Analysis and Optimization of Wireless Cellular Multi-User
Multiple-Input Multiple-Output Communication Systems
Von der Fakultät Informatik, Elektrotechnik und
Informationstechnik der Universität Stuttgart zur Erlangung der
Würde eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte
Abhandlung
Vorgelegt von
Venkata Krishna Chitti
Hyderabad (India)
Hauptberichter: Prof. Dr.-Ing. Stephan ten Brink
Mitberichter: Prof. Dr.-Ing. Andreas Kirstädter
Tag der mündlichen Prüfung: 19. November 2015
Institut für Nachrichtenübertragung der Universität Stuttgart
2015
Acknowledgements
I am ever indebted to my parents for the numerous sacrifices they made for me.
At the INÜ, I firstly thank Professor Joachim Spiedel for providing me a rare op-
portunity to carry out the research by providing necessary financial assistance and
infrastructure. Next, I thank Professor Stephan ten Brink who took over at the last
phase of my research and agreed to be the first supervisor of my work. All my INÜ
colleagues have been more than friendly to me. This made my tenure at the institute
enjoyable.
I thank everyone who has contributed in some way or the other to my journey since
childhood. Above all, I must realize that an ever pervading source that keeps me
connected from within has been guiding me throughout.
ii
Contents
Abbreviations and Notations v
Problem Definitions xi
Abstract xiii
Kurzfassung xiv
1 Introduction 1
1.1 Basics of Wireless Communication . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Frequency Reuse . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Multiple Access . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.4 Multiple-Input Multiple-Output . . . . . . . . . . . . . . . . . . 6
1.2 Multi-User MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Fairness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Transceiver design . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Mathematical Techniques for Optimization 17
2.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Base Station Association . . . . . . . . . . . . . . . . . . . . . . 21
2.3.2 Power Allocation and Beamforming . . . . . . . . . . . . . . . . 23
iii
3 Uplink Resource Allocation under perfect-CSI 27
3.1 Algorithmic Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Rate Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 Iterative Power Allocation . . . . . . . . . . . . . . . . . . . . . 30
3.2.2 Iterative Base Station Association . . . . . . . . . . . . . . . . . 30
3.2.3 Single Stage Formulation . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Power Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.1 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Uplink Power Control under statistical-CSI and imperfect-CSI 41
4.1 Power Control under statistical-CSI . . . . . . . . . . . . . . . . . . . . 42
4.2 Value-at-Risk and Conditional Value-at-Risk . . . . . . . . . . . . . . . 43
4.3 Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.1 Generalized Extreme Value Distribution . . . . . . . . . . . . . 44
4.3.2 Generalized Pareto Distribution for CVaR . . . . . . . . . . . . 48
4.4 Power Control under imperfect-CSI . . . . . . . . . . . . . . . . . . . . 50
5 Downlink Performance Analysis under perfect-CSI 55
5.1 Uplink-Downlink Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Problem Reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Downlink Rate and Power Allocation . . . . . . . . . . . . . . . . . . . 58
6 Conclusion 65
Publications 67
iv
Abbreviations and Notations
3GPP 3rd Generation Partnership Project
AAS Active Antenna System
ACO Ant Colony Optimization
AWGN Additive White Gaussian Noise
BA Bernstein Approximation
BB Branch and Bound
BC Broadcast Channel
BD Bender Decomposition
BER Bit Error Rate
BLAST Bell-Labs Layered Space Time Architecture
BI Bernstein-type Inequality
BF Beamforming
BFGS Broyden-Fletcher-Goldfarb-Shanno
BS Base Station
BSA Base Station Association
cdf Cumulative Distribution Function
CoMP Coordinated Multi-point transmission reception
CCI Cochannel Interference
CDMA Code Division Multiple Access
CSB Coordinated Scheduling and Beamforming
CSI Channel State Information
CVaR Conditional Value-at-Risk
DC Difference in Convex
DoA Domain-of-Attraction
DoF Degrees-of-Freedom
DPC Dirty Paper Coding
DL Downlink
DWD Danzig-Wolfe Decomposition
EVT Extreme Value Theory
FDMA Frequency Division Multiple Access
GA Gaussian Approximation
GAP Generalized Assignment Problem
GeA Genetic Algorithm
v
GEVD Generalized Extreme Value Distribution
GEVP Generalized EigenValue Problem
GFDM Generalized Frequency Division Multiplexing
GP Geometric Programming
GPD Generalized Pareto Distribution
HetNets Heterogeneous Networks
i.i.d. independent and identically distributed
IPM Interior Point Method
IM Interference Management
IP Integer Programming
IMT-A International Mobile Telecommunications Advanced
ITU-R International Telecommunication Union Radiocommunication Sector
JP Joint Processing
KKT Karash-Kuhn-Tucker
KP Knapsack Problem
LICQ Linear Independent Constraint Qualification
LP Linear Programming
LR Lagrange Relaxation
LSE Least Squared Error
LTE-A Long Term Evolution Advanced
MAC Multiple Access Channel
MAI Multiple Access Interference
MC Multi-carrier
MINLP Mixed Integer Non-Linear Programming
MLE Maximum Likelihood Estimator
MMKP Multi-dimension Multi-Choice Knapsack Problem
MSE Mean Squared Error
MLSE Minimum Least Squared Error
MMSE Minimum Mean Squared Error
MUD Multi-User Diversity
MUI Multi-User Interference
MU-MIMO Multi-User Multiple-Input Multiple-Output
MVDR dsdfsdfs
NDMA Network assisted Diversity Multiple Access
NLP Non-Linear Programming
NOMA Non-orthogonal Multiple Access
NP Non-deterministic Polynomial-time
OFDM Orthogonal Frequency Division Multiplexing
OFDMA Orthogonal Frequency Division Multiple Access
pdf Probability Density Function
PoT Peaks over Threshold
PrCons Probabilistic Constraints
vi
PA Power Allocation
PAPR Peak to Average Power Ratio
PC Power Control
PF Proportional Fairness
PSO Particle Swarm Optimization
QoS Quality of Service
r.v. Random Variable
RA Resource Allocation
RR Round Robin
SA Simmulated Annealing
SC-OFDMA Single Carrier OFDMA
SDR Semidefinite Relaxation
SDMA Space Division Multiple Access
SDP Semidefinite Programming
SGINR Signal to Generated Interference Noise Ratio
SIC Successive Interference Cancellation
SINR Signal-to-Interference-Noise-Ratio
SISO Single-Input Single-Output
SNR Signal to Noise Ratio
SO Successive Optimization
SOCP Second Order Cone Programming
SON Self Organizing Networks
SP Signomial Programming
SQP Sequential Quadratic Programming
STBC Space Time Block Code
STTC Space Time Trellis Code
SUS Semi-orthogonal User Selection
TDMA Time Division Multiple Access
THP Tomlinson Harashima Precoding
TS Tabu Search
TSD Transmit Selection Diversity
UE User Equipment
UL Uplink
VaR Value-at-Risk
VUL Virtual-Uplink
ZF Zero-Forcing
M number of BSs
N number of UEs
T number of antenna elements at the BS
R number of antenna elements at the UE, R = 1 assumed throughout
L number of simultaneous data streams or spatial layers for a UE,
vii
L= 1 assumed throughout
αij 0−1 integer BSA variable, no macro-diversity throughout
αj BSA vector of UEj
α overall BSA matrix of all the UEs
αi i
th row of α
pj UL transmit power variable of UEj
qij DL transmit power variable of UEj from BSi
qj DL power vector for UEj across all the BSs
Q overall DL power vector of all the UEs across all the BSs
q active DL power vector of all the UEs across all the BSs
q
i
ith row of Q
hij UL channel vector from UEj to BSi
hHij DL channel vector from BSi to UEj
uij UL receive BF vectors of UEj at BSi
uHij DL transmit BF vectors at BSi for UEj
U overall BF matrix of all the UEs across all the BSs
aj UL transmit symbol of UEj
aˆULj estimated UL transmit symbol of UEj
aij DL transmit symbol of UEj from BSi
aˆDLij estimated DL transmit symbol of UEj
zj UL noise vector for UEj at a BS
σ2j UL AWGN noise variance per-antenna element for UEj at a BS
zj DL noise variable at UEj
σ2j DL AWGN noise variance per-antenna element at UEj
P A primal problem
D Mathematical dual problem of P
L Lagrangian of P
N2 total number of constraints across all the UEs in P
zj primal vector of UEj
cj constraint vector of UEj
sj slack vector for the cj
t iteration index
δ step length in IPM
∆ search directions in IPM
F KKT conditions in vector form
J Jacobian of F w.r.t. the primal and dual variables
µ complementarity measure in perturbed KKT equation
τ parameter for δ calculation
Pmax maximum UL transmit power of each UE
P¯i maximum DL transmit power at BSi
CSIp perfect-CSI
γj UL SINR of UEj under CSIp
viii
rj UL rate of UEj under CSIp
γ
j
DL SINR at UEj under CSIp
rj DL rate at UEj under CSIp
1M (M ×1) vector with all elements equal to unity
rth QoS threshold for each UE on UL and DL
ℜ{·} Real part of a complex number
ℑ{·} Imaginary part of a complex number
(◦) Harammard product
(·)H Hermitian operator
(·)T Transpose operator
(·)−1 Inverse operator
(·)∗ complex conjugate operator
max(·, ·) maximum operator
min(·, ·) minimum operator
tr(·) trace operator
diag(·) diagonal operator
blkdiag(·) block diagonal operator
vec(·) vectorization operator
‖·‖1 ℓ1−norm operation
‖·‖2 ℓ2−norm operation
log2(·) logarithm base−2
ln(·) natural logarithm
erf−1(·) Inverse error function
exp(·) Exponential function
arg-max(·) arg-max operator, chooses the argument that maximizes the function
(·,0)+ non-negative operator, chooses 0 or non-negative argument
K number of i.i.d. random variables in a block for EVT approach
B number of blocks of i.i.d. random variables for EVT approach
η confidence interval for the chance constraints
N (·, ·) normal distribution
CN (·, ·) complex normal distribution
ξ shape parameter of GEVD and GPD
l location parameter of GEVD
s scale parameter of GEVD
l location parameter of GPD
s scale parameter of GPD
CSIs statistical-CSI
γ
(s)
j Statistical UL SINR of UEj with no macro-diversity under CSIs
r
(s)
j Statistical UL rate of UEj with no macro-diversity under CSIs
CSIi imperfect-CSI
γ
(i)
j Imperfect UL SINR of UEj with no macro-diversity under CSIi
r
(i)
j imperfect UL rate of UEj with no macro-diversity under CSIi
ix
θ A large positive constant in P3f
F A single function that evaluates VaR and CVaR together
Q(·, ·) Marcum Q−function
I0(·) 0th order modified Bessel’s function of first kind
b 2rth−1
gijk effective channel gain in the MUI term from UEk at BSi caused to
UEj
Iij MUI power for UEj at BSi
I Identity matrix
χ22 central chi-squared distributed r.v.
nc−χ22 non-central chi-squared distributed r.v.
FY (·) cdf of r.v. Y
F−1Y (·) quantile function of r.v. Y or inverse CDF of r.v. Y
ΨY (·) cdf of r.v. Y
φ(·) A loss function
β
(s)
j CVaR of UEj
ζ
(s)
j VaR of UEj
ζ
(s)
j optimal ζ
(s)
j
ν ℓ1−norm parameter for feasibility handle in PC
x
Problem Definitions
P1 UL sum-rate maximization with QoS under CSIp
P1b BSA sub-problem of P1
P1bn NLP formulation of P1b
P1bs SDP formulation of P1b
P1p PA sub-problem of P1
P1s Single stage NLP formulation of P1
P2 DL sum-rate maximization with QoS under CSIp
P2b BSA sub-problem of P2
P2p PA sub-problem of P2
P2v NLP formulation of P2 with duality constraints
P3 UL sum-power minimization with QoS under CSIp
P3b BSA sub-problem of P3
P3p PA sub-problem of P3
P3s Single stage NLP formulation of P3
P3f P3s with ℓ1−norm feasibility handle
P3c P3s under CSIs
P3w Worst case performance problem of P3s under CSIs
P3i P3s under CSIi
P4 DL sum-power minimization with QoS under CSIp
P4v VUL formulation of P4
P4vb BSA sub-problem of P4v
P4vp PA sub-problem of P4v
P4vm max-min sub-problem equivalent of P4vp
P5 UL max-min problem under CSIp
xi
xii
Abstract
Multi-User Multiple-Input Multiple-Output (MU-MIMO) communication systems are
an attractive solution to the increasing demand in wireless services, limited availability
of resources and increasing user density. MIMO achieves an increased spectral efficiency
and an almost reliable link-quality without the need to expend other valuable network
resources such as bandwidth and power. MIMO exploits the spatial separation among
the network entities and the spatial richness in the environment to achieve this objec-
tive. The effect of shared nature of the wireless channel, i.e., multiple transmissions on
the same resource unit at the same instance, is perceived as the cochannel interference.
This is severe at the cell-edges. Hence the network performance is interference-limited
and the need to optimally allocate resources while providing a reliable service to the
network entities arises.
This thesis focuses on the development of iterative algorithms for the link-level physical
layer abstraction for the task of Interference Management (IM) and Resource Allocation
(RA) in conventional multi-cell multi-user wireless systems. For this, well understood
mathematical methods with some modifications and new optimization principles are
applied. Analysis is carried out separately on both the Uplink and the Downlink.
With a required quality-of-service in a multi-cell MU-MIMO set-up, the problems of
rate maximization and power control under different circumstances and assumptions
are considered. These include analysis in the presence of perfect Channel State Infor-
mation (CSI), statistical-CSI, imperfect-CSI and probabilistic constraints.
To achieve the task of RA, an equivalent mathematical optimization problem, usually
a Mixed Integer Non-Linear Programming (MINLP) problem is formulated. Feasibil-
ity of the MINLP problem is in itself a requirement before well known mathematical
tools can be applied to find the optimum (maxima/minima or global/local). Finding
the optimum for a feasible MINLP problem is mathematically difficult due to non-
convexity and highly coupled nature of the involved functions. The first step towards
improving these results could be to identify a single mathematical framework to solve
a general MINLP problem that performs atleast as good as the existing methods. This
considerably reduces the need to migrate between different methods with a change
in problem. The next step is to apply new mathematical methods and optimization
principles to effectively solve the problem. This is important to improve the quality of
the solution, since the mathematical problem may not be easy to handle due to var-
ious implicit reasons such as non-invertible functions and non-availability of analytic
expressions. For the numerical results, a simplified link-level simulation model that
is an abstraction to the actual system-level model is used. The setup is verified for a
minimum working example and can easily be scaled to match the practical parame-
ters. The setup assumes 2 Base Stations (BSs) each with 2 antennas serving a set of
homogeneous User Equipments (UEs) each with a single antenna. Also each BS serves
xiii
a single cell and all the network entities operate in the same time-frequency slot. In
addition to the thermal noise, the primary source of degradation to a UE signal arises
from the intra-cell and the inter-cell interference. Simulation results at the link-level
compare the performance of the proposed techniques against frequently used methods.
The former outperforms the later in most of the cases and performs at least as good
as the later in some cases. System performance is measured in terms of the sum-rate
and sum-power objectives. Depending on the problem, an improvement could refer to
an increase in the sum-rate, a decrease in sum-power, produce better bounds on the
objective or better handling of the problem.
Kurzfassung
MIMO erreicht eine vergrößerte spektrale Effizienz und eine nahezu zuverlässige Ver-
bindungsqualität ohne andere wertvolle Netzwerkresourcen wie Bandbreite und Leis-
tung zu vergrößern. MIMO nutzt die räumliche Trennung der Netzwerkeinheiten und
die räumliche Fülle der Umgebung aus, um dieses Ziel zu erreichen. Aufgrund der
vorliegenden Natur des geteilten Mediums eines drahtlosen Kanals, d.h. mehrere Über-
tragungen bei Benutzung der gleichen Resource zur gleichen Zeit, wirkt sich dies als
Co-Kanal-Interferenz aus. Dies ist besonders stark an den Zellrändern. Deswegen ist
die Performance des Netzwerks durch die Interferenz begrenzt und das Bedürfnis einer
optimalen Resourcen-Allokation beim Bereitstellen sicherer Dienste für die Netzteil-
nehmer steigt.
Diese Ausarbeitung/Dissertation behandelt die Entwicklung von iterativen Algorith-
men der abstrahierten Bitübertragungsschicht für das Interferenz-Management (IM)
und der Resoucen-Allokation (RA) in konventionellen Mehr-Zellen Mehr-Nutzer Draht-
lossystemen. Dazu werden ausgereifte mathematische Methoden mit ein paar Änderun-
gen und neuartigen Optimierungen angewandt. Untersuchungen wurden getrennt für
den Uplink sowie für den Downlink durchgeführt.
Aufgrund der erforderlichen Dienste-Qualität (quality-of-service) in Mehr-Zellen MU-
MIMO Systemen wurde die Problematik der Maximierung der Rate und Leistungskon-
trolle bei verschiedenen Gegebenheiten berücksichtigt. Dies beinhaltet Untersuchun-
gen der perfekten Kanal-Status-Information (CSI), statistischen CSI, mangelhafter-CSI
und wahrscheinlichkeitstheoretischen Randbedingungen.
Um das Ziel der RA zu erreichen, wurde ein äquivalentes mathematisches Optimie-
rungsproblem, gewöhnlich ein "Mixed Integer Non-Linear Programming"(MINLP) Pro-
blem formuliert. Die Durchführbarkeit des MINLP-Problems ist eine Voraussetzung um
bekannte mathematische Werkzeuge anwenden zu können um das Optimum (Maxi-
mum/Minimum, global/lokal) zu finden. Das Herausfinden des Optimums eines durch-
führbaren MINLP-Problems ist aufgrund von Nicht-Konvexität und der hohen Verkop-
pelung der einbezogenen Funktionen mathematisch anspruchsvoll.
xiv
Der erste Schritt um diese Ergebnisse zu verbessern, könnte durch ein einfaches mathe-
matisches Framework erfolgen um ein allgemeines MINLP-Problem zu lösen, welches
mindestens genau so gute Ergebnisse liefert wie die bereits existierenden Methoden.
Dies vereinfacht drastisch die Notwendigkeit zwischen verschieden Methoden zu wäh-
len wenn sich die Problemstellung ändert.
Der nächste Schritt ist eine neue mathematische Methode und Optimierungsprinzip
anzuwenden, um dieses Problem effektiv zu lösen. Dies ist wichtig, um die Qualität der
Lösungsfindung zu verbessern, da es nicht einfach ist mit dem mathematischen Problem
zu handhaben aufgrund verschiedener Gründe wie nicht invertierbarer Funktionen und
fehlender Existenz von analytischen Ausdrücken.
Für die numerischen Ergebnisse wird ein vereinfachtes Verbindungs-Modell benutzt an-
statt dem aktuellen System-Modell. Diese Ergebnisse werden für eine minimale Anzahl
an lauffähigen Beispielen verifiziert, welche für praktisch benutzten Parameter skaliert
werden können. Der Aufbau enthält zwei Basisstationen (BSs) welche jeweils zwei An-
tennen besitzen, die eine homogene Menge von Benutzer-Geräten (UEs) mit jeweils
einer Antenne bedient. Außerdem bedient jede BS eine Zelle und alle Netzwerkeinhei-
ten operieren im gleichen Zeit-Frequenz-Schlitz. Die hauptsächliche Verschlechterung
eines UE Signals kommt von den Intra-Zell- und Inter-Zell-Interfenzen neben dem ther-
mischem Rauschen.
Simulationsergebnisse der Verbindungsschicht vergleicht die Performance der vorge-
schlagenen Techniken gegenüber den meist genutzten Methoden. Die ehmaligen Me-
thoden, übertreffen die letzten in den meisten Fällen und sind mindestens genau so
gut wie die letzten in manchen Fällen. Die Systemperformance wird anhand von Ge-
samtrate und Gesamtleistung ermittelt. In Abhängigkeit des Problems könnte eine
Verbesserung zur Erhöhung der Gesamtrate, eine Erniedrigung der Gesamtleistung,
bessere Rahmenbedingungen oder bessere Handhabung des Problems führen.
xv
xvi
Chapter 1
Introduction
Meeting the ever-increasing demand for new services and providing better Quality of
Service (QoS) requires higher data rates on the link layer, which basically is limited
by the availability of wireless resources such as bandwidth and power. The specifica-
tions of International Telecommunication Union Radio communication sector (ITU-R)
in its IMT-Advanced are being standardized by the working group 3rd Generation
Partnership Project (3GPP) [1] for the Radio Interface Technology as Long Term Evo-
lution Advanced (LTE-A). With 4G technologies being gradually deployed globally, 5G
technologies are already on the horizon. Several key issues were identified in the stan-
dardization, some requirements include enhancing cell-edge user throughput and cell
spectral efficiency, reduction in latency and cost per data bit, improved coverage, pro-
vide a provision for flexible bandwidth, multi-antenna support and allow compatibility
with existing technologies. Peak data rates as high as 1-Gb/s on the Downlink (DL)
is suggested to meet the increasing wireless traffic, mainly arising from video. This re-
quires evolution of both the wireless cellular technologies and the core networks. To ad-
dress the coverage and capacity issues along with the task of Interference Management
(IM) and Resource Allocation (RA) several candidate technologies have been identified.
These include technologies such as Heterogeneous Networks (HetNets), Self Organizing
Networks (SON), Coordinated Multi Point (CoMP) Transmission and Reception and
3D-MIMO. Affordable wireless devices, increasing demand for new wireless services,
increasing User Equipment (UE) density and limited available resources are driving
the wireless network to a small cell architecture. Small cells with cheaper base stations
are being considered to increase the spectral efficiency. This leads to a heterogeneous
architecture where several low power access points serving a small region, called femto
cells, coexist with the macro base station (eNodeB in LTE) [2]. Inter-networking with
WiFi is also under consideration. So the resource reuse pattern must be carefully ad-
dressed. With femto cells, the operating and maintenance costs are transferred to the
end user with this architecture. SON [3] can be used to optimize the network dynami-
cally. Load balancing, interference control, self-configuring and self-healing properties
are defined to optimize the capacity. With frequency reuse factor one, multiple anten-
nas are being deployed to exploit the space dimension. Massive-MIMO [4], employing
1
large-scale antenna systems with tens and hundreds of antenna elements is also under
consideration. This improves the spectral efficiency and energy efficiency in the sys-
tem. The increase in spectral efficiency is due to spatial-multiplexing. Energy efficiency
increases due to the possibility of focusing energy more precisely into small space re-
gions. 3D-MIMO [5] technology in the Active Antenna Systems (AAS) of LTE-A is an
advanced technique to completely utilize the spatial domain. In addition to expanded
coverage and increase in spectral efficiency, it enables 3D-beamforming where space
domain beamforming in both vertical and horizontal directions is achieved. CoMP [6]
in LTE-A is a technique where more than one distributed serving nodes cooperate to
serve a geographical area. On the DL CoMP [7], if the UE data is available only at the
serving cell then it is categorized as Coordinated Scheduling and Beamforming (CSB)
and if the UE data is available at multiple points it is categorized as Joint Processing
(JP). In JP, UE perceives a virtual antenna array, when more than one access points
are simultaneously serving it. Such fully coordinated array is called the network-MIMO
[8]. In CSB, even though the UE is served by a single Base Station (BS), multiple BSs
coordinate by sharing cell specific and UE specific information to adjust the parameters
in their respective cells to manage the interference levels. CoMP improves cell-edge
spectral efficiency and reduces cost per bit. On the Uplink (UL), CoMP facilitates
joint multi-cell scheduling and joint multi-cell signal processing.
The basic aim of the network operator is to design a cost effective network to maximize
the revenue while providing a quality service to the UEs. There are several design and
implementation issues at every point of a wireless link. The maximum possible theoret-
ical transmission rate is the capacity. Identifying the capacity cells and coverage cells in
a geographical area to carry out the link-budget analysis could be the beginning. The
capacity cells focus on providing high data rates and enhanced services, while the cov-
erage cells mainly focus on providing a minimum service. In a dense UE environment,
capacity cells could be created by cell splitting and using small cell architecture. The
region in which the system operates and the allocated bandwidth in this region impacts
the design of equipment and technology. Wireless transmission is impaired by attenua-
tion, distortion, interference and noise hence it cannot be error-free. Due to the shared
nature of the wireless channel and existence of various wireless services, interference is
the primary source for signal degradation. Cross-layer optimization [9] is a proposed
technique to close the gap between the achieved throughput and the theoretical capac-
ity. This cross-layered architecture simplifies functionality and inter-portability, but a
lack of coordination among the layers limits the performance. Working only over the
physical layer may not satisfy the network throughput requirement. A wireless net-
work design based on cross-layer optimization coordinates the resources allocated to
different layers to improve network performance e.g., reduce handoff latency, improve
energy efficiency, exploit network diversity by Network assisted Diversity Multiple Ac-
cess (NDMA) [10]. The functionality of each layer is still hidden from other layers
but coordination, interaction and joint optimization across them is allowed. Cognitive
Radio (CR) [11] is one of the techniques that aims at efficient utilization of the radio
2
spectrum. With CR spectral holes are identified and the spectrum is opportunistically
used to serve secondary UEs without a drop in primary UE’s QoS. CR senses the sur-
roundings, adapts and reconfigures its parameters for dynamic spectrum utilization.
Software Defined Radio is a practical implementation of a CR.
1.1 Basics of Wireless Communication
1.1.1 Fading
Wireless channel is an unguided transmission medium which is time-varying in na-
ture. This variation could be due to the motion of transmitter or receiver, motion of
the obstacles in the medium and atmospheric changes. These variations cause prop-
agation loss in the received signal. To simplify the propagation analysis and model
the effects of these variations, pathloss, shadowing and multipath fading are usually
defined. Pathloss is the distance-dependent parameter that explains the long term or
average decrease in the received power level with increasing distance between transmit-
ter and receiver. A pathloss exponent determines how fast the received power decays
with increasing distance. This depends on the propagation environment terrain e.g.,
buildings and vegetation. Hence the pathloss is different for urban, rural, indoor and
outdoor environments. Free space pathloss is well modelled in the Friis transmission
equation [12]. Due to the dependence of pathloss on various factors, empirical models
[13] based on practical measurements were developed, such as, Hata and COST models
[14]. Shadowing arises due to the obstacles in the medium. The attenuation of the
signal due to shadowing is statistically described by a log-normally distributed random
variable. Pathloss and shadowing are usually considered as long term fading effects.
Transmitted signal arrives at the receiver from multiple reflected paths due to reflec-
tion, diffraction and scattering. This is a multipath phenomena. The multiple reflected
rays usually have different amplitude, phase and angle of arrival. They combine either
constructively or destructively, leading to severe short term fluctuations in the received
signal. The fluctuations due to multipath is called multipath fading [15]. Frequently
used statistical models for short term fading are the Rayleigh, Rician, Nakagami-m,
Hoyt, Weibull fading models [16]. The fading model name suggests the distribution of
the magnitude of the channel coefficient. In a rich scattering environment, there is a
multitude of independent received signal components. In such a case, if there exists no
line-of-sight communication path between the transmitter and the receiver, the fading
can be statistically modelled by Rayleigh distribution. The mean square value of the
received signal will represent the average long term received power. Rician model is
often used when a line-of-sight exists. Nakagami-m is a more general model that can
capture a variety of fading scenarios. Fading parameter m, captures the severity of
fluctuations. To study the non-homogeneity and the non-linearity of the propagation
medium, η-µ [17], κ-µ [18] and α-µ [19] distributions were proposed. They consider
3
a signal composed of clusters of multipath waves propagating in a non-homogeneous
environment. Within any one cluster, the phases of the scattered waves are random
and have similar delay times, with delay-time spreads of different clusters being rela-
tively large. In κ-µ model there exists within each cluster, a dominant component with
arbitrary power. Similar generalized model in homogeneous medium was suggested by
the λ-µ distribution [20]. To further generalize fading distributions, α-η-µ and α-κ-µ
distributions [21] were proposed.
Together the long term and short term fading are represented by a multiplicative
expression, i.e., the short term fading is superimposed onto the large scale fading.
Short term fading manifests into frequency-flat fading, frequency-selective fading, slow
fading and fast fading. To study and model the time-varying nature of the channel
impulse response, observation-time (t-domain), time-delay (τ -domain), frequency (f -
domain), Doppler-shift (λ-domain) are introduced. To categorize various mechanisms
into their respective domains, the parameters coherence bandwidth Bc, symbol rate
Rs, channel fade-rate Rc, coherence time Tc, delay spread Td and symbol time Ts are
defined. The time-spreading mechanism due to multipath studied in the τ -domain
and f -domain results is frequency-flat fading and frequency-selective fading [15]. In
the τ -domain, Td > Ts results in frequency-selective fading, while Td < Ts results in
frequency-flat fading. In the f -domain, frequency-selective fading results when Bc<Rs
and frequency-flat fading when Bc > Rs. The time-variant mechanism due to motion,
studied in the t-domain and λ-domain, results in fast fading and slow fading [15]. In
the t-domain, fast fading occurs when Ts > Tc, while Ts < Tc results in slow fading. In
the λ-domain, Rc > Rs results in fast fading, while slow fading occurs when Rc < Rs.
Frequency-selective fading causes inter-symbol interference which increases the Bit-
Error Rate (BER) if not proper estimation and compensation is not performed. Fast
fading increases the Doppler-spread and also increases the BER.
1.1.2 Frequency Reuse
Radio spectrum is the fundamental requirement for wireless radio networks and reuse
of resource elements such as time, frequency and space is an important concept. This
directly relates to the network planning and link-budget. In a fractional frequency
reuse scheme [22] each cell is allotted a specific frequency band and adjacent cells are
allotted different bands. The same frequency pattern is repeated over the coverage
area. Each cell gets to use only a fraction of the total spectrum. A minimum distance
called reuse distance must be maintained by the BSs, such that sufficient Signal to
Interference plus Noise Ratio (SINR) is achieved at each UE. Radio links suffer from
cochannel interference (CCI) from cells that share the same frequency. Large frequency
reuse factor reduces the CCI but the bandwidth is not fully utilized, resulting in a
reduced system capacity. Hence a trade-off between the CCI and system capacity. In
a soft frequency reuse [23] scheme the cell-center and cell-edge UEs are first identified.
4
This process varies with the actual traffic distribution. It can be expected that the
received signal at the cell-center UE is stronger with a lower inter-cell interference,
so at the cell-center a frequency reuse factor one, that utilizes complete bandwidth is
used and fractional frequency reuse is adopted for cell-edge UEs. Designing a spatial
reuse pattern could be a marginal solution to optimize network performance. Soft
frequency reuse may also refer to the pattern in which all the BSs operate with reuse
factor one [24]. The CCI could be either intra-cell or inter-cell with full frequency
reuse. Nevertheless by adjusting the power levels of each UE, the reuse pattern and
reuse factor can by dynamically decided with cooperation among various BSs. With a
transmit power limitation on the UL and frequency reuse one, the cell size is reduced,
thereby making the system interference-limited. Inter-cell interference cancellation
techniques [25] become essential.
1.1.3 Multiple Access
To allow multiple UEs to access and share the system resources simultaneously, radio
access technologies are characterized by multiple access schemes. Frequency Division
Multiple Access (FDMA), Time Division Multiple Access (TDMA), Code Division Mul-
tiple Access (CDMA) are well known multiple-access techniques. In FDMA, the total
spectrum is divided into a number of channels and the available channels are allotted
for communication. A guard band and avoiding allocation of adjacent bands in the
same location are usually adopted to prevent power leakage to adjacent bands. UEs
communicate at the same time on their assigned frequency bands. By limiting the
transmit power level, it forms the basis for the frequency reuse in the cellular com-
munication. In TDMA, each UE communicates in the allotted time slot at the same
frequency. This gives access to an increased bandwidth to each UE. In addition, a
proper frame structure, time synchronization between transmitter and receiver, pro-
vision for guard interval must be considered. With CDMA, each UE gets access to
the complete spectrum while communicating at the same time. This is possible due to
a unique spreading code assigned to each UE in the system. Each code must satisfy
some correlation properties and must be orthogonal to the other spreading codes in the
system. In Space Division Multiple Access (SDMA), the angular separation or spatial
separation between the UEs is exploited to simultaneously provide the service on the
same time-frequency unit. It relies on the beamforming technique.
Multi-Carrier (MC) modulation [26] is an attractive solution to achieve high data rates.
Orthogonal Frequency Division Multiplexing (OFDM) [27] is one such scheme that con-
verts the incoming serial data stream into several lower data rate parallel sub-streams
and transmits the sub-streams over orthogonal sub-carriers, one per sub-stream. As a
result a frequency-selective channel is converted to several flat-fading channels. It is
possible to have MC multiple-access schemes such as MC-TDMA and MC-FDMA with
OFDM symbols [28]. Orthogonal Frequency Division Multiple Access (OFDMA) is an
5
extension of OFDM, where OFDM is combined with either TDMA or FDMA to assign
a set of sub-carriers to UEs as required. Nevertheless, OFDMA in general refers to
OFDM-FDMA [29]. OFDMA can be treated as MC-FDMA, where a set of OFDM sub-
carriers are exclusively allocated to a UE. In OFDM-TDMA, the complete bandwidth
is provided to a UE for a number of assigned OFDM symbols. An adaptive technique
[30] which dynamically allocates OFDM sub-carriers and power to UEs could improve
the multiple-access technique. High Peak-to-Average-Power Ratio (PAPR) is one of the
drawbacks of OFDM, hence on the LTE-UL, Single-Carrier OFDMA (SC-OFDMA) is
adpoted. This is attractive since there could be some sub-carriers in that are in deep
fade and not suitable to carry data, while some sub-carriers are more suitable to cer-
tain UEs. A new multiplexing scheme Non-Orthogonal Multiple Access (NOMA) [31]
utilizes a new power-domain. Non-orthogonality is intentionally introduced via power-
domain UE multiplexing. A significant throughput gain over OFDMA is observed.
Generalized Frequency Division Multiplexing (GFDM) [32] proposed to deal with the
heavily fragmented spectrum is another alternative. It is based on the multi-branch,
MC transmission system. When compared to the OFDM, the PAPR is reduced and
each sub-carrier can be individually modulated, also the GFDM sub-carriers are not
orthogonal.
1.1.4 Multiple-Input Multiple-Output
Shannon’s information capacity theorem [33] relates the channel bandwidth and trans-
mission power. For a Single-Input Single-Output (SISO), where there is only one
antenna element at either end of the communication link, the capacity scales logarith-
mically with the Signal to Noise Ratio (SNR) and scales linearly with the bandwidth.
However there is an upper limit to this capacity per unit bandwidth as the SNR→∞.
To meet the ever increasing data rate requirements, another Degree of Freedom (DoF)
must be introduced into the system. Multiple antenna elements at both the transmitter
and the receiver, called Multiple-Input Multiple-Output (MIMO) system, introduces a
new space dimension into the system. The capacity for a MIMO system increases lin-
early with the number of antenna elements [34]. This increase in the capacity, termed
as multiplexing-gain is at the expense of increase in hardware and signal processing
complexity. The effect is similar to an increase in the bandwidth. If the channel is
diagonalized, a number of spatial streams or eigenmodes between the transmitter and
the receiver are created by this new spatial DoF. Data can be parallelly sent over them
and is differentiated at the receiver based on the spatial-signature. The capacity of a
MIMO system can be further increased by spatial waterfilling [35] when the transmitter
has full Channel State Information (CSI), i.e., a closed loop setup where eigenmodes are
accessible. Antenna elements spaced sufficiently wide in a rich-scattering environment
at the receiver may result in each element having independent copy of the transmitted
signal. This is termed as spatial diversity on receive and makes the communication
link more reliable. Assuming no correlation among the antenna elements, it could be
6
expected that at least one antenna element will receive a strong signal component.
This reduces the probability of the received signal being in deep fade when compared
to a SISO link. Receive diversity techniques, such as maximal ratio combining can be
employed to increase the received SNR [36].
Single-UE MIMO is a point-to-point communication where the full spatial dimension is
dedicated to a UE. The capacity scales linearly with min(R,T ), where T is the number
of antenna elements at the BS, R is the number of antenna elements at the UE and
min(·, ·) is the minimum operator. For a MIMO fading channel, the ergodic capacity
is the Shannon capacity averaged over the fading process. Other capacity metrics such
as delay limited capacity, which is the transmission rate possible for any given fading
state of the channel, and outage capacity, which is the rate beyond which the channel is
in outage, are also defined to study the performance. Diversity on transmit is achieved
by spreading the symbols over space and time. Under the open-loop configuration
where there is no feedback from the receiver to the transmitter, space time codes are
employed [37]. They are categorized as Space Time Block Codes (STBC) and Space
Time Trellis Codes (STTC). STTC operate on serial data for encoding and require
memory elements. A multi-dimensional Viterbi decoder is required at the receiver.
The exponential increase in decoding complexity is a drawback of STTC. STBC works
on the orthogonality in the space and time domain for the codewords. This simplifies
the receiver structure by facilitating linear post-processing. Linearity, full-diversity or
orthogonality may have to be compromised for achieving higher data rates. Transmit
Selection Diversity (TSD) is the form of transmit diversity under closed loop setup.
TSD basically selects only a subset of transmit antennas thereby reducing the number
of spatial chains on the link. Reduction in complexity and spatial interference are the
advantages here. Even with reduced transmit antennas, the diversity order remains
the same. A Layered Space Time (BLAST) architecture [38] was designed to provide
spatial multiplexing over MIMO. Data is spatially demultiplexed into sub-streams and
then mapped onto the transmit antennas. Each sub-stream is independently temporal
encoded, interleaved and mapped. The receiver can independently decode the sub-
streams thus avoiding joint decoding.
1.2 Multi-User MIMO
In a single cell, multiple MIMO links can exist, that share the space dimension. The
shared channel is the MU-MIMO channel. Antennas across all the UEs must be consid-
ered for simultaneous scheduling. For understanding, Figure 1.1 shows the system-level
diagram of a multi-cell MU-MIMO setup on the DL with 2 cells. Each cell is served
by a BS, and the UEs are already associated to their respective BSs. All the network
entities operate at a frequency reuse factor one. Each UE has a single antenna, the DL
MIMO channel vector is hH . UE3 and UE4 are almost equidistant from either BSs, i.e.,
they are located at the cell-edge, hence they experience severe Multi-User Interference
7
hH11
UE1 UE2UE3 UE4
BS1 BS2
hH21
desired signal, intra-cell interference, inter-cell interference
Fig. 1.1: Multi-Cell Multi-User MIMO Downlink.
(MUI). As seen, UE1 and UE3 are served by BS1. For UE1, both the desired signal
and the intra-cell interference signal arrive over the channel hH11. The intra-cell inter-
ference signal is the spatially multiplexed data of UE3 with UE1 at BS1. The spatially
multiplexed data of UE2 and UE4 at BS2, arrives over hH21 as inter-cell interference
signal. The contribution to the interference term of UE1 includes the signals of all the
UEs arriving at UE1 over the DL channels between all the BSs and UE1. Similarly,
other UEs receive their signal in the presence of MUI. Thermal noise is added at each
UE receiver.
√
Q1s1UE1 U1
...
...
...
... + H
H
j +
zj
VHj sˆj
√
QNsNUEN UN
Fig. 1.2: Single-Cell, single BS, MU-MIMO Downlink (BC) transmission for UEj .
A Broadcast Channel (BC) in Figure 1.2 is an abstraction of the DL transmission
for N UEs, i.e., a single-point to multi-point transmission [39]. However, it considers
only a single-cell case of Figure 1.1. With each UE equipped with R antennas, it is
possible to schedule L≤R simultaneous data streams to each UE on the DL. Per-layer
processing is considered and in addition to the inter-user interference, the CCI will
have an intra-layer term. The vector sj = [sj1, · · · ,sjL]T is the (L×1) symbol vector
of UEj , {j = 1, ...,N}. The transmitted symbol for UEj on layer l, {l = 1, ...,L} is sjl.
It is normalized such that as E[sjls
∗
jl]= 1, ∀j,∀l. E[·] is the expectation operator, (·)T
is the transpose operator and (·)∗ is the complex conjugate operator. The transmitted
8
symbols are independent, i.e., E[sjls∗jm]= 0, ∀j,∀m 6= j and E[sjls∗kn]= 0, ∀j,∀k. The
matrix Qj = diag(qj) is the (L×L) diagonal power loading matrix for UEj at the
BS. The vector qj = [qj1, · · · , qjL]T is the (L× 1) power vector of UEj . The power
allocated to layer l of UEj is qjl. For a vector input, the diag(·) operator stacks
the vector as the principal diagonal elements of a diagonal matrix and for a matrix
input diag(·) operator extracts the diagonal elements of a square matrix. To effectively
manipulate interference among various entities when transmission occurs on the same
time-frequency unit, a spatial pre-processing called precoding at the transmitter and
a spatial post-processing called decoding at the receiver are employed. This precoding
is not to be confused with the source coding and channel coding, it can be viewed as
changing the effective channel. The suboptimal strategy when the transmitter employs
linear precoding is called Beamforming (BF). It is possible by taking the advantage
of SDMA to schedule more UEs and control the MUI, if the BF vectors are carefully
chosen. The design of precoders and decoders depend on the system objective and
link direction. Uj = [uj1, · · · ,ujL] is the (T ×L) transmit BF matrix of UEj , ujl is
the (T × 1) transmit vector of layer l for UEj . It is required that ‖ujl‖2 = 1, ∀j,∀l,
to satisfy the power constraint at the BS, where ‖·‖2 is the ℓ2−norm. Beamformer
Uj , ∀j, spatially multiplexes the data of UEj . HHj is the (R×T ) DL channel matrix
from the BS to UEj . (·)H is the Hermitian operator. The input to the channel is
the composite signal containing spatially multiplexed data of all the UEs. At the
receiver, Additive White Gaussian Noise (AWGN) gets added to the composite signal,
which is further subjected to a decoding process. zj is the (R× 1) AWGN vector
at UEj . The noise variance per receive antenna element at UEj is σ2, such that,
E[zjz
H
j ]= σ
2IR. IR is the (R×R) Identity matrix. VHj = [vj1, · · · ,vjL]H is the (L×R)
receive BF matrix at UEj . vjl is the (R×1) receive BF vector of layer l for UEj and
‖vjl‖2 = 1, ∀j,∀l. sˆj = [sˆj1, · · · , sˆjL]T is the (L× 1) post-processed received symbol
vector at UEj providing an estimate for sj .
h11
h13
UE1 UE2UE3 UE4
BS1 BS2
h14
h12
desired signal, intra-cell interference for cochannel UEs, inter-cell interference
Fig. 1.3: Multi-Cell Multi-User MIMO Uplink.
Similar to Figure 1.1, Figure 1.3 shows the system-level diagram of a multi-cell MU-
9
MIMO setup on the UL. For UE1, the desired signal is communicated over h11. The
desired signal of UE3 will be the intra-cell interference component of UE1 via channel
h13. Due to reuse factor one, the signals from UE2 and UE4 also contribute to the
inter-cell interference via channels h12 and h14 respectively. The contribution to the
interference term of UE1 includes the signals from all the UEs arriving at BS1 over
different UL channels. A Multiple Access Channel (MAC) in Figure 1.4 models a multi-
point to single-point UL transmission, i.e., a single-cell link-level case of Figure 1.3.
Pj =diag(pj) is the (L×L) diagonal power allocation matrix at UEj . pj = [pj1, · · · ,pjL]
is the (L×1) power vector of UEj . The power allocated to layer l of UEj is pjl. zj is
the AWGN vector for UEj at the BS. The noise variance per receive antenna element
at the BS for UEj is σ2.
√
P1s1UE1 V1 H1 zj
...
...
...
...
... + U
H
j sˆj
√
PNsNUEN VN HN
Fig. 1.4: Single-Cell MU-MIMO Uplink (MAC) transmission for UEj .
In general, the channel matrices, transmit and receive beamformers are different on
both the DL and UL. From mathematical point of view and structure of various ma-
trices, for each UE, the UL channel is the transposed version of the DL channel, the
transposed DL transmit matrix is viewed as the UL receive matrix and the transposed
DL receive matrix is the UL transmit matrix. A duality between the BC and MAC
has been explored and extensively applied for various transceiver optimization prob-
lems. This duality is different from the mathematical duality in optimization theory.
Capacity of a MU-MIMO channel is given by an N -dimensional rate vector where N
is the number of active UEs in the system. This vector is the set of simultaneously
achieved rates of all active UEs. The set of all achievable rate vectors is the capacity
region. Similar SINR and MSE region can also be defined. Performance evaluation by
N−dimensional performance metric vector comparison may not be a suitable criteria.
So a scalar quantity for the system-wide objective such as sum of all UE rates or UE
powers or UE MSEs is usually considered. The objective is optimized w.r.t. various
system and design constraints. By conventional BC-MAC duality [40] under the sin-
gle total transmit sum-power constraint, the SINR region obtained for the BC and
the MAC is the same with conjugate transposed channel and BF matrices. Also, the
noise covariance matrices are identity matrices. Duality based on Mean Squared Error
(MSE) metric [41] is also defined similar to the SINR duality. Under a sum-power con-
10
straint by minimax duality [40], every point on the BC capacity region can be obtained
by solving a MAC minimax optimization problem with the channel transposed but the
MAC noise covariance matrix is an optimization variable.
It is common that multiple BSs serve a geographical area, i.e., a multi-point to multi-
point transmission. With increasing UE density and wireless traffic the need for co-
ordination among the BSs arises. The level of coordination, i.e., the amount of CSI
and UE-data at each BS, determines the scheduling and processing techniques. The
processing could either be centralized or distributed based on how the BSs are con-
nected and the level of side information, i.e., global CSI or local CSI availability. With
multiple BSs, the problem of Base Station Association (BSA) arises. This is crucial,
since the processing node and coordinating nodes are defined at this stage before the
service is provided to a UE. BSA is another DoF introduced into the system that
needs to be optimized. Each UE has a performance metric such as SINR, MSE or rate
that needs to be optimized. Optimizing individual UE performance metric may not
result in optimizing the system-wide metric. So a utility function that is a measure of
satisfaction is assigned to each UE by the operator for RA of certain type.
1.2.1 Fairness
Fading statistics of the UEs are not always similar, so the fairness among the UEs must
also be considered for design. Fairness in RA is important since the UEs are competing
for the finite network resources. In this case, a Round Robin (RR) scheduler is the
simplest way to serve each UE at a time in an ordered manner. It provides equal
opportunity to all the UEs but does not provide any multiplexing gain. A weighted
scheduling can capture certain degree of fairness among UEs e.g., weighted sum-rate
where each UE rate in the sum is multiplied by a weight that reflects certain fairness
criteria. When all the weights are equal to unity, then the obtained scalar objective
is a simple sum-rate. For a simple sum-rate, the BS, in order to improve the system
throughput can select a subset of UEs with favourable channel state. This form of
selection diversity among the UEs is called Multi-User Diversity (MUD) and it may
deprive service to some UEs. The improvement in the performance is the MUD gain.
The sum-capacity of the fading channel is greater than that of a non-fading channel due
to this MUD [42]. The Proportional Fairness (PF) [43] is a weighted scheduling scheme
based on average UE throughput. If the channel statistics of the UEs are similar then
the PF-rates will converge to the same value [42]. While maintaining fairness among
the UEs, it also maintains MUD. A max-min fairness criteria [44] that maximizes the
minimum allocation among the active UEs in the system can be considered in the
interest of the most weak UE in the system. Such a criteria is fair from homogenous
UEs point of view where all the UEs are treated with equal priority. α−fairness [45]
captures a variety of fairness in the system. It uses a generalized utility function for
each UE. α→ 1 is the PF while α→∞ is the max-min fairness. α = 2 corresponds to
11
minimum potential delay fairness which seeks to minimize the total transfer time at a
given rate. Another criteria that focuses on the extent of improvement in the efficiency
and the extent of compromise on fairness is the (α,β)-fairness [46].
1.2.2 Transceiver design
The task of RA is to obtain the desired performance goals with limited resources by
manipulating various DoF in the system. It is challenging since there are many DoF
that need to be simultaneously optimized in a situation where each UE’s performance
is coupled to every other UE’s performance. Also the order in which the updates
must be carried out is also not certain. RA is solved by an equivalent constrained
mathematical optimization problem with various system and design constraints. An
optimal point in certain sense is obtained from the problem. Optimal may refer to the
quality of solution, i.e., local or global optimum or one of the many solutions satisfying
all the constraints in a resonable way. Joint transmit and receive filter design finds a
pair of optimal precoder and decoder. The signal processing and design techniques for
the detection of UE signals in the presence of Multiple Access Interference (MAI) is
called multi-user detection [47]. Joint optimum multi-user detection reduces the overall
BER and has no near-far problem. One advantage of joint optimization is that a series
of relations can be established under some assumptions among various performance
metrics, such as under Linear Minimum Mean Squared Error (LMMSE) transceiver,
maximizing SINR is equivalent to the minimization of normalized MSE [48], minimizing
the product of MSE is equivalent to maximizing the sum-rate [49]. RA in a system is
called Pareto efficient, if any UE’s allocation cannot be improved without reducing any
other UE’s allocation. From Figures 1.2 and 1.4 various optimization problems such as
Power Allocation (PA), that finds power loading matrices Pj and Qj , BF that finds Uj
and Vj , UE selection that finds the required active set of UEs, BSA that finds UE-BS
association and so on arise. For each UE on the DL,
√
Qj ·Uj can be treated as a
single trasmit BF matrix such that ℓ2−norm for each column in the resultant matrix
is equal to qjl. Similarly on the UL,
√
Pj ·Vj can be combined. Depending on the link
direction, parameters and constraints change.
The capacity region of a MIMO BC is achieved by a non-linear precoding scheme
called Dirty Paper Coding (DPC) [50] at the transmitter and Successive Interference
Cancellation (SIC) at the receivers. DPC is an encoding strategy that involves an
ordering of UEs and pre-subtraction of interference. As N > T , a linear increase in
the capacity in terms of T is achieved, irrespective of R. DPC may be impractical
since non-causal knowledge of interference is required and the involved expressions are
non-convex. The capacity region of a MIMO MAC is achieved by performing SIC at
the receiver. The basic aim of precoding at the BS is to simplify the receiver structure
and also achieve the necessary diversity and multiplexing gain. Under the zero-MUI
criteria and N ·R≤ T as a primary requirement, Block Diagonalization (BD) [51] and
12
Zero-forcing (ZF) pre-processing are the simplest form of precoding on the DL. More
often the objective in this case is the weighted sum-rate. The basic idea is that the
precoder of a UE lies in the null space of all other UE channels. BD has capacity loss
due to the nulling of overlapping subspaces of the different UEs [52]. ZF is a special
case of BD when R = 1, ∀j, i.e., each UE has only one spatial layer. With BD, the
MU-MIMO channel is decomposed into several independent parallel channels. To the
interference-free streams, waterfilling can be used to optimally allocate system-wide
power. For N ·L> T , it is not feasible to find null vectors for all the UEs, so a subset
of UEs must be selected for zero-MUI criteria. In general, for N ·L>T , certain amount
of interference is allowed for each UE while providing service as long as the objective
is achieved and the constraints are satisfied. It is possible to have a ZF precoding in
combination with a RR or PF criteria [53]. The scheduler must first select a UE subset
for this. When only one UE with the largest channel gain is selected for transmission,
the sum-rate achieved is called the TDMA rate. The precoders are obtained by channel
inversion and waterfilling is used for optimal power allocation. This cannot achieve
linear increase in the sum-capacity. This TDMA is not to be confused with the multiple
access technique TDMA in section 1.1.3. If an ordering of the UEs is considered and
partial MUI is allowed for the UEs then Successive Optimization (SO) [51] precoding
arises. Instead of having null space requirement of all the UEs, only the null space
requirement of the previous UEs in the order is considered. Random BF [54] is a BF
technique where orthogonal vectors are evaluated and sent to each UE. Each UE then
evaluates the SNR for all the vectors and reports it to the BS scheduler. A UE that
has reported the maximum SNR is provided with a BF vector.
Opportunistic BF [42] is a BF technique that intentionally introduces fluctuations in
the system to exploit the MUD while transmitting to a single UE. It is based on the
fact that the MUD gain is higher with increase in channel randomization. The intro-
duced random coefficients are closely matched to the channel coefficients to achieve
higher SNR. The LMMSE [55] decoding is the Wiener filter that minimizes the overall
MSE. At the optimal solution with the LMMSE beamformers, the minimum individual
normalized MSEs and the maximum SINRs are simultaneously achieved [56], hence the
LMMSE maximizes the SINR. LMMSE transceiver is optimal for total MSE minimiza-
tion [57] under individual power constraint on the UL and maximizing individual SINR
on the DL [58]. Other linear BF designs based on Signal to Generated Interference
Noise Ratio (SGINR) [59] and Minimum Variance Distortionless Response (MVDR)
[60] are also considered is certain cases. SGINR considers the locally available CSI and
self generated CCI for distributed processing while MVDR considers and minimizes
only the CCI without the desired signal power for BF. For the MMSE beamformers, it
is required to know the signal power which may be precisely unavailable. Under such
cases, a Least Squared Error (LSE) based Minimum Least Squared Error (MLSE)
beamformer [61] can be a good alternative. It minimizes the LSE of the observation
and no knowledge of signal power is needed. Thomlinson Harashima Precoding (THP)
[62], a non-linear precoding technique can be used to avoid error propagation. THP
13
can be combined with MMSE and ZF precoders to increase diversity and reduce MUI
[63]. To reduce latency and feedback overhead, a predefined codebook based precoding
can be performed [64].
In a multi-cell setup, the cell-edge UEs perceive severe CCI since the signal power from
more than one BS has almost the same strength. To further increase the system capac-
ity the BSA must also be optimized. In a system withM BSs and a small set of UEs an
exhaustive search with MN combinations could be possible but when N becomes large
this is prohibitive. So finding the optimal BS-UE combination becomes challenging.
There could be per-BS constraints such as available power and load balancing that need
to be considered. BSA can be viewed as a handoff mechanism where UE power levels at
various BSs are compared and the BS resulting in the least power is chosen. Selection
and dropping of UEs, called admission control [65] is also a design criteria since not
every BS-UE combination is feasible or even if feasible, may have inferior performance.
UE selection is often jointly optimized with other DoF, such as BF while optimizing
an objective, such as sum-power or sum-rate. Selecting a UE in a cell that maximizes
the capacity of that cell may not be an optimal choice on the system scale. Dominant
eigenvalue based selection or norm based selection is the simplest form of UE selection.
For ZF-BF, the selected UEs must have a high gain channel and must be nearly or-
thogonal. So a measure of orthogonality must be introduced for the UEs. As N →∞,
ZF-BF with a semi-orthogonal UE selection (SUS) strategy can asymptotically achieve
the capacity as DPC due to the MUD [53]. Construction of the semi-orthogonal group
for SUS is based on the Gram-Schmidt orthogonalization. In Cognitive Radio, a pro-
tection margin that increases the minimum QoS to the operating links can be provided
while admitting secondary UEs. Receive antenna selection [66] could be another DoF
to be optimized. A subset of receive antennas are adaptively activated while retaining
the diversity benefits. Power Control (PC) [67] has been studied to control the CCI
and minimize the total transmit power while the required QoS for each UE is achieved.
If the minimum required QoS, called minimum protection ratio [68], is the same for all
UEs, then the PC technique achieves a balanced QoS across all the active UEs, i.e.,
at the optimal all the active UEs will have the same QoS value. Such a RA is fair
w.r.t. the homogeneous UEs. Also this power minimization problem is coupled to the
max-min problem where the minimum achievable QoS among the UEs is maximized.
The optimal solution for both these problems is the same.
This thesis investigates conventional cochannel multi-cell MU-MIMO communication
systems and optimizes various network DoF. In particular, the problems of Base Station
Association and Power Allocation are solved w.r.t. the objective of sum-rate maximiza-
tion and sum-power minimization. Frequency reuse factor one is applied, such that the
main source of interference to a UE arises from the cochannel UEs. System-level setup
of Figures 1.1 and 1.3 are analysed via the link-level abstractions of Figures 1.2 and
1.4 respectively. Link-level simulations are carried out for the CSB-CoMP based setup
for Rayleigh flat fading channel. For numerical results in each chapter, a minimum
working example is considered. It includes some assumptions, such as T = 2, M = 2,
14
R= 1, unit noise variance at BSs and UEs. Also certain variable values such as, trans-
mit power and number of active UEs, are chosen to just demonstrate the effectiveness
of the applied mathematical methods. These illustrations can be scaled to include the
practical parameter values and analyse the network performance.
Chapter 2 provides the overview of various constraints, mathematical formulations and
mathematical tools considered in various considered optimization problems. Basics
of optimization, problem formulation and problem solving techniques are outlined.
Additional literature results are also mentioned.
Chapter 3 outlines an Interior Point Method (IPM) framework based on numerical
methods for Non-Linear Programming (NLP) techniques. It is applied to solve the
NLP optimization problems in this thesis. A variety of problems can be solved with
the same framework with a little or no modification, thus removing the need to apply
problem specific mathematical tools. The IPM is applied to solve the UL sum-rate
maximization and sum-power minimization problems under perfect-CSI. Though the
results in this chapter appear trivial, they primarily serve as a benchmark for compari-
son with existing methods. All the considered Mixed Integer Non-Linear Programming
(MINLP) problems under various assumptions in further chapters are reformulated as
NLP problems to obtain an optimum. In addition, the feasibility issue is also addressed
in the problem formulation.
Chapter 4 considers the optimization problems on the UL considered in chapter 3,
but under different assumptions such as probabilistic constraints, statistical-CSI and
imperfect-CSI. Under such assumptions, it is difficult to obtain good bounds on the
objective. Concepts from Extreme Value Theory (EVT) and its application to Chance
Constrained optimization are discussed in this chapter. New optimization metrics such
as Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR) from the field of finance
are considered to obtain tighter bounds.With these concepts, the optimization problem
is basically reformulated into an NLP problem with no approximations. The generated
bounds by this approach are tighter when compared to the existing approximation
techniques. Also, these results match with the available analytic expression, which is
not the case with the existing methods. For this, two cases for the UL sum-power
minimization are examined.
Chapter 5 considers the Downlink (DL) setup under perfect-CSI. The expressions on
the DL are highly coupled. This makes the problem difficult to handle, thus the UL-DL
duality is exploited to find the optimum. An UL called virtual-UL (VUL) is formulated,
which decouples the DL expressions and solves for the solution by alternately switching
between the DL and VUL. Under a multi-cell setup, another concept of virtual-noise is
also included. This deviates from the conventional single cell VUL-DL duality. Numer-
ical results based on the implicit Lagrange-duality of the considered IPM show that, if
the VUL-DL single cell duality is included into the constraint set of the optimization
problem then, the IPM is capable of solving the problem without this deviation and
also without the alternate switching.
15
16
Chapter 2
Mathematical Techniques for
Optimization
There is no single method to solve the variety of optimization problems, each with
its own set of objectives, requirements and constraints. Resource Allocation (RA) is
usually a constrained Mixed Integer Non-linear Programming (MINLP) optimization
problem. Non-convexity, highly non-linear functions or a combination of higher order
functions make the problem intractable and closed form solutions may not exist. A
possible local optimum based on numerical methods is obtained and further refined.
Iterative greedy mechanism where an initial random point evolves in multiple stages
hoping to converge to an optimum is usually adopted. Several sub-problems are succes-
sively solved where each sub-problem solves for a single variable while other variables
are fixed. A rearrangement and manipulation of the involved functions and equations
is required to simplify the sub-problems. This helps to identify the problem’s structure
and an appropriate mathematical tool can be applied to obtain the solution. Depend-
ing on the structure of the objective and the constraints, the involved sub-problems
can be identified as Non-Linear Programming (NLP) [69], Integer Programming (IP),
Geometric Programming (GP) [70], Semi-Definite Programming (SDP) [71], Second-
Order Cone Programming (SOCP), Linear Programming (LP) and so on. To solve
the optimization problem, Matlab based modelling language YALMIP [72] is used to
present the problem to a solver. Free solvers based on Interior Point Methods (IPM)
such as SDPT3 [73] an SDP solver, GPPOSY [74] a GP solver, SEDUMI [75] an SOCP
solver, CPLEX [76, academic version] an IP solver can be used in combination with
YALMIP.
2.1 Constraints
For a multi-cell setup, M BSs {i= 1, · · · ,M}, each serving a cell are considered. With-
out loss of generality, R = 1 and L = 1 is assumed throughout. For a UE association
17
with a BS, 0−1 binary integer variable αij is used (different from α in section 1.2.1).
If the UEj is associated with BSi, then αij = 1, else αij = 0, i.e.,
αij = {0,1}, ∀i,∀j. (2.1)
The (M × 1) BSA vector of UEj is αj = [α1j , ...,αMj ]T . The (M ×N) overall BSA
matrix is α = [α1, ...,αN ]. On the UL, every BS receives the signal from all UEs.
Throughout this work, coordinated processing is assumed where the signal of a UE is
decoded only at one assigned BS. Hence there is no macro-diversity and the assignment
constraint is mathematically expressed as
1TMαj = 1, ∀j, (2.2)
where 1M is an (M ×1) vector with all elements equal to unity. For independent and
uncorrelated symbols and noise, equation (2.2) allows the received signal of UEj to be
mathematically expressed as
aˆULj =
M∑
i=1
αiju
H
ijhij
√
pjaj+
M∑
i=1
N∑
k 6=j,k=1
αiju
H
ijhik
√
pkak+
M∑
i=1
αiju
H
ij zij , ∀j. (2.3)
On the right hand side of the equality, the first term in the desired signal of UEj , the
second term in the MUI, the third term in the thermal noise component. In the MUI
term, the intra-cell and the inter-cell interference terms can be identified after a given
BSA. Each UE is assumed to have a maximum available power Pmax. The UL transmit
power is bounded as
pj ∈ [0,Pmax], ∀j. (2.4)
Equation (2.4) is a short term power constraint. For a long term constraint, the
transmit power may exceed the total power for some channel states as long as it is
compensated for other channel states [48]. The UL-SINR γj is expressed as
γj =
pj
M∑
i=1
αij‖uHijhij‖22
M∑
i=1
N∑
k 6=j,k=1
pkαij‖uHijhik‖22+σ2j
, ∀j. (2.5)
The UL-rate for UEj is rj = log2(1+γj), where log2(·) is the logarithm to base-2. This
expression of UE rate is derived by treating the interfering signals as additive Gaussian
noise [77]. For a given α, γj and rj are written as γij and rij respectively. Similarly
the DL received signal at UEj is
aˆDLj =
M∑
i=1
αijh
H
ijuij
√
qijaij+
M∑
m=1
N∑
k 6=j,k=1
αmkh
H
mjumk
√
qmkamk+ zj , ∀j. (2.6)
With maximum available transmit power P¯i at BSi, the DL transmit power qij of UEj
at BSi is bounded as
qij ∈ [0, P¯i], ∀i,∀j. (2.7)
18
The (M × 1) DL power vector for UEj across all the BSs is qj = [q1j , ..., qMj ]T . For
a given α, not every element in qj of UEj will be active, i.e., only one element in qj
corresponding to the associated BS will be equal to 1 due to equation (2.2). The active
DL power vector of all the UEs is q = [1TMq1, ...,1
T
MqN ]. The (M ×N) overall DL
power matrix is Q = [q1, ...,qN ]. The sum of transmit power on the DL allocated to
all the UEs assigned to BSi is constrained by P¯i, i.e.,
α
T
i qi ≤ P¯i, ∀i, (2.8)
where (N×1) vectors αi and qi are the ith rows of α andQ respectively. The DL-SINR
γ
j
of UEj to be expressed as
γ
j
=
M∑
i=1
qijαij‖hHijuij‖22
M∑
m=1
N∑
k=1,k 6=j
qmkαmk‖hHmjumk‖22+σ2j
, ∀j. (2.9)
The DL-rate for UEj from BSi is rj = log2(1+γj). For a given α, γj and rj are written
as γ
ij
and rij respectively. The UL QoS requirement for each UE is the rate metric
given as
rj ≥ rth, ∀j, (2.10)
where rth is the minimum threshold rate. Similarly, the DL QoS requirement is
rj ≥ rth, ∀j. (2.11)
Due to the per unit bandwidth analysis, the rate metric is also the spectral efficiency
which is measured in [bits/sec/Hz]. In equations (2.3), (2.5), (2.6) and (2.9) the BSA
in variable α is still not resolved. Also, the BF matrices Vj , ∀j, shown in Figures
(1.2) and (1.4) do not appear since R= 1. The MUI term in the denominator of SINR
expressions (2.5) and (2.9) includes both the intra-cell and inter-cell interference. Also
an implicit ordering of the UEs is present.
2.2 Problem Formulation
Branch and Bound (BB) methods [78] are frequently used to solve MINLP prob-
lems. BB involves the process of relaxation which makes discrete variables continuous,
branching which solves two sub-problems, one each for a lower bound and an upper
bound on the objective, fathoming that rejects or accepts a solution. In the process
a binary tree evolves and pivoting around various solutions occurs. In the worst case
this enumeration could lead to an exponential computation complexity. So to reduce
the complexity or size of the problem, the integer and the non-linear parts are eval-
uated separately. With this separable approach, applying well known IP and NLP
19
techniques becomes easier. The involved problems and sub-problems solved in this
thesis are outlined next.
The mathematical formulation of a multi-cell multi-user sum-rate maximization with
required QoS under perfect-CSI (CSIp) on the UL is given as
P1: maximize
α,p,U
N∑
j=1
rj (2.12)
subject to: (2.1),(2.2),(2.4),(2.10). (2.13)
P1 for a given (p,U) results in a linear IP sub-problem P1b in α and for a given α
results in an NLP sub-problem P1p in (p,U).
P1b: max
α
N∑
j=1
rj (2.14)
s.t:
(2.1),(2.2),
(2.10).
(2.15)
P1p: max
p,U
N∑
j=1
rj (2.16)
s.t: (2.4),(2.10). (2.17)
The same rate maximization problem on the DL is given as
P2: maximize
α,Q,U
N∑
j=1
rj (2.18)
subject to: (2.1),(2.2),(2.7),(2.8),(2.11). (2.19)
Similarly P2 has two sub-problems, a linear IP sub-problem P2b for a given (Q,U) and
an NLP sub-problem P2p for a given α.
P2b: max
α
N∑
j=1
rj (2.20)
s.t:
(2.1),(2.2),
(2.8),(2.11).
(2.21)
P2p: max
Q,U
N∑
j=1
rj (2.22)
s.t:
(2.7),(2.8),
(2.11).
(2.23)
The problem of simple sum-rate is solved in P1 and P2 for rth =0, where in the interest
of the system, some UEs may be dropped or may have a very low assigned rate. The
sum-power minimization problem on the UL with required QoS under CSIp is given as
P3: minimize
α,p,U
N∑
j=1
pj (2.24)
subject to: (2.1),(2.2),(2.4),(2.10). (2.25)
Similar sub-problems P3b and P3p are formulated for P3.
P3b: min
α
N∑
j=1
pj (constant) (2.26)
s.t:
(2.1),(2.2),
(2.10).
(2.27)
P3p: min
p,U
N∑
j=1
pj (2.28)
s.t: (2.4),(2.10). (2.29)
20
Objective for P3b is a constant where a solution satisfying constraints (2.27) is the only
requirement. The sum-power minimization problem when applied on the DL is given
as
P4: minimize
α,Q,U
M∑
i=1
N∑
j=1
αijqij (2.30)
subject to: (2.1),(2.2),(2.7),(2.8),(2.11). (2.31)
It can be observed that both P1 and P3 have the same constraint set, i.e., equations
(2.13) and (2.25) are equal, only their objective changes. Same is the case for P2 and
P4. Another problem of interest is the UL max-min problem P5 that is related to P3.
P5: maximize
α,p,U
minimize
j
rj (2.32)
subject to: (2.1),(2.2),(2.4). (2.33)
In P3, for homogeneous UEs each with the same QoS requirement rth, the constraint
(2.10) will be active at the optimum, i.e., each UE will achieve the same QoS. At the
optimum, P5 also produces a balanced QoS across all the homogeneous UEs, i.e., the
power vector p is the same for P5 and P3.
2.3 Problem Solving
2.3.1 Base Station Association
BSA sub-problems P1b, P2b and P3b are linear IP problems which are Non-deterministic
Polynomial-time (NP) hard. To improve the bounds on a linear IP, Lagrangian Re-
laxation (LR) [79] and decomposition methods [80] are usually preferred. Primal de-
composition based on Bender’s Decomposition (BD), dual decomposition based on
Dantzig-Wolfe Decomposition (DWD) and cross decomposition [81] are the frequently
used decomposition methods. Decomposition works on the separability of variables
where a master-problem and a slave-problem are successively solved. P2b has a struc-
ture similar to a Generalized Assignment Problem (GAP) [82] which in itself is a special
case of Multi-dimensional Multiple-choice Knapsack Problem (MMKP) [83]. MMKP
and variants are used in the RA problems where several heuristic algorithms exist to
find the solution. Upper bounds are obtained based on the dual LR problems where ei-
ther constraints (2.2) or (2.11) are added to the objective in a weighted manner. These
weights are the Lagrange multipliers. Relaxing assignment constraints (2.2) decom-
poses P2b info M simple 0− 1 KP problems [84]. Non-differential subgradient based
methods [85] or BB methods are employed to further solve this relaxed problem. If the
capacity constraints (2.11) are relaxed, then P2b will possess an integrality property
where a simple linear relaxation as in equation (2.34) for equation (2.1) is sufficient
21
to produce a 0− 1 binary solution. This relaxation produces weaker bounds when
compared to the former relaxation.
To solve P1b and P3b as an NLP problem, integer constraints (2.1) are relaxed as
αij ∈ [0,1], ∀i,∀j, (2.34)
such that, αij is a continuous variable. Other IP to NLP conversion techniques also
exist [86]. In addition to the assignment constraints (2.2), further restrictions on αij
such as
αij(αij−1) = 0, ∀i,∀j, (2.35)
αmj ·αnj = 0, ∀m 6= n,∀j, (2.36)
must be applied to ensure it converges to either a 0 or a 1. The NLP problem is given
as P1bn. Equation (2.35) is an SOCP constraint, hence P1bn is can be solved as an
SOCP problem. For the sake of mathematical formulation, an SDP formulation P1bs
for P1bn is also possible. In the interval (2.34), constraints (2.35) always satisfy
αij(αij−1)≤ 0, ∀i,∀j. (2.37)
By Schur complement [87], a positive semi-definite criteria is obtained as

αij αij
αij 1

≥ 0, ∀i,∀j. (2.38)
P1bn: max
α
N∑
j=1
rj (2.39)
s.t:
(2.2),(2.10),
(2.35) or (2.36).
(2.40)
P1bs: max
α
N∑
j=1
rj (2.41)
s.t:
(2.2),(2.10),
(2.38).
(2.42)
Relaxed constraints (2.34) are not required in P1bn and P1bs. Either P1bn or P1bs
can be used to solve P1b. With only a change in the objective, the same constraint
formulation also applies for P3b.
Neighbourhood search and memory based Tabu Search (TS) algorithm [88] can also be
used to solve IP problems when the search space is very large. Some random solutions,
each an M−dimensional 0− 1 binary vector are initialized in the search space. A
set of solution vectors called neighbours is generated for each obtained solution at
every iteration. A fitness or a cost function is evaluated at each neighbour, i.e., an
optimization problem is solved. For e.g., a utility function based on SINR or rate can
serve as a fitness function and a utility function based on power consumption or BER
can be a cost function. For each solution and for a predefined number of iterations called
tabu-period, a short-term memory called tabu-list maintains the previously visited
solution vectors. Tabu-list solutions must be avoided to prevents cycling, i.e., revisiting
the solution. A best non-tabu move is made from the current state to the neighbour. To
22
escape from getting stuck at a local optima, a move to a lower quality neighbour is also
accepted. At each iteration the process is repeated and the best current solution from
various states is stored globally. A rule called aspiration criteria can override the tabu-
list for diversified solutions. The best available solution after maximum iterations or
any other exit criteria is declared the output. Similar evolution based algorithm called
Ant Colony Optimization (ACO) [89] can also be used to solve the GAP problem. The
mathematical techniques outlined are also applicable to the sub-carrier allocation in
multi-carrier systems if the 0−1 variable αij is identified as a sub-carrier association
variable instead of a BSA variable.
2.3.2 Power Allocation and Beamforming
A beamforming sub-problem for a given p or Q and the power allocation sub-problem
for a given U are further obtained in the NLP sub-problems P1p, P2p and P3p. De-
pending on the problem formulation several iterative methods exist to solve the NLP
sub-problem, few of them are outlined next. With b= (2rth−1) and gijk = ‖uHijhik‖22,
the non-linear UL QoS constraints (2.10) can be rearranged to get a set of linear equa-
tions for LP problem as
pj · gijj
b
= σ2j +
N∑
k 6=j,k=1
pk · gijk, ∀j. (2.43)
Constraints (2.43) can be iteratively solved via Fixed-Point Theory [90]. The total
interference term Iij = (σ2j+
N∑
k 6=j,k=1
pk ·gijk) in equation (2.43) is known as a Standard-
Function [91] and has a Fixed-Point property that allows the iterations to converge to
a solution called Fixed-Point. The same set of equations can also be arranged as a
posynomial (not polynomial) [70] for GP constraints [92] as
b ·σ2j
gijj
p−1j +
N∑
k 6=j,k=1
b · gijk
gijj
p−1j pk ≤ 1, ∀j. (2.44)
A Signomial Programming (SP) [93] objective (2.45) which is the ratio of two posyno-
mials can be obtained for the objective in P1 with constraints (2.44). To solve the SP
objective, an approach called a complementary GP with a technique called a condensa-
tion technique [94] must be employed. Starting from an initial evaluation point pj, ∀j,
an objective (2.46), which is a ratio of a posynomial and a monomial is obtained.
min
p
N∏
j=1
Iij
Sij , (2.45) minp
N∏
j=1
Iij
Sσ
2
j/Sij
ij
N∏
j=1
(cijpj)
cij
. (2.46)
Sij = (Iij + pjgijj), Sij for a given pj , ∀j, is Sij and cij = Sij/(gijjpj). Both these
methods are used for comparison in the numerical results of chapter 3.
23
With Difference in Convex (DC) programming [95], constraints (2.10) can be expressed
as a difference of convex functions. It is expressed as
log2(pjgijj +Iij)− log2(Iij)≥ rth, ∀j. (2.47)
A first order Taylor series is obtained for log2(Iij) and the resultant constraints are
iteratively solved by well known convex techniques. The objective in P1 must also be
changed as equation (2.47) in DC programming.
Population based search methods such as Particle Swam Optimization [96] (PSO), Ge-
netic Algorithms [97] (GeA) find the optimum for NLP by simultaneously solving the
fitness function in different search directions. The initial population must be diverse
to cover most of the search space while iterating. PSO is based on the social behaviour
of a flock of birds searching for food. In PSO, several random solutions called particles
each with a fitness value, velocity and position metric are initialized. The velocity
and position influence the search direction. At each iteration each particle’s and the
populations’ best fitness is evaluated and the corresponding velocity and position are
also updated. The populations’ best fitness and its location in the search space are
declared at convergence. GeA is based on the evolution of biological organisms. In
GeA, several random solutions called chromosomes, each with a fitness value are ini-
tialized. Chromosomes with best fitness called parents are selected and a crossover is
performed to obtain new offsprings each corresponding to a solution. Several selec-
tions and crossovers are performed to obtain diverse solutions. Each iteration is called
a generation which ends at convergence. PSO and GeA can also be used to obtain the
BSA solutions where each particle or chromosome is an M−dimensional 0−1 binary
vector.
Simulated Annealing [98] (SA) based on the heating and gradual cooling in metallur-
gical process can also be used to obtain the solution. In SA, a cost function and a
solution acceptance criteria are defined. A temperature parameter is initialized and
gradually reduced to simulate the cooling process. Trial solutions are generated and
the cost function is evaluated, i.e., an optimization problem is solved. A solution may
be accepted, dropped or may be accepted with a probability. An exponential prob-
ability function is chosen that depends on the temperature and the cost variation in
successive iterations. With every iteration the probability of rejecting bad solutions
is increased. PSO, GeA and SA are evolutionary algorithms where at each iteration
several optimization problems are solved to obtain the overall cost or fitness value.
Sequential Quadratic Programming (SQP) [99] is also well known to handle NLP prob-
lems. SQP replaces the objective with a quadratic approximation and each constraint
with a linear approximation. At each iteration, a quadratic optimization problem is
solved around the trial point obtained from previous iteration along with a search di-
rection and required step size. Positive semi-definiteness of the matrix in the quadratic
term of the objective must be maintained to solve the problem as a convex optimization
problem such as SDP.
24
IPM [100] is the most famous technique to handle NLP problems. Several formulations
of IPM exist, such as potential-reduction methods and central-path methods [101].
In either case a log-barrier function exists which restricts the initial solution and its
iterates to the interior of feasible region. A constrained optimization problem in the
given form is called a primal-problem for which a dual-problem can be formulated.
Depending on the number of constraints, the primal-problem may be easier to handle
in the dual domain. Iterates generated by IPM reduce the gap between the primal
and dual solutions. Another form of IPM called infeasible-IPM, that has a greater
DoF while problem solving also exists, where the initialized solution and subsequent
iterates are not in the primal feasible region. At each iterate a search direction and a
step size is required. More than one way exists to choose these parameters. A primal-
dual based IPM framework used to solve the NLP problems throughout this thesis
is explained in chapter 3. Another mathematical tool recently being used to solve
problems in energy efficient networks is the Fractional Programming [102].
The beamformers that maximize the received SINR are the linear MMSE beamformers
[103]. For a given α they are given as
uij =

σ2jIT +
N∑
k=1,k 6=j
pkhikh
H
ik


−1
hij , ∀i,∀j. (2.48)
The MMSE decoder solution is the Generalized EigenValue Problem (GEVP) solution
[104], i.e., finding the generalized eigenvector of (hijhHij pj ,
N∑
k=1,k 6=j
hikh
H
ikpk+σ
2
jIT ) re-
sults in the MMSE vectors (2.48). The order of matrices must be maintained while
finding the eigenvector. For a joint transmit and receive BF with MMSE precoders
and decoders, a Fixed-Point iteration [57] is also possible, but due to the multi-modal
nature of the MSE function, convergence to a global optimum cannot be guaranteed.
So an iterative technique can be applied to find the beamformers. The PA and BF
sub-problems can be solved in a single step by an SDP approach, if
√
pkuij is con-
sidered a single variable and the substitutions Bijk = pkuijuHij and Rik = hikh
H
ik are
made. Further, the non-convex rank−1 constraints are relaxed to Bijk ≥ pkuijuHij to
obtain a Semi-Definite Relaxation (SDR) problem. By Schur complement these re-
laxed constraints can be expressed as equation (2.50). Additional techniques based on
randomization techniques [105] must be used to extract the rank−1 solution from the
relaxed solution. The overall transceiver constraints are given as
tr(RijBijj)≥
N∑
k=1,k 6=j
b · tr(RikBijk)+ b ·σ2j , ∀j, (2.49)
 Bijk √pkuij√
pkuij IT

≥ 0, ∀j. (2.50)
tr(·) is the trace operator. As mentioned before, several other BF techniques based on
SGINR, MVDR, MLSE exist.
25
26
Chapter 3
Uplink Resource Allocation under
perfect-CSI
In this chapter an Interior Point Method (IPM) framework that is used to solve a
general Non-Linear Programming (NLP) problem is outlined with its application to
sum-rate maximization problem P1 and sum-power minimization problem P3.
3.1 Algorithmic Framework
Consider a constrained optimization problem [106] P in N1 non-negative primal vari-
ables z, and with a scalar objective f0(z). Constraints for UEj are arranged in a
vector as cj(z) such that cj(z) ≥ 0. For each cj(z) a slack vector sj is included such
that sj ≥ 0. The overall constraint vector and slack vector of N UEs are c(z) =
[cT1 (z),c
T
2 (z), ...,c
T
N (z)]
T and s = [sT1 ,s
T
2 , ...,s
T
N ]
T , respectively. c(z) may have both
equality constraints and inequality constraints.
P: minimize
z
f0(z) (3.1)
subject to: c(z)− s= 0. (3.2)
For a maximization problem, −f0(z) is used in the objective of P. Let N2 be the total
number of constraints in equation (3.2). The Lagrangian for P is
L(z,s,λ) = f0(z)−λT (c(z)− s). (3.3)
λ is the non-negative vector of Lagrangian variables corresponding to the dual feasi-
bility. Entities in λ for the corresponding equality constraints in equation (3.2) are
unrestricted. The slack variables converge to values such that the non-negativity is
ensured. For P, a dual problem in dual decision vector λ is defined as
D: maximize
λ
infimum
z, s
L(z,s,λ). (3.4)
27
If P is infeasible, then D is unbounded and vice versa. D provides a lower bound to P
and the difference in the objective values is the duality gap. For a zero duality gap, a
strong duality is said to exist, i.e., P andD converge to the same solution. Let cn(z)−sn
represent an individual constraint from equation (3.2) and λn be its corresponding dual
multiplier. At the optimum (z∗,s∗,λ∗), either the primal constraint cn(z∗) or its dual
multiplier λ∗n is zero, i.e., s
∗
n ·λ∗n = 0, ∀n. This is called complementarity slackness. If
the set ∇zcn(z∗) for the active constraints are independent then the condition is called
Linear Independent Constraint Qualification (LICQ). The gradient operator w.r.t. z is
given as ∇z(·). The implication of LICQ is that it ensures feasibility and the existence
of unique dual multipliers. Also, ∇(z,s,λ)L(z∗,s∗,λ∗) = 0. Together these necessary
conditions called Karash-Kuhn-Tucker (KKT) conditions in vector form are
F=


∇(z,s,λ)L(z,s,λ)
c(z)− s
DλDs1N2

= 0. (3.5)
The first equation set in F is the first-order derivative condition w.r.t. [zT ,sT ,λT ]T .
The second equation set is the primal feasibility condition and the third equation set
is the complementarity slackness condition. Ds and Dλ are the diagonal matrices with
s and λ in their principal diagonal, respectively. For sufficiency condition, the Hessian
∇2zzL(z∗,s∗,λ∗) is verified and a minima occurs for a positive definite Hessian. For
a convex constraint set the KKT conditions are also the sufficient conditions and the
obtained optimum is also the global optimum.
A gradient based iterative algorithm [107] can be derived by perturbing the KKT vector
(3.5). As a result, a feasible instance of P gets solved as a set of linear equations only.
With t as the iteration index and given initial values of z, s and λ the following steps
constitute the algorithm. Search directions [∆Tz ,∆
T
s ,∆
T
λ
]T called the Newton directions
for the next iteration are evaluated as
Jt[∆
T
zt+1
,∆Tst+1 ,∆
T
λt+1
]T = Ft+[0
T
N1,−µt1TN2 ,0TN2 ]T , (3.6)
where J is the Jacobian of F w.r.t. [zT ,sT ,λT ]T , µt =
λ
T
t st
N2
is a complementarity mea-
sure. From a finite value, µ→ 0 during the iterations. The system of equations in
(3.6) form a perturbed KKT system. A certain set of continuous rows and columns in
J represent the Hessian ∇2zzL. To calculate the direction, the steepest descent method
uses the gradient, while the Newton method uses the Hessian. If the Hessian is unavail-
able, then quasi-Newton methods such as Broyden-Fletcher-Goldfarb-Shanno (BFGS)
method [69] can be used, where an approximation of the Hessian is used at each t. The
variables are updated as
[sTt+1,z
T
t+1]
T =
[
sTt ,z
T
t
]T
+ δst+1 [∆
T
st+1
,∆Tzt+1 ]
T , (3.7)
λt+1 = λt+ δλt+1∆
T
λt+1
, (3.8)
28
where δst+1 and δλt+1 are the step sizes that are a solution to
max
{
δst+1|δst+1 ∈ (0,1],st+1 ≥ (1− τ)st
}
, (3.9)
max
{
δλt+1|δλt+1 ∈ (0,1],λt+1 ≥ (1− τ)λt
}
, (3.10)
and the parameter τ ∈ (0,1). Several ways to choose δst+1 and δλt+1 based on line search,
such as Armijo condition, Goldstein condition, Wolfe condition [100], Zjoutendijk con-
dition [108] exist. They are applied for unconstrainted optimization problem and can be
extended to P while optimizing the unconstrained L. The described sequence of steps
are given in Algorithm 1. These iterations called Newton iterations are repeated till
Algorithm 1 : General iteration steps of IPM based on Newton directions.
1: Initialize: t= 0,zt, τ,(λt,st)> 0
2: repeat
3: evaluate L(z,s,λ), ∇(z,s,λ)L, µt, Ft, Jt
4: t= t+1
5: solve (3.6) to obtain new search directions
6: find step size from (3.9) and (3.10)
7: evaluate (3.7) and (3.8) to update variables
8: until convergence
9: Output: z,s,λ.
the required convergence is met. The problem P is assumed feasible and the involved
functions are continuous and continuously differentiable. At (z∗,s∗,λ∗), the conditions
of strict complementarity and LICQ hold. The Lagrangian bound L(z∗,s∗,λ∗) is equal
to the primal optimum f(z∗) as µ→ 0 and ∇(z,s,λ)L→ 0. Hence no duality gap be-
tween P and D. The problem is deemed infeasible if a predefined maximum number
of iterations is exceeded, or when L(z∗,s∗,λ∗)→∞. These conditions can be used as
the convergence and exit criteria for the algorithm. The problem is primal-infeasible
till final convergence. But the iterates are interior to the non-negative orthant (s, λ).
So the IPM is called a primal-dual infeasible interior-point method. It can be applied
to a broad class of utility functions with little or no modification. More often choosing
the initial solution may in itself be as hard as solving the original problem. With the
IPM, the need for such a particular initialization is also eliminated, i.e., it eliminates
the requirement of an initial primal feasible point to begin the algorithm
3.2 Rate Allocation
The explained IPM is applied to solve the sum-rate maximization problem P1 in this
section. Problems P1b and P1p are alternately solved to obtain an optimum. Linear
MMSE beamformers in equation (2.48) are used for beamforming.
29
3.2.1 Iterative Power Allocation
To solve sub-problem P1p by Algorithm 1, the primal variable z is replaced by p and
the constraint vector for UEj is
cj(p) =

 Pmax−pj
rj− rth

 . (3.11)
The first constraint arises due to the upper bound on variable pj. Optimization
w.r.t. variable U is implicit to P1p, since U is updated by the beamformers when-
ever p changes.
3.2.2 Iterative Base Station Association
To solve P1b iteratively, i.e., P1bn, primal variable z is replaced by α. The constraint
vector for UEj is given as
cj(α) =


−αj ◦ (αj−1)
1−1TMαj
rj− rth

 . (3.12)
The constraint (1−αij) due to the upper bound on αij is not added to cj(α) since
the inequality (2.37) is true only in this unit interval. The symbol ◦ is the Hadamard
product. The iterative steps given in Algorithm 2 solve the two stage problem, i.e.,
BSA and PA are solved alternately.
Algorithm 2 : Two stage formulation of P1
1: Initialize: p
2: repeat
3: solve for α by P1bn or P1bs
4: break on convergence
5: solve for p by P1p
6: break on convergence
7: until final convergence
8: Output: p,α.
3.2.3 Single Stage Formulation
Algorithm 1 can be directly applied to solve P1 in a single stage, i.e., to solve P1b and
P1p simultaneously for BSA and PA variables. It is given as
P1s: maximize
α,p,U
N∑
j=1
rj (3.13)
subject to: (2.2),(2.4),(2.10),(2.34),(2.35). (3.14)
30
The primal variables in this case are p and α. Constraint vector cj(α,p) is obtained
by appending constraint [Pmax−pj ] to constraint vector (3.12). Since the problem size
is big when compared to the two stage formulation, both time and space complexity
increase for each iteration.
Figures 3.1 to 3.4 show the convergence of various parameters for a single feasible
instance of P1s under CSIp, i.e., the intermediate Newton-iterations are shown. For
illustration, the results are restricted to a minimum working example. For the link-
level simulations, T = 2, τ = 0.9, Pmax = 0.1,∀j, σ2j = 1, M = 2, N = 5, rth = 0.2
is set. The maximum number of iterations is set to 30. M > 2 can also be set,
but it is easy to visualize a 2 dimensional 0−1 binary convergence plot as in Figure
3.3. With an increase in M or T , the number of active UEs can be increased for a
given rth, or for a given N , the threshold rth can be increased. In either case the
convergence trend remains the same as shown when M = 2, T = 2 is set. On the
dB scale, Pmax = 0.1 is 20 [dBm]. This is close to the actual maximum available UL
power of a UE, which is 23 [dBm]. But 20 [dBm] is good enough for the considered
minimum working example. The channel vector hij ∼CN (0,IT ), ∀i,∀j, i.e., it follows a
complex normal distribution with zero-mean and unit-variance. Elements of hij , ∀i,∀j,
are independent and identically distributed (i.i.d). The absolute value |hij |, ∀i,∀j, is
Rayleigh distributed which is the modelling of the assumed wireless fading channel.
Vectors s and λ are initialized to unit vectors with all elements equal to 1. Initial
values pj = 0.2,∀j, αij = 1.5,∀i,∀j are set, i.e., these variables pj and αij are initialized
to the values outsides their respective intervals (2.4) and (2.34).
iteration index
it
er
at
io
n
va
lu
e
 
 
1 2 3 4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
µ
normalized ∇L
primal-infeasible iterations
Fig. 3.1: Convergence of µ and ∇L.
iteration index
it
er
at
io
n
va
lu
e
 
 
1 2 3 4 5 6 7 8 9 10
−10
−5
0
5
Lagrangian function
sum-rate objective
primal-infeasible iterations
Fig. 3.2: Convergence of L and sum-rate
objective.
Figure 3.1 plots the convergence of the complementarity measure µ and gradient of
the Lagrangian ∇L. Their convergence to the value 0 shows that KKT conditions are
satisfied at the optimal. The convergence of either variable need not be monotonic.
Figure 3.2 plots the convergence of the Lagrangian L and the objective of P1s. Con-
vergence of both plots to the same value shows a zero duality gap between P and D
31
in the problem P1s. Figure 3.3 plots the convergence of the BSA variable α. Since
iteration index
B
S
A
va
ri
ab
le
α
ij
 
 
1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
1.2
α11
α12
α13
α14
α15
α21
α22
α23
α24
α25
primal-infeasible iterations
Fig. 3.3: Convergence of αj, ∀j.
iteration index
in
d
iv
id
u
al
U
E
ra
te
s
[b
it
s/
se
c/
H
z]
 
 
1 2 3 4 5 6 7 8 9 10 11
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
UE1
UE2
UE3
UE4
UE5
primal-infeasible iterations
Fig. 3.4: Convergence of rj , ∀j.
M = 2 one element in each αj converges to the value 1 corresponding to the associated
BS, while the other element in it converges to a 0 thereby satisfying the 0−1 integer
constraint (2.1) and the assignment constraint (2.2). So additional recovery of the 0−1
solution for the BSA variables is not required. Figure 3.3 plots the convergence of UE
rates rj, ∀j. Each UE achieves a rate greater than the required rth. This satisfies the
QoS constraint (2.10). So formulation P1s solves the NP-hard MINLP problem P1
as an NLP problem and obtains a solution to the discrete decision matrix α and the
continuous decision vector p simultaneously. It can be observed that the intermediate
iterates of P1s are all primal infeasible. Hence P1s is primal-infeasible till the final
convergence. Only the non-negativity of s and λ must be ensured throughout the
Newton-iterations.
0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sum-rate [bits/sec/Hz]
cu
m
u
la
ti
ve
d
is
tr
ib
u
ti
on
fu
n
ct
io
n
 
 
(5, .1)
(7, .1)
(5, .01)
(7, .01)
(9, .01)
exhaustive GP
Fig. 3.5: Comparison of P1bs-P1p and
exhaustive-GP.
0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sum-rate [bits/sec/Hz]
cu
m
u
la
ti
ve
d
is
tr
ib
u
ti
on
fu
n
ct
io
n
 
 
(5, .1)
(7, .1)
(5, .01)
(7, .01)
(9, .01)
P1s
Fig. 3.6: Comparison of P1bs-P1p and
P1s.
Figure 3.5 compares the two stage formulation P1bs-P1p with an exhaustive method for
M = 2 and various (N,rth) by plotting the Cumulative Distribution Function (CDF).
32
Other parameters remain the same as before. PA in the P1bs-P1p combination uses
the IPM. In the compared GP formulation [109], the PA uses equations (2.44) and
(2.46), i.e., with single condensation method [110] for all the MN possible combina-
tions of BSA. Not every BSA-UE combination is feasible and more than one feasible
combination is also possible in the exhaustive search. The feasible combination with
the maximum sum-rate objective value is taken as the reference for comparison. It can
be seen that the curves match for both the methods, i.e., same objective value at the
same BSA. Since there is a possibility to achieve the same sum-rate value at more than
one BS, P1bs is also verified by IP solver CPLEX. The obtained α in both the cases is
the same. To reduce the number of outer iterations, implementing P1bs is better than
P1bn. For a fixed N , with increase in rth, the CDF curves shift to the left showing a
drop in the objective. The increased QoS requirement of each UE increases the MUI
at every other UE. As a result, individual power levels are increased to overcome the
MUI and attain the QoS objective. For a fixed rth, with increase in N , the CDF curves
move to upward. This is due to the increase in number of infeasible problem instances,
i.e., increase in the system outage. Figure 3.6 compares the two stage formulation
P1bs-P1p with its single stage formulation P1s for M = 2 and various (N,rth). It can
be seen that either formulation leads to the same objective value. As expected, a right
shift in the curves with increasing N at a fixed rate rth and an upward shift in the
curves with increase in rth for a fixed N is observed. Both these show the competing
nature of the UEs and also the increased interference with QoS and N . There is a
slight variation in the P1s curve for the case (0.5,0.1). This variation is due to the
chosen initial values that resulted in a convergence beyond the predefined maximum
number of iterations. But under the assumption, the algorithm has to terminate if this
number is exceeded. To solve P1 as an NLP problem, a single stage formulation P1s,
or two stage formulations P1bs-P1p, or P1bn-P1p can be chosen.
3.3 Power Control
Similar to P1s, sum-power minimization problem P3 can be solved with only a change
in the objective, it is given as
P3s: minimize
α,p,U
N∑
j=1
pj (3.15)
subject to: (2.2),(2.4),(2.10),(2.34),(2.35). (3.16)
Since the constraint set for P1s and P3s is the same, other aspects applied for solving
P1s are retained, i.e., cj(α,p) fed to the IPM is the same. The problem of sum-power
minimization under SINR constraints is coupled to the problem of maximizing the
minimum SINR, i.e., the max-min problem where the objective is to achieve the same
SINR level across all the UEs. The feasibility of the later leads to formulation of the
former [111]. Solution to the max-min problem is equivalent to solving the PC problem.
33
Some properties of the interference constraints Iij are identified [112] based on Fixed-
Point Theory. This establishes the existence of iterative algorithms to solve the PC
problem. The asynchronous iterative method under constrained power [113], under
sum-power constraint [111] is well explained for the PC problem. However maximum
possible threshold must be evaluated before applying the framework. In this approach,
p is initialized to a random point preferably within its bounds. Then the constraint
(2.43) is evaluated for each UE as pj = Iij/gijj , ∀j, to get one power variable at a time.
These iterations are repeated and will converge if the feasibility is met as defined by
Perron-Frobenius Theory. The matrix form representation of the LP constraints (2.43)
for a given α is given as
(IN −A)p≥ υ. (3.17)
A is an (N ×N) non-negative, irreducible matrix. The diagonal elements Ajj = 0
and other elements Ajk = b · gijk/gijj . Elements in the (N × 1) vector υ are given
as υj = b · σ2j/gijj . The PC sub-problem P3p is feasible only if the eigenvalue of A
is real, positive and less than 1. With these conditions, (IN −A) is invertible and a
positive power vector exists. The transmit power increases without bound as each UE
approaches the maximum attainable threshold. However the upper bound on each pj
must also be taken into consideration to evaluate the feasible QoS. The structure of the
coupling matrix A is different when the sum-power constraint is considered. So either
LP problem (3.17) or the asynchronous iterations based on Fixed-Point Theory can be
used to solve P3p. To extend these to a multi-cell case, they must be solved for every
BSA [114]. As the number of network entities increase, this approach is practically not
viable. PC with macro-diversity [115] is also an interesting problem, however the level
of coordination among the BSs increases to perform joint processing. Other extensions
based on PC are also possible [116].
3.3.1 Feasibility
If the problem instance P3 or P3s is infeasible, then, to satisfy the eigenvalue criteria
of A, either some UEs are dropped or the QoS requirement is reduced. Branch and
Bound techniques to select the QoS-satisfying UEs or dropping QoS-violating UEs exist
with a reasonable performance [117]. It can be viewed as a sub-problem of admission
control in P3. The admission control will maximize the cardinality of the active UE set
and these UEs will be balanced due to homogeneity. Another way to define feasibility
is that if an instance of P3 is infeasible then it is reasonable to serve as many UEs as
possible that satisfy the QoS while still serving the QoS-violating UEs with the best
possible rate capable of being achieved by them instead of dropping them. Again the
selection and the dropping of UEs, which is an IP problem, needs to be evaluated.
This is an admission control problem. To handle the infeasibility of P3, a formulation
based on the ℓ1-norm heuristic [118] is applied. This is a single stage NLP formulation
similar to P1s which also explicitly defines the QoS-satisfying and the QoS-violating
34
UE subsets among the UEs. A new vector ν = [ν1, ...,νN ]T is introduced, such that
νj ≥ 0, ∀j. Further, QoS constraint (2.10) is modified as
νj+ rj ≥ rth, ∀j. (3.18)
The reformulated NLP problem for P3 that handles infeasibility is
P3f : minimize
α,p,U,ν
N∑
j=1
pj+ θ‖ν‖1 (3.19)
subject to: (2.2),(2.4),(2.34),(2.35),(3.18). (3.20)
The parameter θ is a large positive constant, and ‖·‖1 is the ℓ1−norm. The constraint
vector for the IPM is
cj(α,p,ν) =


−αj ◦ (αj−1)
1−1TMαj
νj+ rj− rth
Pmax−pj
pj


. (3.21)
Unlike the other variables, the lower bound on pj has to be included in (3.21). For
a feasible instance, P3f and P3s give the same result with νj = 0,∀j, while for an
infeasible instance the number of QoS violations are minimized due to the sparsity
requirement of the l1-norm. P1f is always feasible and the solution w.r.t. the QoS-
satisfying UEs is balanced w.r.t. rj .
Figures 3.7 to 3.10 plot the convergence of various UE parameters for a feasible channel
instance of P3s as a special case of P3f . For the minimum working example, M = 2,
N = 4, rth = 0.2, Pmax = 20 [dBm], θ = 1e2 is set. Other parameters remain the same
as in P1s. Figure 3.7 plots the convergence of the BSA variable αj , ∀j. As before only
iteration index
B
S
A
va
ri
ab
le
α
ij
 
 
1 4 7 10 13 16
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
α11
α21
α12
α22
α13
α23
α14
α24
primal-infeasible
Fig. 3.7: Convergence of αj, ∀j, in P3f
for a feasible P3s.
iteration index
u
se
r
ra
te
r j
[b
/s
/H
z]
 
 
1 4 7 10 13 16
0
0.5
1
1.5
r1
r2
r3
r4
rth
primal-infeasible
Fig. 3.8: Convergence of rj , ∀j, in P3f
for a feasible P3s.
35
iteration index
u
se
r
p
ow
er
p j
[d
B
m
]
 
 
1 4 7 10 13 16
0
5
10
15
20
25
30
p1
p2
p3
p4
Pmax
primal-infeasible
Fig. 3.9: Convergence of pj, ∀j, in P3f
for a feasible P3s.
iteration index
u
se
r
p
ar
am
et
er
ν
j
 
 
1 4 7 10 13 16
0
0.05
0.1
0.15
0.2
0.25
0.3
ν1
ν2
ν3
ν4
primal-infeasible
Fig. 3.10: Convergence of νj, ∀j, in P3f
for a feasible P3s.
one element in each αj converges to a 1, while other element converge to a 0. Figure
3.8 plots the convergence of the UE rates rj , ∀j. A balanced QoS is achieved across
all the homogeneous UEs, i.e., rj = rth, ∀j. Figure 3.9 plots the convergence of the PA
variable pj, ∀j. Each UE is transmitting at a power level lower than its corresponding
Pmax while satisfying equation (2.10). Hence the objective of minimization of total
transmit power in the system is achieved. Figure 3.10 plots the convergence of the
ℓ1−norm parameter νj , ∀j. Since the problem instance is feasible, νj , ∀j, converges to
zero, i.e., the ℓ1−norm parameter does not affect the objective P3f . Solving P3f is the
same as solving P3s for a feasible case.
Figures 3.11 to 3.14 plot the convergence of various UE parameters for the same setup
as before. The only change is that the threshold is set to rth = 0.3. An increase in
the QoS makes P3s infeasible, however P3f is still feasible based on the new definition
of feasibility. Figure 3.11 shows the convergence of the BSA variable αj , ∀j. Once
iteration index
B
S
A
va
ri
ab
le
α
ij
 
 
1 4 7 10 13 16
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
α11
α21
α12
α22
α13
α23
α14
α24
primal-infeasible
Fig. 3.11: Convergence of αj , ∀j, in P3f
for an infeasible P3s.
iteration index
u
se
r
ra
te
r j
[b
/s
/H
z]
 
 
1 4 7 10 13 16
0
0.5
1
1.5
r1
r2
r3
r4
rth
primal-infeasible
Fig. 3.12: Convergence of rj , ∀j, in P3f
for an infeasible P3s.
again only one element in each αj , will converge to the value 1 while the other element
36
converge to a 0. In Figure 3.12, it can be observed that at convergence, UE2 and
UE3 achieve the required QoS while UE1 and UE4 instead of being dropped receive a
rate value lower than the QoS. So only two UEs satisfy the QoS constraint and the
QoS-violating UEs achieve the best possible rate under infeasibility. It is evident that
formulation P3f , without dropping QoS-violating UEs or reducing the QoS value, han-
dles an infeasible P3s effectively. The initial UE selection sub-problem is eliminated
in P3f . The QoS-satisfying UEs are homogeneous, hence a balanced QoS is achieved
among them. From Figure 3.13 it can be seen that the QoS-violating UEs are trans-
iteration index
u
se
r
p
ow
er
p j
[d
B
m
]
 
 
1 4 7 10 13 16
0
5
10
15
20
25
30
p1
p2
p3
p4
Pmax
primal-infeasible
Fig. 3.13: Convergence of pj, ∀j, in P3f
for an infeasible P3s.
iteration index
u
se
r
p
ar
am
et
er
ν
j
 
 
1 4 7 10 13 16
0
0.05
0.1
0.15
0.2
0.25
0.3
ν1
ν2
ν3
ν4
primal-infeasible
Fig. 3.14: Convergence of νj, ∀j, in P3f
for an infeasible P3s.
mitting at maximum power Pmax, i.e., p1 = p4 = Pmax. This increases the overall power
level in the system when compared to serving only UE2 and UE3 while turning off UE1
and UE4. The QoS-achieving UEs are transmitting at a power level lower than the
maximum value. However it could be possible that for achieving the QoS, a feasible
UE may require Pmax. So focusing only on pj cannot determine the feasibility. Figure
3.14 plots the convergence of the ℓ1−norm vector ν. It has a finite value for the QoS-
violating UEs. The parameter νj for UE2 and UE3 is 0 as in the feasible case whereas
for the infeasible UE1 and UE4 the value is finite. The larger the νj value, the lower the
corresponding received rate value rj. So the set of feasible UEs can be determined by
verifying the values in ν in addition to the corresponding rj . Due to equation (3.18),
P3f achieves a balanced result w.r.t. the sum of rj and νj . Depending on the rj , the
corresponding νj can take any positive value. The actual upper bound on νj is rth and
it is the case when the corresponding rj = 0. Hence an explicit upper limit on νj is
not included in the constraint set. So νj , ∀j, not only identifies the QoS-violating UEs
but also specifies the difference between the achieved QoS and rth, i.e., the amount by
which the infeasibility is caused in the QoS constraint.
Figure 3.15 shows that the increasing average infeasibility with increasing N and also
with rth in P3s. For the same channel realizations, Figure 3.16 compares the average
number of UEs served for P3s and P3f . P3f clearly outperforms P3s by serving more
37
5 6 7 8 9
0.2
0.4
0.6
0.8
1
number of UEs
in
fe
as
ib
il
it
y
p
ro
b
ab
il
it
y
 
 
rth = 0.1
rth = 0.15
Fig. 3.15: Infeasibility increase with N
and rth for P3s.
5 6 7 8 9
0
2
4
6
8
10
number of UEs
av
er
ag
e
nu
m
b
er
of
U
E
s
se
rv
ed
 
 
P3f , 0.1
P3s, 0.1
P3f , 0.15
P3s, 0.15
Fig. 3.16: Average number of UEs served
for P3s and P3f .
QoS-satisfying UEs on an average. A significant performance improvement is observed
for the rth = 0.15 curve.
19 20 21 22 23 24 25 26 27 28 29
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
average minimum sum-power [dBm]
cu
m
u
la
ti
ve
d
is
tr
ib
u
ti
on
fu
n
ct
io
n
 
 
(5, 0.1)
(5, 0.15)
(7, 0.1)
(7, 0.15)
(9, 0.1)
(9, 0.15)
exhaustive
Fig. 3.17: CDF comparision of P3s and
iterative exhaustive case.
19 20 21 22 23 24 25 26 27 28 29
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
average minimum sum-power [dBm]
cu
m
u
la
ti
ve
d
is
tr
ib
u
ti
on
fu
n
ct
io
n
 
 
P3f , (5, 0.1)
P3s, (5, 0.1)
P3f , (9, 0.1)
P3s, (9, 0.1)
P3f , (5, 0.15)
P3s, (5, 0.15)
P3f , (7, 0.15)
P3s, (7, 0.15)
Fig. 3.18: Comparison of P3s and P3f .
Figure 3.17 compares the CDF of P3s and an exhaustive method based on Fixed-
Point Theory for various (N,rth), M = 2, Pmax = 23 [dBm]. For numerical verification,
other arbitrary values of Pmax can also be used. The exhaustive search considers all
possible MN BSA-UE combinations in P3b and solution to P3p is obtained by the
asynchronous iterative approach outlined before. More than one BSA-UE combination
may be feasible and the one with the least sum-power objective in each feasible instance
is taken as the reference to compare the performance of P3s. It can be seen that the
CDF of P3s matches with the CDF obtained by exhaustive search. Also the obtained
α in both cases is the same. With increase in rth, each UE increases its power level
to maintain the required QoS. The increase in the right shift of the curve is reduced
with increase in N or rth due to the increase in the number of infeasible realizations.
Figure 3.18 compares the CDF of P3s and P3f for various (N,rth) and same set of
38
parameters as before. Only the actual transmit sum-power in the P3f objective must
be considered rather than the complete P3f objective. Since the non-binding UEs are
transmitting at full power, and the number of served UEs is more in P3f , an increase
in the objective value of P3f over P3s is observed. This is reflected by a right shift in
the P3f CDF curve corresponding to the P3s curve.
39
40
Chapter 4
Uplink Power Control under
statistical-CSI and imperfect-CSI
When perfect Channel State Information (CSIp) is unavailable or when a small outage is
permitted in the service, the deterministic constraints can be replaced by Probabilistic
Constraints (PrCons) as a compromise. PrCons may at times be simplified to get a
closed form expression [119], however an approximation function is required for this
expression to further solve the problem. Obtaining such function may not be feasible
all the time. Intractability and unavailability of closed-form expressions make PrCons
mathematically difficult to handle. In this chapter, Uplink (UL) power control problems
with PrCons under statistical-CSI (CSIs) and imperfect-CSI (CSIi) are solved for a
multi-cell multi-user scenario. Existing bounds are further improved and the problem is
well handled. To handle the PrCons and analyse the system performance, financial risk
management measures Value-at-Risk and Conditional Value-at-Risk are applied. The
resulting expressions may involve functions that are non-invertible and a combination
of higher order functions, so obtaining closed form solutions may not be possible. To
find the optimal value of the optimization problem Extreme Value Theory is applied.
Worst case performance for a given probability is also evaluated in the process.
The problem of Power Control (PC) under CSIp and deterministic constraints of the
form [φ(y)≤ ζ ] in the decision variable y is well understood [120]. φ(·) is usually a loss
function in SINR and ζ is a constant. Implementing the deterministic constraints may
be too costly or impossible at times. Further, CSI is partially available or is imperfect
due to limited feedback or improper channel estimation. This affects the system’s out-
age performance. In this case, PrCons of the form Pr[φ(y,g)≤ ζ ] ≥ η, called chance
constraints in uncertain variable g, may be implemented as a compromise [121]. Pr[·] is
the probability function and η is the confidence interval. Computationally tractable and
conservative approximations are required to solve the chance constrained optimization
problems. This approximation results in a solvable deterministic optimization prob-
lem with the feasible set contained in the original chance constrained problem. After
obtaining the deterministic equivalent, standard deterministic optimization techniques
41
based on subgradients [122], Fixed-Point Theory based iterative methods [123] or SDR
[124] can be applied for finding the decision vector. In this chapter it is assumed that
for a UEj and a given α, the serving channel hij , ∀j, is perfectly known.
4.1 Power Control under statistical-CSI
Under CSIs, for a UEj and given α, the interfering channels hik, ∀k 6= j, are only
statistically known. Under these assumptions, the BS will have CSIp of the interfering
channels for the UEs with the same BSA as UEj , i.e., {UEn | n 6= j,αin = αij}. But
to generalize the case, it is assumed that it is still statistically known. The uncertain
variable for a given α are the effective channel gains gijk = ‖uHijhik‖22, ∀k 6= j in the
MUI component. So only the distribution of gijk in the MUI term is available. The
statistical-SINR γ(s)j for the deterministic parameter in equation (2.5) is given as
γ
(s)
j =
pj
M∑
i=1
αij‖uHijhij‖22
M∑
i=1
N∑
k 6=j,k=1
pkαijgijk+σ2j
, ∀j. (4.1)
For the deterministic QoS constraints (2.10), the PrCons under CSIs are given by
Pr
[
r
(s)
j > rth
]
≥ η, ∀j, (4.2)
where r(s)j = log2(1+γ
(s)
j ) is the statistical-rate for UEj . With the substitutions
y
(s)
j =
M∑
i=1
N∑
k 6=j,k=1
pkαijgijk, ∀j, (4.3)
ζ
(s)
j =
pj
b
M∑
i=1
αijgijj−σ2j , ∀j, (4.4)
equation (4.2) can be rearranged as constraint (4.5) or (4.6).
F
Y
(s)
j
(ζ(s)j )≥ η, ∀j, (4.5) ζ(s)j ≥ F−1Y (s)j
(η), ∀j. (4.6)
F
Y
(s)
j
is the Cumulative Distribution Function (CDF) and F−1
Y
(s)
j
(η) is the quantile
function of the random variable (r.v.) Y (s)j . The considered multi-cell multi-user
chance constrained MINLP optimization problem under CSIs is given by
P3c: minimize
α,p,U
N∑
j=1
pj (4.7)
s.t: (2.1),(2.2),(2.4),(4.6). (4.8)
The optimal value ζ(s)j = ζ
(s)
j is obtained when equation (4.6) is satisfied with equality,
i.e., ζ
(s)
j is η-quantile of r.v. Y
(s)
j . The analytic expression of the CDF of the r.v. must
exist and be invertible to efficiently solve P3c. Due to the Pr[·] function in constraint
(4.2), the decision vectors α and p must be evaluated simultaneously.
42
4.2 Value-at-Risk and Conditional Value-at-Risk
In the field of finance, risk management measures are required for loss estimation.
Such commonly used measures are Value-at-Risk (VaR) and Conditional Value-at-Risk
(CVaR) [125]. These metrics can be applied to the PC problem, if for each UE, the
loss function φ(·) is mapped to the MUI. Under CSIs and a given η, VaR satisfies the
chance constraint (4.6) and CVaR gives the worst case system performance. Worst case
implies the possible performance loss beyond the evaluated optimal VaR. To evaluate
VaR, the CDF of φ(y,g) is required. It may not be invertible. Further, to evaluate
CVaR, the Probability Density Function (PDF) of φ(y,g) is needed. A closed form
expression of CVaR may be unavailable if the PDF has higher order functions, such as
a combination of Bessel and exponential functions. Approximations only give a bound
which may not be tight enough to predict the exact performance.
For a UE, the MUI or more specifically y(s)j is identified as the loss function. Then by
the definition of VaR, a UE will experience a loss greater than VaR with probability
(1− η). Or the MUI will not exceed VaR with probability η. Mathematically it is
written as
VaR(y(s)j ,η) = inf{y(s)j | Pr[y(s)j ≤ ζ(s)j ]≥ η}, ∀j. (4.9)
Equation (4.5) can be mapped to this VaR definition. Since there still exists a finite
probability of exceedance beyond the maximum evaluated loss, it is further possible
that for a given η, the loss y(s)j exceeds ζ
(s)
j . So a worst case performance loss can be
estimated. However the mathematical formulation of the worst case problem considers
the mean value of the excess loss. The mean value of the exceedance of y(s)j over ζ
(s)
j
is called the CVaR. It is given by the conditional expectation
β
(s)
j = E
[
y
(s)
j |y(s)j ≥ ζ
(s)
j
]
, ∀j. (4.10)
The PDF of r.v. Y (s)j is required to evaluate equation (4.10). Mathematically this worst
case MINLP optimization problem is given as
P3w: minimize
α,p,U
N∑
j=1
pj (4.11)
s.t: (2.1),(2.2),(2.4),(4.10). (4.12)
It can be observed that P3w is not a chance constrained optimization problem. The
outcome of P3c is used as the input parameter to P3w via equation (4.10). It implies,
β
(s)
j ≥ ζ
(s)
j , ∀j, i.e., CVaR can never be less than VaR. So in terms of loss at a given
η, CVaR can be viewed as a loss value exceeding the already evaluated maximum loss,
which is the VaR. Hence CVaR is a worst case loss value in system performance. It is
mathematically possible to calculate ζ
(s)
j and β
(s)
j simultaneously [125] from a single
function
F = ζ(s)j +
1
(1−η)
∫
y
(s)
j >ζ
(s)
j
(y(s)j − ζ(s)j )ΨY (s)j (y
(s)
j )dy
(s)
j . (4.13)
43
Ψ
Y
(s)
j
(·) is the PDF of r.v. Y (s)j . The stationary point of F obtained from its derivative
w.r.t. ζ(s)j is the VaR and F evaluated at the stationary point gives the CVaR. For
stationary point evaluation, equation (4.13) involves differentiation of an integral with
a function of ζ(s)j in the limits, so the fundamental theorem of calculus [126] must
be applied to obtain the result. It could be possible that analytic expressions may
not exist for (4.9) and (4.10) or may be a combination of higher order functions. So
numerical methods based on Extreme Value Theory (EVT) are applied.
4.3 Extreme Value Theory
4.3.1 Generalized Extreme Value Distribution
EVT [127] can be applied to handle the mathematical formulations of VaR and CVaR.
It is based on the properties of maximum order statistics of i.i.d. random variables.
From the Central Limit Theorem it is known that the distribution of the sum of
i.i.d. random variables tends to a Gaussian distribution as the number of r.v.s increase.
This is called Sum-Stability. Similarly a Max-Stability concept exists that handles tail
distributions. Consider a block of K i.i.d. random variables and B such blocks. Let
̺mn, {m= 1, ...,B, n= 1, ...,K} be the nth r.v. in mth block. Let ¯̺m be the maximum
order statistic of each block, i.e., the block-maxima given by ¯̺m =max{̺m1, ...,̺mK}.
max{· · ·} is the maximum operator. The distribution of the set of maximum elements
{ ¯̺m, m= 1, ...,B} converges to a non-degenerate limiting distribution FY (ξ,s, l) called
Generalized Extreme Value Distribution (GEVD) and is given by
Pr[Y ≤ y] = exp

−

1+ ξ (y− l)
s


− l
ξ
+

, (4.14)
where (y)+ =max{0,y} and exp(·) is the exponential function This convergence to a
limiting distribution is called Max-Stability. s is the scale parameter, l is the location
parameter, ξ is the shape parameter. Depending on ξ values, GEVD has three types
of limiting distributions. (1) The Gumbel family, when ξ = 0 and −∞ < y <∞, (2)
The Frechet family, when ξ > 0 and y > −1/ξ, (3) The Weibull family, when ξ < 0
and y <−1/ξ. To get VaR, GEVD from EVT is applied. The block-maxima approach
based on GEVD solves for VaR without actually inverting the CDF. This is the primary
requirement for P3c. Optimal η-quantile [128] after parameter estimation is given by
ζ
(s)
j = lj−sj(1− (−B · ln(η))−ξj)/ξj, ∀j. (4.15)
The natural logarithm is given by ln(·). Algorithm 3 gives the sequence of steps to
obtain η-quantile using GEVD based block-maxima approach. Step.14 estimates the
44
Algorithm 3 : GEVD approach for η-quantile
1: set p, α, K, B
2: repeat
3: for j = 1 :N do
4: ¯̺m = [·] {empty set}
5: for n1 = 1 :K do
6: Ω = [·] {empty set}
7: for n2 = 1 : B do
8: generate hik, ∀k 6= j,∀i
9: for given hik, evaluate loss y
(s)
j , ∀j
10: Ω= Ω∪y(s)j
11: end for
12: ¯̺m = ¯̺m∪max{Ω}
13: end for
14: estimate GEVD parameters ξj, lj ,sj
15: evaluate optimal VaR ζ
(s)
j , ∀j, from equation (4.15)
16: end for
17: solve optimization problem to find p, α by Algorithm 1
18: until convergence
19: output : ζ
(s)
j , ∀j, p, α
GEVD parameters. A parametric approach based on the Maximum Likelihood Esti-
mator (MLE) or a non-parametric approach based on Pickands estimator are usually
used to estimate the GEVD parameters (ξ,s, l). So an optimization in certain sense
needs to be carried out at this step. Step.17 involves solving a determistic optimiza-
tion problem. Algorithm 1 used to simultaneously solve for α and p in chapter 3 is
applied at this point. Under CSIs two different cases are considered for simulation of
a minimum working example. One where analytic expression for VaR exists and the
other where it does not. If hij ∼N (mij ,ρ2ij)+jN (mij ,ρ2ij), ∀i,∀j, then y(s)j for a given
α follows a non-central chi-squared distribution (nc-χ22) with 2 DoF. The CDF and
PDF are respectively given by
F
Y
(s)
j
(y) = 1−Q


√
λ
eij
,
√
y
eij

 , (4.16)
f
Y
(s)
j
(y) =
1
2 · e2ij
· exp

−(y+λ)
2 · e2ij

 · I0


√
λy
e2ij

 . (4.17)
The non-central parameter is given by λ=n2ij+n
2
ij , where nij = (1
T
Tℜ{uij}+1TTℑ{uij})·
N∑
k 6=j,k=1
√
pkmij and nij = (1
T
Tℜ{uij} − 1TTℑ{uij}) ·
N∑
k 6=j,k=1
√
pkmij , ℜ{·} is the real
part, ℑ{·} is the imaginary part, e2ij =
∑
k 6=j pkρ
2
ik, Q(·, ·) is the generalized Marcum’s
45
Q−function [129], I0(·) is the 0th order modified Bessel function of first kind. From
equation (4.16) it can be observed that the analytic VaR expression does not exist. Also
from equation (4.17) the evaluating a closed form expression for CVaR is not possible.
If mij = 0, ∀i,∀j, then y(s)j for a given α follows a central chi-squared distribution (χ22)
with 2 DoF, i.e., an exponential distribution. An analytic expression for VaR exist for
χ22-case and a closed form CVaR expression also exists. Equation (4.6) becomes
ζ
(s)
j ≥−2 · ln(1−η) ·
M∑
i=1
N∑
k 6=j,k=1
ρ2ik ·αijpk, ∀j. (4.18)
The term to the right of inequality (4.18) is the VaR for χ22 case [125].
For comparision, the deterministic equivalent of P3c, i.e., P3s and frequently used
methods to handle chance constraints under CSIs such as Bernstein Approximation
(BA) [121] and Gaussian Approximation (GA) [130] are considered. For GA, the CDF
F
Y
(s)
j
(·) is approximated by Gaussian CDF which is invertible. The GA to equation
(4.2) is
ζ
(s)
j ≥mj+
√
2 ·ρj · erf−1(2η−1), ∀j, (4.19)
where erf−1(·) is the inverse error function [129]. Mean mj , variance ρ2j of the approx-
imated Gaussian distribution must be estimated. GA may not produce tight bounds.
For BA, equation (4.2) for a given α is modified as
f0+
N∑
k 6=j,k=1
µk ·fk+ δk ·
√√√√√ N∑
k 6=j,k=1
σ2k ·f
2
k ≤ 0, ∀j, (4.20)
where f0 = σ
2
j − (pjgijj/b)+
∑
k 6=j εikpk, εik = (dik+ cik)/2, [cik,dik] is the support of
gijk, fk = pkεik, εik = (dik− cik)/2, δk =
√
2 · ln(1−η)−1. Parameters µk and σk are
chosen as described in [121]. BA requires gijk to be normalized such that (gijk−εik)/εik
is supported on [−1,1], it also requires fk to be affine in the decision vector. But
y
(s)
j is bilinear in α and p, so for BA, two stage formulation for PA and BSA is
used. Requirements such as choosing a support for unknown r.v. and formulating
affine functions in decision variables make BA stringent. A support [cik,dik] [122] for
the exponential distribution must be set in BA. Since the χ22 distribution is defined for
the non-negative values of the r.v., cik = 0 can be chosen. Finding the support may
be difficult for most of the distributions and the obtained conservative approximation
could be loose if it is not properly chosen.
P3c is assumed feasible. The considered beamformers are uij =arg-max
uij
‖uijhij‖22, ∀i,∀j
and ‖uij‖22 = 1, ∀i,∀j, i.e., the BF sub-problem is not included in PA. arg-max(·) oper-
ator chooses the decision variable that maximizes the expression it is maximizing. For
BA, an exhaustive search w.r.t. α is performed. The choosen support for gijk is crucial
for BA. No explicit estimation of EVT parameters is performed in Algorithm 3, MLE
46
based function gevfit() in Matlab is used. For simulation, M = 2, N = 4, T = 2,
σ2j = 1, ∀j, Pmax = 23 [dBm], ∀j, B = 1e3, K = 30, mij = 0.6, ρij = 0.5 is set. These
random parameter values are set as a reference to test the performance of Algorithm
3. They can be scaled according to the practical values. Both χ22 and nc-χ
2
2 are in the
Domain of Attraction (DoA) of the Gumbel family where ξ→ 0.
0.02 0.04 0.06 0.08 0.1
14
16
18
20
22
24
26
rth [bits/sec/Hz]
su
m
-p
ow
er
[d
B
m
]
 
 
GEVD , (4, 0.95)
GEVD , (4, 0.99)
analytic , (4, 0.95)
analytic , (4, 0.99)
GEVD , (6, 0.95)
GEVD , (6, 0.99)
analytic , (6, 0.95)
analytic , (6, 0.99)
Fig. 4.1: VaR comparision with analytic
values for χ22 distribution of y
(s)
j .
0.06 0.08 0.1 0.12 0.14
20
21
22
23
24
rth [bits/sec/Hz]
su
m
-p
ow
er
[d
B
m
]
 
 
deterministic
BA, η = 0.95
GA, η = 0.95
GEVD, η = 0.95
BA, η = 0.99
GA, η = 0.99
GEVD, η = 0.99
Fig. 4.2: VaR comparision with other
methods for χ22 distribution of y
(s)
j .
Figure 4.1 compares the numerical bounds obtained by solving P3c with GEVD ap-
proach and its analytic equivalent for the χ22 distribution of y
(s)
j . Two confidence levels
η = 0.99 and η = 0.95 are considered. The plots perfectly overlap showing that the
GEVD approach indeed generates the optimal values. From η = 0.95 to η = 0.99 there
is an outage requirement increase which results in higher objective value of P3c. With
GEVD, generating optimal values is possible without inverting the CDF. Figure 4.2
compares GEVD performance with other existing methods. For η = 0.95 the GA is
underestimating the objective value while BA overestimates it. Neither plot matches
the exact value. For η = 0.99 both GA and BA underestimate the objective, however
the BA plot is very close to the optimal GEVD plot. GA produced weak bounds while
bounds from BA were not tight for different η. An improvement in the bound with
BA could be expected if the support for the unknown r.v. is chosen properly. Under
CSIp the objective value is the least as expected due to more available CSI at the BS.
With the increased QoS, the MUI at each UE increases leading to an increased power
level for each UE to achieve the QoS, hence the objective value increases. Figures 4.1
and 4.2 consider only instances which are feasible for all the methods simultaneously.
Figure 4.3 compares the VaR for the non-invertible nc−χ22 CDF of y(s)j . No analytical
values exist to verify the tightness of the generated numerical bound. The observed
trend in the plots is very simlar to the trend observed in Figure 4.2. GA and BA
underestimate the GEVD bound for η = 0.95. For η = 0.99, BA is close to the GEVD
bound while still underestimating it, GA also generates a weak understimated bound.
So the BA bounds are not consistent with the change in η while the GA bounds are
loose.
47
0.06 0.08 0.1 0.12 0.14
20
21
22
23
24
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[d
B
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]
 
 
deterministic
BA, η = 0.95
GA, η = 0.95
GEVD, η = 0.95
BA, η = 0.99
GA, η = 0.99
GEVD, η = 0.99
Fig. 4.3: VaR comparision for nc−χ22 dis-
tribution of y
(s)
j .
4.3.2 Generalized Pareto Distribution for CVaR
With B = 1, and by setting a threshold t called return value, the exceedance for a
block of i.i.d. random variables over t is evaluated. This is called Peaks-over-Threshold
(PoT). These excess values, i.e., the values exceeding t, also converge to a limiting
distribution FX(ξ,s, l) called Generalized Pareto Distribution (GPD). It is given as
Pr[X = Y − t|Y > t] = 1−
(
1+
ξx
s+ ξ(t− l)
)− 1ξ
+
, (4.21)
where s is the scale parameter, l is the location parameter, shape parameter ξ is
analogous to ξ in GEVD. GPD is analogous to the Max-Stability. To get CVaR we
apply GPD from EVT. CVaR can be evaluated if the threshold is ζ
(s)
j which can be
obtained from P3c. CVaR after parameter estimation is given by
β
(s)
j = ζ
(s)
j +(sj+ ξ(t− lj))/(1− ξ), ∀j. (4.22)
The second term on the right side is the mean excess over the chosen threshold ζ
(s)
j .
Algorithm 4 gives the sequence of steps to obtain CVaR using GPD based PoT ap-
proach. Step.15 involves solving a determistic optimization problem. Algorithm 1 is
again used to find the optimal BSA and PA. The analytic expression for CVaR [128]
for χ22 distribution of the unknown r.v. under the worst case performance is
ζ
(s)
j ≥ 2 · (1− ln(1−η)) ·
M∑
i=1
N∑
k 6=j,k=1
ρ2ikαijpk, ∀j. (4.23)
The terms to the right of inequality (4.23) is the CVaR for χ22 case. For the worst case
performance using GA, equation (4.10) becomes
ζ
(s)
j ≥mj+ρj · exp(−(erf−1(2η−1))2)/(
√
2π(1−η)), ∀j. (4.24)
48
Algorithm 4 : GPD approach for CVaR
1: set p, α, K.
2: solve P3c to get ζ
(s)
j , ∀j, from Algorithm 3.
3: for j = 1 :N do
4: Ω = [·] {empty set}
5: for n1 = 1 :K do
6: generate hik, ∀k 6= j, ∀i.
7: for given hik, evaluate MUI y
(s)
j , ∀j.
8: Ω = Ω∪y(s)j .
9: end for
10: with threshold tj = ζ
(s)
j , ∀j.
11: find PoT, entries greater than ζ
(s)
j in Ω.
12: estimate GPD parameters ξj ,sj , lj .
13: evaluate CVaR β(s)j , ∀j, from equation (4.22)
14: end for
15: solve optimization problem to find p, α by Algorithm 1.
16: output : β(s)j , ∀j, p, α
The terms to the right of inequality (4.24) is the CVaR [125] for GA. For simulation,
same GEVD parameters values are assumed. The same feasible channel realizations are
retained to further evaluate CVaR from the corresponding VaR values. GPD parame-
ters (ξ,s, l) are estimated in Step.12 based on MLE based function gpfit() in Matlab.
Figure 4.4 compares the CVaR bounds with the analytic values and bounds from the
0.06 0.08 0.1 0.12 0.14
21
22
23
24
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]
 
 
VaR, η = 0.95
analytic, η = 0.95
GPD, η = 0.95
GA, η = 0.95
VaR, η = 0.99
analytic, η = 0.99
GPD, η = 0.99
GA, η = 0.99
Fig. 4.4: CVaR comparision for χ22 distri-
bution of y
(s)
j .
0.06 0.08 0.1 0.12 0.14
21
22
23
24
25
rth [bits/sec/Hz]
su
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[d
B
m
]
 
 
VaR, η = 0.95
GPD, η = 0.95
GA, η = 0.95
VaR, η = 0.99
GPD, η = 0.99
GA, η = 0.99
Fig. 4.5: CVaR comparison for nc−χ22
distribution of y
(s)
j .
GA method. Corresponding VaR bounds generated from GEVD are also plotted. As
expected, β(s)j ≥ ζ
(s)
j , ∀j. The GPD bounds are the closest to the analytic values. The
gap between the plots is due to the GPD parameter estimation error. Increasing N
may close this gap thus generating tighter bounds. This gap increases with η, i.e., when
49
compared to analytic values, η = 0.95 produced tighter bounds than the corresponding
η = 0.99 curves. GA produced bounds that are very weak in either case. The values for
η = 0.95 overlap with the deterministic curve, this is not a performance improvement.
GA bounds are underestimating the analytic values. Figure 4.5 compares the CVaR
curves for the nc−χ22 case. A trend similar to Figure 4.4 is observed, with the GPD
bounds greater than the corresponding GEVD values. Clearly EVT outperforms other
methods. The only requirement for EVT approach is that the underlying distribution
is known.
4.4 Power Control under imperfect-CSI
Under CSIi, the interfering channels are modelled by the CSI-error model. This un-
certainty model is given as
hik = hik+eik, ∀i,∀k 6= j, (4.25)
where eik = E
1/2
ik vik, Eik is a known covariance matrix, hik is also a known vec-
tor and vik ∼ CN (0,INr), i.e., unit-variance complex Gaussian vector. Hence hik ∼
CN (hik,Eik),∀i,∀k 6= j. Under CSIi the imperfect-SINR γ(i)j for the deterministic pa-
rameter in equation (2.5) is given as
γ
(i)
j =
pj
M∑
i=1
αij‖uHijhij‖22
M∑
i=1
N∑
k 6=j,k=1
pkαij‖uHij (hik+eik)‖22+σ2j
, ∀j. (4.26)
For the deterministic QoS constraints (2.10), the PrCons under CSIi are given by
Pr
[
r
(i)
j ≥ rth
]
, ∀j, (4.27)
where r(i)j = log2(1+γ
(i)
j ) is the imperfect-rate for UEj . With the substitution
y
(i)
j =
M∑
i=1
N∑
k 6=j,k=1
pkαiju
H
ij (hike
H
ik+eikh
H
ik+eike
H
ik)uij , ∀j, (4.28)
ζ
(i)
j =
M∑
i=1
pjαij
b ‖uHijhij‖22−
M∑
i=1
N∑
k 6=j,k=1
pkαij‖uHijhik‖22−σ2j , ∀j, (4.29)
it is rearranged as equation (4.30) or (4.31)
F
Y
(i)
j
(ζ(i)j )≥ η, ∀j. (4.30) ζ(i)j ≥ F−1Y (i)j
(η), ∀j. (4.31)
y
(i)
j is precisely not the loss for UEj , since the MUI term is present in both y
(i)
j and ζ
(i)
j .
y
(i)
j is quadratic in eik, hence obtaining an invertible expression is almost impossible.
50
Similar to P3c, the chance constrained MINLP under CSIi is given as
P3i: minimize
α,p,U
N∑
j=1
pj (4.32)
s.t: (2.1),(2.2),(2.4),(4.31). (4.33)
Under CSIi, a Bernstein-type Inequality (BI) [131] is frequently used to obtain a con-
servative formulation based on SDP. It is based on identifying a quadratic form in
zero-mean unit-variance Gaussian random variables of the PrCons. BI was also imple-
mented for transmit BF [124] under CSIi, secrecy-rate maximization in the presence of
eavesdroppers [132]. For the BI formulation of PrCons, equation (4.27) must first be
rearranged for a given α.
Wijk = uiju
H
ij pk,
Wij =−blkdiag(Wij1, ...,Wij(j−1),Wij(j+1), ...,WijN ),
Eij = blkdiag(E
1/2
i1 , ...,E
1/2
i(j−1),E
1/2
i(j+1), ...,E
1/2
iN ),
Aj = E
H
ijWijEij ,
gij = [h
H
i1 , ...,h
H
i(j−1),h
H
i(j+1), ...,h
H
iN ]
H ,
cj = σ
2
j −
‖hHijWijj‖22
b
−gHijWijgij ,
bj = E
H
ijWijgij ,
vj = [v
H
i1 , ...,v
H
i(j−1),v
H
i(j+1), ...,v
H
iN ]
H .
(4.34)
is substituted for all UEs and equation (4.27) is rearranged into a known BI form as
equation (4.35). The block diagonal operator that forms a new matrix by placing the
given matrices on the main diagonal of the new matrix is given by blkdiag(·). From
[131], for a zero-mean unit-variance Gaussian vector vj and a positive semidefinite
matrix Aj , the constraint of the form
Pr[vHj Ajvj+2 ·ℜ{vHj bj} ≥ cj ]≥ η, ∀j, (4.35)
is approximated by a conservative deterministic form
tr(Aj)−xj · (−2 · ln(1−η))1/2+dj · ln(1−η) ≥ cj, ∀j, (4.36)
Aj+djI ≥ 0, ∀j, (4.37)

xj
[
vec(Aj)H ,b
H
j
√
2
]
[
vec(Aj)H ,b
H
j
√
2
]H
xj · I

 ≥ 0, ∀j, (4.38)
dj ≥ 0, ∀j, (4.39)
where tr(·) is the trace operator, the vec(·) operator stacks the columns of a matrix into
a vector. The expression inside the Pr[·] operator of equation (4.35) is quadratic in vj .
51
Equations (4.37) and (4.38) are SDP constraints. Implicit constraints include Wij ≥
0,∀j. It can be observed that the problem of transmitter optimization is obtained
by this reformulation, since for each UE, the power variable pj and the beamforming
vector uij are considered as one matrix variable. The approximation of equation (4.27)
by equations (4.36)-(4.39) makes P3i tractable, however it must be evaluated for all
the BSA combinations. Also it might be difficult to expect CSIi to always satisfy the
quadratic form in Gaussian variables. Nevertheless, BI is one of the most frequently
used methods in the presence of CSIi and PrCons. BA, which is applicable to a family
of distributions, is also often used.
For the link-level simulations of a minimum working example, P3i is assumed feasible.
The same set of parameters as in P3c are set, i.e., M = 2, σ2j = 1, ∀j, T = 2, Pmax = 23
dBm. The procedure to obtain the η−quantile by Algorithm 3 is applied without any
change. Step.6 generates the CSI-error model channel as in equation (4.25). For the
MUI channels, hik ∼ CN (0,IT ), ∀i,∀k 6= j, Eik = 0.2 · IT , ∀i,∀k 6= j is set, and for the
deterministic channels, hij ∼ CN (0,IT ), ∀i,∀j is assumed. For a fair comparison of P3i
with P3s, uij = arg-max
uij
‖uijhij‖22, ∀i,∀j, is considered. It can be observed that the
assumed CSI-error model in equation (4.25) gives a complex Gaussian distribution for
hik if it is treated as the sum of complex Gaussian r.v.s. The resulting complex normal
distribution is under the DoA of the Gumbell limiting distribution, i.e., ξ → 0. But
for comparison, a non-zero mean Gaussian distribution is assumed and hik represents
the mean value. A given hik and normally distributed vik, gives a nc−χ22-distribution
for y(i)j . For the BI formulation of P3i, an exhaustive search w.r.t. α is carried out,
where the PA sub-problem is solved at all the MN possible BSA combinations. The
BSA combination which gives the least value of sum-power objective is taken as the
optimal. EVT approach of P3i has no approximations. The CDF of the unknown
r.v. is estimated using VaR and GEVD. Exact quantile function of the non-invertible
CDF is evaluated and the PrCons are now deterministic.
15 17 19 21 23 25 27 29
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sum-power [dBm]
cu
m
u
la
ti
ve
d
is
tr
ib
u
ti
on
fu
n
ct
io
n
 
 
(P3i, 0.95)
(P3bi, 0.95)
(P3i, 0.99)
(P3bi, 0.99)
P3s
Fig. 4.6: Comparision of P3i, P3bi, P3s
with rth = 0.08.
15 17 19 21 23 25 27 29
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sum-power [dBm]
cu
m
u
la
ti
ve
d
is
tr
ib
u
ti
on
fu
n
ct
io
n
 
 
(P3i, 0.95)
(P3bi, 0.95)
(P3i, 0.99)
(P3bi, 0.99)
P3s
Fig. 4.7: Comparision of P3i, P3bi, P3s
with rth = 0.14.
52
Figure 4.6 plots the CDF curves of the sum-power objective obtained by P3s, P3i
and BI-formulation (P3bi) with η = 0.95 and η = 0.99 at a low value of rth = 0.08. As
expected, the CDF plot for P3s is to the left most, giving the least objective value. The
obtained objective at η=0.99 is higher than the objective of the corresponding η= 0.95
for both P3i and P3bi. This is due to the increased outage performance requirement.
Though the performance of both P3i and P3bi is similar, P3bi performs slightly better
than P3i at lower objective values, i.e., a left shift in the CDF curve. Figure 4.7 plots
similar CDF curves when rth is increased to 0.14. Though the CDF of P3i has a right
shift to the corresponding P3bi for η = 0.99, it can be seen that the outage performance
of P3i is better than P3bi, i.e., an increase in the number of feasible instances for P3i.
For η = 0.95, the outage performance is better for P3i and the objective value is also
lower at higher sum-power values. Figure 4.8 plots the average sum-power objective
0.08 0.1 0.12 0.14 0.16
19
20
21
22
23
24
rth [b/s/Hz]
av
er
ag
e
su
m
-p
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[d
B
m
]
 
 
(P3i,0.95)
(P3bi,0.95)
(P3i,0.99)
(P3bi,0.99)
P3s
Fig. 4.8: average sum-power for P3i for
N = 3.
0.08 0.1 0.12 0.14 0.16
20
21
22
23
24
25
26
rth [b/s/Hz]
av
er
ag
e
su
m
-p
ow
er
[d
B
m
]
 
 
(P3i,0.95)
(P1bi,0.95)
(P3i,0.99)
(P1bi,0.99)
P3s
Fig. 4.9: average sum-power for P3i for
N = 4.
vs the QoS for N = 3. As expected, P3s has the least objective value. The plot shows
that for rth≤ 0.1, P3bi performs slightly better than the P3i. But beyond rth =0.1, P3i
outperforms P3bi for both η=0.95 and η=0.99., Figure 4.9 plots the same set of curves
for N = 4. Values of P3i, P3bi and P3s for N = 4 are higher than the corresponding
values in N = 3 i.e., the sum-power increases with increase in N . The plot shows that
the average sum-power for P3i is lower than the value from P3bi for the considered
range of QoS for both η = 0.95 and η = 0.99.
53
54
Chapter 5
Downlink Performance Analysis
under perfect-CSI
From the Downlink (DL) QoS constraint (2.11), it can be observed that the expression
is coupled in the beamformers and power variables. However the Uplink (UL) QoS
constraint (2.10) exhibits a decoupled nature. This forms the basis to solve the DL
problem via the UL. However this UL called a Virtual Uplink (VUL) exists only for
mathematically solving the DL problem via a duality concept, i.e., there is no physical
existence. It is very much the same as the actual UL in terms of the expressions and
parameters. This duality is not the conventional mathematical optimization duality. It
is a duality based on the DL-VUL reciprocity. Such a reciprocity technique is easy to
understand and can be applied to a single-cell case where only a single total sum-power
constraint exists. However it cannot be extended to the multi-cell case where a per-BS
power constraint exists [133]. Even in a single-cell with per-antenna power constraint
the simple DL-VUL duality fails since there is more than one sum-power constraint.
In this case a duality different from the conventional single-cell duality with additional
optimization variables must be formulated. In this chapter, problems P2 and P4 are
solved via the implicit Lagrange-duality in Algorithm 1. This eliminates additional
optimization sub-problems. Also the single-cell duality requirements are included into
the Lagrange-duality.
5.1 Uplink-Downlink Duality
The concept of duality in single-cell is well understood [134]. For duality in a single-cell,
under the same set of beamformers, transposed channels and equal total sum-power,
the same SINR is achieved on both the DL and VUL. Also the noise variance remains
the same in DL and VUL. The parameters and expressions for the actual-UL and VUL
are the same. Understanding the sum-power constraint is important when dealing with
the VUL, since on the DL it indicates the total available power at the BS and on the
55
VUL it is the sum of individual UE powers. So the same UL variables pj , γj, rj ,
σ2j are used as VUL transmit power, VUL-SINR, VUL-rate and VUL-noise variance
respectively with no change. To apply the duality for a single-cell DL based on Perron
Frobienius theory, a non-negative matrix
Γ=

 D ·C D ·1N ·σ2j
1TN ·D ·C/P¯i 1TN ·D ·1N ·σ2j/P¯i

 , (5.1)
is constructed. D is an (N ×N) diagonal matrix with Djj = γj/‖hHijuij‖22, C is an
(N ×N) matrix with Cjj = 0 and Cjk = ‖hHijuik‖22. Even though M = 1, subscript
i is retained to maintain consistency with the previous notations. For a feasible DL
power vector to exist that attains the DL target SINR γ
j
equal to the VUL SINR γj ,
the largest eigenvalue of Γ must be positive, i.e., λ˜max(Γ) > 0, where the maximum
eigenvalue of a given matrix is given by λ˜max(·). Also, λ˜max(Γ)< 1 [135] is needed. The
achievable DL-SINR γ
j
is 1/λ˜max(Γ). The eigenvector corresponding to this λ˜max(Γ) is
a non-negative ((N +1)×1) extended DL power vector q˜ = [qT ,1]T . Such formulation
encompasses the eigenvalue minimization problem [135], where the PA and BF are
alternately optimized to reduce the λ˜max(Γ) to less than 1. Once the these conditions
are satisfied, the power vector on the DL for a fixed beamformer set is given as
q = (IN −D ·C)−1 ·D · (σ21N ). (5.2)
The DL noise variance for all the UEs is the same, i.e., σ2j = σ
2, ∀j. The target SINR
metric can also be replaced by the MSE metric [41], i.e., a duality based on achieving
same MSE on the DL and the VUL can also be used. This can be extended to a joint
transmitter-receiver optimization problem [136] with multiple spatial layers for each
UE. In this case, per-layer UE metric must be considered, where intra-layer interference
is also included into Γ. With such a duality, a simple DL sum-rate maximization
problem can be solved by equivalently solving the problem of minimizing the product
of MSE matrices [49]. In the problem of weighted sum-power minimization with QoS
constraints in SINR, the DL-VUL duality is equivalent to Lagrange-duality when the
considered DL weight vector is the VUL-noise vector [137]. This Lagrange-dual includes
the case where the DL-noise and VUL-noise have the same value and set to unity value.
The coupled max-min problem [138] for transmit BF can also be solved via duality. The
aim of MSE-duality, SINR-duality, max-min duality is to decouple the DL problem.
SinceM =1, this single-cell DL-VUL duality has only one simple total-power constraint
bounded by the available power at the BS.
This single-cell duality from equations (5.1) and (5.2) can be extended to the multi-
cell case only if there is a single total power constraint, i.e., with
M∑
i=1
P¯i as the only
requirement. This may not provide a feasible solution since a possibility of qij > P¯i
can be mathematically obtained for some random UEj at its assigned BS, i.e., the DL
transmit power is greater than the available BS power. A multi-cell BF problem by
56
max-min duality with per-BS power constraints [133] includes VUL-noise as an opti-
mization variable to satisfy the per-BS constraints. Also instead of a single sum-power
constraint, a sum-sum power constraint and a weighted sum of VUL-noise constraint
are implemented to map the DL power constraints. This formulation for the multi-
cell case uses a Lagrange-duality to satisfy the power constraints. In addition, the
VUL-noise is also optimization variable and the value of DL-noise may be different
from its corresponding VUL value. A multi-cell DL throughput maximization problem
with per-BS power constraints by Lagrange-duality [139] also maps the weighted sum
of the thermal noise to the per-BS power constraints. The optimization sub-problem
in the VUL-noise variable is solved via the GP formulation. In most of the cases,
either the BSA is known or only one UE per-BS is considered. The per-antenna power
constraints in a single-cell [140] are similar to the per-BS power constraints. Lagrange-
duality and uncertain VUL-noise is once again the solution approach for transmitter
optimization. The DL-VUL duality in per-BS and per-antenna constrainted problems
is obtained by the primal DL problem’s mathematical dual with uncertain VUL-noise.
For single-cell duality the DL and VUL noise variance remains the same, also the to-
tal sum-power on the DL and VUL are equal. In the multi-cell case with uncertain
noise and Lagrange-duality formulation, the noise variance on DL and VUL may not
be the same, i.e., σ2j 6= σ2j and the total sum-power on the DL and VUL may not
be satisfied with equality, though there is no mathematical violation. The implicit
Lagrange-duality in Algorithm 1 satisfies both these conditions while considering the
per-BS power constraints, i.e.,
M∑
i=1
N∑
j=1
αijqij = pj, (5.3)
σ2j = σ
2
j , ∀j. (5.4)
and eliminates the need to treat VUL-noise as an optimization variable.
5.2 Problem Reformulation
P2p is reformulated such that it includes the Lagrange-duality and single-cell duality.
The reformulation begins with the single-cell duality requirements from equations (5.3)
and (5.4). For a given α, the VUL sum-power is bounded by the total available power
at the associated BS, i.e.,
N∑
j=1
pj ≤
M∑
i=1
P¯i. (5.5)
Equations (5.3) and (5.5) imply that the total available power in the system is constant.
Implicit constraints include
M∑
i=1
N∑
j=1
αijqij ≤
M∑
i=1
P¯i, (5.6) pj ∈ [0,
M∑
i=1
P¯i], ∀j. (5.7)
57
It can be observed from equations (2.7) and (5.7) that each DL power variable qij is
bounded by the per-BS power constraint while each VUL power variable pj is bounded
by the total sum-power across all the BSs. Hence pj in general is not equal to its
corresponding qij and can be greater than the maximum available per-BS power P¯i.
In addition to the equality (5.4), the VUL-noise also remains constant, which is one
of the objectives of the reformulation. For a given α, p, U and achievable γj , ∀j, the
required DL power to satisfy γ
j
= γj , ∀j, is given by equation (5.2). To include the BSA
variable α into the reformulation, the elements of the diagonal desired signal matrix
D are modified as D˜jj = γj/
N∑
j=1
‖αijuHijhij‖22 and elements of crosstalk matrix C are
modified as C˜jk =
M∑
m=1
‖αmkuHmkhmj‖22 and C˜jj = 0. Under the feasibility conditions of
Perron Frobienus theory equation (5.2) changes to
q = (I− D˜ · C˜)−1 · D˜ · (σ21N ). (5.8)
The concept of single-cell duality is included into the constraints of P2p by equation
(5.8). P2b is also solved via VUL, i.e., an approach similar to solving P3b.
5.3 Downlink Rate and Power Allocation
With these additional constraints, P2 is reformulated and the same without any change
applies to P4. To solve P2b, the objective considers VUL-rate rj in equation (2.10)
instead of DL-rate rj in equation (2.11), i.e., the objective has a VUL-rate instead of
DL-rate. It is given as
P2v: maximize
α,q,P,U
N∑
j=1
rj (5.9)
subject to :
(2.1),(2.2),(2.7),(2.8),(2.10),
(5.3),(5.5),(5.6),(5.7),(5.8).
(5.10)
It is not required to include all the constraints in equation (5.10), e.g., equation (5.6)
can be dropped since satisfying equation (2.8) will satisfy equation (5.6). Similarly,
equation (5.5) and the upper bound on pj in equation (5.7) are dropped due to equation
(5.3). For P2v, two sub-problems P2vb and P2vp are formulated to solve for BSA and
PA respectively. The VUL BSA sub-problem is
P2vb: maximize
α
N∑
j=1
rj (5.11)
subject to: (2.1),(2.2),(2.8),(2.10). (5.12)
58
The reformulated VUL PA sub-problem to solve P2vp is
P2vp: maximize
Q,p,U
N∑
j=1
rj (5.13)
subject to :
(2.7),(2.8),(2.10),(5.3),
(5.5),(5.6),(5.7),(5.8).
(5.14)
As before, U is a part of PA sum-problem and is updated whenever there is a change
in PA. Algorithm 5 gives the general sequence of steps to solve P2 via VUL. To solve
Algorithm 5 : General iteration steps to solve DL via VUL
1: Initialize: α, U, IPM parameters
2: repeat
3: solve P2vp by Algorithm-1, Output: Q, p, U
4: evaluate rj , ∀j
5: solve P2b, Output: α
6: until convergence
7: Output: BSA α, DL-power Q, DL beamforming vectors U and VUL-power p.
P4, similar VUL reformulation is obtained. Only the objective in P2 is changed to
solve P4. Due to equation (5.3), the VUL objective can be either equation (5.15) or
(5.16).
M∑
i=1
N∑
j=1
αijqij , (5.15)
N∑
j=1
pj . (5.16)
P4v: minimize
α,p,Q,U
N∑
j=1
pj (5.17)
subject to :
(2.1),(2.2),(2.7),(2.8),(2.10),
(5.3),(5.5),(5.6),(5.7),(5.8).
(5.18)
Similar VUL sub-problems P4vb for DL BSA and P4vp for DL PA are formulated for
P4. Constraints (5.12) and (5.14) are retained without any change to solve P4vb and
P4vp respectively. With P¯i = P¯ , ∀i, the constraint vector for the IPM in Algorithm
1 to solve P2vp and P4vp has a system wide constraint vector cs(p,q,U), a per-UE
constraint vector cj(q,U), ∀j, a per-BS constraint vector ci(q), ∀i. They are given as
cs(p,q,U) =


P¯ ·M − M∑
i=1
N∑
j=1
αijqij
P¯ ·M − N∑
j=1
pj
N∑
j=1
pj−
M∑
i=1
N∑
j=1
αijqij
q− (I− D˜ · C˜)−1 · D˜ · (σ21N )


, (5.19)
59
cj(q,U) =


P¯ −αTj qj
rj− rth
P¯ −qj
qj

 , ∀j, (5.20)
ci(q) =
[
P¯ −αTi qi
]
, ∀i. (5.21)
For link-level simulation of a minimum working example, σ2 = σ21, P¯i = 33 dBm, ∀i,
T = 2 is set. The actual maximum available DL power at the macro cell is 46 [dBm]. So
for the demonstration of the Algorithm via duality, a value higher than the previously
assumed UL values is set for the transmit power. The problem is assumed feasible and
the channel state is perfectly known. hij , ∀i,∀j, is i.i.d. zero-mean complex Gaussian
with variance 0.5 per dimension, i.e., hij = CN (0,IT ). MMSE beamforming vectors
(2.48) that maximize the received SINR are assumed for uij .
2 4 6 8 10 12 14
33.5
34
34.5
35
35.5
36
iteration index
su
m
-p
ow
er
[d
B
m
]
 
 
DL sum−power
VUL sum−power
Fig. 5.1: Convergence of the objective in
P2vp for a feasible realization.
2 4 6 8 10 12 14
0.3
0.6
0.9
1.2
1.5
1.8
iteration index
U
E
-r
at
e
[b
/s
/H
z]
 
 
r1
r2
r3
r1
r2
r3
Fig. 5.2: Convergence of the DL and VUL
UE rates in P2vp for a feasible realization.
Figure 5.1 shows the convergence of the objective for the P2vp sub-problem for a fea-
sible channel instance with N = 3, rth = 0.5, M = 2 and a given random α. Again
these values are random to demonstrate the minimum working example. They can be
scaled to match the actual requirement. It shows that both DL-power and VUL-power
converge to the same value after starting from random initial points. As described in
chapter 3, these initial points can be feasible or infeasible and the problem remains
infeasible until the final convergence. At convergence the total sum-power in DL and
VUL remains the same, which is one of the requirements of the reformulation. For
a given BSA, only one element in qj is finite, this corresponds to the αij = 1 in αj .
The remaining (M − 1) elements in it correspond to the 0−elements. As mentioned
before, in general for a UEj , pj is not equal to the corresponding non-zero qij . Figure
5.2 shows the convergence of the UE rates for the same feasible instance of Figure
5.1. The DL-rate rj and the corresponding VUL-rate rj in P2vp converge to the same
value, i.e., satisfying the definition of DL-VUL duality. With this Lagrange-duality of
60
the IPM the single-cell duality becomes implicit in the formulation. The plots con-
verge as required under the same total sum-power equality on the DL and VUL and
also under the same VUL and DL noise variance.
12 14 16 18 20 22 24 26 28 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sum-power [dBm]
cu
m
u
la
ti
ve
d
is
tr
ib
u
ti
on
fu
n
ct
io
n
 
 
DL-P4vp
VUL-P4vp
DL-P4vm
VUL-P4vm
Fig. 5.3: CDF of VUL and DL sum-power
for P4vp, P4vm.
0 0.05 0.1 0.15 0.2 0.25 0.3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
max-min rate [b/s/Hz]
cu
m
u
la
ti
ve
d
is
tr
ib
u
ti
on
fu
n
ct
io
n
 
 
rj
rj
rmax-min
Fig. 5.4: CDF of max-min rate for P4vp,
P4vm for a UE.
As explained in chapter 3, the sum-power minimization problem with QoS is coupled
to the max-min problem. The feasibility of the later leads to the formulation of the
former. A multi-cell max-min problem on the DL [141] can be formulated similar to P5
on the UL. For a set of homogeneous UEs, the DL max-min problem achieves a balanced
QoS across all the UEs after convergence. The max-min problem with VUL-noise as
an optimization variable (P4vm) [141] is one way to satisfy duality. For comparison,
P4vm is first solved for a randomly assigned BS and then the obtained balanced max-
min rate rmax-min is used as the QoS for P4v. Optimum α is not known for P4vm
until an exhaustive search is carried out. So to avoid an exhaustive BSA search a
random feasible assignment is considered. Figure 5.3 plots the CDF of P4vp and P4vm
for M = N = 3. It is observed that the DL sum-power and the VUL sum-power in
P4vp, the DL sum-power in P4vm converge to the same value. However, the VUL
sum-power of P4vm is shifted the left, i.e., the sum-power on the VUL is lower than
its corresponding sum-power on the DL. VUL-noise σ2j ,∀j, is an optimization variable
in P4vm. It has been observed from the simulation results that σ2j ,∀j, converges to
a value less than its corresponding σ2j = 1,∀j. So the thermal noise variance in the
UE interference term is lower in the VUL and so is the VUL sum-power. Single-cell
duality requirement in equation (5.3) is satisfied by P4vp but not by P4vm. Also for
P4vp, σ2j = σ
2
j = 1, ∀j, a constant and not an optimization variable. Figure 5.4 shows
that the required rmax-min, i.e., rth is achieved by P4vp on both DL and VUL. Since
every UE has the same QoS value, only one UE rate is plotted. Also, P4vb converges
to the same chosen random BSA of P4vm.
Figure 5.5 compares the CDF of P4v with the VUL sum-power objective (5.15) and DL
sum-power objective (5.16) for various (N,rth) and M = 2. As mentioned before, both
converge to the same optimal. An increase in N or rth increases the objective, shown by
61
16 18 20 22 24 26 28 30 32
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sum-power [dBm]
cu
m
u
la
ti
ve
d
is
tr
ib
u
ti
on
fu
n
ct
io
n
 
 
(5, 0.15)
(5, 0.25)
(3, 0.15)
(3, 0.25)
DL-power
Fig. 5.5: CDF of P4v with VUL objective
and DL objective.
2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sum-rate [bits/sec/Hz]
cu
m
u
la
ti
ve
d
is
tr
ib
u
ti
on
fu
n
ct
io
n
 
 
(5, 0)
(5, 0.2)
(3, 0.05)
(3, 0.2)
Fig. 5.6: CDF of sum-rate for P2v.
the right shift in CDF curve. Figure 5.6 plots the CDF of the sum-rate objective in P2v
for various (N,rth) andM =2. For a given N , the CDF curve shifts to left with increase
in rth, i.e., a drop in the objective. This is due to the increased MUI at each UE and
also due to the competing nature of UEs to obtain resources. Maximum throughput
is achieved when rth = 0. This is the simple sum-rate maximization problem. This
increase is due to the MUD, where in the interest of the system, UEs with good channel
conditions are served with a higher resource value. UEs with unfavourable conditions
may have the service deprived or may have a low resource value allocated. There is
also an increase in the outage for this case.
0.1 0.15 0.2 0.25 0.3
21
22
23
24
25
26
27
28
29
rth [b/s/Hz]
av
er
ag
e
su
m
-p
ow
er
[d
B
m
]
 
 
N=5
N=3
Fig. 5.7: Average sum-power vs QoS in
P4v .
0 0.05 0.1 0.15 0.2
4
4.2
4.4
4.6
4.8
5
rth [b/s/Hz]
av
er
ag
e
su
m
-r
at
e
[b
/s
/H
z]
 
 
N=5
N=3
Fig. 5.8: Average sum-rate vs QoS in P2v .
Figure 5.7 plots the average sum-power vs QoS for different N in P4v. An increase in
objective with increase in rth and N is observed. This increment is due to the increase
in MUI at each UE. On the VUL, an increase in the MUI prompts each UE to increase
their respective power level to achieve the same QoS, thereby increasing the overall
power in the system. Figure 5.8 plots the average sum-rate vs QoS for different N in
62
P2v. For a given N , the maximum objective value is achieved for rth = 0 due to MUD.
Also, the sum-rate objective reduces with the increase in rth. The upward shift in the
curve from N = 3 to N = 5 shows an increase in the objective value. But the shift is
reduced with increase in rth. This shows the interference-limited nature of the shared
channel and the competing nature of UEs.
63
64
Chapter 6
Conclusion
In this thesis various multi-cell multi-user resource allocation problems based on rate
maximization and power allocation under perfect Channel State Information (CSI),
statistical-CSI and imperfect-CSI were addressed on both the Uplink and Downlink.
These optimization problems are in general Mixed Integer Nonlinear Programming
problems. A mathematical framework based on Interior Point Method (IPM) was
identified to solve a variety of non-linear optimization problems. This framework
eliminates the need to choose problem specific mathematical tool. However, several
other mathematical tools exist to obtain efficient distributed processing. The Integer
Programming sub-problem of Base Station Association was also solved as a Nonlinear
Programming sub-problem. This IPM facilitates simultaneous solving of both the Base
Station Association and Power Allocation sub-problems. A new definition of feasibility
via ℓ1−norm handle has been implemented for the Power Allocation. Under statistical
Channel State Information, the concept of tail distributions in Extreme Value The-
ory (EVT) has been used to solve the problems involving Probabilistic Constraints.
To handle the Chance Constrained problems, the definitions of risk measurement ele-
ments of Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) from the field of
finance were applied. To find the optimal solution with the Probabilistic Constraints,
the concept of VaR and a block-maxima based Generalized Extreme Value Distribution
in EVT are together implemented. This reformulation has an advantage that the CDF
of the unknown random variable need not be invertible and no approximations were
required. It can be extended to most of the distributions where the quantile function
does not exist. Performance loss based on the finite probability of exceeding the eval-
uated VaR is also considered. While evaluating this, CVaR and block-maxima based
Generalized Pareto Distribution in EVT are combined to overcome the non-availability
of closed form expressions. The VaR concept can also be extended to the resource al-
location problem under imperfect-CSI after identifying the correct distribution of the
imperfect channel. EVT based approach reformulates the problem into a Nonlinear
Programming problem which can be solved by the IPM framework. The generated
bounds clearly outperforms the existing results. On the highly coupled Downlink, the
duality principle has been implemented based on the implicit Lagrange-duality of the
65
IPM. The IPM maintains consistency with the single-cell duality theory and makes no
change to the noise components, which were excluded by the existing methods.
In this thesis, a small fraction of the mathematics as applied to mobile wireless com-
munication has been addressed. The presented results have been numerically solved
for a link-level setup under Rayleigh fading. To demonstrate the effectiveness of the
considered mathematical tools, a minimum working example has been considered in
each case. The analysis and the results contribute to a better understanding of a con-
ventional wireless communication system. It is evident that this work needs a further
investigation. As a part of future work, to realize the effectiveness of the obtained re-
sults, actual channel models and system-level parameters must be considered. Though
in this thesis, a binary 0− 1 problem has been addressed as a Base Station Associa-
tion problem, with an additional power constraint inclusion, a multi-carrier resource
allocation problem can be solved. Multi-carrier analysis is a more practical problem,
since most of the deployed wireless networks are multi-carrier in nature. The cur-
rent research trends demand the network architecture to be more ad-hoc, small-cell
oriented and decentralized. Including the heterogeneous setup, where multiple small
cells coexist in a macro cell is an interesting setup to further extend these results.
The obtained results assume a centralized processing. Decentralized processing, the
actual amount of CSI and processing capability of the network entities must also be
addressed for more robust analysis. The derived results must also include cross layer
architecture, since the functionalities at each layer are interdependent. To include all
these high complexity design considerations, experimental data combined with efficient
simulation environment is a must for the challenging task of Resource Allocation and
Interference Management.
66
Publications
[1] Chitti. K, Tan. X, ten Brink. S.: "Multi-User Downlink Resource Allocation with
per-Base Station Power Constraints", 21st IEEE Symposium on Communications
and Vehicular Technology (IEEE SCVT), Nov 2014, Delft, Netherlands.
[2] Chitti. K, Tan. X, ten Brink. S.: "Multi-User Uplink Power Control un-
der Imperfect-CSI and Probabilistic Constraints", 2014 International Conference
on Wireless Communications and Signal Processing (WCSP), Oct 2014, Hefei,
China.
[3] Chitti, K.; Speidel, J.: "MU-MIMO Power Control under Statistical CSI and
Probabilistic Constraints", 11th International Symposium on Wireless Commu-
nication Systems (ISWCS), Aug 2014, Barcelona, Spain.
[4] Chitti. K, Speidel. J.:"Joint Base Station Association and Power Allocation for
Uplink Sum-Power Minimization", 2013 IEEE 78th Vehicular Technology Con-
ference: VTC 2013-Fall, September 2013, Las Vegas, USA.
[5] Chitti, K; Kuang, Q; Speidel, J.: Joint Base Station Association and Power
Allocation for Uplink Sum-Rate Maximization, 2013 IEEE 14th Workshop on
Signal Processing Advances in Wireless Communications (SPAWC), Darmstadt,
Germany, Jun. 2013.
67
68
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