sRx(t) (G.18) with the transmitted one, i. e.
it is mixed with ATx cos (ϕTx(t) + ϕ0) and ATx sin (ϕTx(t) + ϕ0). We assume the signal to be narrow-
band, i. e. the bandwidth is much smaller than the carrier frequency. Thus the phase of the complex
baseband signal is given as the difference of the Tx phase ϕTx(t) + ϕ0 and the Rx phase ϕRx(t) in
(G.26). Hence the complex baseband signal is
s(t) =
1
2
ARxATx exp ( j [ϕTx(t) + ϕ0 − ϕRx(t) ] )
=
1
2
ARxATx exp
(
j [ϕbase(t) − ϕtarget]
)
= A exp ( jϕbase(t) ) , (G.27)
where we have defined
A =
1
2
ARxATx exp(− jϕtarget). (G.28)
For convenience, we express ϕbase(t) in (G.25) as
ϕbase(t) = ϕbase,a + ϕbase,b(t) + ϕbase,c(t), (G.29)
ϕbase,a = ω0 τ0 (1 − β) − 12 χ τ
2
0 (1 − 2β), (G.30)
ϕbase,b(t) = t [ω0 2β + χ τ0 (1 − 3β)], (G.31)
ϕbase,c(t) = t22 χ β. (G.32)
(G.29) is the same result as the one derived in [Stove, 1992] up to differences in some of the factors
in front of β. The reason is that in [Stove, 1992] a classical derivation is used for the Doppler effect
instead of the relativistic one and therefore it is not exactly correct. Usually these differences do not
matter for the DOA estimation.
In the following, we show that one of the terms in (G.30) is negligible. In order to do that, we have
to compute how much a phase error influences the error in the electrical angle. In Appendix G.3, it
is shown that for a typical radar system with a DOA accuracy of σϑ = 0.1◦ ≈ 1.7 · 10−3 rad and an
observed DOA region with |ϑ| ≤ 30◦, phase errors smaller than 5 · 10−3 are negligible. We compute
the order of magnitude of χ τ20 β in (G.30) using the values given in Tab. 5.1 as maximal values for
the different parameters. For an FMCW radar, χ τ20 β is of order of magnitude 10
−6 and for a chirp
– 163 –
sequence radar 10−5. Note that β and τ0 are given by (5.17) and (5.18), respectively. Therefore, the
term χ τ20 β is negligible and
ϕbase,a = ω0 τ0 (1 − β) − 12 χ τ
2
0. (G.33)
G.3. Influence of the Phase Error on the Error in the Electrical
Angle
In order to know which terms of the phase of the baseband signal are negligible, we have to know
how much a phase error influences the error in the electrical angle. To compute this, we consider two
antennas with an antenna spacing ∆d. The relation between the phase difference ∆ϕ2 antennas between
the two antennas and the electrical angle u of the target is given by
∆ϕ2 antennas = 2 pi
∆d
λ
u (G.34)
with the carrier wavelength λ. Hence
u =
∆ϕ2 antennas
2pi∆d
λ
. (G.35)
The error σu of u due to an error σ∆ϕ2 antennas of the phase difference ∆ϕ2 antennas is therefore
σu =
σ∆ϕ2 antennas
2pi∆d
λ
. (G.36)
The smaller the distance ∆d between two antennas, the larger the error σu due to an error σ∆ϕ2 antennas .
We consider the worst case: the smallest distance ∆d between two antennas is typically λ2 . Hence
σu ≈ σ∆ϕ2 antennas
pi
. (G.37)
Thus, errors σ∆ϕ2 antennas smaller than pi · σu are negligible.
A typical long range radar system achieves a DOA accuracy σϑ = 0.1◦ ≈ 1.7 · 10−3 rad [Reiher,
2012]. Since u = sin(ϑ),
du = dϑ · cos(ϑ). (G.38)
Short, mid, and long range radars for automotive applications have different field of views. A short
range radar observes nearly the whole DOA region, i. e. −90◦ ≤ ϑ ≤ 90◦, for a mid range radar
−50◦ ≤ ϑ ≤ 50◦ and a long range radar observes typically ±10◦ to ±20◦[Schoor, 2010]. A high
DOA accuracy is important for long range radars, because for a target at a large distance, a small
– 164 –
error in the DOA estimation can lead to a large lateral displacement. Because of this we consider a
long range radar and set the observed DOA region to |ϑ| ≤ 30◦ to include the worst case. Hence, in
order to yield at DOA accuracy σϑ = 0.1◦, the accuracy σu of the electrical angle has to satisfy
σu ≤ σϑ · | cos(ϑ)| ∀ |ϑ| ≤ 30◦, (G.39)
and therefore, plugging in the worst case,
σu = σϑ · | cos(30◦)| ≈ 1.5 · 10−3. (G.40)
Plugging this in (G.37) yields
σ∆ϕ2 antennas ≈ 5 · 10−3. (G.41)
Hence, phase errors σ∆ϕ2 antennas smaller than 5 · 10−3 are negligible.
G.4. Phase Difference Between Two Chirps
We calculate the difference ∆ϕ1k of the phases of the baseband signals of the first and k-th chirp at
the time in the middle of the chirps. As explained in Section 5.1.1.2, we investigate chirps which
have the same carrier frequency in the middle of the chirp
ω1(tM,1) = ωk(tM,k) = ω0M (G.42)
as depicted in Fig. 5.4. The subscript i = 1, k, denotes that the corresponding variable belongs to
chirp i. The angular frequencies of the chirps can be expressed as
ωi(t) = ω0M + χi
(
t − tstart,i − 12 Ti
)
, t ∈ [tstart,i, tstart,i + Ti]. (G.43)
We rewrite it as
ωi(t) = ω0,i + χi (t − tstart,i), (G.44)
ω0,i = ω0M − 12 χi Ti. (G.45)
ω0,i is the angular frequency at the start of the i-th chirp. An example of a chirp with these parameters
is depicted in Fig. G.1. Hence it has the same form as (5.7) but with ω0,i instead of ω0 and the start of
the chirp is set to tstart,i. Thus we can use the result (5.13) replacing t by t− tstart,i. Moreover, the chirp
dependent parameters ω0, τ0, χ in (5.14) to (5.16) have to be replaced by ω0,i, τ0,i, χi, respectively.
– 165 –
t
tstart,i tM,i
ωi(t)
ω0M
ω0,i
Ti
Fω,i
Figure G.1.: Example of a linear chirp i and its parameters
Thus, the phase ϕbase,i(t) of the received baseband signal of the i-th chirp is
ϕbase,i(t) = ϕ
(i)
base(t − tstart,i), (G.46)
where
ϕ(i)base(t) = ϕ
(i)
base,a + ϕ
(i)
base,b(t) + ϕ
(i)
base,c(t), (G.47)
ϕ(i)base,a = ω0,i τ0,i (1 − β) −
1
2
χi τ
2
0,i (G.48)
ϕ(i)base,b(t) = t [ω0,i 2β + χi τ0,i (1 − 3β)], (G.49)
ϕ(i)base,c(t) = t
22 χi β. (G.50)
Now we can write the phase difference ∆ϕ1k we want to derive as
∆ϕ1k = ϕbase,k(tM,k) − ϕbase,1(tM,1). (G.51)
We compute ϕbase,i(tM,i) in more detail.
ϕbase,i(tM,i) = ϕ
(i)
base(tM,i − tstart,i)
= ϕ(i)base
(Ti
2
)
= ω0,i τ0,i (1 − β) − 12 χi τ
2
0,i +
Ti
2
[ω0,i 2β + χi τ0,i (1 − 3β)] + 12 T
2
i χi β. (G.52)
Inserting (G.45) results in
ϕbase,i(tM,i) = ω0M τ0,i (1 − β) − 12 χi τ
2
0,i + ω0M βTi − βTi χi τ0,i. (G.53)
By using (5.29), the round trip time τM,i at the middle of chirp i, defined in (5.30), can be written as
τM,i = 2
d0,i + v Ti/2
c
= τ0,i + βTi. (G.54)
– 166 –
Hence
τ0,i = τM,i − βTi. (G.55)
Inserting this in (G.53) yields
ϕbase,i(tM,i) = ω0M τM,i (1 − β) − 12 χi τ
2
M,i + ω0M β
2 Ti +
1
2
χi β
2 T 2i . (G.56)
Using the maximum values of Tab. 5.1, we compute the order of magnitude of the two terms includ-
ing β2: ω0M β2 Ti and 12 χi β
2 T 2i =
1
2 β
2 Fω,i Ti. For an FMCW radar, they are of order of magnitude
10−4 and 10−6, respectively, and for a chirp sequence radar of order of magnitude 10−6 and 10−8,
respectively. Hence these terms are negligible and
ϕbase,i(tM,i) = ω0M τM,i (1 − β) − 12 χi τ
2
M,i. (G.57)
The time delay τM,k can be written as
τM,k =
2
c
dM,k =
2
c
(dM,1 + v ∆tM,1k) (G.58)
with
∆tM,1k = tM,k − tM,1. (G.59)
Thus, the difference of the time delays is
τM,k − τM,1 = 2 β∆tM,1k. (G.60)
Plugging (G.57) in (G.51) yields
∆ϕ1k = ω0M 2 β∆tM,1k (1 − β) −
(
1
2
χk τ
2
M,k −
1
2
χ1 τ
2
M,1
)
. (G.61)
The term ω0M 2 ∆tM,1k β2 is negligible since with the maximum values given in Tab. 5.1 it is smaller
than 3 · 10−3 for both the FMCW and chirp sequence radar. Hence
∆ϕ1k = ωDoppler ∆tM,1k −
(
1
2
χk τ
2
M,k −
1
2
χ1 τ
2
M,1
)
, (G.62)
with
ωDoppler = ω0M 2
v
c
. (G.63)
– 167 –
G.5. Simplification of ∆ϕ1k for a Chirp Sequence Radar
We simplify ∆ϕ1k in (G.62) for a chirp sequence radar. A chirp sequence radar transmits several
equal chirps with χ1 = χk = χ. Thus we can write
∆ϕ1k = ωDoppler ∆tM,1k − 12 χ
(
τ2M,k − τ2M,1
)
. (G.64)
From (G.60) follows τM,k = τM,1 + 2β∆tM,1k and thus
τ2M,k − τ2M,1 = 4 τM,1 β∆tM,1k + 4 β2∆t2M,1k. (G.65)
Hence
∆ϕ1k = ωDoppler ∆tM,1k − 2χ
(
τM,1 β∆tM,1k + β2∆t2M,1k
)
. (G.66)
We use the maximum values given in Tab. 5.1 to compute the maximum value of 2χ β2∆t2M,1k. Con-
sider a measurement with overall measurement time of 20 ms. This time is approximately the
time difference ∆tM,1k between the last and the first chirp. Hence we set ∆tM,1k = 20 ms. Then
2 χ β2∆t2M,1k ≈ 3.6 · 10−3 and can be neglected. The other term 2 χ τM,1 β∆tM,1k is in general larger
and cannot be neglected, but it can be incorporated together with the first term by defining a new
unknown pseudo Doppler frequency
ωpseudoDoppler = ωDoppler − 2 χ τM,1 β
= 2
v
c
(
ω0M − χ τM,1) . (G.67)
Then ∆ϕ1k has the simple form
∆ϕ1k = ωpseudoDoppler ∆tM,1k. (G.68)
Since ωpseudoDoppler contains only the round trip time τM,1 of the first chirp, ωpseudoDoppler stays the same
if we consider the phase difference ∆ϕ1k between any chirp k and the first chirp.
G.6. Estimation of the Error in ∆ϕcorrected,1k
We compute the error in ∆ϕcorrected,1k which arises since not the exact values dM,1, dM,k are used in
(5.34) but their estimates dˆM,1, dˆM,k. Typically, an FMCW radar achieves a distance accuracy σd =
0.1 m [Reiher, 2012]. We define
∆ϕ˜ =
2
c2
(
χ1d2M,1 − χkd2M,k
)
. (G.69)
– 168 –
Then the error σ∆ϕ˜ in ∆ϕcorrected,1k due to the estimation error of dˆM,1 and dˆM,k can be computed by
σ∆ϕ˜ =
∂∆ϕ˜
∂dM,1
σdM,1 +
∂∆ϕ˜
∂dM,k
σdM,k , (G.70)
with σdM,i being the accuracy of dM,i. Hence
∣∣∣σ∆ϕ˜∣∣∣ ≤ ∣∣∣∣∣∣ ∂∆ϕ˜∂dM,1
∣∣∣∣∣∣σdM,1 +
∣∣∣∣∣∣ ∂∆ϕ˜∂dM,k
∣∣∣∣∣∣σdM,k , (G.71)
We set σdM,i = 0.1 m, i = 1, 2. This yields∣∣∣σ∆ϕ˜∣∣∣ = 2c2 (|2 χ1dM,1|σdM,1 + |2 χkdM,k|σdM,k) ≈ 3 · 10−3, (G.72)
where we have plugged in the values of the FMCW radar parameters and the target parameters given
in Tab. 5.1. As shown above, such a phase error is negligible for a typical radar system. If an even
higher chirp rate than the one given in Tab. 5.1 is used and/or the distance estimations dˆM,1, dˆM,k are
very imprecise, the error might have to be considered.
– 169 –
H. Derivation of the Cramer-Rao Bound for
a Planar TDM MIMO Radar with a
Non-Stationary Target
We compute the CRB for the electrical angles and Doppler frequency of a non-stationary target using
a TDM MIMO radar, cf. Section 5.2.1. We have to derive (5.91) and show that C ∈ R3×3. We define
C = C(1) − C(2) (H.1)
with
C(1) = DH D, (H.2)
C(2) = DH Pb D
= DHb
1
‖b‖2 b
HD. (H.3)
Computation of D We compute D defined in (5.86). With b given in (5.56), the derivative with
respect to ω reads
∂b
∂ω
= j tVirt  b. (H.4)
The derivatives with respect to u can be computed by
∂b
∂uT
=
[
∂b
∂ux
∂b
∂uy
]
=
√
ρVirt  exp( j tVirtω)  ∂
∂uT
exp( j DVirtu), (H.5)
∂
∂uT
exp( j DVirtu) =
[
exp( j DVirtu) exp( j DVirtu)
]
 j DVirt, (H.6)
∂b
∂uT
=
[
b b
]
 j DVirt. (H.7)
– 170 –
Plugging this in (5.86) yields
D = j
[
b b b
]

[
DVirt tVirt
]
= j
[
b b b
]
 V, (H.8)
with V =
[
DVirt tVirt
]
. (H.9)
Computation of C(1) From (5.56) follows
b  b∗ = ρVirt, (H.10)
‖b‖2 =
(
1TρVirt
)
. (H.11)
For some matrices X ∈ CNVirt×K1 and Y ∈ CNVirt×K2 holds
[((
b 1TK2
)
 Y
)H ((
b 1TK1
)
 X
)]
kl
=
NVirt∑
m=1
[(
b 1TK2
)H]
km
[
YH
]
km
[(
b 1TK1
)]
ml
[X]ml
=
∑
m
[
YH
]
km
b∗mbm[X]ml
=
[
YH diag(ρVirt)X
]
kl
. (H.12)
Hence, using (H.11) and the definition of CorrWS in (A.2) gives[(
b 1TK2
)
 Y
]H [(
b 1TK1
)
 X
]
= ‖b‖2 CorrWS(X,Y, ρVirt). (H.13)
Thus with (H.8) follows
C(1) = DHD
=
[(
b 1T3
)
 V
]H [(
b 1T3
)
 V
]
= ‖b‖2 CorrWS(V,V, ρVirt) (H.14)
Computation of C(2) C(2) is defined in (H.3). With the help of (H.13), setting Y = 1NVirt , we get
bHD = [b  1NVirt]H j
[(
b 1T3
)
 V
]
= j ‖b‖2 CorrWS(V, 1NVirt , ρVirt)
= j ‖b‖2 EWS(V, ρVirt), (H.15)
where we have used Lemma A.2.2. Putting this in (H.3) yields
C(2) = ‖b‖2 EWS(V, ρVirt)H EWS(V, ρVirt). (H.16)
– 171 –
Plugging the results in C With the property of CovWS in Lemma A.2.12 we get
C = C(1) − C(2)
= ‖b‖2
[
CorrWS(V,V, ρVirt) − EWS(V, ρVirt)H EWS(V, ρVirt)
]
= ‖b‖2 CovWS(V,V, ρVirt)
= ‖b‖2 CovWS(V, ρVirt). (H.17)
Using (H.11) and the definition of ρVirt (5.57) yields
C = NRx
(
1Tρ
)
CovWS(V, ρVirt). (H.18)
Since all variables in (H.18) are real, C ∈ R3×3.
Simplifying C in (H.18)
V =
[
DVirt tVirt
]
=
[
1NPulse ⊗ DRx + DPulse ⊗ 1NRx t ⊗ 1NRx
]
= 1NPulse ⊗
[
DRx 0
]
+
[
DPulse t
]
⊗ 1NRx
= 1NPulse ⊗ V1 + V2 ⊗ 1NRx (H.19)
with
V1 =
[
DRx 0
]
, (H.20)
V2 =
[
DPulse t
]
. (H.21)
Using the properties of CovWS, cf. Appendix A.2.4, we get
CovWS(V, ρVirt) = CovWS(V,V, ρVirt)
= CovWS(1NPulse ⊗ V1 + V2 ⊗ 1NRx , 1NPulse ⊗ V1 + V2 ⊗ 1NRx , ρ ⊗ 1NRx)
= CovWS(1NPulse ⊗ V1, 1NPulse ⊗ V1, ρ ⊗ 1NRx)
+ CovWS(1NPulse ⊗ V1,V2 ⊗ 1NRx , ρ ⊗ 1NRx)
+ CovWS(V2 ⊗ 1NRx , 1NPulse ⊗ V1, ρ ⊗ 1NRx)
+ CovWS(V2 ⊗ 1NRx ,V2 ⊗ 1NRx , ρ ⊗ 1NRx)
= CovS(V1,V1) + 0 + 0 + CovWS(V2,V2, ρ)
= CovS(V1,V1) + CovWS(V2,V2, ρ). (H.22)
– 172 –
Hence
CovWS(V, ρVirt) =

CovS(DRx)
0
0
0 0 0
 +
 CovWS(DPulse, ρ) CovWS(t,DPulse, ρ)CovWS(DPulse, t, ρ) VarWS(t, ρ)
 . (H.23)
Putting this in (H.18) gives (5.91), which we wanted to derive.
– 173 –
I. Proof of Theorem 1
As shown in Section 5.2.2.2, CRBu,SIMO,CRBu and CRBu,stat differ by the terms U given in (5.123)
to (5.126). The expression
(
CovWS(dPulse, t, ρ)
)2
appears in Upenalty (5.126). We show that
(
CovWS(dPulse, t, ρ)
)2 ≤ |VarWS(dPulse, ρ)| · |VarWS(t, ρ)|, (I.1)
for any dPulse, t, ρ. For arbitrary but fixed ρ, we assume w. l. o. g. EWS(dPulse, ρ) = EWS(t, ρ) = 0. This
can always be achieved by shifting the spatial or temporal coordinate system by a constant, which is
physically irrelevant. Then, due to Lemma A.2.12,
CovWS(dPulse, t, ρ) = CorrWS(dPulse, t, ρ), (I.2)
VarWS(dPulse, ρ) = CorrWS(dPulse, dPulse, ρ), (I.3)
VarWS(t, ρ) = CorrWS(t, t, ρ). (I.4)
Note that ρi ∈ R, ρi > 0, i = 1, . . . ,NPulse. For fixed ρ and vectors x, y, ∈ RNPulse , Corr
WS(x, y, ρ) is an
inner product
〈x, y〉 := CorrWS(x, y, ρ), (I.5)
since it fulfills all axioms of an inner product, i. e. it is symmetric in x, y, linear in its first argument
and positive definite [Poole, 2011]. Thus the Cauchy Schwarz inequality holds [Poole, 2011]
|CorrWS(x, y, ρ)|2 ≤ |CorrWS(x, x, ρ)| · |CorrWS(y, y, ρ)|. (I.6)
Therefore
|CorrWS(dPulse, t, ρ)|2 ≤ |CorrWS(dPulse, dPulse, ρ)| · |CorrWS(t, t, ρ)|, (I.7)
from which follows (I.1). Plugging (I.1) in (5.126) yields 0 ≤ Upenalty ≤ VarWS(dPulse, ρ). With
(5.123) to (5.125) follows USIMO ≤ U ≤ Ustat and thus (5.127).
The second part of the theorem (5.128) follows immediately from (5.125). 

– 175 –
J. Proof of Theorem 2: Weighted Mean
Transmit Times and Decoupling of
Electrical Angles from the Doppler
Frequency
We prove Theorem 2 in Section 5.2.3.1. For that purpose, we first state and prove different lemmas
of CovWS.
J.1. Lemmas of the Weighted Sample Covariance of Clustered
Values
Lemma J.1.1 (CovWS of clustered values). Let x, y, ∈ RK and w ∈ CK . Let
x =

x˜1 1M1
...
x˜C 1MC
 =

x(1)
...
x(C)
 ∈ RK ,
with x(i) ∈ RMi and ∑Ci=1 Mi = K. Thus x contains at most 1 ≤ C ≤ K different cluster values
x˜1, . . . , x˜C. Some of them can be identical. Similarly, we write
y =

y(1)
...
y(C)
 , w =

w(1)
...
w(C)
 ,
with y(i) ∈ RMi and w(i) ∈ CMi . Note that in general, only x contains cluster values and y,w do not.
We define
x˜ =
[
x˜1 . . . x˜C
]T
, (J.1)
– 176 –
y˜ =
[
y˜1 . . . y˜C
]T
, (J.2)
y˜i =
1
1T w(i)
w(i)
T
y(i) = EWS
(
y(i),w(i)
)
, (J.3)
w˜ =
[
w˜1 . . . w˜C
]T
, (J.4)
w˜i = 1T w(i). (J.5)
Then
CovWS(x, y,w) = CovWS(x˜, y˜, w˜). (J.6)
Here we assume 1T w , 0 and 1T w(i) , 0, i ∈ {1, . . . ,C}.
This lemma states that due to the clustering x(i) = x˜i1, the weighted covariance of a K-point problem
(x, y,w) can be reformulated as the weighted covariance of a C-point problem (x˜, y˜, w˜). The values
of y corresponding to cluster i are represented by their weighted mean y˜i = EWS(y(i),w(i)) and the
corresponding values of w by the sum of all weights of that cluster w˜i = 1T w(i).
Proof. We define
S = 1T w = 1T w˜. (J.7)
We use Lemma A.2.12 and get
CovWS(x, y,w) = CorrWS(x, y,w) − EWS(y,w) EWS(x,w)
=
1
S
K∑
k=1
wk xk yk −
 1S
K∑
k=1
wk xk
  1S
K∑
k=1
wk yk
 (J.8)
=
1
S
C∑
i=1
w(i)
T
y(i) x˜i −
 1S
C∑
i=1
1T w(i) x˜i
  1S
C∑
i=1
w(i)
T
y(i)

=
1
S
C∑
i=1
w˜i y˜i x˜i −
 1S
C∑
i=1
w˜i x˜i
  1S
C∑
i=1
w˜i y˜i

= CovWS(x˜, y˜, w˜). (J.9)

Lemma J.1.2. Let x˜, y˜ ∈ RC, w˜ ∈ CC, w˜i , 0 and 1T w˜ , 0. Then
y˜ ∝ 1 =⇒ CovWS(x˜, y˜, w˜) = 0. (J.10)
– 177 –
Moreover, if C = 2 and x˜1 , x˜2
y˜ ∝ 1 ⇐⇒ CovWS(x˜, y˜, w˜) = 0. (J.11)
Proof. Since y˜ ∝ 1 implies y˜k = y˜l ∀ k, l ∈ {1, . . . ,C}, (J.10) follows immediately from (J.9). If C = 2
and x˜1 , x˜2,
CovWS(x˜, y˜, w˜) = CorrWS(x˜, y˜, w˜) − EWS(y˜, w˜) EWS(x˜, w˜)
=
1
1T w˜
C∑
k=1
w˜k x˜k y˜k −
 1S
C∑
k=1
w˜k x˜k
  1S
C∑
k=1
w˜k y˜k
 (J.12)
=
1(
1T w˜
)2
 C∑
k=1
w˜k
  C∑
k=1
w˜k x˜k y˜k
 −  C∑
k=1
w˜k x˜k
  C∑
k=1
w˜k y˜k

=
1(
1T w˜
)2 C∑
k=1
C∑
l=1
(w˜k w˜l x˜l y˜l − w˜k w˜l x˜k y˜l)︸                        ︷︷                        ︸
=0 if k=l
=
1(
1T w˜
)2 w˜1 w˜2 (x˜1 y˜1 + x˜2 y˜2 − x˜1 y˜2 − x˜2 y˜1)
=
1(
1T w˜
)2 w˜1 w˜2 (x˜1 − x˜2) (y˜1 − y˜2)
= 0 ⇐⇒ y˜1 = y˜2, (J.13)
since x˜1 , x˜2 and w˜1, w˜2 , 0. 
Combining both lemmas, we get the following lemma.
Lemma J.1.3. Let x ∈ RK , y ∈ RK and the weight vector w ∈ CK . Moreover, let x contain at most
1 ≤ C ≤ K different values. Since CovWS(x, y,w) is invariant w. r. t. a synchronous sorting of the
elements of x, y,w, we sort x, y,w such that
x =

x˜1 1M1
...
x˜C 1MC
 =

x(1)
...
x(C)
 ∈ RK ,
with x(i) ∈ RMi and ∑Ci=1 Mi = K. Hence, we have C clusters, x˜11M1 , . . . , x˜C1MC , where the i-th cluster
of x contains one value x˜i. Some of the values x˜i, i = 1, . . . ,C, can be identical. Similarly
y =

y(1)
...
y(C)
 , w =

w(1)
...
w(C)
 .
– 178 –
Then
EWS
(
y(k),w(k)
)
= EWS
(
y(l),w(l)
)
∀ k, l ∈ {1, . . . ,C}
=⇒ CovWS
(
x, y,w
)
= 0. (J.14)
Moreover, for C = 2, x˜1 , x˜2 and 1T w(k) , 0, k = 1, 2,
EWS
(
y(1),w(1)
)
= EWS
(
y(2),w(2)
)
⇐⇒ CovWS
(
x, y,w
)
= 0. (J.15)
Here we assume 1T w , 0 and 1T w(i) , 0, i ∈ {1, . . . ,C}.
Proof. This lemma follows directly from Lemma J.1.1 and J.1.2. 
J.2. Proof of Theorem 2
We apply Lemma J.1.3 to prove Theorem 2. First we consider the case of a linear array. We set
x = dPulse, y = t, w = ρ. Here, dPulse contains NTx different antenna positions. Hence the synchronous
sorting with C = NTx clusters yields
x˜ = dTx, (J.16)
y(i) = t(i), (J.17)
w(i) = ρ(i). (J.18)
Therefore, for a linear array Theorem 2 follows directly from Lemma J.1.3.
Now we consider the planar array. First we show (5.130) in Theorem 2. Since
CovWS(DPulse, t, ρ) =
CovWS(dPulse,x, t, ρ)CovWS(dPulse,y, t, ρ)
 (J.19)
we have to show
EWS
(
t(k), ρ(k)
)
= EWS
(
t(l), ρ(l)
)
∀k, l ∈ {1, . . . ,NTx} =⇒ CovWS
(
dPulse,x, t, ρ
)
= 0 (J.20)
for the x-coordinate dPulse,x of the Tx antennas and at the same time
EWS
(
t(k), ρ(k)
)
= EWS
(
t(l), ρ(l)
)
∀k, l ∈ {1, . . . ,NTx} =⇒ CovWS
(
dPulse,y, t, ρ
)
= 0 (J.21)
for the y-coordinate dPulse,y. First we prove (J.20). We set x = dPulse,x, y = t, w = ρ and apply
Lemma J.1.3 with C = NTx clusters, where each cluster contains x-coordinate of one Tx antenna.
From that (J.20) follows. Analogously we prove (J.21) in the same way. Hence we have shown
– 179 –
(5.130). The reason is that the MIMO-TDM scheme implies a reuse of individual Tx antenna which
corresponds to a synchronous clustering of the x/y- coordinates of the Tx antennas regardless of the
dimension of the Tx array (1D, 2D).
Now we prove (5.131) in Theorem 2 for 2 Tx antennas. The implication “=⇒” in (5.131) follows
from (5.130). Hence we have to show the other implication “⇐=”:
EWS
(
t(1), ρ(1)
)
= EWS
(
t(2), ρ(2)
)
⇐= CovWS
(
DPulse, t, ρ
)
= 0. (J.22)
Since dTx,1 , dTx,2, the Tx antennas differ in their x-coordinate or their y-coordinate or both. We
assume w. l. o. g. dTx,1x , d
Tx,2
x . We set x = dPulse,x, y = t, w = ρ and use Lemma J.1.3 with C = 2
clusters. From (J.15) follows
EWS
(
t(1), ρ(1)
)
= EWS
(
t(2), ρ(2)
)
⇐= CovWS
(
dPulse,x, t, ρ
)
= 0 (J.23)
and therefore (J.22). Thus we have shown (5.131). 

– 181 –
K. Proof of Theorem 3: Optimal TDM
Schemes for Constrained Transmission
Energy and Constrained Aperture Size
We prove the part which is missing in the proof of Theorem 3 on p. 109 in the following section. In
Section K.2 we state and prove several lemmas which we use for the proof.
K.1. Missing Part of the Proof of Theorem 3
We have to show:
VarWS(dPulse, ρ) is maximal iff
only the two outermost Tx antennas are used and 1Tρ(1) = 1Tρ(2). (K.1)
We abbreviate the assertions by
A : VarWS(dPulse, ρ) is maximal, (K.2)
B : only the two outermost Tx antennas are used and 1Tρ(1) = 1Tρ(2). (K.3)
Thus we have to show
A ⇐⇒ B, (K.4)
where “ ⇐⇒ ” is an abbreviation for “if and only if”. We consider an arbitrary TDM scheme
characterized by dPulse, ρ, NPulse, which does not fulfill B, i. e. it uses not only the two outermost
antennas and/or 1Tρ(1) , 1Tρ(2).
First we consider the case that the TDM scheme does not use only the two outermost antennas.
Afterwards, we consider the case that 1Tρ(1) = 1Tρ(2) is not satisfied.
– 182 –
Case 1: The TDM scheme does not use only the two outermost antennas. In the trivial case,
when the TDM scheme uses only 1 Tx antenna, dPulse ∝ 1 and obviously VarWS(dPulse, ρ) = 0.
If the TDM scheme uses more than 2 Tx antennas, there exists at least one k ∈ {1, . . . ,NPulse} with
dPulsek , d
Pulse
min , d
Pulse
max , where d
Pulse
min = min(d
Pulse) and dPulsemax = max(d
Pulse). Using Lemma K.2.1 we can
increase VarWS(dPulse, ρ) by substituting the element dPulsek in d
Pulse by dPulsemin , appending d
Pulse
max to d
Pulse
while keeping the overall transmitting energy 1Tρ constant. We call the resulting vectors dPulse(a) and
ρ
(a)
, respectively. Hence
VarWS
(
dPulse, ρ
)
< VarWS
(
dPulse(a) , ρ(a)
)
. (K.5)
We do this repeatedly until dPulse(a) consists only of the elements d
Pulse
min and d
Pulse
max , i. e. it uses only 2 Tx
antennas. The resulting vectors are denoted by dPulse(2 Tx) and ρ(2 Tx), respectively. Thus
VarWS
(
dPulse, ρ
)
< VarWS
(
dPulse(2 Tx), ρ(2 Tx)
)
. (K.6)
Now let us assume that the TDM scheme uses 2 Tx antennas, but not the two outermost ones. Hence
all elements in dPulse equal dPulsemin or d
Pulse
max andd
Pulse
min
dPulsemax
 ,
d
Tx
min
dTxmax
 . (K.7)
The pulse energies are denoted by ρ = ρ
(2 Tx)
, since the TDM scheme uses only 2 Tx antennas. Let
dPulse(2 Tx, outermost) be the same as d
Pulse, but with dPulsemin and d
Pulse
max replaced by d
Tx
min and d
Tx
max, respectively.
Hence, the transmission sequence in dPulse(2 Tx, outermost) is the same as in d
Pulse but only the two outermost
antenna positions are used. We can express
dPulse(2 Tx, outermost) = a · dPulse + b · 1 (K.8)
with a ∈ R, a > 1 and b ∈ R chosen appropriately. Hence
VarWS
(
dPulse(2 Tx, outermost), ρ(2 Tx)
)
= VarWS
(
a dPulse, ρ
(2 Tx)
)
+ VarWS
(
b 1, ρ
(2 Tx)
)
= a2 VarWS
(
dPulse, ρ
(2 Tx)
)
> VarWS
(
dPulse, ρ
(2 Tx)
)
. (K.9)
Note that for dPulse(2 Tx) in (K.6),
VarWS
(
dPulse(2 Tx), ρ(2 Tx)
)
≤ VarWS
(
dPulse(2 Tx, outermost), ρ(2 Tx)
)
(K.10)
– 183 –
holds, with the equality iff dPulse(2 Tx) contains only the outermost positions.
Thus we have shown that for any TDM scheme dPulse, ρ, which does not use only the two outermost
antennas
VarWS
(
dPulse, ρ
)
< VarWS
(
dPulse(2 Tx, outermost), ρ(2 Tx)
)
, (K.11)
where dPulse(2 Tx, outermost) and ρ(2 Tx) are constructed from d
Pulse, ρ as described above.
Case 2: 1Tρ(1) , 1Tρ(2). Now let us assume that the TDM scheme uses only the two outer-
most Tx antennas but 1Tρ(1) , 1Tρ(2). We denote dPulse by dPulse(2 Tx, outermost) and ρ by ρ(2 Tx). Since
VarWS(dPulse(2 Tx, outermost), ρ(2 Tx)) is invariant w. r. t. a synchronous sorting of the elements in d
Pulse
(2 Tx, outermost)
and ρ
(2 Tx)
, we sort dPulse(2 Tx, outermost) and ρ(2 Tx) such that we have two clusters in d
Pulse
(2 Tx, outermost)
dPulse(2 Tx, outermost) =
d
Tx
min 1
dTxmax 1
 , (K.12)
ρ
(2 Tx)
=
ρ
(1)
(2 Tx)
ρ(2)
(2 Tx)
 . (K.13)
Now we can apply Lemma J.1.1 for CovWS of clustered values. From that follows
VarWS
(
dPulse(2 Tx, outermost), ρ(2 Tx)
)
= VarWS
(˜
d
Pulse
(2 Tx, outermost), ρ˜(2 Tx)
)
(K.14)
with
d˜
Pulse
(2 Tx, outermost) =
d
Tx
min
dTxmax
 , (K.15)
ρ˜
(2 Tx)
=
1
Tρ(1)
(2 Tx)
1Tρ(2)
(2 Tx)
 . (K.16)
Now we use Lemma K.2.3, which states that VarWS(˜d
Pulse
(2 Tx, outermost), ρ˜(2 Tx)
) is maximal iff
ρ˜
(2 Tx)
∝ [1, 1]T . Since we assumed 1Tρ(1)
(2 Tx)
, 1Tρ(2)
(2 Tx)
VarWS
(˜
d
Pulse
(2 Tx, outermost), ρ˜(2 Tx)
)
< VarWS
(˜
d
Pulse
(2 Tx, outermost), ρ˜(equal energy)
)
(K.17)
with
ρ˜
(equal energy)
= [1, 1]T . (K.18)
– 184 –
Obviously
VarWS
(˜
d
Pulse
(2 Tx, outermost), ρ˜(equal energy)
)
=
1
4
(
dTxmax − dTxmin
)2
. (K.19)
Note that all vectors ρ
(equal energy)
∈ RNPulse which satisfy
1Tρ(1)
(equal energy)
= 1Tρ(2)
(equal energy)
(K.20)
end up in ρ˜
(equal energy)
after sorting and clustering (up to a scaling factor). Here ρ(1)
(equal energy)
and
ρ(2)
(equal energy)
denote the energies of the pulses which the first and second antenna transmits, respec-
tively. Thus we have shown that
VarWS
(
dPulse(2 Tx, outermost), ρ(2 Tx)
)
< VarWS
(
dPulse(2 Tx, outermost), ρ(equal energy)
)
(K.21)
=
1
4
(
dTxmax − dTxmin
)2
, (K.22)
where ρ
(equal energy)
∈ RNPulse is any vector satisfying (K.20).
Result: Summarizing the results of case 1 and 2, we have shown that for any TDM scheme dPulse, ρ
which does not satisfy assertion B (K.3),
VarWS
(
dPulse, ρ
)
< VarWS
(˜
d
Pulse
(2 Tx, outermost), ρ˜(equal energy)
)
(K.23)
holds, where d˜
Pulse
(2 Tx, outermost), ρ˜(equal energy)
is any TDM scheme satisfying B.
From that follows that if B is not satisfied, VarWS(dPulse, ρ) is not maximal. Written in shorter notation
¬B =⇒ ¬A (K.24)
where “¬” means negation and “=⇒” is an abbreviation for “is sufficient for”. This is equivalent to
A =⇒ B. (K.25)
On the other hand, if B is satisfied, VarWS(dPulse, ρ) = 14
(
dTxmax − dTxmin
)2
due to (K.22), i. e. all TDM
schemes satisfying B yield the same value for VarWS(dPulse, ρ). Since all TDM schemes which do not
satisfy B yield a smaller value, VarWS(dPulse, ρ) is maximal, i. e.
B =⇒ A. (K.26)
Thus we have shown (K.4). 
– 185 –
K.2. Lemmas
Lemma K.2.1. Let dPulse, ρ ∈ RNPulse . We define
dPulsemin = min
(
dPulse
)
, dPulsemax = max
(
dPulse
)
. (K.27)
Let dPulsemin < d
Pulse
max . Let there be at least one k ∈ {1, . . . ,NPulse} with dPulsek , dPulsemin , dPulsemax . We can
increase VarWS(dPulse, ρ) by substituting dPulsek in d
Pulse by dPulsemin , adding an additional element d
Pulse
max
and keeping the overall transmission energy constant.
More precisely: Let NPulse be arbitrary and
dPulse =
[
dPulse1 , . . . , d
Pulse
k , . . . , d
Pulse
NPulse
]T
,
ρ =
[
ρ1, . . . , ρNPulse
]
,
ρi > 0, i = 1, . . . ,NPulse.
Then
VarWS(dPulse, ρ) < VarWS(d˜Pulse, ρ˜)
where
N˜Pulse = NPulse + 1, (K.28)
d˜Pulse ∈ RN˜Pulse , (K.29)
d˜Pulsei = d
Pulse
i , i = 1, . . . ,NPulse ∧ i , k, (K.30)
d˜Pulsek = d
Pulse
min , (K.31)
d˜Pulse
N˜Pulse
= dPulsemax , (K.32)
ρ˜ ∈ RN˜Pulse , (K.33)
ρ˜i = ρi, i = 1, . . . ,NPulse ∧ i , k, (K.34)
1Tρ = 1T ρ˜. (K.35)
Proof.
EWS(d˜Pulse, ρ˜) =
1
1T ρ˜
N˜Pulse∑
i=1
d˜Pulsei ρ˜i (K.36)
=
1
1T ρ˜

 NPulse∑
i=1,i,k
dPulsei ρi
 + dPulsemin ρ˜k + dPulsemax ρ˜N˜Pulse
 (K.37)
– 186 –
We choose ρ˜k and ρ˜N˜Pulse such that
dPulsek ρk = d
Pulse
min ρ˜k + d
Pulse
max ρ˜N˜Pulse (K.38)
while
ρk = ρ˜k + ρ˜N˜Pulse (K.39)
in order to fulfill (K.35). This is always uniquely possible since these are two linear independent
equations for the two unknowns ρ˜k and ρ˜N˜Pulse .
Putting (K.38) in (K.37) yields
EWS(d˜Pulse, ρ˜) = EWS(dPulse, ρ). (K.40)
Now we can compute VarWS(d˜Pulse, ρ˜) using (K.40)
VarWS(d˜Pulse, ρ˜) =
1
1T ρ˜
N˜Pulse∑
i=1
[
d˜Pulsei − EWS(d˜Pulse, ρ˜)
]2
ρ˜i (K.41)
=
1
1Tρ
N˜Pulse∑
i=1
[
d˜Pulsei − EWS(dPulse, ρ)
]2
ρ˜i. (K.42)
We compare this to
VarWS(dPulse, ρ) =
1
1Tρ
NPulse∑
i=1
[
dPulsei − EWS(dPulse, ρ)
]2
ρi. (K.43)
We set c = EWS(dPulse, ρ). Note since dPulsemin < d
Pulse
max
dPulsemin < c < d
Pulse
max . (K.44)
Hence
N˜Pulse∑
i=1
(d˜Pulsei − c)2 ρ˜i =
NPulse∑
i=1,i,k
(dPulsei − c)2 ρi + (dPulsemin − c)2 ρ˜k + (dPulsemax − c)2 ρ˜N˜Pulse . (K.45)
Using Lemma K.2.2 yields
N˜Pulse∑
i=1
(d˜Pulsei − c)2 ρ˜i >
NPulse∑
i=1,i,k
(dPulsei − c)2 ρi + (dPulsek − c)2 ρk (K.46)
=
NPulse∑
i=1
(dPulsei − c)2 ρi. (K.47)
– 187 –
Thus
VarWS(d˜Pulse, ρ˜) > VarWS(dPulse, ρ). (K.48)

Lemma K.2.2. Let wsum, w1, w2, a, amin, amax, c ∈ R, then
(a − c)2 wsum < (amin − c)2 w1 + (amax − c)2 w2
with
wsum = w1 + w2,
amin < c < amax,
a wsum = amin w1 + amax w2,
w1, w2 > 0.
Proof.
(a − c)2wsum = (a wsum − c wsum)2 1wsum
= (amin w1 − c w1 + amax w2 − c w2)2 1wsum
= [(amin − c) w1 + (amax − c) w2]2 1wsum .
Since amin < c < amax and w1, w2 > 0, we get
(a − c)2wsum <
{
[(amin − c) w1]2 + [(amax − c) w2]2
} 1
wsum
=
(amin − c)2 w21
w1 + w2
+
(amax − c)2 w22
w1 + w2
< (amin − c)2 w1 + (amax − c)2 w2.

Lemma K.2.3. Let a,w ∈ R2, a1 , a2. Then for fixed a, VarWS(a,w) is maximal iff w ∝ [1, 1]T .
Moreover, max
w
VarWS(a,w) = VarS(a).
Proof. Since
VarWS(a,w) = VarWS(a, s w), (K.49)
where s ∈ R, s , 0, we set w. l. o. g. 1T w = 1. Since VarWS(a − s 1,w) = VarWS(a,w), where s ∈ R,
we set w. l. o. g. a = [0, a2]T , a2 , 0.
– 188 –
Let w = [w1,w2]T . With Lemma A.2.12 follows
VarWS(a,w) = CorrWS(a, a,w) − EWS(a,w)2 (K.50)
=
1
1T w
a22 w2 −
(
1
1T w
a2 w2
)2
(K.51)
= a22 (w2 − w22). (K.52)
Computing the 1. and 2. derivative with respect to w2, we see that VarWS(a,w) has got one global
maximum at w2 = 0.5. With 1T w = 1 follows w1 = 0.5 and thus w = [0.5, 0.5]T . Due to (K.49),
VarWS(a,w) is maximal iff w ∝ [1, 1]T . Moreover,
max
w
VarWS(a,w) = VarWS(a, [1, 1]T ) = VarS(a). (K.53)

– 189 –
L. Proof of Theorem 4: Optimal TDM
Schemes Under Additional Constraints
In the following, we call a TDM scheme which is not balanced an imbalanced TDM scheme.
1. NPulse ∈ 4N. In this case Theorem 3 can be applied, since the conditions 1a to 1d can be
fulfilled, where ρ is already given. Note that ρ is normalized such that 1Tρ = ρmax, i. e. it
fulfills 1c. As already noted, and due to 1a, an optimal TDM scheme has to use only the
two outermost antennas. Since ρ ∝ 1, from 1b follows that both Tx antennas are used equally
often. Moreover, the TDM scheme has to satisfy 1d, ES(t(1)) = ES(t(2)). Since the TDM scheme
is balanced, VarS(dPulse) = 1. Moreover, due to Theorem 2, ES(t(1)) = ES(t(2)) is equivalent to
CovS(dPulse, t) = 0. Thus the optimal value of Ω is
Ω(dPulse,opt)
∣∣∣∣
NPulse ∈ 4N
= 1. (L.1)
2. NPulse ∈ 4N − 2. In this case the maximum value VarS(dPulse) = 1 and CovS(dPulse, t) = 0 in
(5.148) cannot be achieved at the same time. An optimal TDM scheme has to be balanced
and fulfill |CovS(dPulse,opt, t)| = ± 1NPulse . Since it is balanced, VarS(dPulse) = 1. It can be shown
that these TDM schemes yield a lower CRBu than an imbalanced sequence which uses one
antenna twice more often than the other and achieves CovS(dPulse, t) = 0. Any other sequence
either makes the variance smaller and/or the covariance larger or improves the covariance
and is of the second type. Therefore, a sequence of the proposed structure is optimal. Since
VarS(t) = N
2
Pulse−1
12 , the contribution of Ω to the minimal CRBu is
Ω(dPulse,opt)
∣∣∣∣
NPulse ∈ (4N−2)
= 1 − 1
N2Pulse
· 12
N2Pulse − 1
. (L.2)
3. NPulse ∈ 2N + 1. Since NPulse is odd, each TDM scheme is imbalanced. Let the imbalance
be the number of pulses one antenna transmits more often than the other. To maximize
VarS(dPulse), the imbalance has to be as small as possible, i. e. 1. Moreover, the condi-
tion CovS(dPulse, t) = 0 has to be satisfied. Due to Theorem 2 this condition is equivalent
to ES(t(1)) = ES(t(2)). Hence an optimal TDM scheme has an imbalance of 1 and satisfies
ES(t(1)) = ES(t(2)). Any other TDM scheme yields a lower value for Ω since it either makes the
– 190 –
sample variance smaller and/or the covariance larger. The value for Ω of the optimal TDM
schemes is
Ω(dPulse)
∣∣∣∣
NPulse ∈ (2N+1)
= 1 − 1
N2Pulse
. (L.3)
– 191 –
M. Derivation of the Cramer-Rao Bound for
a TDM MIMO Radar with Two
Non-Stationary Targets
The matrix C in (5.171) appears in the CRB of the electrical angles and Doppler frequencies for a
TDM MIMO radar with several non-stationary targets. We give the derivation of C in (5.171), with
C1,C2,C3 defined in (5.172) to (5.174).
We compute a part of the formulas for an arbitrary number of targets NTargets, since we need these
formulas in a further proof as well. According to (5.162)
C = DH P⊥B D
= C(1) − C(2) (M.1)
with the definitions
C(1) = DH D ∈ C2NTargets×2NTargets , (M.2)
C(2) = DH B
(
BH B
)−1
BH D ∈ C2NTargets×2NTargets . (M.3)
First we compute D, then C(1) and C(2) and simplify the formulas afterwards.
Computation of Di From (5.157) follows
∂b(U, ω)
∂U
= b(U, ω)  j dVirt, (M.4)
∂b(U, ω)
∂ω
= b(U, ω)  j tVirt. (M.5)
We define
V =
[
dVirt tVirt
]
∈ RNVirt×2. (M.6)
– 192 –
Then Di in (5.164) equals
Di = j
[
b(Θ(i)) b(Θ(i))
]
 V ∈ CNVirt×2, i = 1, . . . ,NTargets, (M.7)
with Θ(i) defined in (5.160). Thus, with the abbreviation
b(i) = b(Θ(i)), i = 1, . . . ,NTargets, (M.8)
Di can be written as
Di = j
[
b(i) b(i)
]
 V
= j
[
b(i)1T2
]
 V, i = 1, . . . ,NTargets. (M.9)
Computation of C(1) Due to (M.2) and (5.163), we can express C(1) as
C(1) = DH D
=

DH1
...
DHNTargets

[
D1 . . . DNTargets
]
=

DH1 D1 . . . D
H
1 DNTargets
...
. . .
...
DHNTargetsD1 . . . D
H
NTargets
DNTargets
 . (M.10)
Next we simplify the expressions DHq Dr, q, r ∈ {1, . . . ,NTargets}. We investigate the following expres-
sion: for some matrices X ∈ CNVirt×K1 and Y ∈ CNVirt×K2 and q, r ∈ {1, . . . ,NTargets} holds
[((
b(q)1TK2
)
 Y
)H ((
b(r)1TK1
)
 X
)]
kl
=
NVirt∑
m=1
[(
b(q) 1TK2
)H]
km
[
YH
]
km
[(
b(r) 1TK1
)]
ml
[X]ml
=
∑
m
[
YH
]
km
b(q)
∗
mb
(r)
m[X]ml
=
[
YH diag
(
b(q)
∗  b(r)
)
X
]
kl
. (M.11)
Hence, with the definition of CorrWS in (A.2) we get[(
b(q)1TK2
)
 Y
]H [(
b(r)1TK1
)
 X
]
= 1T
(
b(q)
∗  b(r)
)
· CorrWS
(
X,Y, b(q)∗  b(r)
)
. (M.12)
– 193 –
Moreover,
b(q)  b(r)∗ (5.157)=
(√
ρVirt  exp( j tVirtωq)  aVirt(Uq)
)

(√
ρVirt  exp(− j tVirtωr) 
(
aVirt(Ur)
)∗)
(5.75)
= ρVirt  exp( j tVirtωq)  exp( j dVirtUq)  exp(− j tVirtωr)  exp(− j dVirtUr)
= ρVirt  exp( j tVirt∆ωqr)  exp( j dVirt∆Uqr), (M.13)
with the definitions
∆ωqr = ωq − ωr, (M.14)
∆Uqr = Uq − Ur, q, r ∈ {1, . . . ,NTargets}. (M.15)
In addition, we define
b(qr)
∆Tx = ρ  exp( j · t ∆ωqr)  exp( j · dPulse∆Uqr) ∈ CNPulse , (M.16)
b(qr)
∆Rx = exp( j · dRx∆Uqr) ∈ CNRx , (M.17)
b(qr)
∆
= b(qr)
∆Tx ⊗ b(qr)∆Rx ∈ CNVirt , (M.18)
b(qr)
∆sum = 1
T b(qr)
∆
=
(
1T b(qr)
∆Tx
) (
1T b(qr)
∆Rx
)
∈ C. (M.19)
Note that for q = 1, r = 2 these definitions simplify to the definitions without the superscript “(qr)”
given in (5.176) to (5.181). Using these definitions yields
b(qr)
∆
=
(
ρ  exp( j t ∆ωqr)  exp( j dPulse∆Uqr)
)
⊗ exp( j dRx∆Uqr)
=
(
ρ  exp( j t ∆ωqr)  exp( j dPulse∆Uqr)
)
⊗
(
1NRx  1NRx  exp( j dRx∆Uqr)
)
=
(
ρ ⊗ 1NRx
)

(
exp( j t ∆ωqr) ⊗ 1NRx
)

(
exp( j dPulse∆Uqr) ⊗ exp( j dRx∆Uqr)
)
= ρVirt  exp( j tVirt∆ωqr)  exp( j dVirt∆Uqr), (M.20)
where we have used (5.75), (5.78) and (5.79) in the last line. Comparing (M.13) with (M.20) yields
b(q)  b(r)∗ = b(qr)
∆
. (M.21)
Note that
b(rq)
∆
= b(qr)
∆
∗
(M.22)
and
b(qq)
∆
= ρVirt. (M.23)
– 194 –
Plugging (M.21) in (M.12) yields[(
b(q)1TK2
)
 Y
]H [(
b(r)1TK1
)
 X
]
= b(qr)
∆sum
∗
CorrWS
(
X,Y, b(qr)
∆
∗)
, q, r ∈ {1, . . . ,NTargets}. (M.24)
Therefore, using (M.9) results in
DHq Dr =
[(
b(q)1T2
)
 V
]H [(
b(r)1T2
)
 V
]
= b(qr)
∆sum
∗
CorrWS
(
V,V, b(qr)
∆
∗) ∈ C2×2. (M.25)
Note that swapping q and r gives
DHr Dq = b
(qr)
∆sum Corr
WS
(
V,V, b(qr)
∆
)
. (M.26)
Thus (M.10) reads
C(1) =

b(11)
∆sum Corr
WS
(
V,V, b(11)
∆
)
. . . b(1NTargets)
∆sum
∗
CorrWS
(
V,V, b(1NTargets)
∆
∗)
...
. . .
...
b(1NTargets)
∆sum Corr
WS
(
V,V, b(1NTargets)
∆
)
. . . b(NTargetsNTargets)
∆sum Corr
WS
(
V,V, b(NTargetsNTargets)
∆
)
 . (M.27)
For NTargets = 2, keeping (M.23) in mind and using the definition (5.175), we can simplify it to
C(1) =
ρVirtsum CorrWS(V,V, ρVirt) b∗∆sum CorrWS(V,V, b∗∆)b∆sum CorrWS(V,V, b∆) ρVirtsum CorrWS(V,V, ρVirt)
 ∈ C2×2. (M.28)
Computation of C(2) First we compute BH D. With the above abbreviations b(i)
B =
[
b(1) . . . b(NTargets)
]
∈ CNVirt×NTargets . (M.29)
Hence
BH D =

b(1)
H
...
b(NTargets)
H

[
D1 . . . DNTargets
]
=

b(1)
HD1 . . . b(1)
HDNTargets
...
. . .
...
b(NTargets)
HD1 . . . b(NTargets)
HDNTargets
 ∈ CNTargets×2NTargets . (M.30)
Using (M.9) results in
b(q)
HDr = j b(q)
H [(
b(r)1T2
)
 V
]
. (M.31)
– 195 –
This is of the form (M.24) with K2 = 1, Y = 1NVirt . Thus
b(q)
HDr = j b(qr)∆sum
∗
CorrWS
(
V, 1NVirt , b
(qr)
∆
∗)
= j b(qr)
∆sum
∗
EWS
(
V, b(qr)
∆
∗)
, (M.32)
where we have used Lemma A.2.2 in the last line. Hence (M.30) reads
BH D = j

b(11)
∆sum E
WS
(
V, b(11)
∆
)
. . . b(1NTargets)
∆sum
∗
EWS
(
V, b(1NTargets)
∆
∗)
...
. . .
...
b(1NTargets)
∆sum E
WS
(
V, b(1NTargets)
∆
)
. . . b(NTargetsNTargets)
∆sum
∗
EWS
(
V, b(NTargetsNTargets)
∆
∗)
 . (M.33)
For two targets, NTargets = 2, this simplifies to
BH D = j
ρVirtsum EWS(V, ρVirt) b∗∆sum EWS(V, b∗∆)b∆sum EWS(V, b∆) ρVirtsum EWS(V, ρVirt)
 . (M.34)
In the following we stick to the case NTargets = 2. We compute
(
BH B
)−1
.
BH B =
b(1)Hb(2)H
 [b(1) b(2)]
=
 ρVirtsum b∗∆sumb∆sum ρVirtsum
 ∈ C2×2. (M.35)
Thus the inverse is given by
(
BH B
)−1
=
1
ρVirtsum
2 − |b∆sum|2
 ρVirtsum −b∗∆sum−b∆sum ρVirtsum
 ∈ C2×2. (M.36)
From (M.3) follows that
C(2) =
(
C(2)
)H
. (M.37)
Hence it is sufficient to compute the 2 × 2 matrices C(2)1 ,C(2)2 ,C(2)3 , where
C(2) =
 C(2)1 C(2)2(C(2)2 )H C(2)3
 . (M.38)
– 196 –
In the following, we use the abbreviations
τ = ρVirtsum ∈ R, (M.39)
ϕ = b∆sum ∈ C, (M.40)
χ = EWS(V, ρVirt)T ∈ R2, (M.41)
Ψ = EWS(V, b
∆
)T ∈ C2. (M.42)
Note, since V and ρVirt are real, χ is real. Moreover, since V is real
EWS(V, b∗
∆
) = EWS(V, b
∆
)∗ (M.43)
and hence
ΨH = EWS(V, b∗
∆
). (M.44)
Then C(2) in (M.3) reads
C(2) = DH B
(
BH B
)−1
BH D
=
1
τ2 − |ϕ|2
 τχ ϕ∗Ψ ∗
ϕΨ τχ

 τ −ϕ∗−ϕ τ

 τχT ϕ∗ΨH
ϕΨT τχT

=
1
τ2 − |ϕ|2
 τχ ϕ∗Ψ ∗
ϕΨ τχ

 τ2χT − |ϕ|2ΨT τϕ∗ΨH − τϕ∗χT−τϕχT + τϕΨT −|ϕ|2ΨH + τ2χT
 . (M.45)
Hence, using the notation given in (M.38) yields
C(2)1 =
1
τ2 − |ϕ|2
{
τχ
(
τ2χT − |ϕ|2ΨT
)
+ ϕ∗Ψ ∗
(
−τϕχT + τϕΨT
)}
, (M.46)
C(2)2 =
1
τ2 − |ϕ|2
{
τχ
(
τϕ∗ΨH − τϕ∗χT
)
+ ϕ∗Ψ ∗
(
−|ϕ|2ΨH + τ2χT
)}
, (M.47)
C(2)3 =
1
τ2 − |ϕ|2
{
ϕΨ
(
τϕ∗ΨH − τϕ∗χT
)
+ τχ
(
−|ϕ|2ΨH + τ2χT
)}
. (M.48)
We note, that (
C(2)3
)∗
= C(2)1 . (M.49)
C(2)1 can be expressed as
C(2)1 =
1
τ2 − |ϕ|2
{
τ3χ χT − τ|ϕ|2χΨT + τ|ϕ|2
(
−Ψ ∗χT + Ψ ∗ΨT
)}
= τχ χT +
τ|ϕ|2
τ2 − |ϕ|2
(
χ χT − χΨT − Ψ ∗χT + Ψ ∗ΨT
)
. (M.50)
– 197 –
Since χ is real, this can be rewritten as
C(2)1 = τχ χ
T +
τ|ϕ|2
τ2 − |ϕ|2 m˜
∗m˜T (M.51)
with the definition
m˜ = χ − Ψ. (M.52)
Due to (M.49), C(2)3 is given by
C(2)3 = τχ χ
T +
τ|ϕ|2
τ2 − |ϕ|2 m˜ m˜
H. (M.53)
Analog computations yield for C(2)2
C(2)2 =
1
τ2 − |ϕ|2
{
−ϕ∗|ϕ|2Ψ ∗ΨH + τ2ϕ∗Ψ ∗χT + τ2ϕ∗
(
χΨH − χ χT
)}
= ϕ∗Ψ ∗ΨH − τ
2ϕ∗
τ2 − |ϕ|2
(
Ψ ∗ΨH − χΨH − Ψ ∗χT + χ χT
)
= ϕ∗Ψ ∗ΨH − τ
2ϕ∗
τ2 − |ϕ|2 m˜
∗m˜H. (M.54)
Putting the results in (M.38) yields
C(2) =
 τχ χT + τ|ϕ|
2
τ2−|ϕ|2 m˜
∗m˜T ϕ∗Ψ ∗ΨH − τ2ϕ∗
τ2−|ϕ|2 m˜
∗m˜H(
ϕ∗Ψ ∗ΨH − τ2ϕ∗
τ2−|ϕ|2 m˜
∗m˜H
)H
τχ χT + τ|ϕ|
2
τ2−|ϕ|2 m˜ m˜
H

=
 τχ χT + τ|ϕ|2τ2−|ϕ|2 m˜∗m˜T ϕ∗Ψ ∗ΨH − τ2ϕ∗τ2−|ϕ|2 m˜∗m˜H
ϕΨ ΨT − τ2ϕ
τ2−|ϕ|2 m˜ m˜
T τχ χT + τ|ϕ|
2
τ2−|ϕ|2 m˜ m˜
H

=
 τχ χT ϕ∗Ψ ∗ΨH
ϕΨ ΨT τχ χT
 + ττ2 − |ϕ|2
 |ϕ|2 m˜∗m˜T −τϕ∗ m˜∗m˜H−τϕ m˜ m˜T |ϕ|2m˜ m˜H
 . (M.55)
Simplification of the formulas Now we put (M.28) and (M.55) in (M.1). We substitute some of
the used abbreviations:
C =
ρVirtsum CorrWS(V,V, ρVirt) b∗∆sum CorrWS(V,V, b∗∆)b∆sum CorrWS(V,V, b∆) ρVirtsum CorrWS(V,V, ρVirt)

−
ρVirtsum EWS(V, ρVirt)T EWS(V, ρVirt) b∗∆sum EWS(V, b∆)H EWS(V, b∆)∗b∆sum EWS(V, b∆)T EWS(V, b∆) ρVirtsum EWS(V, ρVirt)T EWS(V, ρVirt)

− τ
τ2 − |ϕ|2
 |ϕ|2 m˜∗m˜T −τϕ∗ m˜∗m˜H−τϕ m˜ m˜T |ϕ|2m˜ m˜H
 . (M.56)
– 198 –
Keeping in mind that V is real, with the help of Lemma A.2.12, the equation can be simplified to
C =
ρVirtsum CovWS(V,V, ρVirt) b∗∆sum CovWS(V,V, b∗∆)b∆sum CovWS(V,V, b∆) ρVirtsum CovWS(V,V, ρVirt)
 − ττ2 − |ϕ|2
 |ϕ|2 m˜∗m˜T −τϕ∗ m˜∗m˜H−τϕ m˜ m˜T |ϕ|2m˜ m˜H
 .
(M.57)
We simplify the expressions of the weighted sample covariances. Therefore using (M.6), (5.76) and
(5.79) we compute
V =
[
dVirt tVirt
]
=
[
1NPulse ⊗ dRx + dPulse ⊗ 1NRx t ⊗ 1NRx
]
= 1NPulse ⊗
[
dRx 0
]
+
[
dPulse t
]
⊗ 1NRx
= 1NPulse ⊗ V1 + V2 ⊗ 1NRx , (M.58)
where V1 and V2 are defined in (5.183) and (5.184), respectively. Let c ∈ CNRx and z ∈ CNPulse ,
c, z , 0. Then, using the properties of CovWS, cf. Appendix A.2.4, we get
CovWS(V,V, z ⊗ c) = CovWS(1NPulse ⊗ V1 + V2 ⊗ 1NRx , 1NPulse ⊗ V1 + V2 ⊗ 1NRx , z ⊗ c)
= CovWS(1NPulse ⊗ V1, 1NPulse ⊗ V1, z ⊗ c)
+ CovWS(1NPulse ⊗ V1,V2 ⊗ 1NRx , z ⊗ c)
+ CovWS(V2 ⊗ 1NRx , 1NPulse ⊗ V1, z ⊗ c)
+ CovWS(V2 ⊗ 1NRx ,V2 ⊗ 1NRx , z ⊗ c)
= CovWS(V1,V1, c) + 0 + 0 + CovWS(V2,V2, z)
= CovWS(V1, c) + CovWS(V2, z). (M.59)
Since ρVirt = ρ ⊗ 1NRx and b∆ = b∆Tx ⊗ b∆Rx, we can apply this result to (M.57) and get
C =
 ρVirtsum CovS(V1) b∗∆sum CovWS(V1, b∗∆Rx)b∆sum CovWS(V1, b∆Rx) ρVirtsum CovS(V1)

+
 ρVirtsum CovWS(V2, ρ) b∗∆sum CovWS(V2, b∗∆Tx)b∆sum CovWS(V2, b∆Tx) ρVirtsum CovWS(V2, ρ)

− τ
τ2 − |ϕ|2
 |ϕ|2 m˜∗m˜T −τϕ∗ m˜∗m˜H−τϕ m˜ m˜T |ϕ|2m˜ m˜H
 (M.60)
= C1 + C2 − τ
τ2 − |ϕ|2
 |ϕ|2 m˜∗m˜T −τϕ∗ m˜∗m˜H−τϕ m˜ m˜T |ϕ|2m˜ m˜H
 , (M.61)
– 199 –
where C1 and C2 are defined in (5.172) and (5.173), respectively. Now we express m˜ differently.
With (M.52), (M.41) and (M.42) follows
m˜ = EWS(V, ρVirt)T − EWS(V, b
∆
)T . (M.62)
Let c ∈ CNRx and z ∈ CNPulse , c, z , 0. Using the properties of EWS, cf. Appendix A.2.2, and (M.58)
we get
EWS(V, z ⊗ c) = EWS(1NPulse ⊗ V1 + V2 ⊗ 1NRx , z ⊗ c)
= EWS(1NPulse ⊗ V1, z ⊗ c) + EWS(V2 ⊗ 1NRx , z ⊗ c)
= EWS(V1, c) + EWS(V2, z). (M.63)
Applying this result to (M.62) yields
m˜ = ES(V1)T + EWS(V2, ρ)T −
(
EWS(V1, b∆Rx)
T + EWS(V2, b∆Tx)
T
)
= m, (M.64)
where m is defined in (5.182). We plug this result in (M.61) and substitute the abbreviations τ and
ϕ, defined in (M.39) and (M.40), respectively. This results in
C = C1 + C2 − C3, (M.65)
where C3 is defined in (5.174). This is what we wanted to derive.

– 201 –
N. Proof of Theorem 6: Decoupling of
Electrical Angles and Doppler
Frequencies for Arbitrary Number of
Moving Targets
We prove Theorem 6 as described in the overview of the proof on p. 132. Thus we have to show that
the relevant elements in C(1) and C(2) are 0. We use some of the results derived in Appendix M.
First we derive some relations which we need in the further computations. Since the condition
(5.207) has to be fulfilled, we abbreviate
EWS
(
t(i), ρ(i)
)
= t˜ ∀ i ∈ {1, . . . ,NTx}. (N.1)
With (5.206) and ∆ωqr = 0,
b(qr)
∆Tx = ρ  exp( j · dPulse∆Uqr) (N.2)
follows from (M.16). Defining
c(qr) = exp( j · dPulse∆Uqr) (N.3)
yields
b(qr)
∆Tx = ρ  c(qr). (N.4)
Analogously to t(i) and ρ(i) we define
b(i),(qr)
∆Tx = ρ
(i) · c(i),(qr), (N.5)
c(i),(qr) = exp( j · dTxi ∆Uqr). (N.6)
b(i),(qr)
∆Tx denotes the complex weights of the i-th Tx antenna which transmits at the time instances t
(i)
– 202 –
for the q-th and r-th target. Using these notations
EWS
(
t(i), b(i),(qr)
∆Tx
)
= EWS
(
t(i), c(i),(qr) · ρ(i)
)
. (N.7)
c(i),(qr) is a complex scalar unequal 0. Hence, from the definition of EWS follows
EWS
(
t(i), b(i),(qr)
∆Tx
)
= EWS
(
t(i), ρ(i)
)
= t˜ ∀ i. (N.8)
Next we show that CovWS
(
dPulse, t, b(qr)
∆Tx
)
= 0. We use Lemma J.1.3 with x = dPulse, y = t, w = b(qr)
∆Tx
and with NTx clusters, corresponding to the NTx Tx antennas. Here, we keep q, r fixed. Then, with
the notation of Lemma J.1.3,
x˜ = dTx, (N.9)
y(i) = t(i), (N.10)
w(i) = b(i),(qr)
∆Tx . (N.11)
Due to (N.8)
EWS
(
t(k), b(k),(qr)
∆Tx
)
= EWS
(
t(l), b(l),(qr)
∆Tx
)
∀k, l ∈ {1, . . . ,NTx}. (N.12)
Hence, from Lemma J.1.3
CovWS
(
dPulse, t, b(qr)
∆Tx
)
= 0 (N.13)
follows. Moreover
EWS
(
t, b(qr)
∆Tx
)
=
1
1T b(qr)
∆Tx
b(qr)
∆Tx
T
t
=
1
1T b(qr)
∆Tx
NTx∑
i=1
(
b(i),(qr)
∆Tx
)T
t(i)
=
1
1T b(qr)
∆Tx
NTx∑
i=1
1T b(i),(qr)
∆Tx E
WS
(
t(i), b(i),(qr)
∆Tx
)
(N.14)
and using (N.8) yields
EWS
(
t, b(qr)
∆Tx
)
=
1
1T b(qr)
∆Tx
NTx∑
i=1
1T b(i),(qr)
∆Tx · t˜
= t˜. (N.15)
– 203 –
W. l. o. g. we set the time origin such that t˜ = 0. Hence
EWS
(
t, b(qr)
∆Tx
)
= 0. (N.16)
Due to this
CovWS
(
dPulse, t, b(qr)
∆Tx
)
= CorrWS
(
dPulse, t, b(qr)
∆Tx
)
− EWS
(
t, b(qr)
∆Tx
)
EWS
(
dPulse, b(qr)
∆Tx
)
= CorrWS
(
dPulse, t, b(qr)
∆Tx
)
. (N.17)
With (N.13)
CorrWS
(
dPulse, t, b(qr)
∆Tx
)
= 0 (N.18)
follows.
Investigation of C(1) C(1) is given by (M.10)
C(1) =

DH1 D1 . . . D
H
1 DNTargets
...
. . .
...
DHNTargetsD1 . . . D
H
NTargets
DNTargets
 (N.19)
with DHq Dr computed in (M.25)
DHq Dr = b
(qr)
∆sum
∗
CorrWS
(
V,V, b(qr)
∆
∗) ∈ C2×2, q, r ∈ {1, . . . ,NTargets}. (N.20)
We show that the elements of C(1) at the positions marked with × in (5.208) are 0. Thus we have to
show that the off-diagonal elements of DHq Dr are 0, ∀q, r ∈ {1, . . . ,NTargets}. Plugging in the definition
of V (M.6) yields
CorrWS
(
V,V, b(qr)
∆
)
=
CorrWS
(
dVirt, dVirt, b(qr)
∆
)
CorrWS
(
dVirt, tVirt, b(qr)
∆
)
CorrWS
(
dVirt, tVirt, b(qr)
∆
)
CorrWS
(
tVirt, tVirt, b(qr)
∆
)  . (N.21)
With the definition of b(qr)
∆
(M.18) and some properties of CorrWS given in Appendix A.2.3, the
off-diagonal element can be expressed as
CorrWS
(
dVirt, tVirt, b(qr)
∆
)
= CorrWS
(
1NPulse ⊗ dRx + dPulse ⊗ 1NRx , t ⊗ 1NRx , b
(qr)
∆Tx ⊗ b(qr)∆Rx
)
= CorrWS
(
1NPulse ⊗ dRx, t ⊗ 1NRx , b
(qr)
∆Tx ⊗ b(qr)∆Rx
)
+ CorrWS
(
dPulse ⊗ 1NRx , t ⊗ 1NRx , b
(qr)
∆Tx ⊗ b(qr)∆Rx
)
= CorrWS
(
1NPulse , t, b
(qr)
∆Tx
)
· CorrWS
(
dRx, 1NRx , b
(qr)
∆Rx
)
+ CorrWS
(
dPulse, t, b(qr)
∆Tx
)
– 204 –
= EWS
(
t, b(qr)
∆Tx
)
CorrWS
(
dRx, 1NRx , b
(qr)
∆Rx
)
+ CorrWS
(
dPulse, t, b(qr)
∆Tx
)
. (N.22)
Due to (N.16) and (N.18)
CorrWS
(
dVirt, tVirt, b(qr)
∆
)
= 0. (N.23)
Thus, we have shown that the off-diagonal elements of DHq Dr equal 0 and therefore the relevant
elements in C(1) are 0.
Investigation of C(2) C(2) is defined in (5.211),
C(2) = DH B
(
BH B
)−1
BH D. (N.24)
BH D is given in (M.30)
BH D = j

b(1)
HD1 . . . b(1)
HDNTargets
...
. . .
...
b(NTargets)
HD1 . . . b(NTargets)
HDNTargets
 (N.25)
with (M.32)
b(q)
HDr = j b(qr)∆sum
∗
EWS
(
V, b(qr)
∆
∗) ∈ C1×2. (N.26)
Using the definition of V and b(qr)
∆
yields
EWS
(
V, b(qr)
∆
)
= EWS
(
1NPulse ⊗ V1 + V2 ⊗ 1NRx , b
(qr)
∆Tx ⊗ b(qr)∆Rx
)
= EWS
(
V1, b(qr)∆Rx
)
+ EWS
(
V2, b(qr)∆Tx
)
=
[
EWS
(
dRx, b(qr)
∆Rx
)
EWS
(
0, b(qr)
∆Rx
)]
+
[
EWS
(
dPulse, b(qr)
∆Tx
)
EWS
(
t, b(qr)
∆Tx
)]
=
[
EWS
(
dRx, b(qr)
∆Rx
)
+ EWS
(
dPulse, b(qr)
∆Tx
)
0
]
, (N.27)
where we have used (N.16) in the last line. Thus BH D has the following form
BH D =
[
x1 0 x2 0 . . . xNTargets 0
]
, (N.28)
where x1, . . . , xNTargets are some vectors which are in general unequal 0. Multiplying from the left with
a NTargets × NTargets matrix keeps this structure. Thus
C(2) =
(
DH B
) (
BH B
)−1 (
BH D
)
– 205 –
=

xH1
0T
...
xHNTargets
0T

(
BH B
)−1 [
x1 0 . . . xNTargets 0
]
=

xH1
0T
...
xHNTargets
0T

[
y
1
0 . . . y
NTargets
0
]
=

xH1 y1 0 . . . x
H
1 yNTargets
0
0 0 . . . 0 0
...
...
. . .
...
...
xHNTargetsy1 0 . . . x
H
NTargets
y
NTargets
0
0 0 . . . 0 0

, (N.29)
where y
1
, . . . , y
NTargets
are some vectors which are in general unequal 0. Hence the elements at the
positions marked with × in (5.208) are 0. 

– 207 –
Bibliography
F. Athley. Threshold region performance of maximum likelihood direction of arrival estimators.
IEEE Transactions on Signal Processing, 53(4):1359–1373, 2005. doi: 10.1109/TSP.2005.
843717.
Chen Baixiao, Zhang Shouhong, Wang Yajun, and Wang Jun. Analysis and experimental results on
sparse-array synthetic impulse and aperture radar. In Proc. Radar CIE Int. Conf, pages 76–80,
2001. doi: 10.1109/ICR.2001.984627.
I. Bekkerman and J. Tabrikian. Target Detection and Localization Using MIMO Radars and Sonars.
IEEE Transactions on Signal Processing, 54(10):3873–3883, 2006. doi: 10.1109/TSP.2006.
879267.
J. F. Böhme. Stochastische Signale mit Übungen und einem MATLAB-Praktikum. Teubner, 1998.
Johann F. Böhme. Source-Parameter Estimation by Approximate Maximum Likelihood and Non-
linear Regression. IEEE Journal of Oceanic Engineering, 10(3):206–212, Jul 1985. ISSN 0364-
9059. doi: 10.1109/JOE.1985.1145098.
R. Boyer. Performance Bounds and Angular Resolution Limit for the Moving Colocated MIMO
Radar. IEEE Transactions on Signal Processing, 59(4):1539–1552, 2011. doi: 10.1109/TSP.
2010.2100387.
L. E. Brennan and L. S. Reed. Theory of Adaptive Radar. IEEE Transactions on Aerospace and
Electronic Systems, AES-9(2):237–252, 1973.
I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Mühlig. Taschenbuch der Mathematik. Harri
Deutsch, 2008.
M. Bühren. Simulation und Verarbeitung von Radarziellisten im Automobil. PhD thesis, Universtität
Stuttgart, 2008.
Chun-Yang Chen. Signal processing algorithms for MIMO radar. PhD thesis, California Institute
of Technology, 2009. URL http://thesis.library.caltech.edu/2521/.
C.Y. Chong, F. Pascal, and M. Lesturgie. Estimation performance of coherent MIMO-STAP using
Cramer-Rao bounds. In Radar Conference (RADAR), 2011 IEEE, pages 533 –537, 2011. doi:
10.1109/RADAR.2011.5960594.
– 208 –
M.P. Clark. On the resolvability of normally distributed vector parameter estimates. IEEE Transac-
tions on Signal Processing, 43(12):2975–2981, 1995. ISSN 1053-587X. doi: 10.1109/78.476441.
Harald Cramer. Mathematical methods of statistics. Princeton Univ. Press, 1945.
A. J. den Dekker and A. van den Bos. Resolution: a survey. J. Opt. Soc. Am. A, 14:547–557, 1997.
doi: http://dx.doi.org/10.1364/JOSAA.14.000547.
A. Dogandzic and A. Nehorai. Cramer-Rao bounds for estimating range, velocity, and direction with
an active array. IEEE Transactions on Signal Processing, 49(6):1122–1137, 2001.
J. Dorey, G. Garnier, and G. Auvray. RIAS, synthetic impulse and antenna radar. In International
Conference on Radar, 1989.
M.N. El Korso, R. Boyer, A. Renaux, and S. Marcos. A GLRT-based framework for the Multidi-
mensional Statistical Resolution Limit. In Statistical Signal Processing Workshop (SSP), 2011
IEEE, pages 453–456, 2011. doi: 10.1109/SSP.2011.5967729.
R. Feger, S. Schuster, S. Scheiblhofer, and A. Stelzer. Sparse antenna array design and combined
range and angle estimation for FMCW radar sensors. In Radar Conference, 2008. RADAR ’08.
IEEE, pages 1 –6, 2008. doi: 10.1109/RADAR.2008.4720809.
R. Feger, C. Wagner, S. Schuster, S. Scheiblhofer, H. Jager, and A. Stelzer. A 77-GHz FMCW
MIMO Radar Based on an SiGe Single-Chip Transceiver. IEEE Transactions on Microwave
Theory and Techniques, 57(5):1020–1035, 2009. doi: 10.1109/TMTT.2009.2017254.
E. Fishler, A. Haimovich, R. Blum, D. Chizhik, L. Cimini, and R. Valenzuela. MIMO radar: an idea
whose time has come. In Proc. IEEE Radar Conf, pages 71–78, 2004. doi: 10.1109/NRC.2004.
1316398.
Andrew S Fletcher and F Robey. Performance bounds for adaptive coherence of sparse array radar.
In Proceedings of the 12th Annual Workshop on Adaptive Sensor Array Processing, 2003.
K.W. Forsythe and D.W. Bliss. Waveform Correlation and Optimization Issues for MIMO Radar. In
Conference Record of the Thirty-Ninth Asilomar Conference on Signals, Systems and Computers,
pages 1306 – 1310, 2005. doi: 10.1109/ACSSC.2005.1599974.
B. Friedlander. On the Relationship Between MIMO and SIMO Radars. IEEE Transactions on
Signal Processing, 57(1):394–398, 2009. doi: 10.1109/TSP.2008.2007106.
B. Friedlander. Effects of Model Mismatch in MIMO Radar. IEEE Transactions on Signal Process-
ing, 60(4):2071–2076, 2012. doi: 10.1109/TSP.2011.2179653.
D. R. Fuhrmann and G. San Antonio. Transmit beamforming for MIMO radar systems using partial
signal correlation. In Proc. Conf Signals, Systems and Computers Record of the Thirty-Eighth
Asilomar Conf, volume 1, pages 295–299, 2004. doi: 10.1109/ACSSC.2004.1399140.
– 209 –
D. R. Fuhrmann and G. San Antonio. Transmit beamforming for MIMO radar systems using signal
cross-correlation. IEEE Transactions on Aerospace and Electronic Systems, 44(1):171–186, 2008.
doi: 10.1109/TAES.2008.4516997.
J. Guetlein, J. Bertl, A. Kirschner, and J. Detlefsen. Switching scheme for a FMCW-MIMO radar
on a moving platform. In Radar Conference (EuRAD), 2012 9th European, pages 91–94, 2012.
J. Guetlein, A. Kirschner, and J. Detlefsen. Motion compensation for a TDM FMCW MIMO radar
system. In Radar Conference (EuRAD), 2013 European, pages 37–40, 2013a.
J. Guetlein, A. Kirschner, and J. Detlefsen. Calibration strategy for a TDM FMCW MIMO radar sys-
tem. In Microwaves, Communications, Antennas and Electronics Systems (COMCAS), 2013 IEEE
International Conference on, pages 1–5, Oct 2013b. doi: 10.1109/COMCAS.2013.6685266.
A. M. Haimovich, R. S. Blum, and L. J. Cimini. MIMO Radar with Widely Separated Antennas.
IEEE Signal Processing Magazine, 25(1):116–129, 2008. doi: 10.1109/MSP.2008.4408448.
A. Hassanien and S. A. Vorobyov. Phased-MIMO Radar: A Tradeoff Between Phased-Array and
MIMO Radars. IEEE Transactions on Signal Processing, 58(6):3137–3151, 2010a. doi: 10.1109/
TSP.2010.2043976.
Aboulnasr Hassanien and Sergiy A. Vorobyov. Why the Phased MIMO-Radar Outperforms the
Phased-Array and MIMO Radars. In 18th European Signal Processing Conference (EUSIPCO-
2010), 2010b.
M. Jahn, R. Feger, C. Wagner, Ziqiang Tong, and A. Stelzer. A Four-Channel 94-GHz SiGe-Based
Digital Beamforming FMCW Radar. IEEE Transactions on Microwave Theory and Techniques,
60(3):861–869, 2012. doi: 10.1109/TMTT.2011.2181187.
K-D. Kammeyer and K. Kroschel. Digitale Signalverarbeitung. Springer Verlag, 2012.
S. M. Kay. Estimation Theory, volume I of Fundamentals of Statistical Signal Processing. Prentice
Hall PTR, Upper Saddle River, NJ, 1993.
M. Klemm, J. Leendertz, D. Gibbins, I.J. Craddock, A. Preece, and R. Benjamin. Towards contrast
enhanced breast imaging using ultra-wideband microwave radar system. In Radio and Wireless
Symposium (RWS), pages 516 –519, jan. 2010. doi: 10.1109/RWS.2010.5434240.
H. Krim and M. Viberg. Two decades of array signal processing research: the parametric approach.
IEEE Signal Processing Magazine, 13(4):67–94, 1996. doi: 10.1109/79.526899.
K.S. Kulpa, A Wojtkiewicz, M. Nalecz, and J. Misiurewicz. The simple method for analysis
of nonlinear frequency distortions in fmcw radar. In Microwaves, Radar and Wireless Com-
munications. 2000. MIKON-2000. 13th International Conference on, volume 1, 2000. doi:
10.1109/MIKON.2000.913915.
– 210 –
Moon-Sik Lee, V. Katkovnik, and Yong-Hoon Kim. System modeling and signal processing for a
switch antenna array radar. IEEE Transactions on Signal Processing, 52(6):1513–1523, 2004.
doi: 10.1109/TSP.2004.827204.
N. H. Lehmann, E. Fishler, A. M. Haimovich, R. S. Blum, D. Chizhik, L. J. Cimini, and R. A. Valen-
zuela. Evaluation of Transmit Diversity in MIMO-Radar Direction Finding. IEEE Transactions
on Signal Processing, 55(5):2215–2225, 2007. doi: 10.1109/TSP.2007.893220.
Marc Lesturgie. Some relevant applications of MIMO to radar. In International Radar Symposium,
2011.
J. Li and P. Stoica. MIMO Radar Signal Processing. John Wiley & Sons, Inc., Hoboken, NJ, 2009.
Jian Li and P. Stoica. MIMO Radar with Colocated Antennas. IEEE Signal Processing Magazine,
24(5):106–114, 2007. doi: 10.1109/MSP.2007.904812.
Jian Li, Luzhou Xu, P. Stoica, K. W. Forsythe, and D. W. Bliss. Range Compression and Waveform
Optimization for MIMO Radar: A Cramer-Rao Bound Based Study. IEEE Transactions on Signal
Processing, 56(1):218–232, 2008. doi: 10.1109/TSP.2007.901653.
Z. Liu and A. Nehorai. Statistical Angular Resolution Limit for Point Sources. IEEE Transactions
on Signal Processing, 55(11):5521–5527, 2007. doi: 10.1109/TSP.2007.898789.
U. Lübbert. Target Position Estimation with a Continuous Wave Radar Network. PhD thesis, Tech-
nische Universität Hamburg-Harburg, 2005.
A.-S. Luce, H. Molina, D. Muller, and V. Thirard. Experimental results on RIAS digital beamform-
ing radar. In Proc. Radar 92. Int. Conf, pages 74–77, 1992.
T. McWhorter and L.L. Scharf. Cramer-Rao Bounds for Deterministic Modal Analysis. IEEE
Transactions on Signal Processing, 41(5):1847–1866, May 1993. ISSN 1053-587X. doi: 10.
1109/78.215304.
V. F. Mecca, D. Ramakrishnan, and J. L. Krolik. MIMO Radar Space-Time Adaptive Processing
for Multipath Clutter Mitigation. In Proc. Fourth IEEE Workshop Sensor Array and Multichannel
Processing, pages 249–253, 2006. doi: 10.1109/SAM.2006.1706131.
W. L. Melvin. A STAP overview. IEEE Aerospace and Electronic Systems Magazine, 19(1):19–35,
2004.
Openclipart. https://openclipart.org/, 2014.
S. Pasupathy and A.N. Venetsanopoulos. Optimum Active Array Processing Structure and Space-
Time Factorability. IEEE Transactions on Aerospace and Electronic Systems, AES-10(6):770
–778, 1974. ISSN 0018-9251. doi: 10.1109/TAES.1974.307883.
– 211 –
David Poole. Linear Algebra: A Modern Introduction. Brooks/Cole, Cengage Learning, 3. ed.
edition, 2011. ISBN 978-0-538-73544-5.
K. Rambach and B. Yang. Colocated MIMO Radar: Cramer-Rao Bound and Optimal Time Division
Multiplexing for DOA Estimation of Moving Targets. In Proc. IEEE Int. Conf. Acoustics, Speech
and Signal Processing (ICASSP), 2013. doi: 10.1109/ICASSP.2013.6638411.
K. Rambach and B. Yang. Direction of Arrival Estimation of Two Moving Targets Using a Time
Division Multiplexed Colocated MIMO Radar. In Proc. IEEE Radar Conference (Radarcon),
2014.
K. Rambach and B. Yang. Two-Dimensional TDM-MIMO Radar: Cramer-Rao Bound and De-
coupling of the Direction of Arrivals from the Doppler Frequency of a Moving Target. In to be
published, 2016a.
K. Rambach and B. Yang. MIMO Radar: Time Division Multiplexing vs. Code Division Multiplex-
ing. In to be published, 2016b.
K. Rambach, M. Vogel, and B. Yang. Optimal Time Division Multiplexing Schemes for DOA
Estimation of a Moving Target Using a Colocated MIMO Radar. In Proc. IEEE International
Symposium on Signal Processing and Information Technology (ISSPIT), 2014.
C. R. Rao. Information and the accuracy attainable in the estimation of statistical parameters. Bul-
letin of the Calcutta Mathematical Society, 37:81–91, 1945.
Eckhard Rebhan. Theoretische Physik, Mechanik, Elektrodynamik, Relativitätstheorie, Kosmologie.
Spektrum Akademischer Verlag, Heidelberg, Berlin, 2001.
M. Reiher. Optimierung von Sendesignalen zur Vermeidung von Scheinzielen für frequenzmodulierte
Dauerstrich-Radarsysteme im Automobil. PhD thesis, Universtität Stuttgart, 2012.
M. J. D. Rendas and J. M. F. Moura. Ambiguity in radar and sonar. IEEE Transactions on Signal
Processing, 46(2):294–305, 1998. doi: 10.1109/78.655416.
F. C. Robey, S. Coutts, D. Weikle, J. C. McHarg, and K. Cuomo. MIMO radar theory and experimen-
tal results. In Proc. Conf Signals, Systems and Computers Record of the Thirty-Eighth Asilomar
Conf, volume 1, pages 300–304, 2004. doi: 10.1109/ACSSC.2004.1399141.
G. San Antonio, D. R. Fuhrmann, and F. C. Robey. MIMO Radar Ambiguity Functions. IEEE
Journal of Selected Topics in Signal Processing, 1(1):167–177, 2007. doi: 10.1109/JSTSP.2007.
897058.
A. M. Sayeed and V. Raghavan. Maximizing MIMO Capacity in Sparse Multipath With Recon-
figurable Antenna Arrays. IEEE Journal of Selected Topics in Signal Processing, 1(1):156–166,
2007. doi: 10.1109/JSTSP.2007.897057.
– 212 –
C.M. Schmid, R. Feger, C. Pfeffer, and A. Stelzer. Motion Compensation and Efficient Array Design
for TDMA FMCW MIMO Radar Systems. In Antennas and Propagation (EUCAP), 2012 6th
European Conference on, pages 1746–1750, March 2012. doi: 10.1109/EuCAP.2012.6206605.
M. Schoor. Hochauflösende Winkelschätzung für automobile Radarsysteme. PhD thesis, Universtität
Stuttgart, 2010.
M. I. Skolnik. Introduction to Radar Systems. McGrawHill, 3. edition, 2001.
P. Stoica and A. Nehorai. MUSIC, maximum likelihood, and Cramer-Rao bound. IEEE Transactions
on Acoustics, Speech, and Signal Processing, 37(5):720–741, 1989.
AG. Stove. Linear FMCW Radar Techniques. Radar and Signal Processing, IEE Proceedings F,
139(5), Oct 1992. ISSN 0956-375X.
Hongbo Sun, Frederic Brigui, and M. Lesturgie. Analysis and Comparison of MIMO Radar Wave-
forms. In Internation Radar Conference, 2014.
D. N. Tran, A. Renaux, R. Boyer, S. Marcos, and P. Larzabal. Weiss-Weinstein bound for MIMO
radar with colocated linear arrays for SNR threshold prediction. Signal Processing, 92:1353–
1358, 2012.
S. Uhlich. Parameter Estimation with Additional Information. PhD thesis, Universtität Stuttgart,
2012.
P. P. Vaidyanathan and P. Pal. MIMO radar, SIMO radar, and IFIR radar: a comparison. In Proc.
Conf Signals, Systems and Computers Record of the Forty-Third Asilomar Conf, pages 160–167,
2009. doi: 10.1109/ACSSC.2009.5470139.
H. L. Van Trees. Optimum Array Processing, volume IV of Detection, Estimation, and Modulation
Theory. John Wiley & Sons, Inc., New York, 2002.
Markus Vogel. Optimization of Direction of Arrival Estimation with a Time Division Multiplexed
MIMO Radar. Master’s thesis, University of Stuttgart, 2013.
J. Ward. Cramer-Rao bounds for target angle and Doppler estimation with space-time adaptive
processing radar. In Conference Record of the Twenty-Ninth Asilomar Conference on Signals,
Systems and Computers, volume 2, pages 1198 –1202, 1995. doi: 10.1109/ACSSC.1995.540889.
A. Weiss and E. Weinstein. A lower bound on the mean-square error in random parameter estimation
(Corresp.). IEEE Transactions on Information Theory, 31(5):680–682, 1985. doi: 10.1109/TIT.
1985.1057094.
Hermann Winner. Handbuch Fahrerassistenzsysteme: Grundlagen, Komponenten und Systeme
für aktive Sicherheit und Komfort. Vieweg+Teubner Verlag / Springer Fachmedien Wiesbaden
– 213 –
GmbH, Wiesbaden, Wiesbaden, 2. edition, 2012. ISBN 978-3-8348-8619-4. doi: 10.1007/
978-3-8348-8619-4.
M. Wintermantel. Radar System with Elevation Measuring Capability, 2010.
W. Wirtinger. Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen. Mathe-
matische Annalen, 97(1):357–375, 1927. ISSN 0025-5831. doi: 10.1007/BF01447872.
Ming Xue, Duc Vu, Luzhou Xu, Jian Li, and P. Stoica. On MIMO radar transmission schemes for
ground moving target indication. In Proc. Conf Signals, Systems and Computers Record of the
Forty-Third Asilomar Conf, pages 1171–1175, 2009. doi: 10.1109/ACSSC.2009.5470010.
Ming Xue, W. Roberts, Jian Li, Xing Tan, and P. Stoica. MIMO radar sparse angle-Doppler imaging
for ground moving target indication. In Proc. IEEE Radar Conf, pages 553–558, 2010. doi:
2010.5494560.
B. Yang. Statistical and adaptive signal processing. lecture notes, University of Stuttgart, 2011.
S. F. Yau and Y. Bresler. A compact Cramer-Rao bound expression for parametric estimation of
superimposed signals. IEEE Transactions on Signal Processing, 40(5):1226–1230, 1992. doi:
10.1109/78.134484.
Alex Zwanetski and Hermann Rohling. Continuous wave mimo radar based on time division multi-
plexing. In Radar Symposium (IRS), 2012 13th International, pages 119–121. IEEE, 2012.