1. technology and computing

Coordinated Transmission for
Wireless Interference Networks
HAMED FARHADI
Doctoral Thesis in Telecommunications
Stockholm, Sweden 2014
TRITA-EE 2014:064
ISSN 1653-5146
ISBN 978-91-7595-391-5
KTH, School of Electrical Engineering
Department of Communication Theory
SE-100 44 Stockholm
SWEDEN
Akademisk avhandling som med tillst˚
and av Kungl Tekniska h¨
ogskolan framl¨
agges
till offentlig granskning f¨
or avl¨
aggande av teknologie doktorsexamen i telekommunikation fredagen den 19 December 2014 klockan 13.15 i h¨
orsal F3, Lindstedtsv¨agen 26, Stockholm, Sweden.
© 2014 Hamed Farhadi, unless otherwise noted.
Tryck: Universitetsservice US AB
Abstract
Wireless interference networks refer to communication systems in which multiple
source–destination pairs share the same transmission medium, and each source’s
transmission interferes with the reception at non-intended destinations. Optimizing
the transmission of each source–destination pair is interrelated with that of the other
pairs, and characterizing the performance limits of these networks is a challenging
task. Solving the problem of managing the interference and data communications
for these networks would potentially make it possible to apply solutions to several
existing and emerging communication systems.
Wireless devices can carefully coordinate the use of scarce radio resources in
order to deal effectively with interference and establish successful communications.
In order to enable coordinated transmission, terminals must usually have a certain level of knowledge about the propagation environment; that is, channel state
information (CSI). In practice, however, no CSI is a priori available at terminals
(transmitters and receivers), and proper channel training mechanisms (such as pilotbased channel training and channel state feedback) should be employed to acquire
CSI. This requires each terminal to share available radio resources between channel training and data transmissions. Allocating more resources for channel training leads to an accurate CSI estimation, and consequently, a precise coordination.
However, it leaves fewer resources for data transmissions. This creates the need to
investigate optimum resource allocation. This thesis investigates an informationtheoretic approach towards the performance analysis of interference networks, and
employs signal processing techniques to design transmission schemes for achieving
these limits in the following scenarios.
First, the smallest interference network with two single-input single-output
(SISO) source–destination pairs is considered. A fixed-rate transmission is desired
between each source–destination pair. Transmission schemes based on point-topoint codes are developed. The transmissions may not always attain successful
communication, which means that outage events may be declared. The outage probability is quantified and the ǫ-outage achievable rate region is characterized. Next,
a multi-user SISO interference network is studied. A pilot-assisted ergodic interference alignment (PAEIA) scheme is proposed to conduct channel training, channel
state feedback, and data communications. The performance limits are evaluated,
and optimum radio resource allocation problems are investigated. The analysis is extended to multi-cell wireless interference networks. A low-complexity pilot-assisted
opportunistic user scheduling (PAOUS) scheme is proposed. The proposed scheme
includes channel training, one-bit feedback transmission, user scheduling and data
transmissions. The achievable rate region is computed, and the optimum number
of cells that should be active simultaneously is determined. A multi-user MIMO
interference network is also studied. Here, each source sends multiple data streams;
specifically, the same number as the degrees of freedom of the network. Distributed
transceiver design and power control algorithms are proposed that only require local
CSI at terminals.
Sammanfattning
Tradl¨
˙ osa interferensn¨atverk ¨
ar kommunikationssystem d¨
ar flera par av k¨allor och
destinationer delar pa˙ samma ¨
overf¨
oringsmedium och varje k¨allas s¨andning st¨or
mottagningen vid icke-avsedda destinationer. Optimering av data¨
overf¨oringen f¨or
varje par av k¨allor och destinationer samverkar med optimeringen f¨or de andra
paren, och att karakterisera prestandagr¨
anserna f¨or dessa n¨
atverk ¨ar d¨
arf¨or en utmanande uppgift. Om problemet med hantering av st¨orningar och datakommunikation f¨or dessa n¨
atverk l¨oses, kan dessa l¨osningar potentiellt till¨
ampas pa˙ flera
befintliga och kommande kommunikationssystem.
Tradl¨
˙ osa apparater kan noggrant samordna anv¨andningen av begr¨
ansade radioresurser f¨or att effektivt hantera st¨
orningar och skapa tillf¨
orlitlig kommunikation.
F¨or att m¨ojligg¨
ora samordnad ¨
overf¨
oring maste
˙
terminalerna oftast ha en viss niva˙
av kunskap om utbredningsmilj¨on, dvs. kanaltillstandsinformation
˙
(CSI). I praktiken finns dock a priori ingen CSI pa˙ terminalerna och l¨ampliga kanaltr¨aningsmekanismer (t.ex. pilotbaserad kanaltr¨aning och kanaltillst˙andsaterkoppling)
˙
b¨
or anv¨andas f¨or att f¨orv¨arva CSI. Detta inneb¨ar att varje terminal maste
˙
dela tillg¨angliga
radioresurser mellan kanaltr¨aning och data¨
overf¨oring. Ju fler resurser som tilldelats f¨or kanaltr¨aningen, desto noggrannare blir CSI-uppskattningen och d¨
armed
blir samordningen b¨
attre. Dock ¨
ar f¨
arre resurser kvar f¨
or data¨
overf¨oring. Effektiva
metoder f¨or att genomf¨
ora resursf¨
ordelningen maste
˙
d¨
arf¨or unders¨
okas. En informationsteoretisk strategi f¨
or prestandaanalys av interferenssn¨
atverk utreds och signalbehandlingstekniker anv¨ands f¨
or att utforma ¨overf¨oringssystem som uppnar
˙ dessa
gr¨anser. Denna avhandling behandlar samordnad ¨overf¨oring i f¨oljande scenarier.
F¨orst unders¨
oks det minsta interferensn¨atverket, bestaende
˙
av tva˙ single-input
single-output-par (SISO-par) med k¨allor och destinationer. En ¨overf¨oringsmetod
med fast datatakt ¨
onskas mellan varje par av k¨allor och destinationer, givet ett bivillkor pa˙ anv¨and effekt. ¨
overf¨
oringssystem baserade pa˙ punkt-till-punkt Gaussiska
koder utvecklas d¨
arf¨
or. ¨
overf¨
oringarna uppnar
˙ inte alltid tillf¨
orlitlig kommunikation, och avbrott kan d¨
arf¨
or f¨
orekomma. Avbrottssannolikheten kvantifieras och ǫavbrottsdatataktsregionen karakt¨ariseras. D¨arefter studeras ett fleranv¨andarinterferensn¨atverk med SISO-noder. Ett pilotassisterat system med ergodisk interferensuppr¨atning (PAEIA) f¨
oreslas
˙ f¨
or att genomf¨
ora kanaltr¨aning, kanaltillstands
˙
aterko˙
ppling och datakommunikation. Prestandagr¨anser utv¨
arderas, och problemet med
optimal radioresursallokering studeras. Ett fleranv¨andarinterferensn¨atverk med multiple-input multiple-output-noder (MIMO-noder) studeras ocksa.
˙ H¨
ar s¨ander varje
k¨alla lika manga
˙
datastr¨
ommar som tillats
˙ av frihetsgraderna av n¨
atverket. Distribuerad design av mottagare och s¨andare samt effektstyrningsalgoritmer som enbart kr¨
aver lokal CSI vid terminalerna f¨
oreslas.
˙ Analysen utvidgas ocksa˙ till ett fleranv¨andarinterferensn¨atverk med flera celler. Ett pilotassisterat lagkomplexitetssyst˙
em f¨or opportunistisk anv¨andarschemal¨
aggning f¨oreslas.
˙ Det f¨oreslagna systemet
omfattar kanaltr¨aning, en-bitsaterkoppling,
˙
anv¨andarschemal¨
aggning samt data¨
overf¨oring. Den uppnabara
˙
datataktsregionen ber¨aknas och det optimala antalet celler
som skall vara aktiva samtidigt best¨
ams.
Acknowledgments
It is a pleasant task to express my thanks to all those who contributed in many
ways to the success of my PhD study and made it an unforgettable experience for
me. Foremost, I owe my deepest gratitude to my advisor Prof. Mikael Skoglund.
I am grateful to Mikael for welcoming me to the Department of Communication
Theory, and for giving me the freedom to pursue my research interests. His openness
to ideas, insightful suggestions, and supports enriched my PhD research. I also
wish to thank my co-advisor Prof. Chao Wang for his valuable helps and supports
through different stages of my research and for being a good friend of me all these
years. It was a great pleasure for me to work with Mikael and Chao.
I would like to thank Prof. Vahid Tarokh at Harvard University for giving me
the opportunity to visit his research group. It has been an honor to work with him.
I really appreciate his kind hospitality during my stay at Cambridge. This research
visit has broadened my view on research, tremendously. I wish to thank Dr. Jinfeng
Du at MIT for all his valuable helps and cares during the period of my stay at
Cambridge. I am also thankful to Dr. Mohsen Farmahini and Alireza Mehrtash for
all their kind helps during this visit. I am also grateful to all friends at SEAS who
made my time enjoyable and rewarding. Karin Demin and Kathleen Masse helped
me a lot in the administrative issues of this visit.
I wish to acknowledge the John and Karin Engblom foundation, Knut and Alice
Wallenbergs foundation, Ericsson Research foundation, Qualcomm, and the European School of Antennas for the financial support of the trips for the research visit
and the conferences that I participated. Swedish Foundation for Strategic Research
is acknowledged for the financial support of my PhD study.
I would like to take the opportunity to thank Prof. David Gesbert for acting as
the opponent for this thesis. I also thank Prof. Erik Larsson, Prof. Tommy Svensson,
and Prof. Jeong Woo Cho for acting on the grading committee, and Prof. Joakim
Jaldén for doing the quality review of the thesis.
My sincere thanks also goes to my colleagues whom I had valuable discussions
and collaborations. In particular, I am grateful to Prof. Lars Rasmussen for sharing his valuable international academic career experiences and being a source of
inspiration for good academic practice. I also wish to thank Prof. Carlo Fischione
for collaborations on the problems of mutual interest. I would like to thank Dr.
Per Zetterberg for providing a fantastic wireless test-bed network for the real-time
evaluation of algorithms proposed in this thesis and for his valuable insights on the
practical considerations of wireless system design. Also, I am thankful to Nima Najari Moghadam for test-bed implementation and real-time measurements of some
algorithms proposed in this thesis. It was a pleasure to work with Nima and Per.
I gratefully acknowledge the discussions with my colleagues from KTH, Link¨oping
University, and Ericsson Research within the RAMCOORAN project cooperation.
Prof. Michail Matthaiou and Prof. Mats Bengtsson are acknowledged for valuable
comments on my research. I wish to thank Dr. Majid Nasiri Khormuji for being al-
viii
ways available for fruitful discussions and research collaborations; Peter Larsson for
sharing his valuable industrial research experiences; Dr. Mohammadreza Gholami
for sharing his experiences; and Dr. Themistoklis Charalambous for discussions on
the problems of common interest. Dr. Nafiseh Shariati is acknowledged for being
always available for great conversations on various topics. I acknowledge the great
discussions with Prof. Tobias Oechtering, Frédéric Gilbert Gabry, and Efthymios
Stathakis regards our collaborative efforts on the implementation of interactive
teaching. I am also very thankful to Dr. Ali Zaidi, Hadi Ghauch, and Dr. Nicolas
Schrammar for our scientific collaborations. I wish to thank Nan Li, Hadi Ghauch,
Farshad Naghibi, and especially Chao and Mikael for helping me proofread the
thesis and for their valuable suggestions and feedbacks. I am thankful to Rasmus
Brandt for his kind helps in editing the Swedish parts of this thesis. I also thank
the computer support group for providing reliable resources, and Raine Tiivel and
Dora S¨
oderberg for all their kind helps in administrative issues. I wish to thank all
my past and present colleagues at the Department of Communication Theory and
the Department of Signal Processing for making a friendly working environment.
Especial thanks go to my officemate Iqbal Hussain. It was so nice to share the office
with Iqbal all these years. I enjoyed after work Thai dinner events with Kittipong
and Hieu!
I would like to take this opportunity to acknowledge all individuals who have
inspired or have encouraged me to do research. In particular, I am thankful to
my teachers at KTH, University of Tehran, and Iran University of Science and
Technology from whom I’ve learned a lot during years of study.
My friends Maksym Girnyk, Karina, Farshad Naghibi, Serveh Shalmashi, Euhana Ghadimi, Somayeh Salimi, Elaheh Jafari, Nafiseh Shariati, Amirpasha Shi¨
razinia, Alla,
Arash Owrang, Ehsan Olfat, Majid Gerami, and Ghazaleh Panahandeh in various ways made living in the beautiful city of Stockholm an enjoyable and
memorable experience for me.
I offer my warmest thanks to my parents for all their endless love and supports.
My father was my most inspiring teacher and his caring relationships with students
deeply influenced my view on teaching and education. I wish to thank my brothers
for all their encouragements. Last but not least, I would like to thank my beloved
wife Maryam for all the happiness she brought to my life and all her patience during
my work on this thesis. If it was not her patience, I wouldn’t be able to work on
this thesis.
Hamed Farhadi
Stockholm, November 2014
Contents
Contents
ix
List of Figures
xiii
List of Notations
xv
List of Acronyms
xvii
1 Introduction
1.1 Wireless Network Operations Overview . . . . . . . . . . . . . .
1.2 Thesis Scope and Contributions . . . . . . . . . . . . . . . . . . .
1
5
6
2 Background
2.1 Wireless Interference Networks . . . . . . . . . . . . . . . . . .
2.2 Two-user Interference Networks . . . . . . . . . . . . . . . . . .
2.3 K-user (K > 2) Interference Networks . . . . . . . . . . . . . .
2.3.1 Achievable Degrees of Freedom Region . . . . . . . . . .
2.3.2 Interference Alignment for MIMO Interference Networks
2.3.3 Interference Alignment for SISO Interference Networks .
2.3.4 Ergodic Interference Alignment . . . . . . . . . . . . . .
2.4 Wireless Interference Networks with Imperfect CSI . . . . . . .
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3 Two-user Interference Networks: Point-to-Point Codes
3.1 Two-user SISO Interference Network . . . . . . . . . . . . . .
3.2 Orthogonal Transmission Scheme . . . . . . . . . . . . . . . .
3.3 Non-Orthogonal Transmission Schemes . . . . . . . . . . . . .
3.3.1 Direct Decoding at Both Receivers . . . . . . . . . . .
3.3.2 Successive Interference Cancellation at Both Receivers
3.3.3 Successive Interference Cancellation at One Receiver .
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.A The Proof of Proposition 3.2.1 . . . . . . . . . . . . . . . .
3.B The proof of Proposition 3.3.1 . . . . . . . . . . . . . . . .
3.C The proof of Corollary 3.3.1 . . . . . . . . . . . . . . . . .
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Contents
3.D The Proof of Proposition 3.3.2 . . . . . . . . . . . . . . . . . .
3.E The Proof of Proposition 3.3.3 . . . . . . . . . . . . . . . . . .
61
62
4 K-user SISO Interference Networks: Pilot-assisted Interference Alignment
4.1 Multi-user SISO Interference Network . . . . . . . . . . . . . . .
4.2 Pilot-assisted Ergodic Interference Alignment . . . . . . . . . . .
4.2.1 Pilot Transmission Phase . . . . . . . . . . . . . . . . . .
4.2.2 Feedback Transmission Phase . . . . . . . . . . . . . . . .
4.2.3 Data Transmission Phase . . . . . . . . . . . . . . . . . .
4.3 Analog Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Achievable Rate Region . . . . . . . . . . . . . . . . . . .
4.3.2 The Optimum Power Allocation . . . . . . . . . . . . . .
4.3.3 Achievable Degrees of Freedom Region . . . . . . . . . . .
4.3.4 Numerical Evaluation . . . . . . . . . . . . . . . . . . . .
4.4 Digital Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Power Control Problem . . . . . . . . . . . . . . . . . . .
4.4.2 Throughput Maximization Problem . . . . . . . . . . . .
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.A The Proof of Proposition 4.3.2 . . . . . . . . . . . . . . . . . .
4.B The Proof of Theorem 4.4.3 . . . . . . . . . . . . . . . . . . .
4.C The Proof of Theorem 4.4.4 . . . . . . . . . . . . . . . . . . .
65
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76
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100
5 K-user MIMO Interference Networks: Transceiver Design
and Power Control
5.1 Multi-user MIMO Interference Network . . . . . . . . . . . . . .
5.1.1 Transmitter Structure . . . . . . . . . . . . . . . . . . . .
5.1.2 Receiver Structure . . . . . . . . . . . . . . . . . . . . . .
5.2 Transceiver Design and Power Control . . . . . . . . . . . . . . .
5.2.1 CSI Acquisition, Transceiver Design, and Power Control .
5.2.2 Distributed Power Control . . . . . . . . . . . . . . . . . .
5.3 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . .
5.4 Test-bed Implementation . . . . . . . . . . . . . . . . . . . . . .
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.A The Proof of Theorem 5.2.1 . . . . . . . . . . . . . . . . . . . .
5.B The Proof of Theorem 5.2.2 . . . . . . . . . . . . . . . . . . . .
101
102
102
103
103
104
107
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126
126
6 Multi-cell Interference Networks: Pilot-assisted Opportunistic User Scheduling
6.1 Multi-cell Interference Network . . . . . . . . . . . . . . . . . . .
6.2 Pilot-assisted Opportunistic User Scheduling Scheme . . . . . . .
6.2.1 Pilot Transmission Phase . . . . . . . . . . . . . . . . . .
6.2.2 Feedback Transmission and User Selection Phase . . . . .
6.2.3 Data Transmission Phase . . . . . . . . . . . . . . . . . .
129
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131
131
132
133
Contents
6.3
Achievable Rate Region . . . . . . . . . . .
6.3.1 Achievable Total Degrees of Freedom
6.3.2 Numerical Evaluation . . . . . . . .
6.4 Summary . . . . . . . . . . . . . . . . . . .
6.A The Proof of Theorem 6.3.1 . . . . . . . .
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7 Conclusion
7.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
141
141
143
Bibliography
147
List of Figures
1.1
Operations in a wireless communication network. . . . . . . . . . . .
5
2.1
2.2
2.3
K-user wireless interference network . . . . . . . . . . . . . . . . . .
Interference management schemes . . . . . . . . . . . . . . . . . . .
Interference alignment in a three-user MIMO interference network .
22
24
28
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
Two-user interference network. . . . . . . . . . . . . . . . . . . . . .
Solution of the power control problem for the NOT1 scheme. . . . .
The outage probability of the NOT1 scheme . . . . . . . . . . . . . .
The ǫ-outage achievable rate region of the NOT1 scheme . . . . . . .
Solution of the power control problem for the NOT2 scheme . . . . .
The outage probability of the NOT2 scheme . . . . . . . . . . . . . .
The ǫ-outage achievable rate region of the NOT2 scheme . . . . . . .
Solution of the power control problem for the NOT3 scheme. . . . .
The outage probability of the NOT3 scheme in a symmetric network
The outage probability of the NOT3 scheme in a asymmetric network
The ǫ-outage achievable rate region of the NOT3 scheme . . . . . . .
36
42
43
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54
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57
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
Transmitted symbols’ format in a K-user interference network. .
The optimum power allocation factor. . . . . . . . . . . . . . . .
The achievable rate of the PAEIA and TDMA schemes. . . . . .
The achievable sum-rate versus the number of users. . . . . . . .
Feasibility probability versus transmission rate of each user . . .
Average transmission power versus transmission rate of each user
Feedback bits trade-off in a three-user interference network . . .
Throughput versus power . . . . . . . . . . . . . . . . . . . . . .
Feedback bits allocation trade-off . . . . . . . . . . . . . . . . . .
Throughput of a K-user interference network. . . . . . . . . . . .
Delay-limited throughput of a three-user network . . . . . . . . .
Delay-limited throughput of a K-user interference network . . .
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5.1
The structure of a transmitter and receiver pair . . . . . . . . . . . .
102
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xiv
List of Figures
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
CSI acquisition, transceiver design, and power control procedure . .
Feasibility probability versus transmission rate. . . . . . . . . . . . .
Feasibility probability in a network with maximum power constraint.
The mutual information of the source-destination pairs versus iterations
Transmission power of a source versus number of iterations . . . . .
The mutual information of the source-destination pairs versus iterations
The computed transmission power versus the number of iterations .
The mutual information of the different streams versus iterations . .
Transmission powers versus the number of iterations . . . . . . . . .
Computed power versus the number of iterations . . . . . . . . . . .
104
110
112
119
120
121
122
123
124
125
6.1
6.2
6.3
6.4
6.5
Schematic representation of different phases of the PAOUS scheme
Transmitted symbols within one fading block. . . . . . . . . . . . .
The achievable sum-rate versus power. . . . . . . . . . . . . . . . .
The achievable sum-rate versus the decision threshold. . . . . . . .
The achievable sum-rate versus the power allocation factor. . . . .
130
133
136
137
138
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List of Notations
|x|
CN (m, σ 2 )
f ′ (x)
∅
ν i [X]
≻
E[X]
λmax (X)
N (m, σ 2 )
X−1
ℑ[x]
span(X)
Nf
fX
Q(x)
∆
rank(X)
ℜ[x]
ρ(X)
Tr[X]
XT
X∗
Absolute value of x
Complex Gaussian distribution with mean m and variance σ 2
Derivative of function f (x)
Empty set
Eigenvector corresponding to the ith lowest eigenvalue of matrix X
Element-wise vector inequality
Element-wise strict vector inequality
Expectation of random variable X
The largest absolute eigenvalue of matrix X.
Gaussian distribution with mean m and variance σ 2
Inverse of matrix X
Imaginary part of x
Linear span of columns of matrix X
Number of feedback bits
Probability density function of random variable X
Q-function
Quantization step-size
Rank of matrix X
Real part of x
Spectral radius of matrix X
Trace of matrix X
Transpose of matrix X
Transpose conjugate of matrix X
xv
List of Acronyms
AWGN
CoMP
CSI
CSIR
CSIT
DoF
DSM
EIA
EIA-PC
EIA-RA
FDD
FDMA
KKT
LICQ
M2M
MMSE
MIMO
OT
PAEIA
PAEIA-US
PAOUS
pdf
QoS
RHS
SNR
SIC
SINR
SISO
TDD
TDMA
USRP
WBAN
Additive white Gaussian noise
Coordinated multi point
Channel state information
Channel state information at receiver
Channel state information at transmitter
Degrees of freedom
Discrete superposition model
Ergodic interference alignment
Ergodic interference alignment with power control
Ergodic interference alignment with rate adaptation
Frequency-division duplex
Frequency-division-multiple-access
Karush-Kuhn-Tucker
Linear independent constraint qualification
Machine-to-machine communications
Minimum mean square error
Multiple-input multiple-output
Orthogonal transmission
Pilot-assisted ergodic interference alignment
Pilot-assisted ergodic interference alignment with user selection
Pilot-assisted opportunistic user scheduling
Probability density function
Quality of service
Right hand side
Signal-to-noise ratio
Successive interference cancelation
Signal-to-interference-plus-noise ratio
Single-input single-output
Time-division duplex
Time-division-multiple-access
Universal software radio peripheral
Wireless body area networks
xvii
Chapter 1
Introduction
A
NETWORKED SOCIETY in which everyone and everything, everywhere,
have the potential to benefit from connectivity, will shape the future of human life [Sac03]. Wireless communication systems are expected to play an
important role in the development of the networked society. Fore example, wireless networks will provide ubiquitous access to information, while wireless health
monitoring systems will decentralize medical treatments. Vehicular communication
networks will help make future transportation systems more efficient and safe, and
networked control systems will take advantage of global information fusion in order
to make intelligent decisions.
The trend towards massive usage of wireless technology has led to an explosion in the number of connected devices, and tremendous growth in wireless traffic
volume [Eri12]. However, the scarcity of radio resources and the inherent characteristics of the wireless transmission medium make handling communication in
these conditions a formidable task. The radio spectrum is scarce and is considered to be among the most expensive natural resources [Her85, ZKAQ01]. In addition, the energy budget of mobile terminals is restricted and there are serious
concerns regarding the vast energy consumption of wireless communication systems [CHA+ 11, DCG+ 13]. Consequently, spectral and energy efficient design is essential for emerging wireless technologies. Wireless transmission is generally subject
to two phenomena: fading and interference [PL]. The former is a consequence of
reflectors scattered in the environment surrounding a transmitter and a receiver,
such that the receiver observes a superposition of multiple copies of the transmitted
signal. The superposition of the signals can be either constructive or destructive
depending on the phase shift and the attenuation of received signals from different paths. The randomness of fading may degrade communication quality. Several
effective techniques have been devised in recent decades to overcome the adverse
effects of random fading. For instance, multiple-antenna transmission techniques
have been proposed to realize spatial diversity and to improve the performance of
wireless systems [TJC99, JSO02, GSDs+ 03, LS03].
Another, even more challenging, obstacle in the operation of wireless networks
1
2
Introduction
is interference. Because of the broadcast nature of wireless transmission medium,
a user can interfere with the communication of any other user. This occurs in
some existing and emerging wireless communication scenarios, including inter-cell
interference in cellular networks; ad-hoc networks; interference between neighboring access points in wireless local area networks; inter-user interference between
neighboring devices in machine-to-machine (M2M) communication scenarios; interference between implanted medical sensors in wireless body area networks (WBAN);
and interference between licensed and unlicensed users in cognitive radio networks.
These communication systems, usually composed of multiple information sources
and destinations, can be modelled as an important class of wireless networks: interference networks. In an interference network, each source intends to communicate
with its dedicated destination and all sources share the same transmission medium.
Inter-user interference makes communication of different users interrelated, and
if not properly managed, it can severely degrade communication quality. For instance, in a two-user interference network in which two sources intend to communicate with their respective destinations simultaneously in the same frequency
band [Ahl74], the transmission power of each source affects the signal detection,
not only at its desired destination, but also at the other destination. Therefore, optimizing the performance of each source-destination pair becomes interrelated with
that of the other pair, and it becomes challenging to characterize the performance
limits [AC78,HK81,ETW08,JLD08,LJ08,MK09]. The situation becomes even more
complicated, when the number of users exceeds two.
Wireless devices must carefully coordinate the use of the radio resources in order
to effectively manage interference and enable successful communication [GKGI07].
The term coordination has been widely used in various disciplines, including computer science, sociology, management science, systems theory, and economics [MC90].
The broad definition of coordination according to the American Heritage Dictionary [Dic81] is
“the act of working together harmoniously.”
In the context of wireless interference networks, this implies that the terminals work
together in order to design their transmit signals in such a way that the interference
can be managed properly. Although, they may have conflicts of interest in using
the available radio resources, they intend to cooperate with each other in order to
achieve a reasonably good performance.
Conventional interference management strategies (such as time division multiple
access (TDMA) or frequency division multiple access (FDMA)) tend to divide the
available radio resources and orthogonalize the transmissions of different source–
destination pairs. This requirement causes the subspaces of different interference
signals to be orthogonal to those of the desired signal at each destination, and also
orthogonal to each other. Interference is avoided at the cost of low spectral efficiency.
However, it has been shown recently that transmitters can shape their transmitted signals in an appropriate domain, such as space (by using multiple antennas),
Introduction
3
time (by coding across time-varying channels), or frequency (by coding across different carriers in frequency-selective channels) [MAMK08, CJ08, MGMAK14], or
they may perform opportunistic scheduling in dense networks [JwJS09, YSJP13],
in order to manage interference and conduct communication more efficiently. For
instance, the interference alignment concept [MAMK08, CJ08] reveals that, with
proper coordinated transmission signaling design, different interference signals can
be aligned together, such that more radio resources can be assigned to the desired transmission. Consider a multi-user interference network with more than two
source–destination pairs. At each destination, the interference signals can be aligned
so that so that up to half of the signal space can be left to its desired signal [CJ08].
This means that each user may achieve half of the interference-free transmission
rate, regardless of how many interferers exist. Moreover, in dense communication
networks in which a massive number of users should be served, an opportunistic
scheduling scheme can be deployed to schedule only those users that experience less
interference. This scheme can take advantage of the crowd of users to manage the
interference [JwJS09, YSJP13].
However, the realization of such coordinated transmission schemes can be considerably more challenging than in the case of the conventional orthogonal transmission strategies. For instance, global channel state information (CSI) must usually
be perfectly available at all sources and destinations to conduct the coordinated actions (for example, beamforming, scheduling, and resources allocation). Acquiring
such perfect global CSI is clearly a difficult problem in time-varying environments.
In practice, no CSI is a priori available at terminals and proper channel learning
schemes thus must be applied. The destinations can obtain an estimation of the
CSI through a pilot-based channel training scheme in which each source allocates a
portion of the total transmission time and energy for transmitting pilot symbols,
and the rest for data transmission. The quality of estimated CSI determines how
accurately the coordinated transmission scheme can be conducted and therefore affects the performance of each decoder. In general, each transmitter shares its radio
resources between pilot transmission and data transmission. More accurate channel
estimation can be obtained by allocating more resources for pilots. This implies
that fewer resources are left for data transmission. In such scenarios, the achievable
performance of the network needs to be analyzed carefully and the optimum radio
resource allocation is an important problem to be investigated.
One possible way for other terminals to obtain the CSI is to require the destination to share its estimated channel knowledge with others via channel state
feedback signals. [NJGV09, BT09, KV10, AH12] showed that as long as the capacity
of the feedback channel is sufficiently large – such that the CSI regarding the whole
network obtained by each terminal is accurate enough – interference alignment can
be realized as if perfect global CSI is available. Clearly, this requirement is not usually practical. In most existing systems, the capacity of feedback channels would be
strictly limited. Each terminal can attain only erroneous global CSI (for example,
the quantized global CSI if digital feedback is used [BT09, KV10, KG13], or noisy
global CSI if analog feedback is deployed [AH12]) or the CSI regarding only a part
4
Introduction
of the network (local CSI, for example [GCJ08]). Therefore, it may not be possible to perform interference alignment perfectly. Ideally, the interference signals at
each destination should be aligned together in the same subspace, which is distinguishable from the subspace for its desired signals, so that they can be completely
canceled. However, the limited CSI at each terminal means that it may no longer
be straightforward to perfectly separate these two subspaces. In other words, some
non-negligible interference would leak into the desired signal subspace and it would
not be possible to eliminate it. The communication performance is certainly affected
by such interference leakage. Similar to the above-mentioned case of two-user interference networks, the transmission power of each source influences the signal
detection at all destinations. Optimizing the performance of all source–destination
pairs is interrelated and is a challenging problem.
The objective of this thesis is to investigate the performance limits of wireless
interference networks and to design efficient coordinated transmission schemes to
achieve these limits. These schemes include employing channel training and channel feedback to acquire CSI, and using the estimation of CSI to coordinate data
transmission. We also address the optimum radio resource allocation for channel
training and data communication, and study the existing tradeoff on this matter.
In our designs, we mainly consider two types of networks in terms of the quality of service (QoS) requirements. For the first type, each source must communicate with its destination at a fixed transmit data rate. A power control problem [Zan92, JBS04, FM93a, Yat95, RFLT98, SB04, CHLT08, TCS11, Ngu09]) is studied in order to properly assign transmission power (normally the minimum) to each
source in order to guarantee the transmission’s success. For the second type, each
source’s transmission power is fixed. We investigated a throughput maximization
problem that performs rate adaptation in order to maximize network throughput.
Since interference management and power control (or rate adaptation) are generally highly intertwined in the context of wireless interference networks, our aim is
to efficiently address joint design of transmission strategy and power control (or
rate adaption). Below, we first provide an overview of the operation of wireless
communication systems, and then discuss in more detail how the abovementioned
objectives are investigated in the thesis.
1.1. Wireless Network Operations Overview
5
CH4 CH5
CH4
Power Control
Rate Adaptation
CH4 CH5 CH6
CH3 CH4 CH6
CH5
CH3 CH4 CH5 CH6
Channel training
Channel feedback
Beamforming and Filtering
Data transmission
Scheduling
CH6
Figure 1.1: Operations in a wireless communication network.
1.1
Wireless Network Operations Overview
In wireless communication systems, propagation environment is continuously changing and channels are time-varying. As mentioned in the previous section, coordinated communication techniques (such as interference alignment, coordinated
multi-point (CoMP) transmission, and opportunistic scheduling) usually require
certain CSI to be known at terminals in order to adapt their transmission and
conduct communication [LHL+ 08, MLES13]. Therefore, a mechanism is required
to acquire CSI at terminals and adapt transmission strategy, accordingly. An illustrative representation of this mechanism is shown in Fig. 1.1. In order to estimate the local CSI at each terminal, a channel training scheme can be applied (see
e.g. [HH03,CJKR10,KJC11,ALH12,KRB+ 13,FKS14,Sha14]). One possible way for
the other terminals to acquire such estimated CSI is to share this channel knowledge to the other terminals via channel state feedback schemes (see e.g. [LHL+ 08,
BT09,FWS11,ALH12]). The terminals can then use this estimated CSI to adapt the
appropriate transmission strategy. For instance, in multiple-antenna systems, terminals can compute their beamforming and filtering [LHL+ 08], while in single-antenna
systems they may perform opportunistic transmission (e.g. using the ergodic interference alignment scheme [NGJV12], or user scheduling) in order to manage the interference. Moreover, depending on the system requirements, the transmitters may
apply power control (see e.g. [SB06,CHLT08,FWS14b]) or rate adaptation [FWS11]
in order to minimize transmission power or maximize throughput, respectively. In
large wireless interference networks with many users, a user scheduling can be applied to opportunistically serve a subset of the users that experience the acceptable
channel condition [LTYF03, SN07a, SN07b, SBM09, JwJS09, FGS14]. The decisions
can be made based on available CSI and the terminals then conduct data transmission. In this thesis, these items are investigated for wireless interference networks.
6
Introduction
Table 1.1: Thesis outline.
Chapter
Chapter
Chapter
Chapter
1.2
3
4
5
6
Network size
2×2
K ×K
K ×K
K × KN
Antennas
SISO
SISO
MIMO
SISO
CSI
global
global
local
local
CSIR
perfect
noisy
perfect
noisy
CSIT
perfect
quantized
perfect
quantized
Thesis Scope and Contributions
This thesis investigates coordinated transmission schemes for wireless interference
networks and their performance limits. The thesis is presented in six chapters. A
summary of the system considered in each chapter and the underlying assumptions is shown in Table 1.1. Below, we briefly present the contents along with the
contributions in each chapter.
Chapter 2
This chapter is a review of results, concepts, and definitions that are required for the
presentation of the materials in the following chapters. The presentation starts from
the definition of wireless interference networks in this thesis. We then briefly review
the main research results on seeking the capacity region (i.e., the largest rate region
in which reliable communication is possible) of the two-user interference networks.
For networks with more than two users, we introduce the concept of interference
alignment and some techniques to realize it. The contents of this chapter are in
part based on
• [Far12] H. Farhadi, “Interference alignment and power control for wireless
interference networks,” Licentiate Thesis, KTH Royal Institute of Technology,
Stockholm, Sweden, Sep. 2012.
• [MFZ+ 14] N. N. Moghadam, H. Farhadi, P. Zetterberg, M. Khormuji and
M. Skoglund, “Interference alignment: Practical challenges and test-bed implementation,” book chapter in Contemporary Issues in Wireless Communications, INTECH Open Access Publisher, Nov. 2014.
1.2. Thesis Scope and Contributions
7
Chapter 3
In this chapter, we study a two-user single-input single-output (SISO) interference
network in which each source communicates at a given fixed rate to the intended
destination.
Background
The optimal solution to the throughput maximization problem and the power
control problem for two-user interference networks in general case is unknown.
Indeed, these have been the subject of extensive research. For instance, references [Tun08, WT11, RV11, AB12, EK12] have studied the throughput maximization problem for a two-user fading interference network, in which they deployed the
Han-Kobayashi coding scheme [HK81]. However, the recent development of capacity achieving codes for point-to-point communications has made it attractive to use
such codes for network communications (even though they might be sub-optimal
in these scenarios). Applying point-to-point code between each transmitter–receiver
pair is important from a practical viewpoint as the design and operations of these
codes are relatively less complexity that those designed for multi-user scenarios.
For fading interference networks, the throughput maximization problem – subject
to using Gaussian point-to-point codes – has been addressed in [BGT11] and the
capacity region has been characterized. The power control problem has been studied
only for a class of fading interference networks in which transmitters simultaneously
send their messages and each receiver decodes its message by treating the interference as noise (see e.g. [CHLT08, SWB06] and references therein). Various iterative
power control algorithms have been proposed (see e.g. [Zan92,Yat95,CHLT08]) that
assign power to the transmitters so that every transmitter–receiver pair meets a
desired signal-to-interference-plus-noise ratio (SINR). This keeps the instantaneous
mutual information within each fading block at a value larger than the desired
transmission rate. However, only a few studies have been conducted on the feasibility of the solutions for the power control problems. Specifically, for each fading
block in which channel gains remain constant, [CHLT08, Zen92, HC00a, MDEK10]
have shown whether the power control problem has feasible solutions. Indeed, regardless of how large the powers are, for some channel gains, the power control
problem may not have any solution. When there is no power constraint, a feasibility criteria was derived in [Zen92], [HC00a]. It is possible to use this criteria to
determine whether, for specific channel gains, the power control problem has any
solution to support transmission at desired rates. Nevertheless, the solutions may
require large powers. Reference [MDEK10] has extended the results to the case in
which there are linear power constraints. For a fading channel, when there is a
short-term power constraint in the system, due to the random nature of fading,
for some channel realizations there is no choice of power control that satisfy power
constraint and can guarantee successful communication at the desired rate. In this
case, we define the power control problem to be infeasible, and the system to be in
8
Introduction
outage. At a given transmission rate, the probability that an outage event occurs
is defined as outage probability. In many communication systems, a small value of
outage probability is tolerable. In these cases an ǫ-outage achievable rate region
is defined as the supremum of transmission rates for which the outage probability is less than ǫ [CTB99, VH94, TV05]. The outage probability and the ǫ-outage
achievable rate region for random continuously distributed channels is unknown
(e.g. Rayleigh fading).
Contributions
Chapter 3 considers a two-user Rayleigh block-fading interference network. Each
transmitter utilizes a fixed-rate point-to-point Gaussian code to communicate with
its dedicated receiver. Perfect CSI is globally available at all terminals. Each transmitter is subject to a short-term power constraint. We consider five different transmission schemes. When the two transmitter-receiver pairs are orthogonally activated, inter-user interference can be completely eliminated, possibly at the cost of
spectral inefficiency. When both users non-orthogonally access the available channel, inter-user interference must be taken into account during the decoding process.
This leads to the four following schemes: (1) both receivers directly decode their
desired messages by simply treating interference as noise; (2) both receivers conduct
successive interference cancellation (SIC); (3) the first receiver performs direct decoding and the second receiver performs SIC; and (4) the first receiver performs SIC
and the second receiver performs direct decoding. For each of these five schemes,
we start by finding the solution to the power control problem. Next, we derive a
lower bound and an upper bound on the outage probability, as functions of channel
statistics, desired transmission rates, and power constraints. These results are then
used to find an outer bound and an inner bound on the ǫ-outage achievable rate
region. The contributions of this chapter are based on
• [FWS14a] H. Farhadi, C. Wang, M. Skoglund, “Delay-limited constant-rate
transmission over fading interference channels using point-to-point Gaussian
codes,” submitted to IEEE Trans. Commun., 2014.
• [Far12] H. Farhadi, “Interference alignment and power control for wireless
interference networks,” Licentiate Thesis, KTH Royal Institute of Technology,
Stockholm, Sweden, Sep. 2012.
1.2. Thesis Scope and Contributions
9
Chapter 4
In this chapter, we study a time-varying K-user SISO interference network. We
assume that no CSI is a priori available at terminals, and devise a coordinated
schemes to acquire CSI and also to conduct communication.
Background
The performance limits of K-user interference networks have attracted a great deal
of interest for decades; for example, the capacity region of the two-user interference
networks has been the subject of extensive research. Although various inner bounds
and outer bounds on the capacity region of the two-user interference network have
been proposed, the exact capacity region is still unknown in general [EK11]. Extension of the results to general K-user interference networks is even more complicated. It has been shown that when channels are time-varying, applying an interference management technique called interference alignment [MAMK08, CJ08],
at a given signal-to-noise-ratio (SNR) the sum-rate K
2 log(SNR) + o(log(SNR)) can
be achieved, where limSNR→∞ o(log(SNR))/SNR = 0 [CJ08]. This achievable sumrate linearly scales with the number of users and is substantially higher than the
sum-rate log(SNR) + o(log(SNR)) achieved by the conventional TDMA and FDMA
schemes. Furthermore, an ergodic interference alignment scheme has been developed in [NGJV12] so that, when channel gains are symmetrically distributed, the
2
sum-rate K
2 E[log(1 + 2|h| SNR)] is achievable under an ergodic setting. Such a result implies that the interference network in time-varying fading environments may
not be interference-limited at any SNR. The outstanding performance promised
by the aforementioned schemes is based on underlying assumptions that the CSI
is perfectly known at all terminals, and an asymptotically long delay in transmission (due, for example, to the symbol extension technique in [CJ08], or the channel
pairing technique in [NGJV12]) can be tolerated.
Regarding the delay issue, it has been shown through studying the delay-rate
tradeoff for the ergodic interference alignment scheme [KWG10,MM12,JAP12] that
the delay can be reduced by sacrificing transmission data rate. In general, however,
an asymptotically long delay in transmission is still required in order to exhibit the
advantage of the ergodic interference alignment scheme. It is not known whether
the scheme is capable of providing performance gains over orthogonal transmissions
under finite delay constraint.
Moreover, in practice no CSI is a priori available at terminals. They can deploy
a pilot-based channel training scheme to acquire an estimate of CSI at receivers, and
a channel state feedback scheme to obtain an estimate of CSI at transmitters. Channel training schemes for point-to-point communication systems have been studied
extensively (see, e.g., [HH03, BLM03, LSD04, ZCLB07, Sha14]). However, in multiuser scenarios they are even more challenging and the underlying performance
limits of pilot-based channel training for these systems are less known. Channel
training schemes for broadcast channels have been studied in [CJKR10, KJC11],
10
Introduction
for multiple-access systems in [HKD11], and for certain interference networks in
[ALH12, MGL13, FKS14]. [ALH12] studied channel training for multiple antenna
interference networks and also the impact of the allocated time for channel training on the system performance. Transmission power for pilot transmission was assumed to be the same as that for data transmission. In general, however, they
can certainly be different. A more accurate channel estimation can be obtained
by allocating more power for pilot transmission, which implies that less power is
left for data transmission. The interesting problem of optimum power allocation
to pilot symbols and data symbols in point-to-point communication scenarios was
investigated in [HH03]. In interference networks, finding the solution to this optimum power allocation problem is even more important because the quality of CSI
estimation not only affects the performance of each decoder, but also determines
how accurately the interference alignment can be performed. It is also important to
characterize the performance limits of the pilot-assisted channel training schemes
for interference networks.
Regarding acquiring CSI at the transmitter-side, several references (e.g. [BT09,
KV10, FWS11, NGJV12, RG12, LK12, KMLL12, KLC11, NWHC12, CY14]) have investigated cases in which each destination provides only the quantized version of
its incoming channel gains to the other terminals through channel state feedback.
It has been shown that when the number of quantization bits is proportional to
log (SNR), the achievable rate of interference alignment with perfect CSI at the
high-SNR regime can be preserved [BT09]. However, the capacity of feedback channels is limited in practice and terminals may not be able to attain a sufficiently
accurate CSI estimation. Applying interference alignment based on the imperfect
CSI, inter-user interference can be only partially eliminated so that some residual
interference remains at each destination. This residual interference, if not appropriately managed, will degrade the system performance. On the other hand, instead of
exploiting the available (even imperfect) CSI to solely conduct interference alignment, the sources can also adapt their transmission strategies; for example, by
controlling transmission data rate or power, to compensate for the aforementioned
performance loss and fulfil service requirements.
Contributions
In this chapter, we propose a pilot-assisted ergodic interference alignment scheme.
This scheme deploys pilot-based channel training in order to acquire an estimate
of CSI at destinations. Each destination obtains a noisy estimate of its local CSI
and sends an un-quantized (quantized) version of the estimated CSI to the other
terminals via analog (digital) feedback signals. The ergodic interference alignment
scheme is then applied to perform data transmissions. Each transmitter shares
available radio resources between pilot transmission and data transmission phases.
If analog feedback is deployed, we compute an achievable rate region and investigate the optimum power allocation between channel training and data transmission
phases. We also address the scenario in which digital feedback is used to send quan-
1.2. Thesis Scope and Contributions
11
tized CSI with limited resolution. Two problems are studied in this case. The first
is a power control problem in which each user wishes to successfully transmit information at a fixed rate using minimum power. We use a power control scheme that
adapts transmission power values such that the mutual information corresponding
to each source–destination pair is always larger than the transmission rate, which
means that transmitted codewords can be successfully decoded at the desired destination. We then study a throughput maximization problem in which each source
has a fixed transmission power value and the network throughput is desired to be
maximized [FWS11]. Since each source only knows quantized CSI, it is not aware of
the exact value of mutual information between itself and its intended destination.
Therefore, for some channel realizations, the mutual information may fall below the
transmission rate and communications fail, which leads to an outage event. The outage probability can be used to quantify throughput as a measure of the amount
of information that can be successfully transmitted. We propose a rate adaptation
scheme to maximize network throughput. As noted earlier, most existing works
on ergodic interference alignment assume that the system can tolerate asymptotically long delays. To understand how the considered schemes perform in realistic
situations, we extend our results to communication systems with finite delay constraint, and quantify network throughput in these delay-constrained systems. The
contributions of this chapter are based on
• [FWS14c] H. Farhadi, C. Wang, and M. Skoglund, “Interference alignment
with limited feedback: Power control and rate adaptation,” submitted to IEEE
Trans. Wireless Comm., 2014.
• [FWS11] H. Farhadi, C. Wang, and M. Skoglund, “On the throughput of wireless interference networks with limited feedback,” in Proc. IEEE Int. Symp.
Info. Theory (ISIT’11), St. Petersburg, Russia, Jul. 2011.
• [FWS12] H. Farhadi, C. Wang, and M. Skoglund, “Power control in wireless interference networks with limited feedback,” in Proc. IEEE Int. Symp.
Wireless Comm. Sys. (ISWCS’12), Paris, France, Aug. 2012.
• [FKWS13] H. Farhadi, M. N. Khormuji, C. Wang, and M. Skoglund, “Ergodic
interference alignment with noisy channel state information,” in Proc. IEEE
Int. Symp. Info. Theory (ISIT’13), Istanbul, Turkey, Jul. 2013.
• [FKS14] H. Farhadi, M. N. Khormuji, and M. Skoglund, “Pilot-assisted ergodic interference alignment for wireless networks,” in Proc. IEEE Int. Conf.
Acoustics, Speech and Signal Processing (ICASSP’14), Florence, Italy, May
2014, [Best Student Paper in Signal Processing for Communications and Networking].
• [Far12] H. Farhadi, “Interference alignment and power control for wireless
interference networks,” Licentiate Thesis, KTH Royal Institute of Technology,
Stockholm, Sweden, Sep. 2012.
12
Introduction
Chapter 5
In Chapter 5, we study a K-user multiple-input multiple-output (MIMO) interference network. Each source intends to send multiple independent data streams
to its corresponding destination where the number of data streams coincides with
the achievable DoF of the network. Each data stream is encoded at a fixed data
rate while different streams can have different rates. Only local CSI (that is, the
knowledge related to the channels directly connected to a terminal) is available
at each terminal. We propose iterative algorithms to perform distributed power
control and transceiver design. Transmitter beamforming matrices and receiver filtering matrices are designed to maximize the SINR at each receiver corresponding
to each stream. Power control is conducted to assign the minimum power to each
encoded data stream to guarantee successful communication.
Background
Characterizing the performance limits of K-user MIMO interference networks and
designing efficient interference management schemes to achieve the limits have both
generated a great deal of interest among researchers. As already noted, it is a
challenge to find the capacity region of interference networks even with only two
source-destination. Instead, degrees of freedom (DoF)– defined as the asymptotic
scaling factor of the capacity with respect to the logarithm of SNR– has been investigated recently [CJ08,MAMK08,Jaf11]. Intuitively, DoF represents the number
of independent streams that can be transmitted interference-free in the asymptotically high-SNR regime. The total achievable DoF of the conventional interference
management schemes (for example, TDMA or FDMA) in which the transmissions
of different source-destination pairs are orthogonalized, does not increase with the
number of users. However, as we already mentioned the interference alignment
concept [MAMK08, CJ08] reveals that, with proper transceiver design, different
interference signals at each destination can be aligned together, such that more
radio resources can be assigned to the desired transmission. In certain cases (such
as a three-user MIMO interference network), applying interference alignment can
achieve a total DoF equal to half of the total achievable DoF in the interferencefree network. To do so, the sources perform linear beamforming to simultaneously
transmit multiple independent streams in such a way that, at each destination, interference signals align and span only half of the available signal space (in the spatial
domain). Consequently, the interference can be eliminated with linear zero-forcing
filter at each destination [CJ08]. Thus, a total DoF proportional to the number of
users in the network can be achieved. This idea has inspired several linear beamforming and filtering design solutions for interference networks. For instance, it has
been shown that the beamforming and filtering matrices can be further optimized
to achieve a larger sum-rate at the finite-SNR regime [SPLL10]. Also, for MIMO
interference networks with finite alphabet channel inputs, a linear beamforming
design is proposed in [WXG+ 13] to maximize sum-rate.
1.2. Thesis Scope and Contributions
13
Although interference alignment can achieve a larger DoF than that achieved by
orthogonal transmission strategies, several challenges must be addressed to enable
the deployment of this technique in future wireless networks [APH13, MET13]. To
compute transmitter beamforming matrices and receiver filtering matrices, global
CSI must usually be perfectly known at all terminals [CJ08, SPLL10]. Acquiring
such channel knowledge is clearly a challenging problem in practice. In most cases,
it is more convenient for each terminal to obtain only the CSI corresponding to the
links directly connected to it (that is, the local CSI). For instance, the local CSI can
be the exact channel matrices corresponding to the local links or some function of
these matrices. Accordingly, distributed interference alignment have been proposed
in the literature (e.g., [GCJ11, PH11]).
Consider a MIMO interference network in which each user may send multiple
independent data streams. An iterative algorithm for distributed interference alignment, referred to as the DIA algorithm, is proposed in [GCJ11]. Since only local
CSI is available at terminals, interference signals cannot be perfectly aligned at each
receiver. Therefore, zero-forcing filtering cannot perfectly eliminate the interference
and some leakage interference remains at the receivers. This algorithm iteratively
minimizes the power of the leakage interference. The solution computed by the DIA
algorithm achieves the DoF of the network in the high-SNR regime. However, practical systems have finite SNR and the solutions intended for the high-SNR regime
may not be efficient in this case. Therefore, another iterative algorithm is proposed
in [GCJ11] to maximize the signal-to-leakage interference-plus-noise ratio for each
transmitted data stream. This algorithm, which is referred to as the Max-SINR
algorithm, achieves a larger sum-rate than the DIA algorithm at the finite-SNR
regime while achieving the same DoF at high SNR.
In practical wireless environment, channel variations mean that the equivalent
SINR corresponding to each transmitted data stream changes over time. Thus,
adaptive transmission is required for reliable communication. Depending on the
objectives and the constraints, two different types of adaptive schemes can be applied. In a class of communication systems where the maximum throughput is
desired, similar to the one considered for the DIA algorithm and Max-SINR algorithm in [GCJ11], an adaptive coding and modulation scheme is required to adjust
the data rate based on channel state [GC98]. In such systems, transmission powers might be fixed or can be adapted (for instance, by using an algorithm similar
to the one proposed in [QZH09]) to maximize the throughput. In another group
of applications, such as voice/video communications, or control over communication networks, a fixed-rate data transmission is desired instead [WGM07,CHLT08].
Therefore, a power control problem should be solved. Several iterative power control
algorithms have been proposed for the uplink and downlink transmissions of cellular systems (see e.g. [Zan92, FM93b, Yat95, FJC12]). [Yat95] introduced the family
of standard power control problems and showed the convergence of the corresponding iterative power control algorithm. Furthermore, joint beamforming design and
power control has been studied in some MIMO communication systems (such as the
MIMO downlink channel) where the power control problem is intertwined with the
14
Introduction
transceiver design problem [RFLT98,SB04,CHLT08,TCS11,HZB+ 11,HZB+ 12]. In
interference networks, the SINR corresponding to each source-destination pair depends on the transmission powers of all sources. The solution of the power control
problem for each user is coupled to that for the other existing users. This dependency leads to conflicting goals. Specifically, when each source tries to increase the
transmission power to compensate interference at its intended destination, it also
increases the interference to the other destinations. Ignoring this dependency in
signaling design may cause unnecessarily power demand or unsatisfactory communication quality. With regard to MIMO interference networks, directly applying
conventional power control strategies when multiple source-destination pairs are
non-orthogonally activated is highly likely to be infeasible or lead to large power
requirements. Thus, to guarantee successful fixed-rate transmission for each sourcedestination pair, power control is conventionally carried out when the transmissions
of different sources are orthogonalized. According to the recent intuitions from the
interference alignment concept, this may not be a spectrally efficient transmission
strategy as shown in [Far12, FWS13].
Contributions
In this chapter, we aim to address the problem of transceiver design and power control for MIMO interference networks by further taking into account the achievable
DoF of the network. Specifically, we consider a network where each source sends
multiple data streams: the same number as the corresponding DoF achieved by
interference alignment. We propose two iterative algorithms that compute transmitter beamforming matrices and receiver filtering matrices to maximize the SINR
for each stream, and allocate the minimum powers to realize the desired fixed-rate
communications. In both algorithms, the required power values are computed in a
distributed fashion at each destination and the associated source is informed via
a feedback link. In the first algorithm, the exact value of the computed power is
sent, while only a one-bit feedback signal is transmitted in the second algorithm, via
feedback. The proposed algorithms can provide reliable communication when multiple streams are transmitted, as each is encoded with potentially different rates.
This is particulary useful in wireless networks in which each user intends to communicate multiple multimedia data with diverse contents, each with possibly different QoS requirements. Numerical evaluations confirm that these algorithms require substantially smaller power values compared to the conventional orthogonal
transmission strategies. These algorithms are implemented on KTH’s four-multi
test-bed, which consists of three source-destination pairs of USRP-based terminals [MFZS14, MFZ+ 14]. The experimental measurements in indoor environment
also confirm the promised performance of the proposed algorithms. The contributions of this chapter are based on the following publications.
• [FWS14b] H. Farhadi, C. Wang, and M. Skoglund, “Distributed transceiver
design and power control for wireless MIMO interference networks,” accepted
1.2. Thesis Scope and Contributions
15
for publication in IEEE Trans. Wireless Commun., Oct. 2014.
• [FWS13] H. Farhadi, C. Wang, and M. Skoglund, “Distributed interference
alignment and power control for wireless MIMO interference networks,” in
Proc. IEEE Wireless Commun. and Networking Conf. (WCNC’13), Shanghai,
China, Apr. 2013.
• [FZF+ 13] H. Farhadi, A. Zaidi, C. Wang, and M. Skoglund, “Distributed
interference alignment and power control for wireless MIMO interference networks with noisy channel state information,” in Proc. Int. Black Sea Conf.
Commun. and Networking (BlackSeaCom’13), Batumi, Georgia, Jul. 2013.
• [Far12] H. Farhadi, “Interference alignment and power control for wireless
interference networks,” Licentiate Thesis, KTH Royal Institute of Technology,
Stockholm, Sweden, Sep. 2012.
• [MFZ+ 14] N. N. Moghadam, H. Farhadi, P. Zetterberg, M. Khormuji and
M. Skoglund, “Interference alignment: Practical challenges and test-bed implementation,” book chapter in Contemporary Issues in Wireless Communications, INTECH Open Access Publisher, Nov. 2014.
16
Introduction
Chapter 6
In Chapter 6, we consider a multi-cell interference network with multiple cells,
each of which has a base station and multiple mobile terminals. Each base station
communicates to mobile users in the corresponding cell. We assume that no CSI
is a priori available at terminals. We propose a low-complexity scheme to conduct
channel training and data communication in these networks.
Background
It has been predicted that one of the most typical scenarios in 5G communications systems will be to support an exponentially increasing demand for data rate,
in ultra-dense deployments. Such communication scenarios are characterized by a
high data rate requirement that needs to be sustained, irrespective of the harsh
urban propagation conditions [OBB+ 14, JMZ+ 14]. Moreover, the relatively high
user density in such settings implies that channel training and feedback overhead is
a major challenge. Consequently, spectrally efficient transmission techniques with
low-overhead are greatly desired.
In order to enhance spectral efficiency, the time-varying characteristics of wireless transmission medium can be effectively exploited. It has been shown that
opportunistic transmission can benefit from the time variations of propagation
environment and enhance system performance. These schemes schedule a subset
of the users depending on the instantaneous CSI. Several opportunistic transmission schemes have been developed in the literature; these include opportunistic
scheduling [XCS01, LTYF03, TW08, SBM09], opportunistic beamforming [VTL02],
random beamforming [SH05], and opportunistic interference alignment [PDLC08,
JNPS12, YSJS14, LGLL14]. The early opportunistic schemes were mainly designed
to exploit multi-user diversity in single-cell communication scenarios (e.g. [XCS01,
VTL02,LTYF03,SH05,TW08]). Studies have shown that opportunistic transmission
schemes can also mitigate inter-cell interference, thereby achieving multiplexing gain
in multi-cell communication scenarios (e.g. [PDLC08, JNPS12, YSJS14, LGLL14]).
However, the aforementioned schemes exhibit several characteristics that hinder
their application in dense cellular deployments. For instance, the proposed schemes
in [XCS01, LTYF03, PDLC08] require CSI to be perfectly known at mobile terminals and base stations. while the schemes proposed in [VTL02,SH05,TW08,SBM09,
JNPS12, YSJS14, LGLL14] require only finite rate feedback to acquire CSI at base
stations, they also need CSI to be a priori known at mobile terminals. This effectively limits their scalability since the overhead required for channel estimation at
mobile terminals may degrade the expected delivered performance gains. In practice, CSI is not a priori available at mobile stations and they may only obtain
imperfect CSI via channel training schemes. This has a twofold impact: Firstly,
base stations need to allocate part of their radio resources for channel training,
which means that fewer resources will be available for data transmission; Secondly,
the performance of opportunistic transmission schemes degrades as a consequence
1.2. Thesis Scope and Contributions
17
of imperfect scheduling and erroneous decoding at mobile terminals. Less is known
about how to perform opportunistic transmission when no a priori CSI is available
at terminals, and the performance limits of networks in such scenarios is a high
priority for investigation.
Contributions
We consider a dense cellular communication scenario in which one base station in
each cell serves a large number of mobile terminals, with no a priori CSI available.
We propose a pilot-assisted opportunistic user scheduling (PAOUS) scheme consisting of low complexity channel training and one-bit feedback transmission. We
compute the achievable rate region for the proposed scheme and characterize the
achievable DoF region. Our results reveal that, in a multi-cell network with B base
stations and a coherence time T , the achievable sum-rate increases as the number
of mobile terminals scales and the total DoF Bopt (1 − Bopt /T ) is achievable, given
that the number of mobile terminals in each cell scales is proportional to SNR. This
result indicates that, to maximize the achievable sum DoF only a subset of base
stations should be activated, where the optimum number of active base stations is
Bopt = min {B, T /2}. The contribution of this chapter is based on
• [FGS14] H. Farhadi, H. Ghauch, and M. Skoglund, “Pilot-assisted opportunistic user scheduling for wireless multi-cell networks,” submitted to IEEE
Int. Conf. Commun. (ICC’15), London, UK, Oct. 2014.
Chapter 7
In the last chapter, we summarize our contributions in the thesis, and discuss
potential directions for future research.
18
Introduction
Contributions Outside the Scope of the Thesis
In addition to the contributions listed above, the author of this thesis has also
contributed to some other related works, which are published in the papers listed
below. For consistency of the thesis structure, these are not included in the thesis.
Test-bed Implementation of Transceiver Design and Power Control
Algorithm
We implemented our proposed iterative transceiver design and power control algorithm (presented in Chapter 5) on the KTH four-multi test-bed (see [Zet, ZM12,
Zet14] for more details about this test-bed). The test-bed is composed of three base
stations and three mobile stations. All base stations were transmitting simultaneously and on the same frequency band. The baseband processing at the terminals
was implemented on universal software radio peripheral (USRP) platforms. Each
terminal has two antennas, each of which is connected to one dedicated USRP. The
indoor measurements reveal that, the proposed algorithm can achieve at least 4 dB
reduction in transmission power in 90% of the experiments compared to the case
where MaxSINR algorithm was implemented. The power saving gains as high as 13
dB was also observed in 10% of the measurements. This implementation and the
measurement results are presented in the following publications:
• [MFZS14] N. N. Moghadam, H. Farhadi, P. Zetterberg and M. Skoglund,
“Test-bed implementation of iterative interference alignment and power control for wireless MIMO interference networks,” in Proc. IEEE Int. Workshop on Signal Proc. Advances in Wireless Commun. (SPAWC’14), Toronto,
Canada, June 2014.
• [MFZ+ 14] N. N. Moghadam, H. Farhadi, P. Zetterberg, M. Khormuji and
M. Skoglund, “Interference alignment: Practical challenges and test-bed implementation,” book chapter in Contemporary Issues in Wireless Communications, INTECH Open Access Publisher, Nov. 2014.
Multi-user Relay Networks
We have studied the achievable sum DoF of a class of multi-user SISO relay networks. In these networks, the communications between K unconnected source–
destination pairs are provided by a large number of half-duplex relays. When the
number of relays is sufficiently large, we show that the sum DoF of this network
is K. This can be achieved through the combination of spectrally efficient relaying and interference alignment. This result implies that allowing only distributed
processing and half-duplex operation can provide performance that is similar to
permitting joint processing and full-duplex operation in wireless relay networks at
a high-SNR regime. This material has been published in:
1.2. Thesis Scope and Contributions
19
• [WFS12] C. Wang, H. Farhadi, M. Skoglund, “Achieving the degrees of freedom of wireless multi-user relay networks,” IEEE Trans. Commun., vol. 60,
no. 9, pp. 2612-2622, Sept. 2012.
• [WFS10] C. Wang, H. Farhadi, and M. Skoglund, “On the degrees of freedom
of parallel relay networks,” in Proc. IEEE Global Commun. Conf. (GLOBECOM’10), Miami, USA, Mar. 2010.
Cognitive Interference Network
We have investigated the throughput of a K-user cognitive fading interference network. Specifically, we have considered a cognitive radio network consisting of one
primary and multiple secondary source–destination pairs. The secondary sources
have non-causal knowledge of the message of the primary user. We have found
a tuple of achievable rates by utilizing the discrete superposition model (DSM),
which is a simplified deterministic channel model. The coding scheme devised for
the DSM can be translated into a coding scheme for the additive white Gaussian
noise (AWGN) channel model, where the rate achieved in the AWGN channel model
is at most a constant gap below the one achieved in the DSM. We have also derived
the average throughput of the secondary users under Rayleigh fading environments.
Our results show that the sum-throughput of the proposed scheme increases with
the number of secondary pairs when the interference is weak. This material has
been published in:
• [SFRS12] N. Schrammar, H. Farhadi, L. K. Rasmussen, M. Skoglund, “Average throughput in AWGN cognitive fading interference channel with multiple
secondary pairs,” in Proc. Int. Conf. Cognitive Radio Oriented Wireless Net.
and Comm. (CROWNCOM’12), Stockholm, Sweden, Jun. 2012.
Copyright Notice
As specified in Section 1.2, parts of the material presented in this thesis are partly
verbatim, based on the thesis author’s joint works, which have previously been
published or submitted to conferences and journals held by or sponsored by the
Institute of Electrical and Electronics Engineer (IEEE). The IEEE holds the copyright of the published papers and will hold the copyright of the submitted papers
if they are accepted. Materials (such as figures, graphs, tables, or textual material)
have been reused in this thesis with permission.
Chapter 2
Background
I
N this chapter we will review some concepts, definitions and results which are
required for the presentation of the material in the following chapters. First,
we will describe wireless interference networks and some of the current research
challenges. Next, we will briefly review the results on the capacity region characterization of the two-user interference networks. We will explain the concept of
interference alignment for larger networks with more than two source-destination
pairs. Also, we will present some interference alignment techniques and will review
the results on interference alignment with imperfect/partial CSI.
2.1
Wireless Interference Networks
The wireless interference network is a model for communication systems composed
of multiple sources and destinations. Each source intends to communicate with its
dedicated destination and all sources share the same transmission medium. Because
of the broadcast nature of wireless medium, each destination also overhears the signals from the unintended sources. Hence, each destination observes a noisy combination of the transmitted signals from the desired and undesired sources, weighted
by the corresponding channel gains. Figure 2.1 shows a K-user wireless interference
network with sources and destinations denoted as Sk and Dk (k ∈ {1, 2, ..., K}),
respectively. Many practical wireless communication scenarios can be modelled as
in Figure 2.1. Examples include cellular networks, device-to-device communication
systems, ad-hoc networks, wireless local area networks, and cognitive radio networks.
An increasing demand for wireless data traffic in the future has been forecasted.
For instance, in cellular networks an exponential data traffic growth has been reported by Ericsson [Eri12]. As a consequence, current wireless networks will expand,
more wireless infrastructures will be deployed, and more wireless devices will operate in such networks. This will lead to an increasing demand for radio resources
such as spectrum and energy. However, the radio spectrum is scarce and is considered as one of the most expensive natural resources. Also, there are serious concerns
regarding vast energy consumption and the energy budget of mobile terminals is
21
22
Background
z1
h11
S1
+
D1
h2
h K1
1
z2
2
S2
h1
h22
+
D2
hK
h
2K
h1
K
2
SK
hKK
zK
+
DK
Figure 2.1: K-user wireless interference network
restricted due to the limited battery storage capacity. Thus, spectral and energy
efficient design is essentially required for emerging wireless systems.
Nevertheless, finding the optimum transmission schemes and characterizing the
best performance in wireless networks is in general a challenging problem. To efficiently utilize radio resources, proper interference management and resource allocation techniques are required. Sufficient CSI knowledge at each terminal would be
important. Such knowledge can be obtained through coordinations among users,
e.g. in the form of pilot transmission and channel estimation at the receivers, and
feedback from destinations to the sources. In practice, perfect coordination may
be difficult to be guaranteed, due to different reasons such as limited resources for
channel training, and limited bandwidth and noise/delay in the feedback channels.
Thus, designing transmission schemes with imperfect or only partial CSI is required.
In the next sections, we will briefly review some of the key transmission techniques
for wireless interference networks.
2.2
Two-user Interference Networks
The basic interference network is composed of two source-destination pairs. Characterizing the capacity region (the closure of the set of rate vectors for which jointly
reliable communications are possible with independent sources [Car78]) of this network has been the subject of research for many years. The two-user interference network was first studied by Ahlswede, who established basic inner and outer bounds
on the capacity region [Ahl74]. Some achievable rates and upper bounds have been
further proposed in later literatures [EK11]. However, except in some special scenarios such as when the inter-user interference is very strong [Sat81, CE87], the
capacity region in general case is still unknown.
2.3. K-user (K > 2) Interference Networks
23
Despite the intuition that interference always degrades the network’s performance, it has been shown that in certain cases the capacity region does not shrink
due to interference. For example, Carleial showed that in Gaussian interference
networks when the interference is very strong, each destination can first decode
the message of the unintended source and subtract it from the received signal before decoding its own message [Car75]. In this way, the capacity region would not
be affected by interference. The scheme was extended to the strong interference
scenario and the capacity region was established by Sato [Sat81]. Costa and El
Gamal further generalized this result to the discrete memoryless interference channel model [CE87].
When the interference between users is moderate or weak, destinations may not
be able to decode the message of the interfering sources. Characterizing the capacity region is more challenging compared to the strong interference scenario and the
very strong interference scenario. Carleial applied the superposition coding technique, which was originally developed for broadcast channels by Cover in [Cov72],
to the two-user interference network. An inner bound of the capacity region was
established through data splitting at the sources and successive decoding at destinations [Car78]. This inner bound was further improved via joint decoding and
coded time sharing by Han and Kobayashi [HK81]. An equivalent characterization
with a reduced set of inequalities was presented in [CGG08].
Some outer bounds to the capacity region have also been derived. For instance,
a genie-based outer bound was presented by Kramer in [Kra04]. Etkin, Tse, and
Wang used a variant of this genie-based outer bound and the Han-Kobayashi inner
bound to establish the capacity region of the two-user Gaussian interference network
within one-bit [ETW08]. Also, incorporating a tight outer bound, the sum capacity
of the two-user Gaussian interference network with weak interference has been
derived in [MK09], [SKC09], [AV09]. It has been shown that in a weak interference
regime, where the channel gains between undesired source-destination pairs are
below certain thresholds, using Gaussian codebooks and performing decoding by
treating the interference as noise can achieve the sum capacity.
2.3
K-user (K > 2) Interference Networks
Applying the above techniques developed for two-user interference networks to
larger networks is not straightforward. Three major approaches to dealing with
interference in multi-user interference networks are displayed in Figure 2.2. In Figure 2.2 (a) all sources simultaneously transmit in the same frequency band. Each
source applies single-user coding techniques. At each destination, the desired signal
cannot be distinguished from interference signals. Hence, the destination performs
decoding by directly treating the interference signals as noise. In the low-SNR region, the level of interference may be limited by proper power control techniques.
However, when SNR is high, inter-user interference would be dominant. Power control alone does not suffice to manage the interference and this transmission strategy
24
Background
H11
S1
D1
H21
H31
H12
H22
S2
D2
H32
H13
H23
S3
D3
H33
(a)
H11
S1
D1
H21
H31
H12
H22
S2
D2
H32
H13
H23
S3
D3
H33
(b)
signal
subspace
H11
S1
interference
subspace
D1
H21
H31
H12
S2
H22
D2
H32
H13
H23
S3
D3
H33
(c)
Figure 2.2: Transmission schemes in three-user interference networks: (a) nonorthogonal transmission and decdoing by treating interference as noise, (b) orthogonal transmission, and (c) interference alignment.
2.3. K-user (K > 2) Interference Networks
25
may not lead to a good performance.
To avoid interference at the destinations, the conventional approach is to orthogonalize the transmissions of different users. Each source-destination pair has
access to only a portion of the available channel, as shown in Figure 2.2 (b). Although signal reception at each destination does not directly suffer from inter-user
interference, this scheme may not be spectrally efficient. This is because at each
destination the interference signals span a large dimension of the received signal
space, since they are unnecessarily orthogonal to each other.
Clearly, if at each destination the dimension of the subspace occupied by only the
interference signals can be reduced, a larger interference-free subspace would be left
for desired transmission. In fact, this can be realized using a new technique called
interference alignment [MAMK08]. Specifically, interference alignment for interference networks refers to “a construction of signals in such a manner that they cast
overlapping shadows at the receivers where they constitute interference while they
remain distinguishable at the intended receivers where they are desired” [CJ08]. In
general, two conditions should be satisfied. The first is to align interference signals
at the same subspace, termed interference subspace. The second is that the subspace left for the desired signal, called desired subspace, should be independent from
the interference subspace. Both conditions are essential to interference alignment
techniques. An illustrative representation of this concept is shown in Figure 2.2 (c).
Interference alignment can be performed in different domains such as space
(across multiple antennas [MAMK08, CJ08]), time (exploiting propagation delays
[MJS12, MAT10] or coding across time-varying channels [CJ08, NJGV09]), frequency (coding across different carriers in frequency-selective channels [CJ08]), and
code (aligning interference in signal levels [MGMAK14]). For different system models with different assumptions on the available CSI, different interference alignment
techniques have been developed in the literature. In this section, we briefly review
some of them.
2.3.1
Achievable Degrees of Freedom Region
Consider a K-user interference network. Source Sk (k ∈ {1, 2, ..., K}) intends to
send an independent message wk ∈ Wk to its destination, where Wk denotes the
corresponding message set. The message wk is encoded to a codeword of length n.
k|
, where |Wk | denotes the cardiThus, the corresponding code rate is Rk = log |W
n
nality of Wk . The rate tuple (R1 , R2 , ..., RK ) is said to be achievable if a sequence
of codebooks exists, such that the probability that each destination incorrectly decodes its message, can be arbitrarily small, by choosing long enough codewords.
The capacity region of the network is the closure of the set of all achievable rates.
In Gaussian interference networks where the noise is additive white Gaussian, the
capacity region depends on the transmission powers of sources, the noise powers
and channel gains. Since the exact capacity region is difficult to find, as a starting
point one can use the degrees of freedom (DoF) region to characterize/approximate
the capacity/achievable rate region in the high-SNR region (where interference is
26
Background
the dominant phenomenon that degrades system performance). The achievable DoF
region is defined as follows
Rk
D = (d1 , ..., dK ) ∈ R+ |∃(R1 , ..., RK ) ∈ C(p), dk = lim
, k ∈ {1, ..., K} ,
p→∞ log p
(2.1)
where C(p) denotes the capacity region, and p is the transmission power of each
source. The DoF can be seen as the pre-log factor of the achievable rate and the
DoF region describes how the capacity region expands as transmission power increases. Another more ‘practical’ interpretation of DoF per user, dk , is the number
of interference-free signaling dimensions.
2.3.2
Interference Alignment for MIMO Interference Networks
In this section, we show how to align interference signals at each destination in the
spatial domain, through the scheme proposed in [CJ08]. Consider a network with
three source-destination pairs (K = 3). Each terminal is equipped with M antennas.
For presentation simplicity here we assume M to be even. The transmission scheme
when M is odd is also provided in [CJ08]. Let Sk (k ∈ {1, 2, 3}) send vector xk .
Then, the channel output at destination Dk is
yk = Hk1 x1 + Hk2 x2 + Hk3 x3 + zk ,
(2.2)
where xl is an M ×1 transmitted signal vector of source Sl , Hkl is the M ×M channel
matrix between Sl and Dk , and zk is an M × 1 AWGN noise vector. The channels
are time-invariant and do not change during the transmission. The goal is to show
that the achievable DoF for each source-destination pair is M/2. Sk (k ∈ {1, 2, 3})
transmits M/2 independent codeword streams, denoted as xik (i ∈ {1, 2, ..., M/2}),
by modulating vectors vik as follows:
M/2
xk =
X
xik vik = Vk xk ,
(2.3)
i=1
M/2
where xk = [x1k x2k · · · xk ]T , and Vk is the beamforming matrix of Sk . Therefore,
according to (2.2) the received signal of Dk can be represented as follows
yk = Hk1 V1 x1 + Hk2 V2 x2 + Hk3 V3 x3 + zk .
(2.4)
Each destination tries to recover the desired message from the received signal. The
receiver can perform linear filtering as follows
yk = U∗k yk ,
(2.5)
and decode its desired messages. There are two interference signals at each destination. The interference signals at each destination will be aligned if we can design
2.3. K-user (K > 2) Interference Networks
27
the beamforming matrices V1 , V2 and V3 such that they satisfy the following conditions:
Alignment at D1 :
Alignment at D2 :
span(H12 V2 ) = span(H13 V3 ),
H21 V1 = H23 V3 ,
Alignment at D3 :
H31 V1 = H32 V2 ,
(2.6)
where span(A) is the space spanned by the column vectors of matrix A. Thus,
the interference signals only occupy an M/2-dimensional subspace and the first
requirement of interference alignment is satisfied. Since the elements of matrices
Hkl (∀ k, l ∈ {1, 2, 3}) are randomly chosen from continuous distributions, then the
latter has a full rank of M , almost surely. Therefore, the above set of equations can
be re-written as follows:
span(V1 ) =
V2 =
V3
=
span(EV1 ),
FV1 ,
GV1 ,
(2.7)
where
E
−1
−1
, H−1
31 H32 H12 H13 H23 H21 ,
F
G
, H−1
32 H31 ,
, H−1
23 H21 .
(2.8)
This problem has different solutions. One of the solutions is
V1
=
[e1 , e2 , ..., eM/2 ],
V2
V3
= F[e1 , e2 , ..., eM/2 ],
= G[e1 , e2 , ..., eM/2 ],
(2.9)
where ei (i ∈ {1, ..., M }) are the eigenvectors of matrix E. It can be verified that
the solution in (2.9) satisfies all the alignment conditions in (2.6).
To retrieve the desired message from the received signal, the second condition
for interference alignment, which requires the desired signal subspace and the interference subspace to be linearly independent, must be satisfied. This requirement
can be fulfilled if the following conditions are satisfied:
rank ([H11 V1 , H12 V2 ]) = M
rank ([H22 V2 , H21 V1 ]) = M
rank ([H33 V3 , H31 V1 ]) = M.
(2.10)
It has been shown in [CJ08] that these conditions are almost surely satisfied. Therefore, sum DoF 3M/2 is achievable in this network. Figure 2.3 shows an illustrative
28
Background
H13 V3
V1
H11
S1
D1
H12 V2
H11 V1
H21
H31
H22 V2
H12
V2
H22
S2
D2
H21 V1
H23 V3
H32
H13
H31 V1
H23
V3
S3
D3
H33
H32 V2
H33 V3
Figure 2.3: Interference alignment in a three-user MIMO interference network with
two antennas (M = 2) at each terminal.
example of the solution of this scheme when M = 2. In this network each source can
transmit one stream performing interference alignment. The receivers can retrieve
the desired message from the received signal by zero-forcing filtering. If two conditions of the interference alignment are fulfilled, the transmitter-side beamforming
matrices and the receiver-side filtering matrices satisfy the following conditions:
U∗k Hkj Vj
=
rank(U∗k Hkk Vk ) =
0, ∀j 6= k : j, k ∈ {1, 2, 3},
M
, ∀k ∈ {1, 2, 3}.
2
(2.11)
It is clear that the solution requires global CSI to be available at each terminal.
Specifically, matrix E is required for the calculation of the solution in (2.9) and
also finding the zero-forcing receiver at each destination.
2.3.3
Interference Alignment for SISO Interference Networks
When there is only one antenna at each terminal, interference signals cannot be
aligned in the spatial domain. The techniques mentioned in the previous sections
cannot be applied. However, it has been shown that in time-varying or frequencyselective fading environments, interference alignment is still possible. We use an
example to reveal the basic idea of interference alignment for time-varying SISO
interference networks. Consider a three-user interference network (K = 3). The
received signal at Dk (k ∈ {1, 2, 3}) is:
yk (t) = hk1 (t)x1 (t) + hk2 (t)x2 (t) + hk3 (t)x3 (t) + zk (t),
(2.12)
2.3. K-user (K > 2) Interference Networks
29
where xl (t) is the transmit symbol of Sl at time instant t, and hkl (t) is the channel
coefficient between Sl and Dk at time instant t.
The channel coherence time is assumed to be one time slot (channel gains remains constant within one time slot, but change independently across different time
slots). Global CSI is perfectly known at all terminals. Since each terminal has only
one antenna, at each time slot there are not enough spatial dimensions to separate
interference subspace with desired signal subspace. This problem can be resolved
using the symbol extension technique proposed in [CJ08]. We define the q symbols
transmitted over q time slots by Sk as a vector:
xk (t) , [xk (q(t − 1) + 1), xk (q(t − 1) + 2), ..., xk (qt)]T .
(2.13)
Similarly, denote the q symbols received over q time slots by Dk as a vector:
yk (t) , [yk (q(t − 1) + 1), yk (q(t − 1) + 2), ..., yk (qt)]T .
(2.14)
The noise vector zk (t) is the similar expansion of the noise over q time slots. Thus,
the received signal at Dk can be expressed as follows
yk (t) = Hk1 (t)x1 (t) + Hk2 (t)x2 (t) + Hk3 (t)x3 (t) + zk (t).
(2.15)
In this equation Hkl (t) (k, l ∈ {1, ..., K}) is a diagonal extended channel matrix
defined as follows


hkl (q(t − 1) + 1)
0
···
0


0
hkl (q(t − 1) + 2) · · ·
0 


Hkl (t) , 
(2.16)
..
..
.. 
..
.
.

.
.
. 
0
0
· · · hkl (qt)
This is called the extended interference channel model where each destination has
a q-dimensional received signal space. The goal of interference alignment design is
to align all the interference signals at every destination within one half of the total
received signal space, leaving the other half interference-free for the desired signal.
Let q = 2n + 1, where n is a positive constant. The source S1 encodes its message to
m
n + 1 independent data streams xm
1 (t) (m = 1, 2, ..., n + 1). Each data stream x1 (t)
m
is sent along a q × 1 vector v1 . Therefore, x1 (t) can be represented as follows:
x1 (t) =
n+1
X
m
xm
1 (t)v1 = V1 x1 ,
(2.17)
m=1
(t)]T and V1 , [v11 , v21 , · · · , vn+1
]. Similarly, S2
where x1 , [x11 (t), x21 (t), · · · , xn+1
1
1
and S3 encode their messages to n independent data streams as follows:
x2 (t) =
x3 (t) =
n
X
m=1
n
X
m
xm
2 (t)v2 = V2 x2 ,
m=1
m
xm
3 (t)v3 = V3 x3 .
(2.18)
30
Background
Therefore, the received signal at Dk can be expressed in the following matrix format
yk (t) =
3
X
Hki (t)Vi xi + zk (t).
(2.19)
i=1
As we mentioned in Section 2.3 two conditions should be satisfied to realize
interference alignment. First, the interference signals should be aligned at each
destination such that interference occupies a subspace with dimensions less than
the total dimensions of the available signal space. The second condition is that the
interference and desired signal subspaces should be independent. To obtain the (n+
1)-dimensional interference-free desired signal subspace from the received (2n + 1)dimensional signal y1 (t), the number of dimensions of the interference subspace
must not be larger than n. This can be achieved by aligning the interference signals
from S2 and S3 as follows
H12 (t)V2 = H13 (t)V3 .
(2.20)
To have an n-dimensional interference-free subspace at D2 , the interference signals
from S1 and S3 must be aligned as follows
span(H23 (t)V3 ) ⊂ span(H21 (t)V1 ).
(2.21)
Similarly, the alignment condition at D3 is
span(H32 (t)V2 ) ⊂ span(H31 (t)V1 ).
(2.22)
This set of equations have more than one solution. One of them is
V1
= [u, Tu, ..., Tn u],
V2
V3
n−1
= H−1
u],
32 (t)H31 (t)[u, Tu, ..., T
−1
2
n
= H23 (t)H21 (t)[Tu, T u, ..., T u],
(2.23)
−1
−1
where T = H12 (t)H−1
21 (t)H23 (t)H32 (t)H31 (t)H13 (t) and u is a (2n + 1) × 1 all-one
vector. To check the second condition, it has been shown in [CJ08] that the columns
of matrix
[H11 (t)V1 , H12 (t)V2 ]
(2.24)
are linearly independent, with probability one. Thus, the desired signal and interference subspaces at D1 can be separated, almost surely. The same results hold for
the following matrices corresponding to the other destinations
[H22 (t)V2 , H21 (t)V1 ],
(2.25)
[H33 (t)V3 , H31 (t)V1 ].
(2.26)
2.3. K-user (K > 2) Interference Networks
31
Consequently, S1 , S2 and S3 can transmit n + 1, n and n independent
messages,
n+1
n
n
respectively over 2n + 1 channel uses. Thus, the DoF tuple 2n+1 , 2n+1 , 2n+1
is
achievable in the asymptotic case where
the transmission power goes to infinity.
As n → ∞, the DoF tuple 12 , 21 , 12 is achievable. Every user can achieve half of
its interference-free DoF (i.e., DoF one, since only one user is present). This result
generalizes to the K-user case, where each user is still able to achieve 1/2 DoF.
2.3.4
Ergodic Interference Alignment
The interference alignment scheme described in the previous section promises that
each user can achieve 1/2 DoF only as SNR goes to infinity. For a time-varying
fading environment and when the fading coefficients follow symmetric distributions,
an ergodic interference alignment scheme is proposed in [NJGV09] to align the
undesired signals and eliminate interference at any SNR. We briefly review the idea
of the ergodic interference alignment in this part, and will provide more details
about this scheme in Chapter 4. Consider a K-user SISO interference network.
Assume that Sk (k ∈ {1, ..., K}) transmits signal xk at time slot t. Then, the
received signal at Dk is
yk (t) = hkk xk +
K
X
hkl xl + zk (t),
(2.27)
l=1,l6=k
where hkl is the channel fading coefficient between Sl and Dk and is perfectly known
at all terminals. Let channel matrix at time t be denoted as follows


h11
h12
···
h1K


h22
···
h2K 
 h21

.
H(t) , 
(2.28)
..
..
..
..

.


.
.
.
hK1
hK2
···
hKK
Then, the complement channel matrix to H(t) can be defined as follows


h11
−h12
···
−h1K


h22
···
−h2K 
 −h21
.
H(tc ) , 
..
..
..
..


.


.
.
.
−hK1
−hK2
···
hKK
(2.29)
Assume that in time slots t and tc , the channel matrices are complement to each
other. If the sources transmit the same signals in time slot tc , the received signal
at Dk is:
yk (tc ) = hkk xk −
K
X
l=1,l6=k
hkl xl + zk (tc ).
(2.30)
32
Background
If Dk adds the two signals it received in time slots t and tc , we have
yk = yk (t) + yk (tc ) = 2hkk xk + zk (t) + zk (tc ).
(2.31)
Therefore, an interference-free channel can be provided at a price of two channel
uses. We will refer to this as the channel pairing technique throughout the thesis. For
symmetric fading distributions (e.g. Rayleigh fading distribution), the complement
channel matrices H(t) and H(tc ) occur with the same probability. Since the channel
gains are drawn from a continuous distribution, the probability that each individual
channel matrix happens is equal to zero. To handle this issue, the channel pairing
can be performed based on quantized channel gains. Thus, if the time duration
of interest is long enough, for every channel matrix, a corresponding complement
channel matrix can be found, almost surely [NJGV09]. To take this advantage, let
each source transmit an independent codeword to its destination in every time slot.
If in one time slot, the channel matrix is complement to that in a previous slot,
all sources repeat their codewords. In this way, if the time duration of interest is
long enough, it is as if that the sources use half of the total time slots to send
independent signals to their destination, and the transmissions do not experience
inter-user interference. This means the average rates Rk = 21 E[log(1 + 2|hkk |2 SNR)]
(k ∈ {1, 2, ..., K}) are achievable. Each user can achieve half its interference-free
data rate at any SNR, no matter how many users exist in the network. This rate
is archived at the cost of asymptotic delay assuming that perfect CSI is globally
available at terminals. The data transmission employing this technique for delaylimited systems for which no CSI is a priori available at terminals is investigated
in Chapter 4.
2.4
Wireless Interference Networks with Imperfect CSI
To achieve the outstanding performance promised by the aforementioned schemes,
in general perfect global CSI is required at all terminals. In practice, no CSI is a
priori available at terminals. An estimate of CSI can be acquired through a pilotbased channel training scheme in which each source allocates a portion of the total
radio resources for transmitting pilot symbols and the rest for data transmission.
The pilot-based channel training schemes have been investigated in the literature
for several multiple antenna communication systems including point-to-point communication systems (see e.g. [HH03, KRB+ 13, SWB14]), and multi-user communication systems (e.g. broadcast channel [CJKR10,KJC11], uplink systems [HKD11],
interference channel [XYMN11, ALH12, MGL14]).
In [ALH12], the impact of the allocated time for channel training on the performance of interference alignment for multiple antenna systems has been addressed.
It has been assumed that transmission power for pilot transmission is the same as
the one for data transmission. In general, they can be different. A more accurate
channel estimation can be obtain by allocating more power for pilot transmission
which implies a lower power is left for data transmission. The interesting problem
2.4. Wireless Interference Networks with Imperfect CSI
33
of optimum power allocation to pilot symbols and data symbols in point to point
communication scenarios has been investigated in [HH03]. In multi-user interference
networks, finding the optimum power allocation to pilot symbols and data symbols
is even more important because of the fact that the quality of CSI estimation not
only affects the performance of each decoder, but also determines how accurately
the interference alignment can be performed. In Chapter 4, we will investigate the
performance limits of the PAEIA scheme for multi-user interference networks, and
will study this resource allocation problem. Also, similar problem is addressed in
Chapter 6 for the PAOUS scheme proposed for multi-cell interference networks.
After the estimation of CSI, the destinations can transmit either un-quantized
CSI using analog feedback or quantized CSI employing digital feedback to inform
the other terminals about the CSI. In [AH12] an analog channel state feedback
scheme for interference alignment is proposed. Destinations directly transmit the
channel coefficients as un-coded quadrature and amplitude modulated symbols. It
has been shown that no loss in achievable DoF is incurred, if the transmission power
of the analog feedback signals scales similar to the actual transmission power of the
sources.
Digital channel state feedback strategies for the interference alignment scheme
mentioned in Section 2.3.3 are provided in [BT09, KV10, FWS11, NGJV12, RG12,
LK12,KMLL12,KLC11,NWHC12,CY14]. It has been shown that the same DoF as
when perfect CSI is available can be obtained, provided that the feedback signals’
rate is proportional to log P , where P is the transmission power of each source. This
result is also observed in multiple-antenna cases in [KV10]. A continuous tradeoff
between the scaling of feedback rate and achievable DoF has also been demonstrated. This feedback strategy is further improved in [KMLL12] by introducing
proper filtering before quantizing the channel gains.
Clearly, sufficiently accurate CSI estimates can still guarantee the performance
of interference alignment. However, in practice the radio resources (e.g. bandwidth
and power) available for feedback channels are limited so that terminals may not
be able to attain CSI estimates of sufficient quality. The issue will be considered in
this thesis. Specifically, in Chapter 4, we will study the performance of the ergodic
interference alignment scheme, when the quantization resolution is strictly limited.
Chapter 3
Two-user Interference Networks:
Point-to-Point Codes
T
HIS chapter studies a two-user interference network where each transmitter
intends to communicate to the corresponding receiver at a fixed rate. The
channels follow a block-fading model with Rayleigh distributed amplitudes.
We assume that perfect CSI is globally available at all terminals. Each transmitter
is subject to a short-term power constraint. We consider the use of a point-topoint Gaussian code to conduct communication between each transmitter-receiver
pair, and investigate five different transmission schemes. When the two transmitterreceiver pairs are orthogonally activated, inter-user interference can be completely
eliminated, with the possible price of spectral inefficiency. When both users nonorthogonally access the available channel, inter-user interference must be taken
into account at the decoding process. This leads to four schemes: 1) both receivers
directly decode their desired messages by simply treating interference as noise;
2) both receivers conduct successive interference cancellation (SIC); 3) the first
receiver performs direct decoding and the second receiver performs SIC; 4) the first
receiver performs SIC and the second receiver performs direct decoding. For each
of these five schemes, we find the solution of the power control problem. For some
channel realizations this problem may not have a feasible solution. In these cases,
an outage event will be declared. We compute a lower bound and an upper bound
on the outage probability, as functions of channel statistics, desired transmission
rates, and power constraints. These results are then used to find an outer bound
and an inner bound on the ǫ-outage achievable rate region.
The structure of this chapter is as follows. In Section 3.1, we present the two-user
interference network model and define the performance metrics which will be investigated in this chapter. Section 3.2 addresses the orthogonal transmission scheme.
The non-orthogonal transmission schemes are described and their performance limits are analyzed in Section 3.3. Finally, Section 3.4 summarizes this chapter.
35
36
Two-user Interference Networks: Point-to-Point Codes
H
m1
E1
{x1 (t)}nt=1
P1
{z1 (t)}nt=1
n
√
p1 x1 (t) t=1
+
h11
{y1 (t)}nt=1
H
m
ˆ1
D1
1
h2
H
{z2 (t)}nt=1
E2
{x2 (t)}nt=1
P2
h1
2
m2
n
√
p2 x2 (t) t=1
h22
+
{y2 (t)}nt=1
H
m
ˆ2
D2
Figure 3.1: Two-user interference network.
3.1
Two-user SISO Interference Network
We consider a single-antenna fading interference network with two transmitterreceiver pairs as shown in Figure 3.1. The channels follow a Rayleigh block-fading
model, in which the channel gains remain constant within a coherent interval (the
time slots in which the considered communications occur and their number is denoted as n), and independently change across different intervals. We denote the
fading coefficient of the link between the source Sk (k ∈ {1, 2}) and the destination
Dk (direct link) as hkk ∼ CN (0, σS2 ), and denote that of the link between Sk and Dl
(l ∈ {1, 2}, l 6= k) (interference link) as hlk ∼ CN (0, σI2 ). The parameters σS2 and σI2
are the variances of the direct links and the interference links, respectively. These
generally can have different values and their ratio is denoted as
ρ , σS2 /σI2 .
(3.1)
The channel gains are mutually independent. We denote the network channel matrix
as H whose element on the ith row and the jth column is hij . In this chapter, we
assume that perfect channel knowledge to be globally available, i.e. H is known
at every terminal. During one fading block, each source sends one message to its
destination. The message from Sk , mk , is encoded using a point-to-point Gaussian
codebook with fixed data rate Rk (bits/channel use).
Definition 3.1 (Encoders). The source Sk (k ∈ {1, 2}) has an encoding function,
Ek : Mk → Cn , that maps its
message mk, which is independently and uniformly
chosen from the set Mk = 1, 2, ..., 2nRk , into a length-n codeword {xk (t)}nt=1 .
The codeword satisfies the power constraint
n
1X
2
|xk (t)| ≤ 1.
n t=1
(3.2)
Each encoder is concatenated with a power controller as shown in Figure 3.1.
3.1. Two-user SISO Interference Network
37
Definition 3.2 (Power controllers). The power controller associated with Sk (k ∈
n
{1, 2}) applies a function, Pk : Cn → Cn , that scales the codeword {xk (t)}t=1 ac√
n
cording to the channel gains to
pk xk (t) t=1 , where pk denotes the average transmission power of Sk . The assigned power obeys short-term constraint, i.e. within
each fading block it satisfies
pk ≤ pmax,k ,
(3.3)
where pmax,k is the maximum transmission power of Sk .
Definition 3.3 (Decoders). Each receiver has a decoding function, Dk : Cn → Mk ,
that maps its observed length-n channel output {yk (t)}nt=1 to an estimate m
ˆ k of
the transmitted message mk .
We address the power control problem for different transmission schemes in
the considered network. Specifically, in each fading block, transmitters seek proper
transmission powers which meet the short-term power constraints in (3.3) and also
guarantee successful communications for both transmitter-receiver pairs. In the
following, we provide some definitions and performance metrics regarding the considered system.
Definition 3.4 (Achievable rate region). Within each fading block, for a given
transmission power vector p (p = [p1 p2 ]T ) and channel matrix H, applying a
transmission scheme ‘A’, the average probability of error is defined as
n
o
ˆ 1, M
ˆ 2 ) 6= (M1 , M2 ) ,
(3.4)
Pe(n) = Pr (M
ˆ k (k ∈ {1, 2}) denote a randomly transmitted message and the
where Mk and M
corresponding decoded message, respectively. If there exist channel encoding and
(n)
decoding functions such that limn→∞ Pe = 0, then we say that a rate pair (R1 , R2 )
is achievable. We denote the corresponding achievable rate region as C A (p, H).
Definition 3.5 (Feasible transmission scheme). If the achievable rate region corresponding to a power vector p, where 0 p pmax , and pmax = [pmax,1 pmax,2 ]T ,
includes the required transmission rates of the network, i.e. (R1 , R2 ) ∈ C A (p, H),
then the power control problem for a transmission scheme ‘A’ has a solution, and we
say that the transmission scheme ‘A’ is feasible. We define the set of these vectors
as follows
A
PH
((R1 , R2 ) , pmax ) , p : 0 p pmax , (R1 , R2 ) ∈ C A (p, H) .
(3.5)
A
Using ∅ to denote an empty set, PH
6= ∅ means that the transmission scheme ‘A’
is feasible.
In fact, due to the random nature of fading, for some channel realizations, a
transmission scheme ‘A’ may not be feasible. In such cases, the messages cannot
be successfully transmitted, and the system is said to be in outage. We define the
outage probability as follows.
38
Two-user Interference Networks: Point-to-Point Codes
Definition 3.6 (Outage probability). The outage probability of a transmission
scheme ‘A’ is defined as follows
A
A
Pout
((R1 , R2 ) , pmax ) , Pr PH
((R1 , R2 ) , pmax ) = ∅ .
(3.6)
Some wireless applications can tolerate a certain amount of outage probability.
Thus, it is desirable to characterize the rate region for which an outage probability
less than the maximally tolerable outage probability can be attained. In the following we provide the definition of a related performance measure which will be
investigated for the considered system in this chapter [TV05].
Definition 3.7 (ǫ-outage achievable rate region). The ǫ-outage achievable rate
region of a transmission scheme ‘A’ is
A
CǫA (pmax ) , (R1 , R2 ) : Pout
((R1 , R2 ) , pmax ) ≤ ǫ ,
(3.7)
where ǫ is the maximum outage probability that a specific application can tolerate.
In the following sections, we study the performance of different transmission
schemes. Our analysis starts from the orthogonal transmission scheme.
3.2
Orthogonal Transmission Scheme
In wireless networks, inter-user interference may significantly degrade the communication system’s performance. As discussed in Chapter 2, one intuitive solution to
deal with interference is to orthogonalize different users’ operations. Since each user
has access to only a fraction of the available channel, this can provide interferencefree communication for each user. We term this transmission scheme orthogonal
transmission (OT) throughout the chapter. Using δ (0 < δ < 1) to denote the
channel-sharing factor, the fractions of the total channel used by the first and the
second transmitters are δ and (1 − δ), respectively. In a fading block, the achievable
rate region C OT (p, H) includes the rate pairs (R1 , R2 ) which satisfy
|h11 |2 p1
(3.8)
R1 ≤ δ log2 1 +
δN0
|h22 |2 p2
R2 ≤ (1 − δ) log2 1 +
,
(3.9)
(1 − δ)N0
where N0 is the noise power at each receiver. The set of solutions of the corresponding power control problem is
OT
PH
((R1 , R2 ) , pmax ) = {p : nT p pmax },
(3.10)
(1 − δ)N T
δN 0
0
R2 /(1−δ)
R1 /δ
2
−1
.
nT = 2
−1
2
2
|h11 |
|h22 |
(3.11)
in which
3.2. Orthogonal Transmission Scheme
39
Clearly, the minimum required transmission powers of the OT scheme are
δN
0
pOT
=
2R1 /δ − 1
(3.12)
1
|h11 |2
(1 − δ)N
0
.
(3.13)
pOT
=
2R2 /(1−δ) − 1
2
|h22 |2
This is similar to the channel inversion method in single-user point-to-point communication systems [GV97]. The solutions in (3.12) and (3.13), however, may violate the maximum power constraints in (3.3). Thus, with a certain probability, the
scheme is infeasible and outage events occur. The following proposition characterizes the outage probability.
Proposition 3.2.1. The outage probability of the OT scheme is
OT
Pout
((R1 , R2 ) , pmax ) = 1 − e
−
N0
σ2
S
(1−δ)
δ
+(2R2 /(1−δ) −1) p
(2R1 /δ −1) pmax,1
max,2
Proof. See Appendix 3.A.
.
(3.14)
The channel sharing factor δ can be carefully selected to minimize the outage
OT
probability. It can be shown that Pout
((R1 , R2 ) , pmax ) is a convex function of δ.
The optimum choice of δ, denoted as δopt , can be found by solving the following
equation:
pmax,1 R2 /(1−δopt )
R1 ln 2
R2 ln 2
1−
−1 − 2R1 /δopt 1−
2
−1 = 0.
pmax,2
(1−δopt)
δopt
For instance, if pmax,1 = pmax,2 and R1 = R2 , then δopt = 0.5 is the solution. The
result in Proposition 3.2.1 can be used to obtain the boundary of the ǫ-outage
achievable rate region CǫOT (pmax ) by solving
OT
Pout
((R1 , R2 ) , pmax ) = ǫ,
for (R1 , R2 ). The solution is
(
R1 = t
δ
×
R2 = (1 − δ) log2 1 − 2t/δ − 1 1−δ
where
pmax,2
pmax,1
(3.15)
−
pmax,2
1−δ
×
σ2
pmax,1
× S ln(1 − ǫ) .
0 ≤ t ≤ δ log2 1 −
δ
N0
2
σS
N0
,
ln(1 − ǫ)
(3.16)
(3.17)
Each (R1 , R2 ) corresponding to one particular t shows one point on the boundary
of CǫOT (pmax ).
As mentioned earlier, the OT scheme eliminates inter-user interference with the
possible price of spectral inefficiency. In fact, depending on the values of σS2 , σI2
and pmax , permitting both transmitters to send messages non-orthogonally may
outperform the OT scheme. In the next section, we will focus on non-orthogonal
transmission schemes.
40
Two-user Interference Networks: Point-to-Point Codes
3.3
Non-Orthogonal Transmission Schemes
We permit the two transmitters to transmit non-orthogonally. The source Sk (k ∈
{1, 2}) sends a unit-power codeword xk as described in Section 3.1. The channel
outputs at time t (t ∈ {1, ..., n}) are
√
√
y1 (t) =
p1 h11 x1 (t) + p2 h12 x2 (t) + z1 (t),
√
√
y2 (t) =
p1 h21 x1 (t) + p2 h22 x2 (t) + z2 (t),
(3.18)
where zk (t) is the AWGN with power N0 . Each destination may either directly
decode its intended message by treating interference as noise, or it may first decode the message of the unintended source and next decode its intended message
after removing the interference. Thus, depending on the decoding strategy, we have
four different transmission schemes. We study their power control strategies and
performance in what follows.
3.3.1
Direct Decoding at Both Receivers
In this part, we require each receiver to directly decode its desired message by
treating the interference as noise. We term this transmission scheme NOT1 . For a
fading block, the achievable rate region C NOT1 (p, H) includes the rate pairs (R1 , R2 )
which satisfy
|h11 |2 p1
,
(3.19)
R1 ≤ log2 1 +
|h12 |2 p2 + N0
|h22 |2 p2
R2 ≤ log2 1 +
.
(3.20)
|h21 |2 p1 + N0
After some mathematical manipulations, we can present these conditions as the
following power constraints
2R1 − 1
p1 ≥
|h12 |2 p2 + N0 ,
(3.21)
2
|h11 |
2R2 − 1
|h21 |2 p1 + N0 .
(3.22)
p2 ≥
|h22 |2
The constraints in (3.21) and (3.22) can be shown in the matrix format
p DS FS p + nS ,
where
DS =
"
nS =
0
2R1 − 1
0
2R2 − 1
2
R1
N0
−1
|h11 |2
2
#
, FS =
R2
(3.23)
"
0
|h21 |2
|h22 |2
N0
−1
|h22 |2
T
.
|h12 |2
|h11 |2
0
#
,
(3.24)
3.3. Non-Orthogonal Transmission Schemes
41
The matrix DS is related to the transmission rates, while the matrix FS depends
only on the channel gains.
Power Control Solution
The positive element-wise minimum transmission power vector among those which
satisfy the constraint in (3.23) - if there is any such vector - is [CHLT08]
pNOT1 = (I − DS FS )−1 nS ,
(3.25)
where I is the 2 × 2 identity matrix. Therefore, the minimum required powers are
|h21 |2
N0 2R1 − 1 |h
2 + l
|
11
1
=
pNOT
,
(3.26)
1
|h21 |2 (1 − l)
|h12 |2
N0 2R2 − 1 |h
2 + l
|
22
1
,
(3.27)
pNOT
=
2
|h12 |2 (1 − l)
|h12 |2 |h21 |2
where l = 2R1 − 1 2R2 − 1 |h
2
2 . For some channel realizations, there is no
11 | |h22 |
positive power vector that satisfies the constraints in (3.3) and (3.23). In these cases,
the power control problem does not have any feasible solution, and transmissions
will cause outage events. In the following, we will investigate the outage probability.
Outage Probability Analysis
For a given channel H, the set of feasible solutions of the power control problem is
NOT1
PH
((R1 , R2 ), pmax ) , {p : p DS FS p+nS , 0 p pmax } .
(3.28)
Figure 3.2 illustrates this set when there is at least one vector p that satisfies the constraints in (3.3) and (3.23). The red region in the figure indicates the
powers which satisfy the maximum power constraint in (3.3), and the green region
illustrates the powers which satisfy the inequality in (3.23). The intersection of the
NOT1
two regions indicates the powers within the set PH
((R1 , R2 ) , pmax ). For some
channel realizations, there is no positive power vector which satisfies (3.23), or such
a positive power vector exists but does not satisfy the maximum power constraints
NOT1
in (3.3). In this case, PH
= ∅, i.e. the power control problem is infeasible, and
an outage event occurs. We aim to characterize the outage probability. For this
purpose, we first present the following lemma to provide a necessary and sufficient
condition for the existence of a positive vector that satisfies (3.23).
Lemma 3.1. There exists at least one positive vector p (p ≻ 0) that satisfies the
inequality in (3.23), if and only if λmax (DS FS ) < 1, where λmax (DS FS ) denotes
the largest magnitude of the eigenvalues of matrix DS FS given in (3.24).
Proof. The proof is similar to that of Theorem 5 in [HC00b].
42
Two-user Interference Networks: Point-to-Point Codes
p2
pmax,2
1
pNOT
2
nS,2
nS,1
1
pNOT
1
pmax,1
p1
Figure 3.2: Solution of the power control problem for the NOT1 scheme.
Now, we use the above result to characterize a lower bound and an upper bound
NOT1
on the outage probability of the NOT1 scheme denoted as Pout
((R1 , R2 ) , pmax ).
Proposition 3.3.1. The outage probability of the NOT1 scheme is bounded as
o
o
n
n
NOT1
NOT1
NOT1
NOT1 NOT1
(3.29)
.
≤ Pout
((R1,R2) , pmax) ≤ min 1, 2−PF,1
−PF,2
1−min PF,1
,PF,2
In these equations
γβk σS4 −(bk +ρak ) βk ak σS4
k
ak ρ ak ρ(1−γ)−γb
γ
e
e
+
E1
ρ(1−γ)
1−γ
γ
4 −bk
βk γσS e
ak ρ ak ρ(1−γ)
γ
+
e
E
(a
ρ)
−
E
, k ∈ {1, 2}, (3.30)
1 k
1
ρ(1 − γ)2
γ
NOT1
PF,k
, −
where
The function E1 (x) ,
γ
,
ak
,
bk
,
βk
,
R∞
x
Proof. See Appendix 3.B.
e−t
t dt
ρ2 / 2R1 − 1
2R2 − 1 ,
N0 / σS2 pmax,k ,
N0 2Rk − 1 / σS2 pmax,k ,
ρ (ak ρ−bk )
e
.
σS4
(3.31)
denotes exponential integral as defined in [AS64].
3.3. Non-Orthogonal Transmission Schemes
100
ldbc
ldbc
ldbc
ldbc
43
bcld
ldbc
ldbc
ldbc
bcld
bc
ld
10−1
bc
ld
bc
ld
bc
bc
A
Pout
ld
bc
ldbc
10−2
bc
bc
bc
ld
10−3
ld
ld
10−4
0
10
NOT1
Pout
,
NOT1
Pout,u
,
NOT1
Pout,l ,
NOT1
Pout
,
NOT1
Pout,u
,
NOT1
Pout,l ,
OT
Pout
20
ld
ρ = 30
ρ = 30
ρ = 30
ρ = 40
ρ = 40
ρ = 40
dB
dB
dB
dB
dB
dB
bc
bc
bc
bc
bc
bc
bc
bcbc
bcbc
bc
bc
ld
ld
ld
ldld
ld
ld
ld
ld
ld
ld
ld
30
40
50
ld
60
ldld
70
SNRmax (dB)
A
Figure 3.3: Pout
versus SNRmax , R1 = R2 = 5 (bits/channel use).
In general, deriving the closed-form of outage probability for the NOT1 scheme
is involved. However, in the following two special cases, the exact value of this
probability can be found. The first case is when one transmitter has sufficiently
large maximum transmission power. Specifically, if pmax,1 → ∞, then condition
1
< pmax,1 in (3.79) almost surely holds and comparing (3.79) and (3.81) in
pNOT
1
Appendix 3.B we can see that
NOT1
NOT1
Pout
((R1 , R2 ) , pmax ) = 1 − PF,2
.
(3.32)
Similarly, if pmax,2 → ∞, then
NOT1
NOT1
Pout
((R1 , R2 ) , pmax ) = 1 − PF,1
.
(3.33)
The next case is when one of the transmission rates is sufficiently small. Specifically,
1
1
if R1 → 0, then according to (3.26) pNOT
→ 0, and condition pNOT
< pmax,1 in
1
1
(3.79) almost surely holds. Comparing (3.79) and (3.81) in Appendix 3.B it can be
concluded that the outage probability is that in (3.32). Also, it can be shown that
if R2 → 0, then the exact outage probability is the probability given in (3.33). In
other cases, we can verify the tightness of the bounds through numerical evaluation.
44
Two-user Interference Networks: Point-to-Point Codes
NOT1
NOT1
Figure 3.3 shows Pout,l
((R1 , R2 ) , pmax ) and Pout,u
((R1 , R2 ) , pmax ) given in
Proposition 3.3.1 versus SNRmax (SNRmax = pmax,1 /N0 = pmax,2 /N0 ) for different values of ρ. Also, the simulation results of the exact outage probability
NOT1
OT
Pout
((R1 , R2 ) , pmax ) and the value of Pout
((R1 , R2 ) , pmax ), computed according to Proposition 3.2.1, are shown for comparison. In this example, we set σS2 = 1
and R1 = R2 = 5 (bits/channel use). We can observe that in a certain range of
SNRmax , NOT1 outperforms OT by attaining lower outage probabilities. The range
increases when ρ becomes larger. However, when SNRmax is selected to be sufficiently large, the outage probability is no longer sensitive to the change of power
constraints, i.e. an error floor exists. In this region, the OT scheme can attain better
performance in terms of outage probability. The reason is that at high SNRmax , the
constraint in (3.23) is the dominant constraint which is required to be satisfied for
successful communication. Since according to Lemma 3.1, whether this constraint
is satisfied does not depend on pmax , in fact, it only depends on the channel gains
and the rates. Thus, the outage probability saturates at certain level as SNRmax increases. This figure also shows that as ρ increases, the NOT1 scheme attains a lower
outage probability at asymptotically high SNRmax . This is because as ρ increases,
the interference links become statistically weaker compared to the desired links and
the outage probability decreases. We drive the outage probability at asymptotically
high SNRmax to investigate how it depends on ρ.
Corollary 3.3.1. For the network presented in (3.18), the outage probability of the
NOT1 scheme at asymptotically high SNRmax is
(
γ ln(γ)
γ
+ (γ−1)
1 − γ−1
2 γ 6= 1
NOT1
.
(3.34)
Pout ((R1 , R2 ), ∞) =
0.5
γ=1
Proof. See Appendix 3.C.
It is worth mentioning that, although Proposition 3.3.1 provides a lower
bound on the outage probability, Corollary 3.3.1 gives the exact value of the
outage probability at high SNRmax . This result shows that at this regime, the
outage probability only depends on parameter γ, which is a function of R1 , R2 , and
the channel parameter ρ.
The ǫ-outage Achievable Rate Region
We can use the lower bound in Proposition 3.3.1 to obtain an outer bound on
NOT1
NOT1
CǫNOT1 (pmax ) denoted as Cǫ,out
(pmax ) by solving Pout,l
((R1 , R2 ) , pmax ) = ǫ for
(R1 , R2 ). This equation has different solutions for (R1 , R2 ); each of them denotes
NOT1
one point on the boundary of Cǫ,out
(pmax ). Similarly, we can characterize an inner
NOT1
NOT1
(pmax ) denoted as Cǫ,in
bound on Cǫ
(pmax ) by solving
NOT1
Pout,u
((R1 , R2 ) , pmax ) = ǫ
(3.35)
3.3. Non-Orthogonal Transmission Schemes
45
11
NOT1
Cǫ,out
,
NOT1
Cǫ,in ,
NOT1
Cǫ,out
,
NOT1
Cǫ,in ,
NOT1
Cǫ,out
,
NOT1
Cǫ,in ,
CǫOT
10
9
R2 (bits/channel use)
8
7
ρ = 20
ρ = 20
ρ = 30
ρ = 30
ρ = 40
ρ = 40
dB
dB
dB
dB
dB
dB
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
R1 (bits/channel use)
Figure 3.4: The inner and the outer bounds on the ǫ-outage achievable rate region of the NOT1 scheme and ǫ-outage achievable rate region of the OT scheme
(ǫ = 0.001, SNRmax = 50 dB).
for (R1 , R2 ). Figure 3.4 shows the inner bounds and the outer bounds on the ǫoutage achievable rate regions of the NOT1 scheme for different values of ρ. We
also plot the ǫ-outage achievable rate regions of the OT scheme for comparison. In
this example, we set ǫ = 0.001, and SNRmax = 50 dB. This figure shows that for
these parameters, if ρ is sufficiently large, then the achievable rate region of the
NOT1 scheme is larger than that of the OT scheme.
Although the outage probability of the NOT1 is small when the interference
links are relatively weak compared to the desired links (i.e. when ρ is large), it is
large when the interference links become relatively stronger than the desired links
(i.e. when ρ is small). Therefore, for such channels, decoding the desired message
by treating interference as noise may not be the best decoding strategy to apply. In
the next section, we consider another scheme which is suitable for such scenarios.
3.3.2
Successive Interference Cancellation at Both Receivers
If each receiver knows the codebooks of both transmitters, it can perform SIC by
decoding the message of the interfering transmitter, removing the interference, and
46
Two-user Interference Networks: Point-to-Point Codes
then decoding the message of the desired transmitter. We term this transmission
scheme NOT2 . The receivers would be able to decode the messages of the interfering
transmitters in a fading block, if the following conditions are satisfied:
|h21 |2 p1
R1 ≤ log2 1 +
,
(3.36)
|h22 |2 p2 + N0
|h12 |2 p2
R2 ≤ log2 1 +
.
(3.37)
|h11 |2 p1 + N0
The equivalent power constraints are
p1
p2
2R1 − 1
|h21 |2
≥
2R2 − 1
|h12 |2
≥
|h22 |2 p2 + N0 ,
(3.38)
|h11 |2 p1 + N0 .
(3.39)
The corresponding power constraint in matrix format is
p DI FI p + nI ,
(3.40)
where
DI
,
"
nI
,
2R1 − 1
0
2R1 − 1
0
R2
2 −1
N0
|h21 |2
#
, FI ,
2R2 − 1
"
|h22 |2
|h21 |2
0
|h11 |2
|h12 |2
N0
|h12 |2
T
0
.
#
,
(3.41)
The matrix DI depends on the rates and the matrix FI is a function of only the
channel gains. After interference cancellation, each receiver can successfully decode
its desired message if the following conditions are satisfied:
|h11 |2 p1
R1 ≤ log2 1 +
,
(3.42)
N0
|h22 |2 p2
R2 ≤ log2 1 +
.
(3.43)
N0
Therefore, the transmission powers should also satisfy the following condition
p nF ,
(3.44)
where
T
nF , [nF,1 nF,2 ] =
2
R1
N0
−1
|h11 |2
2
R2
N0
−1
|h22 |2
T
.
(3.45)
3.3. Non-Orthogonal Transmission Schemes
47
Power Control Solution
The positive element-wise minimum transmission power vector among those satisfying the constraints in (3.40) and (3.44) - if there is any such vector - can be found as
described in the following. Let’s define pI , (I − DI FI )−1 nI where pI , [pI,1 pI,2 ]T .
Depending on the channel gains and transmission rates, the minimum required
powers can be found according to one of the four cases that will be mentioned in
the following. Figure 3.5 shows four plots each corresponding to one possible case.
The red region denotes powers which satisfy the power constraint in (3.3); the green
region shows the powers which satisfy (3.40); the blue region illustrates the powers which satisfy the constraint in (3.44). The intersection of these regions shows
powers that can provide successful transmission. There are four different possible
cases to allow this to happen. In each case, we find the minimum required powers
as follows.
Case 1 If nF pI pmax , as shown in Figure 3.5 (a), the minimum power
solution of the power control problem is
pNOT2 = pI .
(3.46)
This solution is marked by a small circle in the figure.
Case 2
If the following conditions are satisfied
nF,2
pI,1 ≤ nF,1 ≤ pmax,1 ,
≤ 2R1 2R2 − 1 |hN120|2 ≤ pmax,2 ,
(3.47)
as shown in Figure 3.5 (b), the minimum required powers are
N0
|h11 |2
N0
.
−1
|h12 |2
2
pNOT
1
=
nF,1 = 2R1 − 1
(3.48)
2
pNOT
2
=
2R1 2R2
(3.49)
The solution is shown by a small circle in the figure.
Case 3
If
nF,1 ≤ 2R2 2R1 − 1
N0
|h21 |2
≤ pmax,1
pI,2 ≤ nF,2 ≤ pmax,2 ,
(3.50)
as shown in Figure 3.5 (c), the minimum required powers are
N0
|h21 |2
N0
.
−1
|h22 |2
2
pNOT
1
= 2R2 2R1 − 1
(3.51)
2
pNOT
2
= nF,2 = 2R2
(3.52)
48
Two-user Interference Networks: Point-to-Point Codes
p2
p2
pmax,2
pmax,2
pI,2
pI,2
nF,2
nF,2
nF,1 pI,1
pmax,1
p1
pI,1 nF,1
(a) Case 1
pmax,1
p1
(b) Case 2
p2
p2
pmax,2
pmax,2
nF,2
nF,2
pI,2
pI,2
nF,1
pI,1
pmax,1
p1
(c) Case 3
pI,1 nF,1
pmax,1
p1
(d) Case 4
Figure 3.5: Solution of the power control problem for the NOT2 scheme. In these
plots, the red region denotes powers satisfying the maximum power constraint in
(3.3); the green region shows the powers satisfying the constraint in (3.40); the blue
region illustrates the powers which satisfy the constraint in (3.44). The intersection
of these regions shows powers which can provide successful transmission. The small
circle in each plot indicates the minimum required powers in each case.
This solution is shown in the figure by a small circle.
Case 4
If the following conditions satisfy
pI,1 ≤ nF,1 ≤ pmax,1
2
11 |
R2
−1 +1 ≤ nF,2
2R1−1 |hN120|2 |h
|h22 |2 2
3.3. Non-Orthogonal Transmission Schemes
nF,2 ≤ N0
1
|h11
|2
49
|h21 |2
1
− |h21
2
|
|h22 |2 ,
(3.53)
as shown in Figure 3.5 (d), the minimum required powers are
N0
|h11 |2
N0
.
−1
|h22 |2
2
pNOT
1
=
2R1 − 1
(3.54)
2
pNOT
2
=
2R2
(3.55)
This solution is marked in the figure by a small circle.
Outage Probability Analysis
The set of feasible solutions of the power control problem for the NOT2 scheme is
NOT2
PH
((R1 , R2 ) , pmax ) = {p : p DI FI p + nI , nF p pmax } .
(3.56)
This set is illustrated in Figure 3.5 as the intersection of the green region,
the red region, and the blue region. The red region in this figure denotes powers
which satisfy the constraint in (3.3); the green region shows the powers satisfying (3.40); the blue region illustrates the powers which satisfy the constraint in
(3.44). Therefore, if these three regions have no common intersection region, then
NOT2
PH
((R1 , R2 ) , pmax ) = ∅ and an outage event happens. In the following corollary, we provide a lower bound on the outage probability of the NOT2 scheme.
Proposition 3.3.2. For the network described in (3.18), the outage probability of
the NOT2 scheme is bounded as follows:
n
o
NOT2
NOT2
NOT2
NOT2
1 − min PF,D,1
, PF,D,2
, PF,I
≤ Pout
((R1 , R2 ) , pmax ) .
(3.57)
In this equation
a (1−γ ′ )−γ ′ρb
k
k
ak
γ ′ ρβk′ σS4 − ρbkρ+ak βk′ ak σS4
ργ ′
e
E
e
+
1
′
′
′
(1−γ )
1−γ
ργ
a (1−γ ′ ) ′
′ 4 −bk
k
β ργ σS e
ak
ak
+ k
−
E
E
e ργ ′
1
1
′
2
′
(1 − γ )
ρ
ργ
R
R
2
1
(2 −1) + (2 −1)
N
− 0
NOT2
PF,D,k
=−
NOT2
PF,I
=e
σ2
S
pmax,1
pmax,2
k ∈ {1, 2},
(3.58)
where
γ ′ = 1/ ρ2 2R1 −1 2R2 −1 ,
ak = N0 / pmax,k σS2 ,
bk = N0 2Rk − 1 / pmax,k σS2 ,
1 (ak −ρbk )/ρ
βk′ =
e
.
ρσS4
(3.59)
50
Two-user Interference Networks: Point-to-Point Codes
100
rSbC
rSbC
rSbC
rSbC
rSbC
bCrS
10−1
bCrSbC
bC
rSbC
bC
10−2
bC
bC
bCbC
bC
A
Pout
rS
bCbC
bC
bC
bC
rS
10−3
rS
rS
bC
bC
rS
10−4
rS
10−5
0
10
NOT2
Pout
,
NOT2
Pout,l
,
NOT2
Pout ,
NOT2
Pout,l
,
OT
Pout
20
ρ = −30
ρ = −30
ρ = −40
ρ = −40
30
dB
dB
dB
dB
rS
rS
rS
rS
40
50
rS
rS
60
rS
70
SNRmax (dB)
A
Figure 3.6: Pout
versus SNRmax , R1 = R2 = 5 (bits/channel use).
Proof. See Appendix 3.D.
NOT2
Figure 3.6 plots the lower bound on Pout
versus SNRmax for different values
NOT2
OT
of ρ. Also, Pout and the simulation results of the exact values of Pout
are shown
for comparison. This shows that in a certain range of SNRmax , NOT2 outperforms
the OT scheme. We can also see that the outage probability of NOT2 saturates at
high SNRmax . Thus, if SNRmax is larger than a certain value, OT attains a lower
outage probability and outperforms NOT2 . This is similar to the behaviour of the
NOT1 scheme. The reason is that, at high SNRmax , the dominant factor leading
to an outage event is that there is no positive power vector to satisfy (3.40) rather
than the violation of the power constraints. However, in contrast to the NOT1
scheme, the performance of the NOT2 scheme improves as ρ decreases. The reason
is that for the smaller values of ρ, the interference links are most likely stronger than
the desired links. Thus, it becomes easier to decode the message of the interfering
transmitter and remove the interference.
Even at high SNRmax , the outage event may occur due to the fact that no positive power vector may exist to satisfy (3.40). The following corollary characterizes
the outage probability of the NOT2 scheme at asymptotically high SNRmax .
3.3. Non-Orthogonal Transmission Schemes
51
Corollary 3.3.2. For the network presented in (3.18), the outage probability of the
NOT2 scheme with asymptotically high SNRmax is
(
′
′
′
)
′
1 − γ ′γ−1 + γ(γ ′ln(γ
NOT2
−1)2 γ 6= 1
Pout ((R1 , R2 ), ∞) =
,
(3.60)
0.5
γ′ = 1
where γ ′ , 1/ ρ2 2R1 − 1 2R2 − 1 and ρ is given in (3.1).
Proof. The proof is similar to that of Corollary 3.3.1.
This result confirms our observation in Figure 3.6, regarding the saturation of
the feasibility probability at high SNRmax .
The ǫ-outage Achievable Rate Region
NOT2
We can characterize an outer bound on CǫNOT2 (pmax ) denoted as Cǫ,out
(pmax )
using the lower bound given in Proposition 3.3.2 by solving
NOT2
Pout,l
((R1 , R2 ) , pmax ) = ǫ,
(3.61)
for (R1 , R2 ). Each solution of this equation for (R1 , R2 ) denotes one point on the
NOT2
boundary of Cǫ,out
(pmax ). The outer bound on CǫNOT2 (pmax ), for different values of
ρ, and the ǫ-outage achievable rate region of the OT scheme are shown in Figure 3.7.
In this particular example, we set ǫ = 0.001 and SNRmax = 50 dB. It can be seen
that, when ρ has a small value, NOT2 achieves a larger rate region compared to
OT. Indeed, non-orthogonal transmission and SIC at both receivers is beneficial in
this case.
If only one of the receivers observes strong interference, SIC at both receivers
may not be the best decoding strategy. Instead, SIC can be employed at the receiver
which observes strong interference and direct decoding by treating the interference
as noise can be performed at the other receiver to achieve a better performance.
We investigate this scheme in more detail in the next part.
3.3.3
Successive Interference Cancellation at One Receiver
The receivers may implement different decoding strategies. For instance, the first
receiver can perform SIC, while the second receiver decodes its message directly by
treating the interference as noise. We refer to this scheme as NOT3 . The following
conditions should be satisfied for a successful transmission in a fading block:
|h12 |2 p2
,
(3.62)
R2 ≤ log 1 +
|h11 |2 p1 + N0
|h11 |2 p1
,
(3.63)
R1 ≤ log 1 +
N0
|h22 |2 p2
R2 ≤ log 1 +
.
(3.64)
|h21 |2 p1 + N0
52
Two-user Interference Networks: Point-to-Point Codes
10
NOT2
Cǫ,out
, ρ = −20 dB
NOT2
Cǫ,out , ρ = −30 dB
NOT2
Cǫ,out
, ρ = −40 dB
OT
Cǫ
9
R2 (bits/channel use)
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
R1 (bits/channel use)
Figure 3.7: The outer bounds on the ǫ-outage achievable rate region (ǫ =
0.001, SNRmax = 50 dB).
Therefore, the powers should satisfy
p nR ,
(3.65)
where

nR = 
2R2−1 N0 max
N0
|h11 |2
2
2 R1
2R1 |h11 | +|h21 | (2 −1)
,
|h12 |2
|h22 |2 |h11 |2
2R1−1

.
(3.66)
Power Control Solution
The minimum powers which satisfy the constraint in (3.65) are
3
pNOT
1
=
3
pNOT
2
=
N0
,
|h11 |2
(
)
N0 2R1 N0 |h11 |2+|h21 |2 2R1−1
R2
.
,
2 −1 max
|h12 |2
|h22 |2 |h11 |2
2R1−1
(3.67)
(3.68)
3.3. Non-Orthogonal Transmission Schemes
53
Based on the same argument, we can show that the minimum required powers
for the successful transmission of the NOT4 scheme (performing SIC at the second receiver, while the first receiver directly decodes its message by treating the
interference as noise) are
(
)
N0 2R2 N0 |h22 |2+|h12 |2 2R2−1
NOT4
R1
,
p1
= 2 −1 max
|h21 |2
|h11 |2 |h22 |2
4
pNOT
= 2R2−1
2
N0
.
|h22 |2
(3.69)
However, for some channel realizations, the calculated powers in (3.67) and
(3.68) for NOT3 or those in (3.69) and (3.69) for NOT4 may violate the power constraint in (3.3); thus, an outage event may happen. In the next part, we investigate
the outage probability in more details.
Outage Probability Analysis
The set of the feasible solutions of the power control problem for the NOT3 scheme
is
NOT3
PH
((R1 , R2 ) , pmax ) , {p : nR p pmax } .
(3.70)
This set is illustrated in Figure 3.8 as the intersection region of the red and
the blue regions. The red region shows powers which satisfy the maximum power
constraint in (3.3) and the blue region denotes the powers which satisfy the constraint in (3.65). If the red and the blue regions do not overlap, then an outage
event occurs. The outage probability is characterized as follows.
Proposition 3.3.3. For the network presented in (3.18), the outage probability of
the NOT3 scheme is bounded as follows
o
o
n
n
NOT3
NOT3
NOT3
NOT3 NOT3
.
≤ Pout
((R1 , R2 ) , pmax ) ≤ min 1,2−PF,1
−PF,2
1−min PF,1
,PF,2
(3.71)
In these equations
e−(2 −1)N0 /(pmax,1 σS ) ,
c1 b2 c1ρb2
c1 b 2
NOT3
PF,2
=
e−b2 −
e−(c1 +1)b2 ρ ,
e
E1
ρ
ρ
where c1 = 2R1 − 1 and b2 = N0 2R2 − 1 /pmax,2 σS2 .
NOT3
PF,1
=
Proof. See Appendix 3.E.
R1
2
(3.72)
(3.73)
54
Two-user Interference Networks: Point-to-Point Codes
p2
pmax,2
nR,2
nR,1
pmax,1
p1
Figure 3.8: Solution of the power control problem for the NOT3 scheme.
Similarly, we can determine upper and lower bounds on the outage probability
of the NOT4 scheme as mentioned in the following proposition.
Proposition 3.3.4. For the network presented in (3.18), the outage probability of
the NOT4 scheme is bounded as
n
o
n
o
NOT4
NOT4
NOT4
NOT4
NOT4
1−min PF,1
, PF,2
≤ Pout
((R1 , R2 ),pmax ) ≤ min 1,2−PF,1
−PF,2
.
(3.74)
In these equations,
c2 b 1
c2 b1 c2ρb1
NOT4
e−(c2 +1)b1 ρ ,
e
E1
PF,1
= e−b1 −
ρ
ρ
R2
2
P NOT4 = e−(2 −1)N0 /(pmax,2 σS ) ,
F,2
where c2 = 2R2 − 1, and b1 = N0 2R1 − 1 /pmax,1 σS2 .
(3.75)
(3.76)
NOT3
Figure 3.9 shows the lower bound and the upper bound on Pout
versus
SNRmax for different values of ρ. The figure shows that in this specific setting
the OT scheme outperforms NOT3 . To obtain a better evaluation of this scheme,
the performance of NOT3 and OT have been shown for a sample asymmetric
network in Figure 3.10. In the considered network, the users have different rates
(R1 = 7, R2 = 2). It can be seen that in this case NOT3 outperforms OT.
3.3. Non-Orthogonal Transmission Schemes
100
uTbCrS
uTbCrS
uTbCrS
uTbCrS
uTbCrS
uTbCrS
uTbC
55
uTbC
uT
rS
uT
bC
uT
rS
bC
rS
uT
rS
10−1
bC
uT
rS
bC
rS
10−2
A
Pout
rS
rS
bC
10−3
bC
bC
uT
uT
10−4
uT
10−5
0
10
NOT3
Pout,l
,
NOT3
Pout ,
NOT3
Pout,u
,
NOT3
Pout,l ,
NOT3
Pout
,
NOT3
Pout,u ,
NOT3
Pout,l
,
NOT3
Pout ,
NOT3
Pout,u
,
OT
Pout
20
uT
rS
ρ = 0 dB
ρ = 0 dB
ρ = 0 dB
ρ = 10 dB
ρ = 10 dB
ρ = 10 dB
ρ = 20 dB
ρ = 20 dB
ρ = 20 dB
30
bCbC
uT
rS
bC
rS
bC
rS
rS
40
50
60
70
SNRmax
A
Figure 3.9: Pout
versus SNRmax , R1 = R2 = 5 (bits/channel use).
The ǫ-outage Achievable Rate Region
Similar to the approach that we have considered for the other schemes, the lower
bound on the outage probability in Proposition 3.3.3 can be used to obtain an
NOT3
outer bound on CǫNOT3 (pmax ), denoted as Cǫ,out
(pmax ), by solving the equation
NOT3
Pout,l ((R1 , R2 ) , pmax ) = ǫ for (R1 , R2 ). This equation may have many solutions
for (R1 , R2 ); each of them denotes one point on the boundary of the outer bound
NOT3
region. Similarly, an inner bound on CǫNOT3 (pmax ), denoted as Cǫ,in
(pmax ), can
NOT3
be found by solving Pout,u ((R1 , R2 ) , pmax ) = ǫ for (R1 , R2 ). Also, the results of
Proposition 3.3.4 can be used to find an inner bound and an outer bound on
CǫNOT4 (pmax ).
Figure 3.11 shows the inner bound and the outer bound on the ǫ-outage achievable rate regions of the NOT3 scheme and the ǫ-outage achievable rate region of the
OT scheme when ρ = 0 dB. In this example, we set ǫ = 0.001 and SNRmax = 50 dB.
The figure shows that, the achievable rate region of none of these schemes is strictly
larger that the other. In fact, each of them can achieve certain rates that may not
be achievable by the other scheme. Similar observations can be obtained for the
NOT4 scheme.
56
Two-user Interference Networks: Point-to-Point Codes
100
rS
rS
rS
rS
rS
rS
rS
rS
10−1
rS
A
Pout
rS
10−2
rS
rS
rS
10−3
rS
10−4
0
10
rS
OT
Pout
NOT3
Pout,u
, ρ = 0 dB
20
30
40
50
60
70
SNRmax
A
Figure 3.10: Pout
versus SNRmax , R1 = 7, R2 = 2 (bits/channel use).
3.4
Summary
In this chapter, we have studied transmission schemes for fixed-rate transmission
over a two-user Rayleigh block-fading interference network. The proposed schemes
use a point-to-point encoder at each transmitter in concatenation with a power
controller. The power controllers adjust transmission powers during each fading
block, considering a short-term individual power constraint, to successfully transmit at desired rates. We have considered orthogonal and non-orthogonal transmission schemes. In the latter case, we have investigated different decoding schemes
in which each receiver either directly decodes its desired message by treating the
interference as noise, or performs SIC. For each of these schemes, we have found
the solution of the power control problem to assign the minimum required power
to each transmitter. Since the power control problem for some channel realizations
may not have any feasible solution, transmission may be in outage. We have computed lower bounds and upper bounds on the probability of such events for these
schemes. Using these results, we have characterized an inner bound and an outer
bound on the ǫ-outage achievable rate region for each transmission scheme.
3.A. The Proof of Proposition 3.2.1
57
8
CǫOT
NOT3
Cǫ,out
NOT3
Cǫ,in
7
R2 (bits/channel use)
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
R1 (bits/channel use)
Figure 3.11: The inner bound and the outer bound on the ǫ-outage achievable rate
region. (ǫ = 0.001, ρ = 0 dB, SNRmax= 50 dB).
3.A
The Proof of Proposition 3.2.1
The outage probability of the orthogonal transmission scheme can be found as
follows
OT
OT
Pout
((R1 , R2 ), pmax ) = 1−Pr PH
((R1 , R2 ), pmax ) 6= ∅
δN
0
< pmax,1 ,
= 1 − Pr 2R1 /δ − 1
|h11 |2
(1 − δ)N
0
R2 /(1−δ)
2
−1
< pmax,2
|h22 |2
δN
(a)
0
2
R1 /δ
< |h11 |
= 1 − Pr 2
−1
pmax,1
(1 − δ)N
0
R2 /(1−δ)
2
×Pr 2
−1
< |h22 |
pmax,2
N
(1−δ)
− 20 (2R1 /δ −1) p δ
+(2R2 /(1−δ) −1) p
(b)
σ
max,1
max,2
= 1−e S
(3.77)
58
Two-user Interference Networks: Point-to-Point Codes
where (a) follows the independence of the channel gains, and (b) follows the fact
that Xk = |hkk |2 (k ∈ {1, 2}) has an exponential distribution with a cumulative
distribution function (cdf) as follows:
2
FXk (x) = 1 − e−x/σS , x > 0.
3.B
(3.78)
The proof of Proposition 3.3.1
NOT1
The probability Pout
((R1 , R2 ) , pmax ) is
n
o
NOT1
NOT1
Pout
((R1 , R2 ),pmax ) = 1−Pr PH
((R1 , R2 ),pmax ) 6= ∅
= 1−Pr λmax (DS FS ) < 1, pNOT1≺pmax ,
(a)
(3.79)
1
1
where pNOT
and pNOT
are the minimum required transmission powers for the
1
2
NOT1 scheme calculated in (3.26) and (3.27), and λmax (DS FS ) is the largest magnitude of the eigenvalues of matrix DS FS given in (3.24). The equality (a) follows
NOT1
the definition in (3.28) and Lemma 3.1. Defining the probabilities PF,1
and
NOT2
PF,1
as
o
n
NOT1
1
PF,1
, Pr λmax (DS FS ) < 1, pNOT
< pmax,1
1
n
o
NOT1
1
PF,2
, Pr λmax (DS FS ) < 1, pNOT
<
p
,
max,2
2
(3.80)
(3.81)
we can bound the feasibility probability as follows
(a)
NOT1
NOT1
PF,1
+PF,2
−1 ≤ Pr λmax (DS FS )<1, pNOT1 ≺ pmax
n
o
(b)
NOT1
NOT1
≤ min PF,1
, PF,2
,
(3.82)
n T o
T
where the equality (a) follows the fact that Pr {A B} = 1 − Pr A B = 1 −
S Pr A B ≥ 1 − Pr A − Pr B = Pr {A} + Pr
T {B} − 1, where A and B are
random events; and the equality (b) follows Pr {A B} ≤ Pr {A}. The probability
NOT1
PF,1
can be further simplified as follows
NOT1
PF,1


21 |2 
|h |2 |h |2 γ N0 2R1−1 |h
+l
2
|h11 |
12
21
,
<
,
<p
= Pr
max,1

|h11 |2 |h22 |2 ρ2
|h21 |2 (1 − l)
(3.83)
3.B. The proof of Proposition 3.3.1
59
2
2
2
2
|h12 | |h21 |
ρ
ρ
where γ = (2R1 −1)(2
R2 −1) , and l = |h |2 |h |2 × γ . After some mathematical ma11
22
nipulations we have
)
(
2
2
|h22 | 2
2
NOT1
2
2 ρ
PF,1
= Pr
2 |h11 | −b1 σS > |h12 | +a1 σS
γ
|h21 |


 |h21 |2 |h12 |2+a1 σS2
o
n
γ
< 2
= Pr b1 σS2 < |h11 |2 ×Pr 0<
 |h |2 |h |2−b σ 2
ρ 
11
1 S
22
=
where Q = Pr 0 <
e−b1 × Q
(3.84)
2
|h21 |2 (|h12 |2 +a1 σS
)
2
|h22 |2 (|h11 |2 −b1 σS
)
<
γ
ρ2
, a1 =
N0
,
2p
σS
max,1
and b1 =
N0 (2R1 −1)
.
2p
σS
max,1
To calculate Q, we derive the probability density function (pdf) of the random
|h21 |2 (|h12 |2 +a1 σ2 )
variable M , |h |2 |h |2 −b σ2S . Let Xkl , |hkl |2 and Ykk , |hkk |2 (∀k, l ∈ {1, 2},
1 S)
22 ( 11
k 6= l), we have
1 −x/σI2
x ≥ 0,
e
σI2
2
1
fYkk (y) = 2 e−y/σS y ≥ 0.
σS
fXkl (x) =
(3.85)
(3.86)
Let
|h21 |2
X21
=
,
|h22 |2
Y22
X12 + a1 σS2
|h12 |2 + a1 σS2
=
,
,
2
2
|h11 | − b1 σS
Y11 − b1 σS2
Z21 ,
Z12
using the fact that the pdf of a random variable Z =
we can show that the pdf of Z21 and Z12 are
X
Y
R +∞
is fZ (z)= −∞ |y|fXY (zy, y)dy,
ρ
, z ∈ R,
(1+ρz)2
1+ρz
β1 σS4
β1 a1 σS4
e−a1( z ) , z ∈ R,
fZ12 (z)=
+
(1+ρz)z (1+ρz)2
fZ21 (z)=
where ρ =
2
σS
σI2
and β1 =
ρ (a1 ρ−b1 )
.
4e
σS
(3.87)
(3.88)
(3.89)
The pdf of the random variable
2
|h12 | + a1 σS2
= Z12 Z21
M,
|h22 |2 |h11 |2 − b1 σS2
2
|h21 |
(3.90)
60
Two-user Interference Networks: Point-to-Point Codes
R +∞ 1
fZ12 Z21 (t, m
cumulative districan be derived as fM (m) = −∞ |t|
t )dt. Since the
Rm
bution function (cdf) of the random variable M is FM (m) = x=−∞ fM (x)dx, we
have
Z m Z ∞
β1 σS4 ρ(a1 t2+(1+a1ρ)t) −(a1 ρ+a1 t)
FM (m)=
e
dtdx
2
2
x=−∞ t=0 (1 + ρxt) (ρ + t)
1 −ρa1
β1 σS4
=−
−
a
E
(a
ρ)
e
1
1
1
1 − ρ2 m ρ
a1 (1−ρ2 m)
β1 ρmσS4
β1 a1 σS4
a1
ρm
.
−
e
×
E
(a
ρ)
−
E
−
1
1
1
1 − ρ2 m (1 − ρ2 m)2
ρm
(3.91)
Therefore,
Q = FM (γ/ρ2 ) − FM (0)
a1 ρ(1−γ)
a1 ρ
γβ1 σS4 −ρa1 β1 a1 σS4
e γ
+
e
E1
=−
ρ(1 − γ)
1−γ
γ
4
a1 ρ(1−γ)
a1 ρ
β1 γσS
γ
e
E1 (a1 ρ)−E1
+
ρ(1 − γ)2
γ
(3.92)
NOT1
NOT1
By plugging (3.92) in (3.84), we can find PF,1
. It is possible to find PF,2
with similar approach. By plugging these in (3.82), the lower bound and the upper
NOT1
bound on Pout
((R1 , R2 ) , pmax ) can be obtained. This completes the proof.
3.C
The proof of Corollary 3.3.1
We have
(a)
NOT1 (b)
NOT1
lim
P NOT1 = 1− lim PF,2
=1− lim PF,1
pmax,1 ,pmax,2 →∞ out
pmax,2→∞
pmax,1 →∞
NOT1
= 1− lim PF,1
= 1−(A+B +C),
a1,b1 →0
(3.93)
NOT1
where (a) and (b) follows the equality in (3.79), and the definitions of PF,1
and
NOT1
PF,2
in (3.80) and (3.81), respectively. The values A, B and C can be calculated
as follows
γ
γβk σS4 −(bk +ρak )
e
=−
ak ,bk →0
ρ(1 − γ)
1−γ
ak ρ(1−γ)−γbk (a)
βk ak σS4
ak ρ
γ
B = lim
e
= 0
E1
ak ,bk →0 1 − γ
γ
a1 ρ a1 ρ(1−γ)
βk γσS4 e−bk
γ
e
E1 (a1 ρ)−E1
C = lim
a1 ,b1 →0 ρ(1 − γ)2
γ
A = lim
−
(3.94)
(3.95)
3.D. The Proof of Proposition 3.3.2
61
a1 ρ
γ
(1+a1q+· · · )
lim E1 (a1 ρ) − E1
=
(1 − γ)2 a1 →0
γ
(c)
γ
a1 ρ
=
lim E1 (a1 ρ) − E1
(1 − γ)2 a1 →0
γ
Z ∞ −t
Z ∞ −t !
e
γ
e
(d)
lim
dt −
dt
=
a1 ρ
(1 − γ)2 a1 →0
t
t
a1 ρ
γ
Z aγ1 ρ −t
γ
e
=
lim
dt
(1 − γ)2 a1 →0 a1 ρ t
Z aγ1 ρ
(1 − t + t2 /2 − · · · )
γ
(e)
lim
dt
=
2
(1 − γ) a1 →0 a1 ρ
t
γ
=−
ln γ
(1 − γ)2
(b)
(3.96)
where q = (1 − γ)ρ/γ. The equalities (a) and (c) follow the fact that limx→∞ x ·
E1 (x) = 0; (b) follows the Taylor series expansion of ea1 ρ(1−γ)/γ ; (d) follows the
definition of E1 (x); and (e) follows the Taylor series expansion of e−t . Therefore,
lim
pmax,1 ,pmax,2 →∞
3.D
NOT1
Pout
=1−
γ
γ
ln γ.
+
γ − 1 (γ − 1)2
(3.97)
The Proof of Proposition 3.3.2
We have
NOT2
Pout
((R1 , R2 ),pmax ) ≥ 1 − Pr{λmax (DI FI ) < 1, pI pmax ,nF pmax } . (3.98)
Defining
PF,I
,
PF,D,k
,
Pr{nF pmax }
Pr{λmax (DI FI ) < 1, pI,k ≤ pmax,k } k ∈ {1, 2},
(3.99)
(3.100)
we can show that
Pr {λmax (DI FI ) < 1, pI pmax , nF pmax }
(a)
≤ min {PF,I , PF,D,1 , PF,D,2 } .
T T
The inequality (a) follows Pr {A B C} ≤ Pr {A}. We have
PF,I =Pr {nF pmax }
N0
N0
R2
−1
<
p
,
2
<
p
=Pr 2R1 −1
max,1
max,2
|h11 |2
|h22 |2
(
)
(
)
N0 2R1 −1
N0 2R2 −1
2
2
=Pr |h11 | <
×Pr |h22 | <
pmax,1
pmax,2
(3.101)
62
Two-user Interference Networks: Point-to-Point Codes
=e
=e
−(2R1 −1)
−
N0
σ2
S
N0
σ2 pmax,1
S
×e
−(2R2 −1)
(2R1 −1) + (2R2 −1)
pmax,1
pmax,2
N0
σ2 pmax,2
S
(3.102)
and
PF,D,1 = Pr {λmax (DI FI ) < 1, pI,1 ≤ pmax,1 }
(
|h11 |2 |h22 |2
= Pr 2R1 − 1 2R2 − 1
< 1,
|h12 |2 |h21 |2

|h22 |2 N0
R2
R1

−
1
1
+
2
−
1
2
|h21 |2
|h12 |2
<
p
max,1
2
2
11 | |h22 |

1 − (2R1 − 1) (2R2 − 1) |h
|h12 |2 |h21 |2
)
(
|h21 |2 − b1 σS2 |h12 |2
2
2
2
= Pr a1 σS + |h11 | |h22 | <
(2R1 − 1) (2R2 − 1)
(
)
|h22 |2 |h11 |2 +a1 σS2
2
2 ′
= Pr b1 < |h21 | ×Pr 0 <
<ρ γ ,
|h12 |2 (|h21 |2 −b1 σS2 )
(3.103)
where a1 = N0 / σS2 pmax,1 , and b1 = N0 2R1−1 /σS2 pmax,1 . Comparing the equality
in (3.103) and (3.84), we can obtain (3.58) by substituting σS2 , 1/ρ, and σI2 , instead
of σI2 , σS2 , and ρ in (3.30), respectively.
3.E
The Proof of Proposition 3.3.3
NOT3
The probability Pout
((R1 , R2 ) , pmax ) can be simplified as
n
o
NOT3
NOT3
Pout
((R1 , R2 ) , pmax )= 1−Pr PH
((R1 , R2 ) , pmax ) 6= ∅
o
n
NOT3
3
= 1−Pr pNOT
<p
,
p
<p
max,1 2
max,2 .
1
Defining the probabilities
we have
NOT3
PF,1
NOT3
PF,1
,
NOT3
PF,2
,
(3.104)
NOT3
PF,2
and
as follows
n
o
3
Pr pNOT
<
p
max,1
1
o
n
3
Pr pNOT
< pmax,2 ,
2
o
n
(a)
NOT3
NOT3
NOT3
3
2 − PF,1
− PF,2
≤ Pr pNOT
<
p
,
p
<
p
max,1
max,2
1
2
o
n
(b)
NOT3
NOT3
,
≤ min PF,1
, PF,2
(3.105)
(3.106)
(3.107)
3.E. The Proof of Proposition 3.3.3
63
where the equality (a) follows the fact that
\ n \ o
n [ o
Pr A B = 1−Pr A B = 1 − Pr A B
≥ 1−Pr A −Pr B = Pr {A}+Pr {B}−1,
where A and B are random events; and the equality (b) follows Pr {A
We have
o
n
NOT3
3
PF,1
= Pr pNOT
< pmax,1
1
N0
2
R1
2 −1
= Pr |h11 | >
pmax,1
N0
R1
−
−1)
2 (2
= e pmax,1 σS
,
T
(3.108)
B} ≤ Pr {A}.
(3.109)
and
NOT3
PF,2
= Pr 0 < M < 1/ bσS2
(3.110)
o
n R
N0
1
|h21 |2
1
2R1 − 1 + 1 , and b = 2R2 − 1 pmax,2
where M , max |h212 |2 , |h22
2.
σS
|2
|h11 |2
Therefore, we require to find the cdf of random variable M . We define
!
2
|h21 |
X ,
c+1 ,
|h11 |2
1
Y ,
(3.111)
2,
|h22 |
where c = 2R1 − 1 . It can be shown that
fX (x)
=
fY (y)
=
ρc
x ≥ 1,
(c − aρ + ρx)2
1 − yσ12
S y ≥ 0.
e
2
y σS2
(3.112)
(3.113)
We define Z , XY , and we have
Z ∞
Z ∞
z − x
ρcx
1
σ2 z
S dx.
dx =
e
fX (x)fY
(3.114)
fZ (z) =
2z2
2
(c
−
ρ
+
ρx)
|x|
x
σ
S
1
−∞
The cdf of Z is
FZ (z) =
Z
z
t=−∞
fZ (t)dt = e
−
1
σ2 z
S
c
c
2
− 2 e ρσS z E1
σS zρ
c
ρσS2 z
.
(3.115)
64
Two-user Interference Networks: Point-to-Point Codes
Now, defining W ,
c+1
,
|h12 |2
we have
FW (w) = e
− c+1
2
σ w
I
w > 0.
(3.116)
Therefore, for the random variable M = max{Z, W }, since Z and W are independent, we have
FM (m) = FZ (m)FW (m).
(3.117)
Then,
NOT3
PF,2
= Pr 0 < M < 1/ bσS2
= FM 1/ bσS2 − FM (0)
bc
bc bc
−b
ρ
e−(c+1)bρ .
= e − e E1
ρ
ρ
(3.118)
Chapter 4
K-user SISO Interference Networks:
Pilot-assisted Interference Alignment
I
N the previous chapter we studied transmission schemes for the smallest (twouser) interference networks. We have assumed that global CSI is a priori available at terminals (sources and destinations). These transmission schemes may
not be effective for larger networks; furthermore, usually no a priori CSI is available at terminals. Conducting coordinated transmission in such networks when no
a priori CSI is available at terminals is a difficult task. In this chapter, we study
K-user (K > 2) SISO interference network with time-varying channels and no a
priori CSI available at terminals. We propose the pilot-assisted ergodic interference
alignment (PAEIA) transmission scheme. The proposed scheme consists of three
phases: pilot transmission, feedback transmission, and data transmission phase. We
consider orthogonal pilot transmission and minimum mean square error (MMSE)
channel estimation in the pilot transmission phase. We study analog and digital
feedback schemes for channel state information feedback. In the data transmission
phase, an ergodic interference alignment (EIA) is applied. In addition, transmitters
perform rate adaptation or power control based on the feedback signal to adapt the
transmission strategy according to the channel state. We evaluate the performance
limits of the proposed scheme, and investigate radio resource allocation problems.
We first consider networks with analog feedback, and derive an achievable rate region. Moreover, we find the optimum power allocation for pilot transmission and
data transmission. To gain insight into the performance of the system at high-SNR
regime, we derive the achievable DoF region. Next, we study networks with digital
feedback. We investigate two problems: first, we study a power control problem,
and we propose a power control scheme that adapts transmission powers such that
the mutual information corresponding to each source-destination pair is always
larger than the transmission rate, and thus, transmitted codewords can be successfully decoded at the desired destination. Next, we study a throughput maximization
problem for communication systems in which the transmission powers are fixed and
cannot be adjusted. We propose a rate adaptation scheme to maximize network
65
66
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
KTτ = αT
Td = (1 − α)T
Tτ
z1
h11
S1
+
D1
h2
h K1
1
z2
2
S2
h1
h22
+
D2
hK
h
2K
h1
K
2
SK
hKK
zK
+
DK
Figure 4.1: Transmitted symbols within one fading block in a K-user interference
channel. The crosshatched red slot, the plain green slot, and the blue angle lined
slots denote no transmission, pilot symbols, and data symbols, respectively.
throughput. This chapter is organized as follows. Section 4.1 describes the considered multi-user SISO interference network. We present the pilot-assisted ergodic
interference alignment scheme in Section 4.2. Section 4.3 presents the achievable
performance when analog feedback is deployed. In this section, an achievable rate
region is presented, and the achievable degrees of freedom region is computed. We
also compute the optimum power allocation between channel training phase and
data transmission phase. For networks with digital feedbacks, we present power control algorithm and rate adaptation scheme in Section 4.4. Finally, we summarize
this chapter in Section 4.5.
4.1
Multi-user SISO Interference Network
We consider an interference network composed of K single-antenna source–destination pairs, as shown in Figure 4.1. The sources and the destinations are denoted by
Sl and Dk (k, l ∈ {1, 2, ..., K}), respectively. The channels are time-varying and
the channel gain from Sl to Dk at time t is denoted as htkl . The channels follow
block fading model in which the channel gains are constant over one fading block.
nT +i
At fading block n, we have hnT
(i = 1, ..., T − 1), where T is channel cokl = hkl
herence time. The channel gains are ergodic time-varying and have independent
and identical distribution across different fading blocks. The channel gains are independently drawn from a zero-mean unit-variance complex Gaussian distribution,
i.e. hnT
kl ∼ CN (0, 1).
4.2. Pilot-assisted Ergodic Interference Alignment
4.2
67
Pilot-assisted Ergodic Interference Alignment
As shown in Figure 4.1, transmission is performed in three phases: pilot transmission, feedback transmission and data transmission phase. The pilot transmission
phase and the data transmission phase are conducted within each fading block and
have the duration of αT and (1 − α)T , respectively. The channel sharing factor α
(K/T ≤ α ≤ 1) is a design parameter. A frequency-division duplex (FDD) transmission is considered for pilot and data transmission from sources to destinations
and feedback transmission form destinations to sources.
4.2.1
Pilot Transmission Phase
Channel training is performed in an orthogonal fashion in which the training period
is divided into K equal time slots (each has the duration of Tτ = αT /K), as shown
in Figure 4.1. Each destination estimates the gain of the corresponding direct link
and interference links. Let source Sl (l ∈ {1, ..., K}), at fading block n, sends Tτ
known pilot symbols with power Pτ as follows
p
i
Xτ,l
= Pτ i = nT + (l − 1)Tτ + 1, ..., nT + lTτ ,
then the received signals at Dk is
i
Yτ,k
=
p
i
Pτ hnT
kl + Zk .
(4.1)
The MMSE estimate of the channel gain between Sl and Dk is obtained as follows
˜ nT =
h
kl
Pτ
N0 + Tτ Pτ
nTX
+lTτ
i
Yτ,k
.
(4.2)
i=nT +(l−1)Tτ +1
The following equation holds
nT
˜ nT
hnT
kl = hkl + εkl ,
(4.3)
nT
˜ nT
where εnT
kl is the channel estimation error. The random variables hkl and εkl
2
are independent zero mean Gaussian distributed with variances 1 − σε and σε2 ,
respectively, where
σε2 =
1
.
1 + Tτ Pτ /N0
(4.4)
It is clear that the variance of channel estimation error decays by allocating more
time or power to channel training.
4.2.2
Feedback Transmission Phase
As we have discussed in the previous part, each destination can acquire a noisy
estimate of the local CSI through a pilot-based channel training scheme. The destinations then can send the estimated CSI to the other terminals via channel state
68
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
feedbacks. They can transmit either un-quantized CSI (analog feedback) or quantized CSI (digital feedback) via feedback channels. The feedback signal correspond˜ t can be denoted as follows
ing to the channel gain h
kl
ˆ t , f (h
˜ t ),
h
kl
kl
(4.5)
where the function f (.) is the feedback signal transmit function. We consider orthogonal feedback channels and there is no interference in the transmission of feedback
signals. In the scenarios that analog feedback is applied, we assume that f (x) = x,
i.e. destinations send the estimated channel gains to the other terminals.
In the scenarios that digital feedback is deployed, the estimated channel gains are
first quantized, and then the index of the quantized value is sent over the feedback
channel. To be able to analytically investigate the problem and gain insight on the
performance of interference networks with limited feedback, we consider a uniform
quantization scheme in this chapter. It can be conjectured that using more sophisticated quantization schemes (such as the quantizers mentioned in [GN98]) may lead
to an even better performance. We deploy a two-dimensional vector quantizer to
quantize each complex-valued channel gain. The channel gains are unbounded, but
the number of quantization regions is limited. To resolve this problem, only channel gains within a bounded region will be quantized. Specifically, we only quantize
those channel gains for which the corresponding channel matrix Ht (the element
on the ith row and the jth column is denoted as htij ) belongs to the following set
H = H ∈ CK×K |hmin < |Re [hkk ]| , |Im [hkk ]| < hmax .
(4.6)
The constants hmin and hmax are quantizer design parameters, and in Section 4.4.2
we will discuss how to select these parameters. If the channel matrix does not belong
to this set, no transmission occurs during this fading block. We refer to this event
as channel outage event, and channel outage probability is defined as follows:
out
Pch
, Pr {H ∈
/ H} .
(4.7)
The complex plane is divided into multiple equal-sized (∆ × ∆) square regions. Each
of these regions is called a quantization cell, and ∆ is termed quantization step-size.
The quantizer maps channel coefficients within a quantization cell to the centroid
of the corresponding cell, i.e. the quantized value. We represent the quantization of
˜t .
ˆ t , and the corresponding quantization error is denoted as δ t = h
ˆt − h
˜ t as h
h
kl
kl
kl
kl
kl
The number of bits associated to the quantization of direct links and interference
links can be different in general. Therefore, we consider two types of quantizers
at each destination with possibly different resolutions. A direct link quantizer is
associated to the direct link and uses 2NI bits, and an interference link quantizer is
associated to the interference links and uses 2NII bits for the quantization of each
interference link’s gain. The step size of the direct (interference)
link quantizers is
∆I = (hmax − hmin ) /2NI −1 ∆II = (hmax − hmin ) /2NII −1 .
4.2. Pilot-assisted Ergodic Interference Alignment
69
Before data transmission starts, each destination broadcasts the estimated channel gains (when analog feedback is deployed), or sends
Nf = 2NI + 2(K − 1)NII
(4.8)
bits through feedback channels to all other terminals (when there is digital feedback). The feedback channels are in general subject to errors. Since we intend to
investigate the impact of quantized CSI in the case of digital feedback and channel
estimation error in the case of analog feedback on network performance, we assume
that feedback channels are error-free.
4.2.3
Data Transmission Phase
For data transmission, we consider a multiplexed coding scheme similar to the
one proposed in [GV97], where there are multiple codebooks each associated with
a specific channel state. For a given channel state, Sk encodes message mk to a
′
Td
′
length N ′ Td codeword {Xki }N
i=1 , where N is the number of fading blocks with the
same channel state and Td = (1 − α)T is the duration
of data transmission within
each block. There is a power constraint E |Xk |2 < Pd . In fading block n, Sk sends
(n+1)T
{Xki }i=nT +KTτ +1 during Td data transmission time slots. The channel output at
Dk is
i
i
Yd,k
= hnT
kk Xk +
K
X
i
i
hnT
kl Xl + Zk , i = nT + KTτ + 1, ..., (n + 1)T
(4.9)
l=1,l6=k
where Zki ∼ CN (0, 1). We apply the EIA scheme proposed in [NGJV12], but we
assume that only the estimated channel gains are available at terminals. Thus, if
the quantized channel gains at fading blocks n and np (np > n) satisfy
ˆ np T = h
ˆ nT
h
kk
kk
np T
ˆ
ˆ nT , (∀k, l ∈ {1, 2, ..., K}, k 6= l),
hkl
= −h
kl
(4.10)
then Sk at fading block np retransmits the signal which was transmitted at fading
n T +i
block n, i.e. Xk p
= XknT +i (i = KTτ + 1, ..., KTτ + Td ). To avoid measure zero
events, the channel pairing in (4.10) in the analog feedback case can be performed
based on a quantized version of the estimated channel gain using sufficiently fine
quantization [NGJV12]. It can be shown that if channel gains have symmetric
distribution and long delay can be tolerated, with high probability the complement
channels can be found. The destination Dk receives the following signals
nT +i
nT +i
Yd,k
= hnT
+
kk Xk
K
X
nT +i
hnT
+ZknT +i
kl Xl
(4.11)
l=1,l6=k
K
X
np T +i
np T
n T
n T +i
nT +i
Yd,k
=hkk Xk
+
hklp XlnT +i+Zk p
.
l=1,l6=k
(4.12)
70
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
Then, it combines the received signals and forms the following signal
nT +i
n T +i
Y d,k = YknT +i + Yk p
˜ nT + εnT + εnp T + δ nT + δ np T
= 2h
XknT +i
kk
kk
kk
kk
kk
+
K X
np T
n T
n T +i
nT
εnT
+ δkl
+ δklp XlnT +i+ ZknT +i+Zk p
. (4.13)
kl +εkl
l=1,l6=k
The receiver decodes its message after receiving all N ′ segments of the transmitted
codeword.
Each source transmits at power P . The transmission power of pilot symbols
(Pτ ) and the one for data symbols (Pd ) can be different in general. Let Pd = βP ,
where 0 ≤ β ≤ 1/(1 − α) is a power allocation factor. A large value of β implies
that a large power is allocated to data transmission and less power is left for pilot
transmission. Because of energy conservation, we have
αT Pτ /K + (1 − α)T Pd
= T P.
(4.14)
Therefore, Pτ = K ((1 − (1 − α) β)/α) P . Clearly, there is a trade-off between the
power allocation for channel training and the one for data transmission.
4.3
Analog Feedback
In this part, we evaluate the performance of the proposed scheme in the previous section when analog feedback is deployed. We also study the optimum power
allocation between channel training and data transmission.
4.3.1
Achievable Rate Region
In this section, we present an achievable rate region.
Proposition 4.3.1. In the K-user interference network presented in Section 4.1,
a rate tuple (R1 , R2 , ..., RK ) is achievable, where
Rk =
i
1−α h e , ∀k ∈ {1, 2, ..., K}
E I Xk ; Y d,k | H
2
(4.15)
e is the estimated
and random variable Y d,k is given in (4.13). In this equation H
channel matrix.
Proof. The proof follows that of Theorem 2 in [NGJV12]. The difference is that
in [NGJV12] the EIA scheme is applied based on the assumption that each destination has perfect knowledge of its incoming channel gains, but here only imperfect
estimations of the channel gains are available.
4.3. Analog Feedback
71
We next present a closed-form inner bound on the achievable rate region in
(4.15).
Proposition 4.3.2. An inner bound on the achievable rate region in (4.15) is
"
!#
2
hkk Pd
2 ˜
1−α
Rk ≥
∀k ∈ {1, 2, ..., K}.
(4.16)
Eh˜ kk log 1 +
2
1 + (Kσε2 ) Pd
e in (4.15) can be lower bounded as
Proof. The term I Xk ; Y k | H
e (a)
e − h Xk | H,
e Yk
I Xk ; Y k | H
= h Xk | H
(b)
e Yk
= h (Xk ) − h Xk | H,
(c)
bk | H,
e Yk
= h (Xk ) − h Xk − X
(d)
bk | H,
e Yk
= log 2πePd − h Xk − X
(e)
≥ log 2πePd − log 2πeσ 2
(4.17)
where (a) follows the definition of the conditional mutual information; (b) holds
since the transmitted codeword is chosen independent of the noisy CSI; (c) follows
bk to be a function of H,
e and Y k , which will be specified in the below;
by defining X
(d) follows by the assumption that Xk is a complex Gaussian random variable; (e)
follows by [CT91, Theorem 8.6.5] that shows the entropy of random variables with
bounded variance is upper bounded by that of a random variable with Gaussian
distribution. To obtain a tight bound on the achievable rate in (4.17), we choose
bk to be an MMSE estimate of Xk ; that is
X
i
h
∗
e Yk
E Xk · Y k | H,
bk = h
iY k
X
∗
e Yk
E Y k · Y k | H,
∗
e
hkk Pd
(4.18)
=
2 Y k
1 + Kσε2 + 2 e
hkk Pd
which yields
σ2 =
1+
Pd
.
˜ kk |2 Pd
2|h
(4.19)
1+(Kσε2 )Pd
The details of the derivation of σ 2 are presented in Appendix 4.A. The proof is
completed by substituting (4.19) in (4.17).
72
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
Theorem 4.3.1. In the considered K-user interference network in Section 4.1, the
rate-tuple (R1 , R2 , ..., RK ) is achievable where
"
!#
˜ kk |2 P
1−α
2β|h
Rk =
,
(4.20)
Eh˜ kk log 1 +
βKP
2
N0 + 1+(1−β(1−α))T
P /N0
˜ kk ∼ CN 0, T P (1−β(1−α))/N0 .
and h
1+T P (1−β(1−α))/N0
Proof. The proof follows from Proposition 4.3.2 by using the fact that the estimation error of an MMSE estimator is uncorrelated with the estimated channel.
The variance of the estimation error given in (4.4) can be simplified by substituting
Tτ = αT and Pτ given in (4.14).
This achievable rate region can be further simplified by computing the expectation in (4.20) and using equation (34) in [AG99] as presented in the next corollary.
Corollary 4.3.1. The achievable rate Rk given in (4.20) can be simplified as follows
Rk =
where E1 (x) =
R∞
1
1−α
log2 (e) exp (1/SNReq )E1 (1/SNReq )
2
1 −xt
dt,
te
SNReq =
N02
(4.21)
x > 0, and
2(1 − (1 − α)β)T βP 2
.
+ (1 − (1 − α)β)T P N0 + KβP N0
(4.22)
We use this result in the next part to find the optimum power allocation between
channel training phase and data transmission phase.
4.3.2
The Optimum Power Allocation
If transmitters are capable to transmit at different powers during the channel training and the data transmission phases, then power can be allocated such that the
achievable rate region in Corollary 4.3.1 be enlarged.
Proposition 4.3.3. In the considered interference network, for a given α, the
optimum power allocation between pilot transmission phase and data transmission
phase is
Pd,opt
Pτ,opt
= βopt P,
= K ((1 − (1 − α) βopt )/α) P,
(4.23)
where
βopt
1
1+
=
1−α
s
1 + KP/(1 − α)
1 + P T /N0
!−1
.
(4.24)
4.3. Analog Feedback
73
Proof. Since Rk in (4.21) is a monotonic increasing function of SNReq , it is sufficient
to maximize SNReq . We can prove that SNReq is a strictly concave function of β.
Therefore, a unique β that maximizes Rk can be found by solving KKT conditions
[BV04].
Corollary 4.3.2. If P ≫ 1, then the optimum power allocation factor is approximately as follows
βopt ≈
1/(1 − α)
p
.
1 + KN0 /T (1 − α)
(4.25)
This can be further simplified for sufficiently large or sufficiently small networks:
( p
T /((1 − α)KN0 )
K≫T
.
(4.26)
βopt ≈
1/(1 − α)
T ≫K
This result shows that at high-power regime, in sufficiently large networks, βopt
depends on T , α, and K; while in sufficiently small networks, βopt is approximately
equal to its maximum possible value which is 1/(1 − α). It can be seen that, in
sufficiently large networks, βopt increases as T increases, thus, by increasing T
more power should be allocated for data transmission. Also, βopt decays as the size
of the network K increases. This implies that as the size of the network increases,
less power should be allocated for data transmission, and instead channel training
most be performed more accurately. In all cases, βopt decays as α decreases. The
reason is that the allocated power for pilot transmission should be increased to
compensate the rate loss due to the shorter period of pilot transmission.
4.3.3
Achievable Degrees of Freedom Region
In this section, investigate the performance of the proposed scheme at high-SNR
regime. The following theorem presents the achievable DoF region.
Theorem 4.3.2. In the K-user interference network with coherence time T , if
no CSI is a priori available at terminals, then the DoF region (d1 , d2 , ..., dK ) is
achievable where
(
1
1− K
K<T
2
T
dk =
.
(4.27)
0
K≥T
Proof. Using (4.20), we have
dk = lim Rk /log P
P →∞


˜ kk |2 P
(a) 1 − α
2β|h
Eh˜ kk lim log1+
=
P →∞
2
1 + T P βKP
1+ N (1−β(1−α))
0
1−α
,
=
2


/log P 
(4.28)
74
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
where (a) follows the monotone convergence theorem [MW12]. To maximize the
achievable DoF in (4.28) we set α equal to its minimum possible value which is
K/T .
According to Theorem 4.3.2, if T ≫ K, then the achievable DoF by the EIA
scheme with perfect CSI can be preserved. However, the achievable DoF for each
user decays by increasing the number of users. The achievable total DoF depends
on both the number of the users and the achievable DoF for each user. A specific
number of users maximizes the achievable total DoF. Thus, we select only a subset
of users called active users to transmit.
Theorem 4.3.3. In the K-user interference network with ergodic block fading
Gaussian channel and coherence time T , the achievable total DoF is
K
dΣ = 12 Kopt 1 − Topt ,
(4.29)
where Kopt is the number of the active users
Kopt = min T2 , K .
(4.30)
Proof. Let K ′ < T denote the number of the active users, then according to Theorem 4.3.2 we have
′
dΣ =
K
X
i=1
2
′
1
dk = 12 K ′ 1− KT = − 2T
K ′ − T2 +
T
8.
We can observe that dΣ is maximized when K ′ = min K, T2 .
To maximize the total DoF (and the network throughput at high SNR), in
large networks (K > T /2), Theorem 4.3.3 suggests to first apply a user selection
scheme, and then perform channel training and interference alignment only within
the subset of active users. Since the network is symmetric, a random user selection
is sufficient. In addition, this theorem crystallizes the dependency of the optimum
number of active users to be selected on the coherence time of channel.
4.3.4
Numerical Evaluation
In this section, we numerically evaluate the analytical results presented in the previous sections when analog feedback is deployed. Figure 4.2 shows the optimum
power allocation factor βopt given in (4.24) as a function of the number of users for
different values of T . We set P/N0 = 20 dB, and α = 0.1. It can be observed that
βopt decays as the number of users increases. The intuition behind this observation
is that in large networks the impact of residual interference due to imperfect interference alignment become more important, thus, it is recommended to allocate
more power to pilot symbols in order to acquire CSI more accurately. Also, we can
4.3. Analog Feedback
75
1.4
bC
1.2
rS
ut
qp
qp
qp
ut
qp
ut
ut
rS
1
qp
qp
ut
= 100
= 1000
= 10000
= 100000
qp
rS
rS
ut
bC
T
T
T
T
qp
βopt
0.8
rS
ut
bC
qp
0.6
bC
rS
ut
qp
0.4
bC
rS
ut
bC
qp
rS
ut
qp
0.2
bC
rS
ut
bC
rS
bC
0
101
102
103
104
ut
bC
105
bC
rS
rS
bC
106
K
Figure 4.2: The optimum value of power allocation factor (β) versus the number of
users (K) for different coherence time (T ).
observe that βopt increases by increasing T . This implies that as the channel coherence time increases, a larger power should be allocated for data transmission. It is
clear from (4.4) that to preserve a given variance of the channel estimation error, a
lower Pτ is required as T and consequently Tτ increases. Thus, a larger power can
be devoted for data transmission.
Figure 4.3 shows the achievable rate per user of the PAEIA scheme in both
cases that power allocation is optimized (β = βopt ) and when there is no power
optimization (β = 1). The network parameters are K = 40, T = 1000, and we
set α = 0.04. We plot the achievable rate of the TDMA with pilot-based channel
training scheme and that of the EIA scheme with perfect CSI for comparison. This
figure shows that, for the given parameters, the PAEIA scheme can achieve almost
the same (slightly less) DoF as the EIA scheme with perfect CSI. This confirms the
result in Proposition 4.3.2 when K is sufficiently smaller than T . A large gap
between the achievable rate of the EIA scheme and that of the TDMA scheme can
be seen. Furthermore, 2 dB gain can be seen using the optimum power allocation
compared to the case with uniform power allocation.
Figure 4.4 shows the achievable sum-rate of the PAEIA scheme and that of the
PAEIA with user selection (PAEIA-US) for different number of users. In this exam-
76
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
7
rS
Achievable rate [bits/channel use]
6
bC
uT
5
PAEIA, β = βopt
PAEIA, β = 1
EIA, perfect CSI
TDMA
rS
bC
rS
bC
rS
bC
4
rS
bC
rS
3
bC
rS
bC
2
rS
bC
1
rS
bC
rS
bC
0
uT
uT
uT
uT
0
5
10
15
uT
uT
uT
uT
uT
20
25
30
35
40
P/N0 [dB]
Figure 4.3: The achievable rate of the PAEIA and TDMA schemes.
ple, T = 100, P/N0 = 10 dB, and β = βopt . For the PAEIA scheme, we set α = K/T
and for the PAEIA-US scheme we set α = Kopt /T , where Kopt is given in (4.30).
It can be seen that the achievable sum-rate of the PAEIA scheme is maximized for
a specific number of users. This observation coincides with Theorem 4.3.3. The
intuition behind this result is that, by increasing the number of the active users,
in one hand the number of the independent transmitted symbols increases, and on
the other hand the achievable rate per each user decreases due to the less available
resources for the channel training and consequently more interference. It can be
seen that the PAEIA-US scheme outperforms the PAEIA scheme in large networks
(K > T /2).
4.4
Digital Feedback
Since the bandwidth of the feedback channel is in general limited, the destinations
can compress the estimated CSI and send quantized CSI instead through digital feedback signals. Applying interference alignment based on the quantized CSI,
inter-user interference can be only partially eliminated so that some residual interference remains at each destination. This residual interference, if not appropriately
managed, will degrade the system performance. Hence, the sources should exploit
4.4. Digital Feedback
77
200
PAEIA-US, no a priori CSI
PAEIA, no a priori CSI
EIA, perfect CSI
Achievable sum-rate [bits/channel use]
180
160
140
120
100
80
60
40
20
0
0
10
20
30
40
50
60
70
80
90
100
K
Figure 4.4: The achievable sum-rate versus K.
the quantized CSI not only to partially eliminate the interference, but also to adapt
their transmission strategies, e.g. by adapting transmission rate or controlling transmission power, to fulfil certain service requirements, i.e. to increase throughput or
to transmit successfully at given rates with minimum power, respectively.
In this part, we investigate the impact of quantized CSI on the performance of
the EIA and propose adaptive transmission schemes to enhance the performance of
the EIA with digital feedback: we first study a power control problem in which each
user desires to successfully transmit information at a fixed rate using the minimum
transmission power. We propose a power control scheme which adapts transmission power values such that the mutual information corresponding to each sourcedestination pair is always larger than the transmission rate, and thus, transmitted
codewords can be successfully decoded at the intended destination.
Next, we study a throughput maximization problem for adaptive communication
systems in which each user desires to maximize the throughput. In contrary to the
previous case, transmission power values are fixed. Since each source only knows
quantized CSI, it is not aware of the exact value of mutual information between
itself and its intended destination. Therefore, in certain channel realizations, the
mutual information may fall below the transmission rate and communications fails
which leads to an outage event. The outage probability can be used to quantify
78
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
throughput as a measure of the amount of information that can be successfully
transmitted. The impact of the transmission rate on throughput is twofold: On
the one hand, increasing the rate tends to improve the throughput; but on the
other hand, a higher rate leads to a higher outage probability which results in a
throughput loss. It can be shown that, for given channel gains, there is a specific
rate which maximizes network throughput. We propose a rate adaptation scheme
to maximize network throughput.
Moreover, the EIA scheme presented in Chapter 2 requires asymptotically long
delay to achieve the promised performance. It has been shown in [JAP12] that the
required delay can be reduced by sacrificing transmission rate, but still asymptotically long delay is needed to achieve the promised rates. The performance limit of
the EIA scheme in delay-limited communication systems is in general unknown. In
the following parts of this chapter, we extend our results on the throughput maximization problem to communication systems with delay constraints. In the case
of analog feedback, as discussed in the first part of the chapter, there is a natural
connection between estimating the CSI at receivers and sending analog feedback. I
n this part we however look at digital feedback, and thus there is no natural way
of connecting the CSI estimation at the receivers with a corresponding feedback
mechanism. Since we wish to focus on the impact of the feedback quantization itself, and not the specific implementation of estimator algorithms at the receivers,
we now assume that the receivers have perfect CSI a priori (εnT
kl = 0). Hence no
radio resources are explicitly used for channel training, while in practice the receivers would still need to get CSI information via a training phase. In addition,
sources may have different transmission powers and pk denotes the transmission
ˆ n and δ n to denote h
ˆ nT and δ nT ,
power of Sk . For the sake of simplicity, we use h
kl
kl
kl
kl
respectively. In the following, we first study the power control problem and next
address the throughput maximization problem.
4.4.1
Power Control Problem
In certain communication systems, fixed-rate transmission is desired [CHLT08]. In a
time-varying environment, channel quality is changing over different blocks. Therefore, radio resources (e.g. power) can be adaptively allocated to support successful
transmission. Consider the time-varying K-user interference network presented in
Section 4.1 where transmission rates are fixed. We desire to allocate powers such
that the mutual information corresponding to each source-destination pair remains
larger than a certain level.
Assume that the channels with block indices m and mp are complement. To
study the power control problem, we assume that a complement channel corresponding to each channel realization can be found after sufficiently many fading
blocks to occur. According to the transmission scheme presented in Section 4.2,
we require each source to repeat the same codeword over blocks m and mp . Following the input-output relation in (4.13) and the assumption that perfect CSI is
a priori available at destination, the SINR of the equivalent received signal at Dk
4.4. Digital Feedback
79
(k ∈ {1, 2, ..., K}) is
SINRym
k
=
2
ˆm
m m
+ δkkp pk
2hkk + δkk
.
P
δ m + δ mp 2 p l
2+ K
l=1,l6=k
kl
(4.31)
kl
This SINR value is random and depends on the quantization errors which are
≥ SINRmin
unknown to the sources. This value is lower bounded as SINRym
, where
ym
k
k
min
SINRym can be computed as follows:
k
h
i h
i
ˆ m 2
2
ˆ m + Im h
ˆ m pk
2 h
+∆
−2∆
h
Re
I
I
kk
kk
kk
.
(4.32)
SINRmin
=
P
ym
K
k
1 + ∆2II l=1,l6=k pl
Therefore, the mutual information
of the
source-destination pair Sk − Dk can be
min
1
lower bounded by 2 log2 1 + SINRym where the factor 21 comes from the fact
k
that two blocks are used to transmit one codeword. In order to guarantee successful
transmission at the fixed rate Rk , the following condition should be satisfied:
1
log2 1 + SINRym
> Rk .
(4.33)
k
2
Clearly, if the sources compute their transmission powers to meet the following
condition, we can ensure that the inequality in (4.33) is satisfied:
1
m
log 1 + SINRymin
> Rk .
(4.34)
k
2
According to (4.32), the condition in (4.34) can be rewritten as a power constraint
pk > Ik (p), where
PK
22Rk − 1 1 + ∆2II l=1,l6=k pl
Ik (p) , 2
(4.35)
h
i h
i ,
ˆ m 2 − 2∆
m +
m ˆ
ˆ
2 h
+
∆
h
h
Re
Im
I
kk
kk
kk
I
and p , [p1 · · · pK ]T . Thus, we define feasibility set as follows:
P EIA , {p|pmax p 0, p ≻ I(p)},
(4.36)
where I(p) , [I1 (p) I2 (p) ... IK (p)]T is the interference function and Ik (p) is
defined in (4.35). The notation a ≻ b (a b) means that every element of vector a
is larger than (larger than or equal to) the corresponding element of vector b. The
power control problem can be formulated as the following optimization problem:
minimize
s.t.p∈P EIA
K
X
pl .
(4.37)
l=1
If P EIA 6= ∅, then this problem is said to be feasible, otherwise the problem has no
solution and terminals stop transmission during the corresponding fading block.
80
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
Feasibility of the Power Control Problem
In this part, we study the feasibility of the power control problem in (4.37). This
problem has a feasible solution if the set in (4.36) is non-empty. The set P EIA given
in (4.36) can be represented in the following equivalent form
P EIA = {p|pmax p 0, p ≻ Ap + b},
where the element of the kth row and the lth column of the matrix A is
22Rk − 1 ∆2II
A(k, l) , 2
h
i h
i ,
ˆ m 2 − 2∆
ˆ m + Im h
ˆ m 2 h
+
∆
h
Re
I
kk
I
kk
kk
(4.38)
(4.39)
and the kth element of the vector b is
22Rk − 1
b(k, 1) , 2
h
i h
i .
ˆ m 2 − 2∆
m +
m ˆ
ˆ
2 h
+
∆
h
h
Re
Im
I
kk
kk
kk I
(4.40)
We use this representation of the feasibility set to study the condition under which
the problem has a feasible solution. Since this set depends on transmission rates,
the number of quantization bits, and the number of users, the feasibility of the
power control problem will be affected by all these parameters. The next theorem
presents the condition under which this problem has a feasible solution.
Theorem 4.4.1. Assume that quantized channel gains are given, and the maximum
transmission power is asymptotically large (i.e., pmax → ∞). The power control
problem in (4.37) is feasible (P EIA 6= ∅) if and only if λmax (A) < 1, where λmax (A)
is the Perron-Frobenius eigenvalue of the matrix A, and the matrices A is defined
in (4.39).
Proof. Since it is assumed that pmax → ∞, we need to show that there is a positive
power vector p satisfying p ≻ Ap+b if and only if λmax (A) < 1. For K > 2, the matrix A is a regular matrix (a matrix X is called a regular matrix, if all the entries of
some power of the matrix X are positive) because all the entries of A2 are positive.
Therefore, the Perron-Frobenius Theorem [PSS05] guarantees the existence of a
positive eigenvalue λmax (A) and the corresponding positive right and left eigenvectors pr ≻ 0 and pl ≻ 0 which satisfy λmax (A)pr = Apr and λmax (A)pTl = pTl A,
respectively. To prove the necessary condition, assuming p1 ∈ P EIA we have
p1 ∈ P EIA
(a)
⇒
p1 ≻ Ap1 + b
⇒
p1 − Ap1 ≻ 0
(b)
(c)
⇒
pTl (p1 − Ap1 ) > 0
⇒
(1 − λmax (A))pTl p1 > 0
⇒
λmax (A) < 1,
(d)
(e)
(4.41)
4.4. Digital Feedback
81
where (a) follows from the definition in (4.38); (b) follows from the fact that b ≻ 0;
(c) follows from the positivity of the left eigenvector (pl ≻ 0); (d) follows from
the characteristic of eigenvectors and (e) follows from the fact that pl ≻ 0 and
p1 ≻ 0, and consequently pTl p1 > 0. To prove the sufficient condition, assuming
λmax (A) < 1 we have
(a)
λmax (A)pr = Apr ⇒ p2 = Ap2 + b + a(1 − λmax (A))Apr − b
(b)
⇒ p2 Ap2 + b
(c)
⇒ p2 ∈ P IA
⇒ P IA 6= ∅,
(4.42)
where (a) holds if we set p2 = aλmax (A)pr ; (b) holds if we set
a = max
i
b(i, 1)
+ ǫ,
(1 − λmax (A))A(i, :)pr
(4.43)
where b(i, 1) is the ith element of b, A(i, :) is the ith row of A, and ǫ > 0 is a
constant, then since λmax (A) < 1 we have a(1 − λmax (A))Apr − b ≻ 0; (c) follows
from the definition in (4.38) and the assumption that pmax → ∞.
Figure 4.5 shows the asymptotic feasibility probability, when NI = NII = 8 and
pmax → ∞, versus rate for different number of users (K ∈ {3, 4, 5, 6}). This figure
shows that the feasibility probability monotonically decreases as transmission rate
increases. In a communication system, it is desired to have the feasibility probability
larger than a certain threshold. These curves can be used to find the maximum
transmission rate such that feasibility probability can be maintained at the desired
level. Also, we can see that as the number of users increases the probability that
the power control problem has a feasible solution decays. The reason is that by
increasing the number of users, each user experience more severe interference, in
addition the number of constraints that need to be satisfied for a feasible power
control problem increases.
In a fading block m, terminals stop transmission either if channel outage event
occurs (i.e. Hm ∈
/ H), or if the power control problem does not have a feasible
solution i.e. P EIA = ∅ . Therefore, the probability that no transmission occurs in
fading block m is
PnoT X
=
=
Pr{H ∈
/ H} + Pr{H ∈ H} × Pr{P EIA = ∅}
out
out
× Pr{P EIA = ∅},
Pch + 1 − Pch
(4.44)
out
where Pch
is defined in (4.7). The channel outage probability and the feasibility probability both are affected by the quantizer parameters hmin and hmax . A
larger hmax reduces the channel outage probability, but also reduces the feasibility
probability of the power control problem because of larger quantization regions. In
82
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
uTrSbC
100
uTrSbC
uTrSbC
uTrSbC
uTrSbC
uTrSbC
uTrSbC
uTrSbC
uT rS bC
uT rS bC
uT rS bC
uT rS bC
uT rS bC
uT rS bC
uT rS bC
uT rS bC
uT rS bC
uT rS bC
uT Sr
Cb
uT rS
bC
uT
rS
bC
uT
rS
bC
uT
rS
uT
bC
K
K
K
K
uT
Feasibility probability
10−1
rS
bC
=3
=4
=5
=6
uT
rS
bC
uT
rS
bC
uT
rS
uT
bC
rS
uT
rS
bC
10−2
uT
bC
rS
10−3
rS
10−4
bC
0
1
2
3
4
5
uT
rS
6
Rate per user (bits/channel use)
Figure 4.5: Feasibility probability versus transmission rate of each user in a K-user
interference network, NI = NII = 8.
contrast, a larger hmin increases the channel outage probability, and increases the
feasibility probability of the power control problem. There are specific values for
m
these parameters which minimize PnoT
X . In the numerical evaluations, we select
these parameters by searching for the values which minimize this probability. If the
power control problem has feasible solutions, the next question to be answered is
how to find the solution which is corresponding to the minimum required transmission power. In the next subsection, we present an iterative power control algorithm
to address this question.
Iterative Power Control Algorithm
We present an iterative power control algorithm, shown in Algorithm 1, to solve
the power control problem in (4.37). In each iteration, Dk (k ∈ {1, 2, ..., K}) comˆ m and the total transmission
putes function Ik (p) given in (4.35) according to h
kk
powers of the other sources in the previous iteration. Next, it updates the transmission power of the corresponding source following Algorithm 1. As we will show in
Section 4.4.1, this algorithm converges to the optimum solution if the power control
problem in (4.37) has a feasible solution.
4.4. Digital Feedback
83
Algorithm 1 Iterative Power Control
Initialize: p1 (0), ..., pK (0), maxitr
for t = 1 : maxitr do
for k = 1 : K do
Sk updates its transmission power:
P
pl (t−1)
(22Rk −1) 1+∆2II K
l=1,l6=k
pk (t) = Ik (p(t − 1)) = ˆ m 2 2
.
ˆ m ]|+|Im[h
ˆ m ]|)
2|hkk | +∆I −2∆I (|Re[h
kk
kk
end for
t=t+1
end for
Convergence of the Power Control Algorithm
There are two main questions regarding the convergence of Algorithm 1. The
first question is whether the sequence of the computed powers by the algorithm
converges to a fixed-point. If the answer to this question is positive, we need to
answer whether the fixed-point is corresponding to the minimum required powers.
In the following theorem we answer these questions.
Theorem 4.4.2. If P EIA 6= ∅, for any initial power vector p(0), Algorithm 1
converges to a unique fixed-point p∗ which corresponds to the solution of the problem
in (4.37).
Proof. To provide the convergence proof of the proposed algorithm, we refer to a
family of functions defined in [Yat95]. I′ (p) is called a standard interference function
if for all vectors p, p′ 0, it satisfies the following conditions:
1) Positivity condition : I′ (p)≻0,
2) Monotonicity condition : I′ (p)I′ (p′ ) ∀ p p′ ,
3) Scalability condition : αI′ (p)≻I(αp) ∀ α > 1.
(4.45)
We now show that the function I(p) given in (4.35) satisfies these conditions. For
the simplicity of presentation, we rewrite Ik (p) as follows


K
X
(4.46)
pl  ,
Ik (p) = Lk 1 + ∆2II
l=1,l6=k
where
22Rk − 1
Lk = h
i h
i > 0
ˆ m 2
2 − 2∆
ˆ kk + Im h
ˆ kk 2 h
+
∆
h
Re
I
kk
I
is a constant.
(4.47)
84
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
1) Positivity condition:

Ik (p) = Lk 1 + ∆2II
K
X
l=1,l6=k

pl  ≥ L k > 0
(4.48)
Thus, the interference function given in (4.35) satisfies the positivity condition.
2) Monotonicitycondition:
PK
PK
If p p′ , then 1+∆2II l=1,l6=k pl ≥ 1+∆2II l=1,l6=k p′l and since Lk > 0 we
have Ik (p) ≥ Ik (p′ ). Therefore, the monotonicity condition is satisfied.
3) Scalability condition:
If α > 1, then


K
X
pl 
Ik (αp) = Lk 1 + α∆2II
<

αLk 1 + ∆2II
and the scalability condition is satisfied.
l=1,l6=k
K
X
l=1,l6=k

pl  = αIk (p),
These conditions are satisfied for each k (k ∈ {1, ..., K}), and we can conclude
that the function I(p) given in (4.35) is a standard interference function. Therefore,
according to Theorem 2 in [Yat95] for any initial power vector p(0) Algorithm 1
is a standard power control algorithm and converges to a unique fixed-point p∗ .
Lemma 1 in [Yat95] implies that this fixed-point corresponds to the solution with
minimum powers.
In this part, we numerically evaluate the performance of the proposed interference alignment and power control (EIA-PC) scheme. Figure 4.6 shows the required
powers of the IA-PC scheme for each user, averaged over different channel realizations, in a K-user interference network (K ∈ {3, 5, 7}, NI = NII = 8) versus transmission rate of one user (R1 = R2 = R3 = R). Also, the performances of TDMA
with power control denoted as TDMA-PC (with the same number of feedback
bits), and the EIA scheme with perfect CSI are shown for comparison. It can be
seen in Figure 4.6 that even with only limited feedback bits, applying the EIA-PC
scheme outperforms the TDMA-PC scheme in the intermediate rate region. When
the interference alignment scheme is performed based on perfect CSI, the required
powers for transmission at a given rate do not increase by increasing the number of
users. However, they considerably increase for the TDMA-PC scheme, especially at
high transmission rates. If only limited feedback bits are available and EIA-PC is
performed, at the low-rate regime, increasing the number of users does not significantly increase the required powers. However, if the transmission rate is high, the
4.4. Digital Feedback
85
uT
40
rS
Tu uT
uT
rS
uT
uT
35
bC
rS
bC
30
uT
bC
bC
bC
rS
rS
bC
uT
bC
bC
bC
rS
bC
rS
uT
bC bC rS
rS
bC
Average power [dB]
rS
bC
25
bC
uT
rS
uT
20
bC
rS
15
uTrS
bC
bC
K = 3, EIA-PC, quantized CSI
K = 3, TDMA-PC, quantized CSI
K = 5, EIA-PC, quantized CSI
K = 5, TDMA-PC, quantized CSI
K = 7, EIA-PC, quantized CSI
K = 7, TDMA-PC, quantized CSI
EIA-PC, perfect CSI
bC
rS
uTrSbC
bC
bCuTrSbC
uT
bCuT
rS
rS
rS
bC
uT
5
uT
0
bCrS
−5
0
bC
rSuTbC
rSuTbC
10
rS
uT
1
2
3
4
5
6
7
Rate [bits/channel use]
Figure 4.6: Average transmission power versus rate of one user in a K-user interference channel, NI = NII = 8.
performance is more severely affected by the residual interference; consequently,
the required powers notably increase as the number of users increases.
Figure 4.7 illustrates the performance of the EIA-PC scheme for different strategies to allocate feedback bits to direct link and interference links when the total
number of feedback bits from each destination is fixed at Nf = 42. It can be seen
that the best feedback allocation strategy depends on the transmission rates. This
figure reveals that in the low-rate regime, allocating more bits to the quantization
of the direct links is preferred, while in the high-rate regime, it is more efficient
to allocate more bits to the quantization of the interference links instead. This is
because when the desired transmission rate is low, the network operates in the
noise-limited regime and accurate power control is more effective. However, to ensure successful transmission at high-rate regime, the users are required to transmit
with large powers, and the network becomes interference-limited. Therefore, at this
regime, it is preferred to more precisely eliminate the interference by allocating
more feedback bits to the interference links.
86
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
45
bCbC
rS
bC
rS
uT
rS
rS
bC
uT
uT
35
Average total power [dB]
uT
bC
40
uT
uT
rS
uT
uT bC
uT
uT
uT
uT
uT
30
uT
rS
bC
25
rS
rS
rS
20
rS
rS
rS
bC
bC
15
bC
uT
10
bC
rS
bC
5
0
0
2
4
6
8
10
EIA-PC, perfect CSI
EIA-PC, NI = 5, NII = 8
EIA-PC, NI = 7, NII = 7
EIA-PC, NI = 9, NII = 6
12
14
16
18
20
Sum-rate [bits/channel use]
Figure 4.7: Different strategies for sharing total feedback bits (Nf = 42) among
direct link quantizer and interference link quantizer in a three-user interference
channel.
4.4.2
Throughput Maximization Problem
Considering adaptive wireless communication systems, another group of systems, in
contrast with the ones addressed in the previous section, may not have capability to
adaptively adjust transmission powers and successful fixed-rate transmission is not
their primary concern. Instead, they can adapt their transmission rate according to
the channel conditions in order to maximize network throughput. In this section,
we discuss this group of systems and address a throughput maximization problem.
We propose a rate adaptation scheme to maximize network throughput. The transmission scheme mentioned in Section 4.2 is employed to remove interference at each
destination. However, due to the limited resolution of the quantized CSI at sources,
interference alignment cannot be performed accurately and some interference remains at destinations. The exact value of each source-destination pair’s mutual
information is unknown at the corresponding source. Thus, for some channel realizations the instantaneous mutual information might become lower than the data
transmission rate, and consequently the transmission fails. In addition, terminals
stop transmission if channel gains do not belong to the set H defined in (4.6). Out-
4.4. Digital Feedback
87
age probability is defined as the probability that transmission fails because of either
of these events [TV05]. The outage probability increases as we increase data transmission rate. Network throughput, defined as the expected rate of successful data
transmission over the network, depends on both data transmission rate of each user
and the corresponding outage probability. To maximize network throughput we use
the fact that, for a given quantized CSI available at terminals, there is a specific
data transmission rate for each source which maximizes the network throughput. In
this section, we first compute a closed-form upper bound on the outage probability,
and then we propose a rate adaptation scheme based on the quantized CSI at each
source which maximizes the lower bound on the throughput.
Outage Probability Analysis
Terminals can have a successful communication only if the channel matrix belongs
to H defined in (4.6) (otherwise no transmission occurs) and the mutual information
is larger than the transmission rate. Otherwise, an outage event will be declared. Let
channels with block indices m and mp belong to H and be complement according
to (4.10). Requiring each source to transmit the same codeword over these blocks,
the SINR of the equivalent received signal at Dk is given in (4.31). The mutual
. If
information between the source–destination pair Sk − Dk is 21 log2 1 + SINRym
k
the transmission rate Rkm is smaller than this mutual information, decoding error
probability can be made arbitrary small by choosing proper code with sufficiently
long codewords. Otherwise, the channel between the source–destination pair Sk −Dk
is said to be in outage. The outage probability given that quantized CSI is available
at terminals is
1
out,m
out,m
m ˆ m
, (4.49)
<
R
h
log2 1 + SINRym
Pkout,m , Pch
+ 1 − Pch
× Pr
k kk
k
2
out,m
where Pch
is channel outage probability defined in (4.7). Finding the exact
expression for the outage probability in (4.49) is a challenging task. Therefore,
instead we present a closed-form upper bound on Pkout,m in the following theorem.
m
Theorem 4.4.3. If Rkm < Rmax,k
, then the outage probability Pkout,m defined in
(4.49) can be upper bounded as follows
out,m
out,m
out,m
Pkout,m ≤ Pch
+ 1 − Pch
× Pup,k
(4.50)
where
out,m
Pup,k
=
m
Rmax,k
=
1
2,
1 + (rkm )


ˆ m |2 + ∆2 pk
12|
h
kk
I
1
,
log2 1 +
PK
2
∆2II i=1,i6=k pi + 6
(4.51)
(4.52)
88
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
ˆ m |2 +∆2 pk
12|h
kk
I
rkm
s
=
out,m
Pch
3
2Rm
2 k −1
7
4
90 ∆II
K
P
i=1,i6=k
p2i
−
+
∆2II
3
8
3
K
P
i=1,i6=k
2 2
7
∆I + 90
∆4
|hˆ m
kk |
2 I
2Rm
2
k
−1
2K 2
1 − (2Q (hmin ) − 2Q (hmax ))
=
pi − 2
,
(4.53)
p2k
,
(4.54)
otherwise Pkout,m ≤ 1.
out,m
Proof. The probability Pch
defined in (4.7) can be computed as follows
out,m
Pch
= 1−
K Y
K
Y
k=1 l=1
= 1−
K Y
K
Y
m
(Pr{hmin < |Re[hm
kl ]| < hmax }×Pr{hmin < |Im[hkl ]| < hmax })
2
(2Q(hmin)−2Q(hmax))2 = 1−(2Q(hmin )−2Q(hmax))2K (4.55)
k=1 l=1
where Q(x) =
Pr
√1
2π
R∞
x
2
e−u
/2
du. For the other term in (4.49) we have
)
(
ˆ m |2 pk
1
4|h
m
m
m
kk
ˆ
ˆ
log2 1 + SINRym
− 2 hkk , (4.56)
< Rk hkk = Pr Y ≥ 2Rm
k
2
2 k −1
ˆ | p
4|h
ˆ m is given, is a constant
where the quantity 2Rkkm k − 2, conditioned on that h
kk
2 k −1
value. The random variable Y is
m 2
K
X
Y =
l=1,l6=k
−
−
m
m
m 2
m 2
pl
+ Im δkl
+ δkl p
Re δkl
+ δkl p
h
i h
i ˆ m Re δ m + δ mp + Im h
ˆ m Im δ m + δ mp pk
4 Re h
kk
kk
kk
kk
kk
kk
m
22Rk − 1
m
m
mp 2
m 2
Re δkk + δkk
+ Im δkk
+ δkkp
pk
m
22Rk − 1
.
(4.57)
The mean and the variance of Y are (computed in Appendix 4.B)
µY
=
σY2
=
∆2II
3
K
X
∆2I pk
,
m
3 22Rk − 1
l=1,l6=k
2
7
8 ˆ m 2
4
∆
+
p2k
h
∆
K
I
kk
3
90 I
7∆4II X 2
.
pl +
2
m
90
22Rk − 1
l=1,l6=k
pl −
(4.58)
(4.59)
4.4. Digital Feedback
89
2
Let X be a random variable with mean µX and variance σX
. For any real value
r > 0, the Cantelli inequality implies Pr {X − µX ≥ rσX } ≤ 1/ 1 + r2 [MR69].
ˆ m |2 pk / 22Rm
k − 1 − 2, and applying the
Setting X = Y and rkm σY + µY = 4|h
kk
Cantelli inequality lead to the value of rkm that is given in (4.53), and the upper
bound on Pkout,m shown in (4.50).
Theorem 4.4.3 clarifies how the upper-bound on the outage probability of each
source–destination pair depends on different parameters such as the transmission
rate of the source, the transmission powers of all sources, the number of users,
the quantization resolutions, and the quantized direct link’s gain. We will use this
result in Section 4.4.2 to design a rate adaptation scheme which maximizes the
lower bound on network throughput. In the special case that quantization bits are
asymptotically large, we have the following corollary of Theorem 4.4.3.
ˆ m |2 pk , then high-resolution quantizers
Corollary 4.4.1. If Rkm < 21 log2 1+2|h
kk
lead to
lim
∆I ,∆II →0
Proof. If Rkm <
1
2
Pkout,m = 0.
ˆ m |2pk , then
log2 1+2|h
kk
(4.60)
lim rkm = +∞ and the upper bound
∆I ,∆II →0
on Pkout,m approaches 0. Since, Pkout,m is lower bounded by zero, we can conclude
that lim Pkout,m = 0.
∆I ,∆II →0
This result shows that when the quantizers are sufficiently fine, if the transmis2
sion rate Rkm is less than 21 log(1 + 2|hm
kk | pk ), then reliable transmission is possible.
1
Indeed, the average rate of 2 Eh log2 (1 + 2|h|2 pk ) is achievable, where Eh [·] is the
expectation over direct link’s gain. This coincides with the results in [NGJV12],
where perfect CSI is available at terminals.
Network Throughput Maximization
For any pair of complement blocks m and mp , we define the throughput of source–
destination pair Sk − Dk as follows
Tkm , 1 − Pkout,m Rkm .
(4.61)
The network throughput T m can be represented as the summation over the throughput of the individual users, i.e.,
Tm =
K
X
k=1
Tkm =
K X
1 − Pkout,m Rkm .
k=1
(4.62)
90
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
m
m
According to Theorem 4.4.3, for Rkm ≤ Rmax,k
where Rmax,k
is given in (4.52),
out,m
we can use the upper bound on Pk
in (4.50) to find a lower bound on the
throughput as follows
Tm
≥
out,m
1 − Pch
K X
k=1
out,m
1 − Pup,k
Rkm ,
(4.63)
m
while for Rkm > Rmax,k
the lower bound on the throughput is the trivial value of
zero. For each complement block pair, the lower bound (4.63) can be maximized by
a proper rate adaptation. This network throughput maximization problem can be
formulated as follows
m
Topt
,
max
0≤Rm ≤Rm
i
max,k
i∈{1,...,K}
K X
out,m
out,m
1 − Pch
1 − Pup,k
Rkm .
(4.64)
k=1
In the following, we discuss the solution of this problem. For a given transmisout,m
sion power vector p, since Pup,k
only depends on variable Rkm , the optimization
problem in (4.64) can be decomposed to individual optimization problems, each
corresponding to one source, i.e.,
K
X
out,m
m
= 1 − Pch
Topt
k=1
maxm
m
0≤Rk ≤Rmax,k
out,m
Rkm .
1 − Pup,k
(4.65)
out,m
Substituting Pup,k
given in (4.50), each source solves the following optimization
problem
(rkm )2
out,m
m
Tk,opt = 1 − Pch
×
maxm
Rkm ,
(4.66)
0≤Rm
≤Rmax,k
1 + (rkm )2
k
m
where Rmax,k
and rkm are given in (4.52) and (4.53), respectively. After certain
mathematical manipulations and introducing an auxiliary optimization variable x,
we have the following equivalent optimization problem:
out,m
m
Tk,opt
= 1 − Pch
×
max
m
2R
k −1
x=2
0≤x≤xmax , 0≤Rm
k
Ax2 + Bx + C m
R ,
Dx2 + Ex + F k
where

A = −
B=
2
3
∆2II
3
K
X
i=1,i6=k
2
pi − 2  ,

ˆ m |2 + ∆2 − ∆II
12|h
kk
I
3
K
X
i=1,i6=k

pi − 2  pk ,
(4.67)
4.4. Digital Feedback
91
1 ˆm 2
12|hkk | + ∆2I p2k ,
9

2
K
K
2 X
X
7 4
∆
∆
p2i + − II
pi − 2  ,
D=
90 II
3
i=1,i6=k
i=1,i6=k


K
X
2
ˆ m |2 + ∆2 − ∆II
E=
12|h
pi − 2  pk ,
kk
I
3
3
i=1,i6=k
8 ˆm 2 2
7 4 2
1 ˆm 2
2
2
12|hkk | + ∆I pk +
|h | ∆I + ∆I pk ,
F =
9
3 kk
90
C=
m
xmax = 22Rmax,k − 1.
(4.68)
This problem is not convex. However, it can be proved that the feasible set of this
problem satisfies the linear independent constraint qualification (LICQ) conditions
[Hen92], and its duality gap is zero. Therefore, any pair of the primal and the
dual optimal points of this problem must satisfy the Karush-Kuhn-Tucker (KKT)
conditions [BV04]. Solving the KKT conditions, the necessary condition on the
optimal solution (Rkm∗ ,x∗ ) is
f (x∗ ) = ln(x∗ +1)+
(x∗ +1)−1 (Ax∗ 2 +Bx∗ +C)(Dx∗ 2 +Ex∗ +F )
= 0.
(AE −DB)x∗ 2 +2(AF −DC)x∗ +(BF −EC)
(4.69)
The Newton method iteratively solves (4.69) as follows:
x(i + 1) = x(i) −
f (x(i))
,
f ′ (x(i))
(4.70)
where i is the iteration index and f ′ (x) is the derivative of f (x) with respect to
x. The solution for the optimum rate is Rkm∗ = 21 log(1 + x∗ ), where x∗ is the
convergence point of series {x(i)}∞
i=1 .
In general, hmax and hmin are design parameters that can be optimized; a larger
(smaller) value of hmax (hmin ) reduces the channel outage probability defined in
(4.7); but, on the other hand it increases (decreses) the upper-bound on the outage
probability given in (4.51) because of the larger (smaller) quantizer step-size. In the
simulations, we numerically measure the throughput for different values of these
parameters, and select the ones which maximize the network throughput.
We numerically evaluate the throughput of the ergodic interference alignment
and rate adaptation (EIA-RA) scheme. Figure 4.8 shows the lower bound on the expected throughput of a three-user interference network, where pk = p (k ∈ {1, 2, 3})
and NI = NII = N . As demonstrated in this figure, by increasing the number of
quantization bits (N ), the throughput approaches the one when perfect CSI is available at each terminal. Also, it can be seen that for a given value of N , the throughput saturates at the high-SNR regime. The reason is that the residual interference
power is proportional to the transmission power p and according to (4.31), as p
increases, the SINR converges to a limited value.
92
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
25
N = 3, EIA-RA
N = 4, EIA-RA
N = 5, EIA-RA
N = 6, EIA-RA
N = 7, EIA-RA
N = 8, EIA-RA
N = 9, EIA-RA
N = 10, EIA-RA
Perfect CSI, EIA
Network throughput [bits/channel use]
bC
rS
20
r
uT
u
15
l
ld
*
10
*
*
ld
ld
*ld
l
l
l
*ldl
u
u
u
u
ld
*
l
u
uT
*u
uT
uT
uT
uT
ld
l
uT
*lulduT
r
r
r
r
5
*lduTlur
luduT
0 *lduTrSbC
−20
r
r
r
*lurdTSbC
*lduTrSbC
−10
*rrSbC
*lurlduTrS
rS
rS
rS
rS
rS
rS
bC
bC
bC
bC
bC
bC
rS
bC
rS
bC
bC
0
10
20
30
40
SNR [dB]
Figure 4.8: The lower bound on the throughput of a three-user interference network
with limited feedback when the EIA-RA scheme is applied, NI = NII = N .
For a fixed number of total feedback bits broadcasted by each destination
(Nf = 30), the lower bound on the throughput is shown in Figure 4.9. We consider three different scenarios of feedback bits allocation similar to the ones mentioned for the power control problem in Figure 4.7. In the first one, we equally
allocate bits to the direct and interference link quantizers (NI = 5, NII = 5), in
the second scenario we allocate more bits to the quantizer dedicated to direct link
(NI = 7, NII = 4), and finally in the third scenario we allocate more bits to the
quantizer associated to the interference links (NI = 3, NII = 6). It can be seen in
Figure 4.9 that at the low-SNR regime the second scenario outperforms the others,
while at the high-SNR regime, allocating more bits to the quantization of the interference links provides a larger throughput. The intuition behind this result is that
noise is the dominant factor which degrades network throughput in the low-SNR
regime, and in this regime it is recommended to have more accurate information
about the direct link’s gain to perform rate adaptation more accurately. On the
other hand, at the high-SNR regime, interference instead of noise is the dominant
phenomenon which degrades network throughput, thus, in this regime it is preferred
to have more accurate information about the interference links to more precisely
eliminate interference. This coincides with our observations in Figure 4.7 where we
4.4. Digital Feedback
93
9
rS
Network throughput [bits/channel use]
8
bC
uT
7
rS
rS
rS
NI = 3, NII = 6, EIA-RA
NI = 5, NII = 5, EIA-RA
NI = 7, NII = 4, EIA-RA
rS
rS
bC
bC
bC
bC
uT
uT
uT
uT
bC
bC
6
rS
bC
5
uT
uT
rS
uT
4
bC
uT
3
rS
uT
bC
2
rS
uT
bC
1
rS
uTbC
uT
0 rSbC
−20
uTrSbC
rS
−10
0
10
20
30
40
SNR [dB]
Figure 4.9: The lower bound on network throughput under different strategies for
sharing the total feedback bits (Nf = NI + 2NII = 30) among direct link quantizer
and interference link quantizer in a three-user interference network.
studied feedback bits allocation in the power control problem.
The lower bound on the throughput of K-user (K ∈ {3, 5, 10}) interference
networks for different number of feedback bits is shown in Figure 4.10. Also, the
performance of the TDMA scheme is shown assuming perfect CSI is available at
the sources. This figure shows that the IA-RA scheme can outperform the TDMA
scheme even with only limited feedback given that the number of feedback bits is
sufficiently large. Specifically, for a given number of users and number of feedback
bits, there is a specific SNR below which the EIA-RA scheme outperforms TDMA.
This result together with our observation in Figure 4.6 confirm that, in a certain
operating region, the interference alignment based transmission schemes (e.g. EIAPC and EIA-RA) can outperform the TDMA scheme even if the number of feedback
bits are limited.
Delay-Limited Throughput
So far, we have studied the cases that the delay required for finding the complement
fading blocks in the EIA scheme can be asymptotically long. But in many practical
94
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
20
uT
uT
bC
bC
uT
uT
uT
Network throughput [bits/channel use]
18
EIA-RA, N = 3,
EIA-RA, N = 3,
EIA-RA, N = 3,
EIA-RA, N = 6,
EIA-RA, N = 6,
EIA-RA, N = 6,
TDMA, K = 3
TDMA, K = 5
TDMA, K = 7
rS
bC
16
uT
rS
14
bC
12
uT
rS
10
bC
uT
8
K
K
K
K
K
K
=3
=5
=7
= 3 uT
=5
=7
uT
bC
bC
bC
rS
bC
uT
bC
bC
rS
uT
bC
rS
uT
bC
rS
rS
rS
uT
uT
bC
uT
rS
rS
rS
uT
bC
rS
rS
uT
bC
bC
rS
6
uT
rS
uT
uT
bC
4
rS
rS
bC
uT
uT
uT
uT
uT
rS
bC
bC
bC
bC
bC
rS
rS
rS
rS
rS
bC
uT
uT
uT
bC
uTrSuT
bC
2
bCbC
0
−20
bC
uTuTbCrS
uTuTbCbCrS
rS
−10
bC
rS
rS
rS
bC
uT
uTbCrS
rS
rS
uTuTrS
bC
rS
0
10
20
30
40
SNR [dB]
Figure 4.10: Throughput of a K-user interference network for different number of
feedback bits.
applications only limited communication delay can be tolerated. To incorporate this
requirement in the considered rate adaptation scheme, in this subsection we consider
that terminals can accept only a limited delay in terms of the number of fading
blocks needed to perform EIA, denoted by T . Let m and mp to be the block indices
of two subsequent complement channels. In the considered transmission scheme, the
waiting time for the complement channel to occur may pass the acceptable delay,
i.e. mp − m > T . In this case, communication fails and an outage event will be
declared. We define delay outage probability as follows
o
n
ˆm ,
(4.71)
PDout,m , Pr mp − m > T |H
ˆ m is the network quantized channel matrix at fading block m. The following
where H
proposition quantifies this probability.
Theorem 4.4.4. The delay-limited outage probability is approximately
PDout,m
≈
!T
√
2
2
(2hmax /(σ π))K − 12 PK PK |hˆ m
|
kl
l=1
k=1
e σ
,
1−
2KNf /2
(4.72)
4.4. Digital Feedback
95
Delay-limited throughput [bits/channel use]
6
N=3
N=4
N=5
N=6
5
4
3
2
1
0
100
101
102
103
104
105
106
107
T
Figure 4.11: Delay-limited throughput of a three-user network versus the maximum
acceptable delay (NI = NII = N ).
where σ 2 is the variance of channel gains.
Proof. See Appendix 4.C.
Theorem 4.4.4 shows that, for a given T , PDout,m is an increasing function of
Nf , and a decreasing function of hmax . This implies that a more accurate channel
quantization increases the delay-limited outage probability in (4.71). In addition, it
can be observed that limT →∞ PDout,m = 0 which coincides with the intuition. In the
following, we define the delay-limited throughput of the network for transmission
over blocks m and mp .
Definition 4.1. The delay-limited throughput of the network, at blocks m and
mp , is defined as follows
TDm , 1 − PDout,m × T m ,
(4.73)
where T m and PDout,m are defined in (4.61) and (4.71), respectively.
Figure 4.11 shows the numerical expectation of the delay-limited throughput of
a three-user network as a function of the maximum acceptable delay (T ) for different number of quantization bits NI = NII = N . In the numerical evaluations, we set
96
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
pk /N0 = 1000 [dB] (k ∈ {1, 2, 3}). It can be observed that when the maximum acceptable delay is large, a higher throughput can be achieved by a finer quantization.
In contrast, when acceptable delay is limited to small values, coarse quantization
should be applied instead to achieve a larger throughput. The reason is that using
fine quantizer incurs a larger expected delay for a complement channel to happen
which increases delay outage probability defined in (4.71). On the other hand, more
accurate quantization leads to less residual interference, and less uncertainty about
the actual mutual information value, and consequently outage probability defined
in (4.49) decays. Depending on the value of the maximum acceptable delay, one
of the mentioned phenomena is dominant. Specifically, in applications with strict
delay constraints, outage events due to delay constraints are dominant and it is recommended to use coarse quantization, while in the applications which are robust
to delay, residual interference due to the lack of perfect CSI is the main cause of
the outage events and a fine quantization is suggested to be deployed instead.
Figure 4.12 shows the delay-limited throughput of a K-user interference network
as a function of the maximum acceptable delay for different number of users in
the network. In the simulations, we set pk /N0 = 1000 [dB] (k ∈ {1, 2, 3}), and
NI = NII = N = 3. It can be observed that for large values of T , the throughput
increases by increasing the number of users. The trend is different for small values of
T : the throughput decays as the number of users increases. This is because for small
values of T , the outage events due to delay constraint are the dominant reason of
failure in transmission. By increasing the size of the network, users are required to
wait longer for the complement channels to happen; consequently, the delay outage
probability defined in (4.72) increases and delay-limited throughput decays.
4.5
Summary
In this chapter, we have studied SISO interference networks. In the first part of
the chapter, we looked at networks where no a priori CSI is available at the terminals. We proposed a pilot-assisted ergodic interference alignment scheme in order
to obtain CSI and conduct data communications. We considered pilot-based channel training and investigated both analog and digital feedback cases. When there
is analog feedback from destinations to sources, we computed an achievable rate
region. Our study reveals that the total DoF Kopt (1−Kopt/T )/2 is achievable when
the number of the active users is selected to be Kopt = min{T /2, K}. Thus, it can
be recommended that, in large networks (K > T /2), to perform a user selection,
and to apply interference alignment only within the set of the active users. In addition, we have derived the optimum problem allocation to the channel training
and the data transmission. Our results show that to increase the achievable rate in
large networks more power should be allocated to the channel training instead of
the data transmission.
In the second part of the chapter, we looked at using digital feedback to convey CSI to the transmitters. Here, we however assumed that the receivers have
4.5. Summary
97
Delay-limited throughput [bits/channel use]
1.4
K
K
K
K
1.2
1
=3
=4
=5
=6
0.8
0.6
0.4
0.2
0
100
101
102
103
104
105
106
107
T
Figure 4.12: Delay-limited throughput of a K-user interference network versus the
maximum acceptable delay.
perfect CSI. Two problems have been addressed. First, we have studied a power
control problem to compute the minimum required power to guarantee that each
source-destination pair can successfully communicate at the desired fixed rate. We
proposed a power control algorithm, and investigated its convergence behavior.
Next, we have addressed a throughput maximization problem when transmission
powers are fixed. We derived an upper bound on the outage probability, and using that we proposed a rate adaptation scheme to maximize the lower bound on
network throughput. Finally, we have studied the impact of the constraint on the
maximum acceptable delay on the throughput for delay sensitive applications.
Our study shows that with a proper power control or rate adaptation, interference alignment can outperform conventional orthogonal transmission schemes, even
if only quantized CSI with limited resolution is available at the sources. For a given
number of total feedback bits, we have investigated feedback bits allocation to the
quantization of direct links or interference links. For the power control scheme, to
decrease transmission power, our study reveals that for transmission at high rates
more bits should be allocated to interference links; on the other hand, more bits
should be allocated to direct links for transmission at low rates. For the rate adaptation scheme, our results recommend to allocate more bits to interference links at the
98
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
high-SNR regime, and to allocate more bits to direct links at the low-SNR regime.
When there is a maximum acceptable delay, we have shown that if the maximum
acceptable delay is large, the throughput increases as the number of feedback bits
increase; However, for a small maximum acceptable delay, increasing the number
of feedback bits leads to a decrease of throughput. Furthermore, we have seen that
when the maximum acceptable delay is large, the throughput increases by increasing the number of users; while the throughput decays by increasing the number of
users when the maximum acceptable delay is low.
4.A
The Proof of Proposition 4.3.2
The variance in (4.19) can be derived as follows
σ2
=
=
(a)
=
=
(b)
=
(c)
=
=
i h
i2
∗ h
e Y k bk H,
e Y k − E Xk − X
bk H,
b k Xk − X
Xk − X
i
∗ h
e Yk
bk H,
b k Xk − X
E Xk − X
∗ h
i
e Yk
bk H,
E Xk · Xk − X
h
∗ i
e Yk
bk H,
Pd − E Xk · X
i
h
˜ kk Xk Y k ∗ H,
e Yk
E Pd h
Pd −
2 hkk Pd
1 + Kσε2 +2 ˜
2 2
hkk Pd
2 ˜
Pd −
2 hkk Pd
1 + Kσε2 + 2 ˜
E
1+
Pd
˜ kk |2 Pd
2|h
(4.74)
1+Kσε2 Pd
where (a) follows from the orthogonality of the estimated signal to the estimation
bk given in (4.18);
error of the MMSE estimator; (b) follows by the substitution of X
and (c) follows substituting Y k given in (4.13), and noting that Xk is mutually
m
independent of Zkm , Zk p and Xl (∀l ∈ {1, 2, ..., K}, l 6= k).
4.B
The Proof of Theorem 4.4.3
In this part, we compute the mean and the variance of the random variable Y
defined in (4.57). We exploit the property that the quantization error of a uniform
quantizer quantizing a Gaussian random variable with variance σ 2 is uniformly
distributed with an acceptable approximation when σ/∆ ≥ 1, where ∆ is the
quantizer step size [SS77]. The parameter hmin is set to be zero in the throughput
4.B. The Proof of Theorem 4.4.3
99
maximization problem. Therefore, we assume uniform distribution for variables
m
m
m
m
], Im[δkl p ] ∼ U (−∆II /2, ∆II /2), where ∆II = hmax /2NII −1
Re[δkl
], Im[δkl p ], Re[δkl
for NII bit quantization when the magnitude of the real and imaginary parts of
ℜ
ℑ
the channel gains are limited to hmax . We define random variables gkl
and gkl
as
follows:
m
m
ℑ
m
ℜ
m
= Im[δkl
+ δkl p ] ∀k, l ∈ {1, 2, ..., K}, k 6= l.
gkl
= Re[δkl
+ δkl p ] , gkl
(4.75)
Assuming uniform distribution for the quantization errors, since fX+Y = fX ∗ fY ,
where ‘*’ is the convolution operation, the pdfs of these variables are given by:
fgℜ (x) = fgℑ (x) =
kl
kl
1
(∆II − |x|) 0 < |x| < ∆II .
∆2II
(4.76)
The mean and the variance of these random variables are as follows:
ℑ ∆2II
ℜ
ℑ
ℜ
=
= var gkl
= 0 , var gkl
= E gkl
E gkl
.
6
(4.77)
ℑ
Also, we define the random variables sℜ
kl and skl as follows:
m
p
m
sℜ
kl = Re[δkl + δkl ]
If Y = g(X), then
n(y)
fY (y) =
X
k=1
2
mp 2
m
.
, sℑ
kl = Im[δkl + δkl ]
1/ g ′ (gk−1 (y)) .fX (gk−1 (y)),
(4.78)
(4.79)
where n(y) is the number of solutions in x for the equation g(x) = y, and gk−1 (y) is
the kth solution. Exploiting (4.76), the pdfs of these random variables are
fsℜ (x) = fsℑ (x) =
kl
kl
1
1
√ −
∆II x ∆II 2
0 < x < ∆2II .
(4.80)
Also, the mean and the variance of these random variables are
ℑ 7∆4II
ℑ ∆2II
, var sℜ
.
E sℜ
kl = var skl =
kl = E skl =
6
180
(4.81)
According to (4.57), (4.81), and (4.77), we have the mean and the variance values
which are given in (4.58).
100
K-user SISO Interference Networks: Pilot-assisted Interference Alignment
4.C
The Proof of Theorem 4.4.4
The delay outage probability defined in (4.72) can be computed as follows
o
o
n
n
ˆ m+T 6= H
ˆ mp
ˆ m+1 6= H
ˆ mp , ..., H
ˆ m = Pr H
Pr mp − m > T |H
oT
n
(a)
ˆ m+1 6= H
ˆ mp
= Pr H
(b)
=
1−
n
K i
io
h
n Y
h h mi
ˆm
ˆ
Pr Re hm+1
∈ Re h
kk − ∆I /2, Re hkk + ∆I /2
kk
k=1
h
i
io
h h mi
ˆ
ˆm
× Pr Im hm+1
∈ Im h
kk − ∆I /2, Im hkk + ∆I /2
kk
K Y
K n h
i
io
Y
h h mi
ˆ
ˆm
Pr Re hm+1
∈ Re −h
×
kl − ∆II /2, Re −hkl + ∆II /2
kl
k=1
l=1
l6=k
h
io
i
h h mi
ˆ
ˆ m + ∆II /2
∈
Im
−
h
−
∆
/2,
Im
−
h
× Pr Im hm+1
II
kl
kl
kl
n
!T
(4.82)
where (a) follows the fact that channels are i.i.d; (b) follows the independence of the
elements of channel matrix. We have hm+1
∼ CN (0, σ 2 ), and hm+1
∼ CN (0, σ 2 ),
kk
kl
therefore
n
Pr Re
hm+1
kk
i
io Z
h
h h mi
ˆ m +∆I /2 =
∈ Re ˆ
hkk −∆I /2, Re h
kk
ˆ m ]+∆I /2
Re[h
kk
ˆ m ]−∆I /2
Re[h
kk
x2
1
√ e− σ2 dx
σ π
2
(Re[hkk ])
∆I
σ2
.
≈ √ e−
σ π
ˆm
Similarly, we can compute the other items. We can conclude that
n
ˆm
Pr mp − m > T |H
o
!
(K 2 −K)
PK PK ˆ m 2 T
1
∆K
∆
−
|
h
|
≈ 1 − I √II K 2 e σ2 k=1 l=1 kl
(σ π)
!T
√
2
2
(2hmax /(σ π))K − 12 PK PK |hˆ m
|
kl
l=1
k=1
e σ
(4.83)
= 1−
2KNf /2
where Nf = 2NI + 2(K − 1)NII .
Chapter 5
K-user MIMO Interference Networks:
Transceiver Design and Power Control
I
N the previous chapter we focused on SISO interference networks, where time
variations of the channel have been exploited to partially eliminate the interference. In this chapter, we consider MIMO interference networks in which multiple
antennas at terminals can be exploited to manage the interference. We aim to address the power control problem for these networks, by further taking into account
the achievable DoF of the network. Specifically, we consider a network in which
each source sends multiple data streams the number of which is the same as the
corresponding DoF achieved by interference alignment. We propose two iterative
algorithms that compute transmitter beamforming matrices and receiver filtering
matrices to maximize the SINR for each stream, and allocate the minimum powers
to realize the desired fixed-rate communications. In both algorithms, the required
power values are computed in a distributed fashion at each destination and the
associated source is informed via a feedback link. In the first algorithm, the exact
value of the computed power is sent, and in the second algorithm only a one-bit
feedback signal is transmitted via feedback. The proposed algorithms can provide
reliable communication when multiple streams are transmitted each being encoded
with potentially different rates. Numerical evaluations show that these algorithms
require substantially smaller powers, when compared to the conventional orthogonal transmission strategies. Our test-bed implementation of these algorithms in a
network consisting three source-destination pairs is reported in [MFZS14]. The experimental measurements in indoor environment confirm the promised performance
of the proposed algorithms.
The structure of this chapter is as follows. Section 5.1 presents the considered
multi-user MIMO interference network. In Section 5.2, we elaborate on the iterative
algorithms to conduct transceiver design and power control. Section 5.3 provides
numerical performance evaluations. Finally, Section 5.5 summarizes this chapter.
101
102
K-user MIMO Interference Networks: Transceiver Design and Power Control
mdkk
Ekdk
cdkk
Pkdk
m1k
Ek1
q
pdkk cdkk
..
.
..
.
Sk
xdkk =
c1k
Pk1
nS
Vk
p
x1k = p1k c1k
nD
xk k
yk k
..
.
..
.
x1k
yk1
y dkk
Dkdk
m
ˆ dkk
..
.
Uk
y 1k
Dk1
Dk
m
ˆ 1k
Figure 5.1: The structure of transmitter and receiver corresponding to the sourcedestination pair Sk − Dk .
5.1
Multi-user MIMO Interference Network
We consider a MIMO interference network consisting of K source-destination pairs.
The sources and the destinations are denoted as Sk and Dk (k ∈ {1, 2, ..., K}), and
they are equipped with nSk and nD
k antennas, respectively. Each source intends to
communicate to the corresponding destination. The structure of transmitter and
receiver corresponding to the source-destination pair Sk −Dk is shown in Figure 5.1,
and will be described in more details in what follows.
5.1.1
Transmitter Structure
The source Sk sends dk independent messages mdk (d ∈ {1, ..., dk }). The value of dk
is selected according to the DoF region of the network characterized in [CJ08,GJ10,
GCJ11] 1 . The encoder Ekd encodes mdk to a unit-power codeword cdk selected from
d
d
a Gaussian codebook
q with code rate Rk . Next, the power controller Pk scales this
codeword to xdk =
pdk cdk , where pdk is the power of transmitted signal and satisfies
h
iT
a maximum power constraint, i.e. pdk < pmax . The dk × 1 vector xk = x1k , ..., xdkk
denotes these scaled codewords. The transmitted signal of Sk is
xk = Vk xk ,
h
(5.1)
i
where Vk = v1k , ..., vdkk is the nSk × dk beamforming matrix, and vdk denotes the
unit-norm beamforming vector corresponding to the dth transmitted codeword.
1 In this chapter, the number of streams sent by the sources is selected as the DoF value that
can be maximally achieved through linear interference alignment. In fact, this value is unknown
in the case of general MIMO interference channels. However, due to the rapid development of the
interference alignment concept in recent years, the maximal achievable DoF for several network
structures has been discovered. For instance, in three-user MIMO interference channels with M
antennas at each terminal the available DoF has been characterized in [CJ08], and subsequently
that of K-user interference channels in which each transmitter has M antennas and each receiver
has N antennas has been characterized in [GJ10]. For networks with an arbitrary number of
antennas at each terminal, the necessary condition for the achievability of a tuple of DoF has
been discussed in [YTJK10], and the achievability can be numerically checked using the method
proposed in [GCJ11]. The results presented in these references can be used to determine the
number of streams sent from each source.
5.2. Transceiver Design and Power Control
5.1.2
103
Receiver Structure
The received signal at the destination Dk is
yk = Hkk Vk xk +
K
X
Hkl Vl xl + zk ,
(5.2)
l=1,l6=k
where Hkl is the channel matrix corresponding to the link from Sl to Dk , and zk is
additive noise. Throughout this chapter, the link from Sk to Dk is called desired link
and those from Sl to Dk (l 6= k) are referred to as interference links. The channel
gains are constant during the transmission of each codeword, but independently
change across time. The channel gains follow a complex
Gaussian distribution, i.e.
hkl ∼ CN (0, σS2 InS ×nD ) where InS ×nD is a nSk × nD
×
nSk × nD
identity matrix,
l
l
k
l
k
l
and hkl denotes vector representation of the elements of matrix Hkl . The noise
has complex Gaussian distribution, i.e. zk ∼ CN (0, N0 InD ) in which N0 is noise
k
D
power at each destination, and InD is an nD
k × nk identity matrix. Let Uk denote an
k
d
nD
k ×dk receiver filtering matrix with unit-norm column vectors uk (d ∈ {1, ..., dk }).
h
iT
The filter output of Dk , yk = y 1k , ..., ydkk , is
yk = U∗k yk ,
(5.3)
where A∗ denotes the conjugate transpose of matrix A. When the sources are
non-orthogonally activated, the filter output of Dk generally contains interference,
i.e.,
yk = U∗k Hkk Vk xk +
K
X
U∗k Hkl Vl xl + U∗k zk .
(5.4)
l=1,l6=k
The decoder Dkd (d ∈ {1, ..., dk }) decodes the received signal y dk to an estimate of
the transmitted message m
ˆ dk by treating the remaining interference as noise. In the
considered network, it is desired that source Sk reliably communicates dk independent streams with its intended destination Dk while stream d (d ∈ {1, ..., dk }) is
encoded at a given fixed rate Rkd . This requires Vk , Uk , and pdk , to be properly
designed.
5.2
Transceiver Design and Power Control
In this section, we address transceiver design and power control which are performed
before the actual data transmission starts. The transceiver design and power control
are performed in an iterative fashion and each iteration occurs during one training
slot. There are N training slots, and within each slot different tasks are performed
as depicted in Figure 5.2. The quantity N is a design parameter and the accuracy
of system design can be improved by setting N with a larger value. The computed
104
K-user MIMO Interference Networks: Transceiver Design and Power Control
Training slot 1
forward
training
...
receivers
optimization
Training slot n
power
updating
...
reverse
training
Training slot N
transmitters
optimization
Figure 5.2: CSI acquisition, transceiver design, and power control.
beamforming vector, filtering vector, and power corresponding to the lth stream of
the source-destination pair Sk − Dk at the nth training slot are denoted as vlk (n),
ulk (n), and plk (n), respectively.
In this chapter, the term CSI is used to denote channel knowledge in the sense
of the following discussion. To compute receiver filtering matrix
and transmission
q
d
power, in training slot n, the destination Dk needs to know pj (n − 1)Hkj vdj (n−1)
(j ∈ {1, ..., K}, d ∈ {1, ..., dj }). These can be obtained using training sequences sent
by the sources over forward channels (channels from sources to destinations). Similarly, to compute transmitter
beamforming matrix, in training slot n, the source
q
Sk requires to know pdFj Hrkj udj (n) (j ∈ {1, ..., K}, d ∈ {1, ..., dj }), where Hrkj de-
notes the channel matrix from Dj to Sk . These can be obtained at each source using
training sequences transmitted over reverse channels (channels from destinations to
sources). We assume that reverse channels are separated from forward channels in
time via time-division duplexing (TDD), and the reciprocity assumption holds, i.e.,
Hrkl = H∗lk (∀l, k ∈ {1, 2, ..., K}). This is the channel knowledge required to conduct
the transceiver design and power control. Since each terminal (source/destination)
needs to know the knowledge related to local links only, we refer to this channel
knowledge as local CSI in this chapter.
5.2.1
CSI Acquisition, Transceiver Design, and Power Control
In the following, we briefly present the process of iterative CSI acquisition, transceiver
design, and power control. Before the first training slot starts, Sk (k ∈ {1, 2, ..., K})
initializes power plk (0) = pmax (l ∈ {1, ..., dk }) and randomly selects unit-norm
beamforming vector vlk (0). The nth (n ∈ {1, 2, ..., N }) training slot is composed of
the following phases:
1) Forward training phase: During this phase, the destination Dk estimates
5.2. Transceiver Design and Power Control
105
interference-plus-noise covariance matrix
Qlk (n)
,
dj
K X
X
∗
∗
pdj (n − 1)Hkj vdj (n − 1) vdj (n − 1) (Hkj )
j=1 d=1
∗
∗
−plk (n − 1)Hkk vlk (n − 1) vlk (n − 1) (Hkk ) + N0 InD ,
k
(5.5)
and effective desired channel matrix
l
Hkk (n) , Hkk vlk (n − 1)
(5.6)
corresponding to the lth (l ∈ {1, ..., dk }) stream. To obtain these matrices, an
orthogonal training
PK scheme can be deployed. Let the length of the forward training
phase be L × k=1 dk . Each source subsequently sends L pilot symbols for each
data stream modulated with the corresponding beamforming vector. The other
sources remain silent during thePpilot transmission of one source.
Pj−1Specifically, the
j−1
source
S
at
time
slot
t
=
L
d
+
(d
−
1)L
+
1,
...,
L
j
i
i=1
i=1 di + dL sends
q
pdj (n − 1)vdj (n − 1), where d ∈ {1, ..., dj }. The destination Dk receives
yk (t) =
q
pdj (n − 1)Hkj vdj (n − 1) + zk (t),
(5.7)
where zk (t) ∼ CN (0, N0 InD ) is the receiver noise. If L ≫ 1, then according to the
law of large numbers we have
Pj−1
L
d +dL
q
i=1
Xi
1
yk (t) ≈ pdj (n − 1)Hkj vdj (n − 1).
(5.8)
L
Pj−1
t=L
i=1
di +(d−1)L+1
Since the destination Dk (k ∈ {1, ..., K}) knows its noise power N0 , it can use
the estimated quantities in (5.8) for different values of j ∈ {1, ..., K} and d ∈
{1, ..., dj } to compute the interference-plus-noise covariance matrix given in (5.5).
Also, since Dk knows the transmitted power pdk (n − 1) of the corresponding source
(in Section III.B.5 we will propose algorithms to allow each destination to compute
the required power for the corresponding source, and inform the source about the
computed value via a feedback signal), it can compute the effective desired channel
matrix in (5.6) using the estimated values in (5.8). In [MFZS14], a similar channel
training approach is used in a test-bed implementation of Algorithm 2 proposed
in Section 5.2.2 and demonstrates good performance. Also, we have investigated the
impact of channel estimation errors on system performance in [FZF+ 13, FKWS13].
Since channel training is not the concern of this chapter, we assume that perfect
estimation of these matrices are available at terminals. The interested readers are
referred to [KRB+ 13] for more thorough discussion on channel training design for
MIMO systems.
106
K-user MIMO Interference Networks: Transceiver Design and Power Control
2) Receivers optimization phase: Using the interference-plus-noise covariance matrix in (5.5) and the effective desired channel matrix in (5.6) estimated in
the forward training phase, the receiver filtering vector ulk (n), similar to the one
for Max-SINR algorithm in [GCJ11], is selected to be an MMSE filter as follows
ulk (n) =
(Qlk (n))−1 Hkk vlk (n − 1)
.
k(Qlk (n))−1 Hkk vlk (n − 1)k2
(5.9)
This filter maximizes the received SINR of the lth stream at destination Dk denoted
as SINRlk .
3) Power updating phase: In this phase, the power values are updated using a distributed power control scheme that will be described in Section 5.2.2. In
Section 5.2.2, we will first consider an iterative method. In each iteration, each
destination Dk computes the required transmission powers of Sk , and then inform
Sk about these values via feedback signals. We will prove the convergence of this
approach. Since conducting an iterative scheme in the power updating phase may
increase system complexity and communication delay, we next simplify the process
by considering two methods that update powers only once, i.e., no iteration is performed. In the first method, each destination sends back the exact values to which
the source transmission powers should be updated. In the second method, each destination uses only a one-bit feedback signal associated with each stream to inform
the corresponding source to scale the transmission power by a certain factor. The
convergence behaviour of the overall iterative process, considering the transceiver
design and power control, can be confirmed by simulations shown in Section 5.3.
4) Reverse training phase: To acquire CSI at the sources, the destinations orthogonally broadcast training sequences. The destination Dk beamforms its training
sequences with a fixed power pF uniformly allocated to different sequences, using
r
an nD
k × dk matrix Vk (n − 1) = Uk (n). The source Sk (k ∈ {1, 2, ..., K}) estimates
the reverse interference-plus-noise covariance matrix corresponding to the lth data
stream
Qr,l
k (n) =
dj
K X
X
pF
j=1 d=1
−
dj
∗ r ∗
r,d
Hrkj vr,d
j (n − 1) vj (n − 1) (Hkj )
∗ r ∗
pF r r,l
Hkk vk (n − 1) vr,l
,
k (n − 1) (Hkk ) + N0 InS
k
dk
(5.10)
r
where vr,d
j (n − 1) denotes the dth column of matrix Vj (n − 1). In addition, Sk estimates the effective reverse desired channel matrix corresponding to the lth stream
of the kth source-destination pair:
r,l
Hkk (n − 1) = Hrkk vr,l
k (n − 1).
(5.11)
The similar channel training scheme as the one mentioned in the forward training
phase can be deployed for reverse channel training as well to acquire Qr,l
k (n) and
r,l
Hkk (n − 1).
5.2. Transceiver Design and Power Control
107
5) Transmitters optimization phase: In the reverse training phase, the
source Sk applies an nSk × dk filtering matrix Urk (n) to its received signal. Similar to the receivers optimization phase, ur,l
k (n) (∀l ∈ {1, ..., dk }) is chosen as follows
to maximize the received SINR of the lth stream received at Sk
ur,l
k (n) =
−1
(Qr,l
Hkk vlk (n − 1)
k (n))
−1 H vl (n − 1)k
k(Qr,l
kk k
2
k (n))
.
(5.12)
The source can compute these vectors using the effective reverse desired channel
matrix in (5.11) and the interference-plus-noise covariance matrices in (5.10) estimated in the reverse training phase. Next, the source Sk sets Vk (n) = Urk (n) as its
updated beamforming matrix in this training slot.
An illustrative summary of the five phases above is shown in Figure 5.2. The
whole process repeats in the subsequent training slots until the whole training is
completed (i.e., after N training slots). In the following subsection, we will explain
the principles of the distributed power control scheme, performed during the power
update phase, in more details.
5.2.2
Distributed Power Control
In the power updating phase of the nth training slot, the transmitter beamforming vectors and receiver filtering vectors at Sk and Dk are vlk (n − 1) and ulk (n)
(k ∈ {1, 2, ..., K}, l ∈ {1, 2, ..., dk }), respectively. For the simplicity of presentation,
we omit the slot index and let ulk , vlk , and plk denote ulk (n), vlk (n−1), and plk (n−1),
respectively. Then, the SINR of the lth stream at Dk is
2
l ∗
uk Hkk vlk plk
l
,
(5.13)
SINRk =
ϕlk (p) + N0
where
ϕlk (p) =
dj K X
2
2
X
∗
l ∗
uk Hkj vdj pdj − ulk Hkk vlk plk ,
(5.14)
j=1 d=1
iT
h
P
K
and p = p11 , ..., pd11 , ..., p1K , ..., pdKK
is a
k=1 dk × 1 power vector. The mutual
information
corresponding
to the lth stream of the source-destination pair Sk − Dk
is log2 1 + SINRlk . To guarantee successful transmission, the following condition
should be satisfied:
log2 1 + SINRlk ≥ Rkl .
(5.15)
According to (5.13), (5.15) can be rewritten as a power constraint in the following
format:
plk ≥ Ikl (p),
(5.16)
108
K-user MIMO Interference Networks: Transceiver Design and Power Control
where
l
2Rk − 1 ϕlk (p) + N0
.
Ikl (p) =
2
l ∗
uk Hkk vlk (5.17)
Therefore, all power constraints can be represented as
p I(p),
(5.18)
where the operator denotes an element-wise vector inequality, and
h
iT
dK
1
I(p) , I11 (p), ..., I1d1 (p), ..., IK
(p), ..., IK
(p)
(5.19)
is called interference function.
Definition 5.1. For given transmitter beamforming matrices and receiver filtering
matrices, the set of the positive power vectors, which satisfies (5.18) and the power
constraint, is defined
PH = {p|0 p pmax , p I(p)},
(5.20)
where I(p) is the interference function defined in (5.19), and pmax is the maximum
transmission power vector whose elements are set to be pmax .
Therefore, the power control problem can be formulated as the following optimization problem:
min
s.t. p∈PH
dk
K X
X
plk .
(5.21)
k=1 l=1
In the remaining parts, we first discuss the existence of a feasible solution for this
problem. Next, we present an iterative algorithm to find the solution of this problem.
Finally, we study the convergence of the presented algorithm.
Feasibility of the Power Control Problem
The power control problem in (5.21) has a feasible solution when the set PH is
nonempty. We can rewrite the condition in (5.18) as a linear power constraint in
matrix form as follows:
p DFp + n,
(5.22)
K
matrix F is the normalized gain matrix. Its
where the
k=1 dk
k=1 dk ×
P
P
k−1
j−1
elements on the
m=1 dm + l th row and the
m=1 dm + d th column (k, j ∈
P
P
K
5.2. Transceiver Design and Power Control
109
{1, ..., K}, l ∈ {1, ..., dk }, d ∈ {1, ..., dj }) are
F
k−1
X
dm + l,
j−1
X
!
dm + d
m=1
m=1
 |(ul )∗ H vd |2
kj j
k


 |(ul )∗ Hkk vl |2
k
=
j 6= k
k
2
|(ulk )∗ Hkk vd
k|
l )∗ H
l |2
|(u
v

 k kk k
j = k, d 6= l .

(5.23)
0
j = k, d = l
P
K
The
k=1 dk matrix D is a diagonal matrix and its elements on the
k=1 dk ×
P
P
k−1
j−1
d
+
l
th
row
and
the
d
+
d
th column (k, j ∈ {1, ..., K}, l ∈
m
m
m=1
m=1
P
K
{1, ..., dk }, d ∈ {1, ..., dj }) are
D
k−1
X
dm + l,
m=1
m=1
The
P
k−1
m=1
j−1
X
!
dm + d
=
( l
2Rk − 1
j = k, l = d
0
otherwise
.
(5.24)
dm + l th element (k ∈ {1, ..., K}, l ∈ {1, ..., dk }) of the vector n is
n
k−1
X
m=1
dm + l, 1
!
=
l
2Rk − 1 N0
|(ulk )∗ Hkk vlk |2
.
(5.25)
Therefore, we can represent the set PH defined in (5.20) as follows:
PH = {p|0 p pmax , p DFp + n}.
(5.26)
Since this set depends on the channel matrices, the maximum transmission powers,
the transmission rates, the transmitter beamforming matrices, and the receiver
filtering matrices, all these parameters affect the feasibility of the problem in (5.21).
The following theorem characterizes the condition under which the power control
problem is feasible.
Theorem 5.2.1. Assume that the transmitter beamforming matrices and the receiver filtering matrices for the scheme considered in Section 5.2.1 are given as
Vk and Uk (∀k ∈ {1, 2, ..., K}), and the maximum transmission power is asymptotically large (i.e., pmax → ∞). The power control problem in (5.21) is feasible
(PH 6= ∅) if and only if λmax (DF) < 1, where λmax (DF) is the Perron-Frobenius
eigenvalue of the matrix DF, and the matrices D and F are defined in (5.23) and
(5.24), respectively.
Proof. See Appendix 5.A.
For given channel matrices, beamforming matrices, and receiver filtering matrices, the condition given in Theorem 5.2.1 determines whether the power control
problem has feasible solutions. For random channel matrices, with a certain probability, the power control problem has solutions. This probability is referred to as
110
K-user MIMO Interference Networks: Transceiver Design and Power Control
uTbCrS
100
uTbCrS
rSbC
uT
rS
uTbC
rS
rS
bC
uT
rS
bC
uT
rS
bC
rS
uT
bC
rS
bC
uT
rS
Feasibility Probability
10−1
bC
rS
uT
bC
uT
10−2
bC
M
M
M
M
rS
10−3
bC
uT
=2
=4
=6
=8
uT
uT
10−4
0
2
4
6
8
10
12
R [bits/channel use]
Figure 5.3: Feasibility probability versus transmission rate in a three-user interference network with M antennas at each terminal.
feasibility probability. In wireless communication systems, the feasibility probability
should be larger than a certain threshold. The value of this threshold depends on
the reliability requirements of the target application. In sensitive applications such
as industrial control systems, this threshold can be high; while for applications such
as voice transmission, a relatively low threshold can be accepted.
Figure 5.3 shows the asymptotic feasibility probability, when pmax → ∞, versus
transmission rate for different number of antennas at the terminals. We consider
a three-user interference network and assume that each terminal is equipped with
M antennas (M ∈ {2, 4, 6, 8}). The achievable DoF of each source-destination pair
is M/2, and each source intends to send M/2 independent data streams each with
rate R. It can be observed that as the number of antennas increases the feasibility probability decays. The reason for this observation is that in this simulation
set up, as the number of antennas increases we also increase the number of transmitted streams in each source-destination link. This leads to a larger inter-stream
interference and consequently a lower feasibility probability.
Remark 5.1. The necessary and sufficient condition for the feasibility of the considered power control problem given in Theorem 5.2.1 is valid in the asymptoti-
5.2. Transceiver Design and Power Control
111
cally high-SNR regime. If the power budget is finite, the condition λmax (DF) < 1
is only a sufficient condition but not a necessary condition. In other words, for
finite values of pmax , if the considered power control problem is feasible, then
λmax (DF) < 1. However, the reverse statement does not apply.
Figure 5.4 shows the feasibility probability versus rate for different values of
pmax . In this example, we consider a three-user interference network where each
terminal is equipped with two antennas, and each source sends one data stream. It
can be observed that for a fixed transmission rate, when the power budget decreases
the feasibility probability decreases. The feasibility probability under unlimited
power budget is the same as the probability that the condition λmax (DF) < 1
is satisfied, and serves as the upper bound of the case under finite power limit.
Clearly, if the power budget level is relatively large compared to the transmission
rate demand, the feasibility probability is close to the upper bound. The effect of
limited power budget is small. Thus if a system has a large power budget, with a high
probability the condition given by Theorem 5.2.1 can be used to correctly predict
whether it is possible to guarantee successful communication at the demanded fixed
rate. Note that a large power budget does not necessarily mean a large power
consumption. The convergence behavior of our proposed algorithms shows that the
lowest level of power consumption can be reached, which exhibits the advantages
of the proposed algorithms. If the power control problem has feasible solutions,
the next question to be answered is how to find the solution corresponding to
the minimum required transmission power. In the next subsection, we present an
iterative power control scheme to address this question.
Iterative Power Control
Consider an iterative power control where the maximum number of iterations is set
to be T . Recall that the transmitter beamforming matrices and the receiver filtering
matrices are Vk and Uk (k ∈ {1, ..., K}), respectively. We use plk [i] (l ∈ {1, ..., dk })
to denote the computed power at the ith iteration. In training slot n, before the
first iteration starts, Sk sets its initial power values plk [0] = plk (n − 1). At the ith
iteration (i = 1, 2, ..., T ), Dk computes Ikl (p[i − 1]) using (5.17). Next, it sends these
values to Sk . The source then updates its power values as follows:
plk [i] = min{Ikl (p[i − 1]) , pmax }.
(5.27)
After the last iteration, the source Sk updates the power of the lth data stream at
the nth training slot as plk (n) = plk [T ].
Convergence
The iterative power control algorithm in (5.27) generates a sequence of vectors
p[1], p[2], ..., p[T ]. In this part, we answer two questions regarding the convergence
of this sequence. The first is whether this sequence converges to a fixed-point p∗
112
K-user MIMO Interference Networks: Transceiver Design and Power Control
*uTrSbC
100
*uTrSbC
uTrSbC
*
uTrSbC
uTrSbC
*
bCuTrS
*
rSbC
uT
rSbC
rSbC
bC
bC
rS
uT
bC
rS
uT
*
rS
uT
Feasibility Probability
*
10−1
uT
*
bC
10−2
rS
uT
*
pmax /N0
pmax /N0
pmax /N0
pmax /N0
pmax /N0
→∞
= 50 dB
= 40 dB
= 30 dB
= 20 dB
*
uT
10−3
0
2
4
6
8
10
12
R [bits/channel use]
Figure 5.4: Feasibility probability versus transmission rate in a three-user interference network with two antennas at each terminal and maximum transmission power
pmax .
(i.e., p∗ = I(p∗ )) for sufficiently large T . If yes, the second question is whether
fixed-point p∗ reflects the minimum required transmission powers. The answers to
these questions are summarized in the following theorem.
Theorem 5.2.2. If the problem in (5.21) is feasible (PH 6= ∅), for any initial power
vector p[0], the recursive equation in (5.27) generates a sequence of vectors which
converges to a unique fixed-point p∗ . The fixed-point p∗ corresponds to the solution
of the problem in (5.21). Thus the proposed iterative power control scheme in the
power updating phase can attain the solution of the power control problem (5.21).
Proof. See Appendix 5.B.
In the distributed implementation of the iterative power control scheme, the corresponding convergence rate directly affects the radio resources required for training. Therefore, in the next part we study the convergence rate of the above scheme.
5.2. Transceiver Design and Power Control
113
Convergence Rate
In this part, we find the convergence rate of the iterative power control scheme.
This can be used to determine the minimum number of iterations which is required
to have the computed powers sufficiently close to the solution of the power control
problem. For this propose, we investigate geometrical convergence defined as follows
[CHLT08].
Definition 5.2. A sequence a[1], a[2], ... is said to geometrically converge to a∗ at
rate α (a smaller rate is corresponding to faster convergence) if there exist nonnegative constants A and α (α < 1) such that
ka[n] − a∗ k ≤ Aαn ∀n ∈ {1, 2, ...},
(5.28)
where kxk denotes a general norm of the vector x.
Let weighted maximum norm of a vector x = [x1 , ..., xm ]T be defined as follows:
xj ,
kxkw
=
max
(5.29)
∞
j wj where w = [w1 , ..., wm ] is an element-wise positive vector. This definition induces
a matrix norm [HJ85]
kAkw
∞ = max
x6=0
kAxkw
∞
.
kxkw
∞
(5.30)
The following lemma characterizes the connection between the weighted maximum norm and the Perron-Frobenius norm of a non-negative square matrix.
Lemma 5.1. [BT89, Corollary 6.1] For a non-negative square matrix A, there
exists a vector w 0 such that kAkw
∞ < 1 if and only if λmax (A) < 1.
Theorem 5.2.3. If λmax (DF) < 1 and pmax is asymptotically large, then the
convergence rate of the power control algorithm in (5.27) with respect to the weighted
maximum norm defined in (5.29) is α = kDFkw
∞.
Proof.
kp[i] − p∗ kw
∞
=
=
=
(a)
≤
≤
kI(p[i − 1]) − I(p∗ )kw
∞
k(DFp[i − 1] + n) − (DFp∗ + n)kw
∞
kDF(p[i − 1] − p∗ )kw
∞
∗ w
kDFkw
∞ kp[i − 1] − p k∞
i
∗ w
(kDFkw
∞ ) kp[0] − p k∞ ,
(5.31)
where (a) follows from Theorem 5.6.2 in [HJ85]. Since λmax (DF) < 1, according to
Lemma 5.1 there exists a vector w allowing us to have kDFkw
∞ < 1. Consequently,
114
K-user MIMO Interference Networks: Transceiver Design and Power Control
the upper-bound on kp[i] − p∗ kw
∞ converges to zero as i → ∞. This result together
with the fact that norm of a vector is always non-negative implies the convergence
of p[i] to p∗ . The convergence rate of the scheme according to Definition 5.28
is α = kDFkw
∞.
If the matrix DF is irreducible (which it is likely to be since the channel matrices
are random), using Proposition 6.6 in [BT89] it can be shown that for the positive
right Perron-Frobenius eigenvector of DF denoted as v we have kDFkv∞=λmax (DF),
and thus the convergence rate is α = λmax (DF). In the case that all transmission
rates are the same, i.e., Rkl = R (k ∈ {1, 2, ..., K}, l ∈ {1, 2, ..., dk }), the convergence
rate is
α = 2R − 1 λmax (F).
(5.32)
Using the bounds on Perron-Frobenius eigenvalue given in [HJ85], the convergence
rate can be bounded as follows
αmin < α < αmax ,
(5.33)
where
αmin
αmax
2
dj ul ∗ H vd K X
k
X
kj j = (2R − 1) min min
2 − 1,
∗
k
l
l
j=1 d=1 ul
H
v
kk k k
2
dj ul ∗ H vd K
XX
kj j k
= (2R − 1) max max
2 − 1.
∗
k
l
l
j=1 d=1 ul
H
v
kk
k
k
(5.34)
It can be seen that by decreasing the transmission rate the convergence rate exponentially decreases and consequently the sequence converges faster.
This result can be used to find the minimum number of iterations required
to have the calculated powers sufficiently close to the solution of power control
problem (p∗ ). It can be shown that if T ≥ log(ǫ)/ log(λmax (DF)), where ǫ < 1 is a
constant, then kp[T ] − p∗ kv∞ /kp[0] − p∗ kv∞ ≤ ǫ. Therefore, the minimum number
of required iterations (Tmin ) to have kp[Tmin ] − p∗ kv∞ /kp[0] − p∗ kv∞ ≤ ǫ is
Tmin = log(ǫ)/ log 2R − 1 λmax (F) .
(5.35)
Clearly, Tmin decreases by decreasing transmission rate R.
Iterative Transceiver Design and Power Control
The presented power control scheme requires iterative computations to be performed during the power updating phase of each training slot. In each iteration,
5.2. Transceiver Design and Power Control
115
Algorithm 2 Transceiver Design and Power Control
Initialize: Vk (0), plk (0) = pmax (k ∈ {1, ..., K}, l ∈ {1, 2, ..., dk }).
n=1
repeat
Update receiver filtering matrix:
ulk (n) =
(Qlk (n))−1 Hkk vlk (n − 1)
k(Qlk (n))−1 Hkk vlk (n − 1)k2
where Qlk (n) given in (5.5) is obtained in the forward training phase.
Update transmission power:

 l
∗

 2Rk − 1
ulk (n) Qlk (n)ulk (n)
.
,
p
plk (n) = min
max
∗


| ulk (n) Hkk vlk (n − 1)|2
Set reverse beamforming matrix: Vrk (n − 1) = Uk (n).
Update transmitter beamforming matrix:
ur,l
k (n) =
−1 r
(Qr,l
Hkk vr,l
k (n))
k (n − 1)
−1 Hr vr,l (n − 1)k
k(Qr,l
2
k (n))
kk k
,
where Qr,l
k (n) given in (5.10) is obtained in the reverse training phase.
Set beamforming matrix: Vk (n) = Urk (n).
Increase the training slot index n = n + 1.
until n = N
updated interference-plus-noise covariance matrices, effective desired channel matrices, reverse interference-plus-noise covariance matrices, and effective reverse desired channel matrices need to be acquired, in order to update power. This can
increase the complexity of the transmission scheme. Nevertheless, this scheme can
be simplified such that during the power updating phase, each source updates its
transmission powers only once (the number of iterations set to be one). Although we
do not prove the convergence of the new approach, it can be confirmed by extensive
numerical evaluations, as shown in Section 5.3.
Now, we present the whole process of the iterative transceiver design and power
control in each training slot. As mentioned above, there are five phases. The first
two and the last two phases are dedicated for distributed design of transmitter
beamforming and receiver filtering, and the third one is used to find the minimum
powers to support successful communication. The complete procedure is conducted
in an iterative fashion across the N training slots. Two algorithms are proposed,
shown in Algorithm 2 and Algorithm 3. Both of them set the iteration number
116
K-user MIMO Interference Networks: Transceiver Design and Power Control
Algorithm 3 Transceiver Design and Power Control: one-bit feedback signal for
each power value
Initialize: Vk (0), plk (0) = pmax (k ∈ {1, ..., K}, l ∈ {1, 2, ..., dk }).
Set 0 < γ < 1, n = 1.
repeat
Update receiver filtering matrix:
ulk (n) =
(Qlk (n))−1 Hkk vlk (n − 1)
,
k(Qlk (n))−1 Hkk vlk (n − 1)k2
where Qlk (n) given in (5.5) is obtained in the forward training phase.
Update transmission
l power:
∗
R
2 k −1 ((ulk (n)) Qlk (n)ulk (n))
l
if pk (n − 1) <
, then
|(ulk (n))∗ Hkk vlk (n−1)
|2
plk (n) = min γ −1 plk (n − 1), pmax
else
plk (n) = min γplk (n − 1), pmax
end if
Set the reverse beamforming matrix: Vrk (n − 1) = Uk (n).
Update transmitter beamforming matrix:
ur,l
k (n) =
−1
(Qr,l
Hkk vlk (n − 1)
k (n))
−1 H vl (n − 1)k
k(Qr,l
kk k
2
k (n))
,
where Qr,l
k (n) given in (5.10) is obtained in the reverse training phase.
Set the beamforming matrix: Vk (n) = Urk (n).
Increase the training slot index: n = n + 1.
until n = N
of power control in each power updating phase as one. The difference between them
lies in different requirements regarding the feedback channels which are used to send
the computed power values from each destination to the corresponding source. In
Algorithm 2, during a training slot each destination first updates its receiver filtering matrix and computes the required transmission powers of the corresponding
source. Next, it feeds the values of the computed powers back to the corresponding
source. Each source updates its transmission powers accordingly. This algorithm requires a perfect feedback channel from each destination to the corresponding source.
A simplified updating procedure is proposed in Algorithm 3, and requires only a
one-bit feedback signal corresponding to each data stream. Thus, each destination
informs the corresponding source to either increase or decrease the transmission
powers. The updated power is the scaled version of the power in the previous training slot. Depending on the value of feedback signal, the scaling factor is either γ
5.2. Transceiver Design and Power Control
117
or γ −1 . The design parameter γ (0 < γ < 1) controls the convergence speed of
the algorithm and the accuracy of the computed solution. On one hand, decreasing this parameter increases the convergence speed of the algorithm. On the other
hand, decreasing this parameter to small values may lead to large fluctuations in
the computed powers and consequently large fluctuations of mutual information
around the desired rates in the steady state (i.e. when the training slot index n is
large).
Remark 5.2. Although in this chapter the number of streams delivered between
the sources and destinations is selected to be the maximal DoF value of the networks, the system model is not limited by this assumption. The proposed algorithms
can be readily applied to scenarios in which the number of streams is smaller than
the available DoF (i.e. any value within the achievable DoF region).
Complexity Analysis
To investigate the complexity of the proposed algorithms, we follow reference [SYY12]
and use the number of complex multiplication operations as the complexity criterion to evaluate the proposed algorithms. For the sake of simplicity we consider
D
S
S
a symmetric network where dk = d, nD
k = n , and nk = n , ∀k ∈ {1, ..., K}. In
other words, d codeword streams are delivered between each source-destination pair.
The sources have the same number of nS antennas and the destinations have the
same number of nD antennas. During the nth training slot, the main computation
operations that each destination performs are listed in the following:
• The destination Dk needs to compute Qlk (n)
q for each l ∈ {1, ..., d}. AsD
sume that Dk estimates the n × 1 matrix plj (n − 1)Hkj vlj (n − 1) using
the method presented in Section III.A. Then, it can compute matrix
∗
∗
plj (n − 1)Hkj vlj (n − 1) vlj (n − 1) (Hkj )
(5.36)
2
by a matrix multiplication of nD ×1 and 1×nD matrices, with nD complexvalued multiplications. The interference-plus-noise covariance matrix corresponding to the lth stream can be computed as follows
∗
∗
Qlk (n) = Qk (n) − plk (n − 1)Hkk vlk (n − 1) vlk (n − 1) (Hkk ) , (5.37)
where
Qk (n) ,
K X
d
X
j=1 l=1
∗
∗
plj (n − 1)Hkj vlj (n − 1) vlj (n − 1) (Hkj ) + N0 InD . (5.38)
Therefore, the total complexity (i.e. the number of multiplications) of computing the interference-plus-noise covariance matrices at each destination is
2
equal to that required for the computation of Qk (n), which is Kd nD .
118
K-user MIMO Interference Networks: Transceiver Design and Power Control
• To compute the receiver vector ulk (n) corresponding to the lth stream, the des−1
tination needs to perform multiplication of the nD ×nD matrix (Qk (n)) and
2
the nD ×1 matrix Hkk vlk (n−1) with nD multiplication operations, one ma
3
trix inverse of complexity nD and one norm computation
of nD multiplica
2
3
tion operations. Each destination needs to perform d nD + nD + nD
complex-valued multiplications.
• To compute power corresponding to each stream using the computed nD × nD
interference-plus-noise covariance matrix Qlk (n) and the computed nD × 1
∗
receiver filter ulk (n), the destination Dk can compute ulk (n) Qlk (n)ulk (n)
2
by performing nD + nD multiplication operations. Using the estimated
nD × 1 effective desired channel matrix Hkk vlk (n − 1), and the computed
ul (n) ∗ Hkk vl (n −
nD × 1 receiver filter ulk (n), the receiver can
compute
k
k
2
2
1) using nD + 1 multiplications. In total d nD + 2nD + 1 multiplication
operations is needed to compute power values at each destination.
3
2
In total, d nD + (K + 2) nD + 3nD + 1 complex-valued multiplication
are performed at each destination. Similarly, using the fact that there is no power
control in the reverse training phase it can be shown that the totalnumber of multi3
2
plications needed at each source is d nS + (K + 1) nS + nS . Comparing the
number of multiplications needed for power updating with the total multiplication
required for each source-destination link, it is clear that the power control has low
overhead to the complexity of the transmitter and receiver design.
5.3
Performance Evaluation
In this section, we numerically evaluate the performance of Algorithm 2 and
Algorithm 3. We also provide simulation results for TDMA to compare its performance with that of the proposed algorithms. In the orthogonal scheme, the
sources are individually activated to avoid interference. Consequently, the transmitter beamforming matrix design and the receiver filtering matrix design for each
source-destination pair is simplified to that for point-to-point MIMO systems. Thus,
the solution of the power control algorithm for each interference-free channel can be
computed by the water-filling approach. We consider two representative examples
to evaluate the performance of the proposed algorithms.
In the first example, we consider a three-user network in which each terminal
is equipped with two antennas. When the orthogonal transmission is performed,
only one source is active at a time, and the active source transmits two independent data streams. However, it has been known that the degrees of freedom
(d1 , d2 , d3 ) = (1, 1, 1) is achievable in this network [CJ08]. Therefore, in our proposed algorithms, all sources are concurrently activated while each of them trans-
5.3. Performance Evaluation
119
4
bC
rS
bC
bC
3.5
rS
bC
Mutual Information
rSrS
bC
rS
bC
3
user 3
bC
rSbC
bC
bC
bC
bC
rS
rSbC
rS
rSbC
rSbC
rSbC
rS
bC
rS
2.5
bC
rS
rS
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
bC
rS
user 2
bC
rS
bC
2
rSbC
bC
rS
rSbC
rS
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rS
bC
1.5
rS
user 1
bC
rS
1
0.5
2
4
6
rS
bC
rS
8
10
rSbC
12
rSbC
14
16
18
20
Number of training slots
Figure 5.5: Mutual information of source-destination pair Sk − Dk (Ik ) versus the
number of training slots using Algorithm 2 (−) and Algorithm 3 with γ = 0.5
(◦) and γ = 0.4 ().
mits one data stream. The rates are fixed as (R11 , R21 , R31 ) = (1, 2, 3) [bits/channel
use], and plk (0)/N0 = pmax /N0 = 20 [dB]. The number of different channel realizations is set to 108 , and we evaluate Algorithm 3 for γ = 0.5 and γ = 0.4. Figure 5.5
displays the mutual information of each of the three source-destination pairs averaged over channel realizations. It can be seen that, using either of the proposed
algorithms, the mutual information of each pair converges to the corresponding desired transmission rate as the number of training slots increases. Figure 5.6 shows
the average computed powers in each training slot. We can see that by increasing
the number of training slots computed power values by these algorithms converge.
The computed power values by Algorithm 2 converge faster compared to the ones
computed by Algorithm 3. The computed powers (relative to the noise power N0 )
for sources S1 , S2 , and S3 using TDMA scheme are 6.2 dB, 12 dB, and 19 dB, respectively. The proposed algorithms compared to TDMA converge to substantially
lower power for each user. These two figures illustrate how the proposed algorithms
reduce transmission powers to the minimum required levels while successful transmissions at given rates are still possible for all source-destination pairs. Also, it
can be seen that in Algorithm 3, by decreasing the parameter γ the algorithm
120
K-user MIMO Interference Networks: Transceiver Design and Power Control
18
bC
16
rS
bC
bC
rS
bC
14
rS
bC
rS
bC
pk /N0 [dB]
12
bC
rS
bC
user 3
rS
rS
bC
bC
10
bC
rS
rS
rS
bC
bC
rS
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rS
rS
bC
rSbC
rS
4
rSbC
rSbC
rSbC
rSbC
rSbC
rS
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
bCrS
bCrS
bCrS
bCrS
rSbC
bC
bC
user 1
rS
bC
2
rS
0
rSbC
rSbC
user 2
bC
rSbC
6
rSbC
rS
8
bC
2
4
6
rS
bC
rS
8
rSbC
rSbC
10
rSbC
rSbC
12
rSbC
rSbC
14
rSbC
16
18
20
Number of training slots
Figure 5.6: Computed pk /N0 versus the number of training slots using Algorithm 2
(−) and Algorithm 3 with γ = 0.5 (◦) and γ = 0.4 ().
converges faster. However, very small values of γ may lead to large fluctuations in
the computed powers and corresponding mutual information in the steady state
(large number of training slots). This shows a trade-off between convergence speed
of Algorithm 3 and accuracy of the computed solutions.
Figure 5.7 and Figure 5.8 show simulation results of Algorithm 2 for similar set
up as the one in the previous example. Computed mutual information and computed
power versus the number of training slots are shown for different values of pmax /N0
(pmax /N0 ∈ {15 dB, 20 dB, 30 dB}). It can be seen that in each of these different
settings, mutual information values converge to the desired rates, and the computed
power of each user converges to a certain value. This shows that when the initial
power is set to a value larger than the one required for reliable communication, the
algorithm reduces power to the required level. However, further simulations with
smaller values of pmax /N0 show that the mutual information may not converge
to the desired rates. This coincides with the observation in Figure 5.4 and is a
consequence of the fact that in the presence of power constraint the power control
problem may not be always feasible.
In the second example, we evaluate the performance of the proposed algorithms
in a three-user network in which each terminal is equipped with four antennas.
5.3. Performance Evaluation
121
4.5
bC
bC
Mutual information [bits/channel use]
4
user 3
rS
3.5
bC
rS
bC
3*
bC
rS
*bC
*
bC
bC
rS
rS
*
*
rS
*
rSbC
*
rSbC
*
rSbC
*
rSbC
*
rSbC
*
rSbC
*
rSbC
rSbC
*
*
rSbC
*
rSbC
*
rSbC
*
rSbC
*
rSbC
*
rSbC
*
rS
2.5
bC
2
user 2
rS
bC
*
rS
bC
*
*
*
rS
bC
rS
*
bC
1.5
0.5
rSbC
*
rSbC
*
rSbC
*
rSbC
*
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
rSbC
*
*
*
*
*
*
*
*
*
*
bC
*rS
bC
*rSbC
*rS
bC
*rSbC
*rSbC
*rSbC
*rSbC
*rSbC
*rSbC
rS
*
1
rSbC
*
user 1
rS
*
2
bC
rS
*
bC
rS
*
4
rSbC
*
rSbC
*
6
rSbC
*
bC
*rS
8
bC
*rS
bC
*rS
10
*rS
12
14
16
18
20
Number of training slots
Figure 5.7: Mutual information of source-destination pair Sk − Dk (Ik ) versus
the number of training slots using Algorithm 2 for different values of pmax /N0 :
pmax /N0 = 30 dB (◦), pmax /N0 = 20 dB (), pmax /N0 = 15 dB (∗).
Since the degrees of freedom (d1 , d2 , d3 ) = (2, 2, 2) are achievable [CJ08], each
source transmits two independent data streams. We consider a scenario that each
user encodes transmitted data streams with different transmission rates. This represents communication scenarios in which each user desires to send different type of
data while the transmission of each of them should satisfy a specific level of quality of service. In this particular example, (R11 , R12 , R21 , R22 , R31 , R32 ) = (1, 2, 3, 4, 5, 6)
[bits/channel use], and plk (0)/N0 = pmax /N0 = 20 [dB]. The number of different
channel realizations is set to 108 .
Figure 5.9 shows the mutual information corresponding to each data stream of
the source S1 averaged over different channel realizations. It can be seen that the
mutual information of each stream converges to the transmission rate. Therefore,
each of the streams can be reliably transmitted. Figure 5.10 shows the computed
power associated to each stream of S1 . The powers computed according to Algorithm 3 converge to approximately similar values as the ones by Algorithm 2.
These numerical results indicate that the algorithms converge to the desired solution. It can be observed that each computed power compared to the corresponding
mutual information in Figure 5.9 has a slower convergence. The reason is that
122
K-user MIMO Interference Networks: Transceiver Design and Power Control
35
30
bC
25
pk /N0 [dB]
bC
20
bC
user 3
bC
rS
15
*
5
user 2
rS
*
rS
10 *
bC
bC
*bCrS
rS
user 1
rS
*
rS
*
bC
*bC
*bC
rS
0
2
bC
bC
bC
bC
bC
bC
bC
bC
bC
bC
bC
bC
bC
bC
bC
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
rS
rS
*
bC
*
rS
*bC
rS
rS
rS
rS
rS
bC
*
rS
bC
*
rS
bC
*
rS
bC
*
*
bC
rS
bC
*
*
4
rS
rSbC
*
6
rSbC
*
rS
bCrS
rS
rSbC
*
8
rS
*
rSbC
*
bCrS
*
rSbC
*
10
rS
bCrS
*
bCrS
*
rS
bCrS
*
bCrS
*
12
rS
bCrS
rS
bCrS
*
*
bCrS
bCrS
*
*
14
rS
bCrS
*
bCrS
*
rS
rSbC
*
rSbC
*
16
rS
rSbC
*
rSbC
*
rS
rSbC
*
rSbC
*
18
rS
bC
rSbC
*
rSbC
*
rS
*
rSbC
*
20
Number of training slots
Figure 5.8: Computed pk /N0 versus the number of training slots using Algorithm 3
for different values of pmax /N0 : pmax /N0 = 30 dB (◦), pmax /N0 = 20 dB (),
pmax /N0 = 15 dB (∗).
multiple sets of power values can provide the desired mutual information for each
stream. However, only one of them is corresponding to the lowest transmission
power. Therefore, mutual information may converges to the desired value while
transmission power values are still larger than the possible lowest values.
Figure 5.11 compares the computed power values in two different scenarios: in
the first scenario the number of transmitted streams is set according to the DoF of
the network, while in the second scenario the number of transmitted streams is set to
a number smaller than the DoF of the network. Specifically, we consider a three-user
interference channel in which each terminal has four antennas. In the first scenario,
each source sends two streams while in the second scenario each source transmits a
single stream. The computed transmission power values for each stream are shown.
This result shows that the algorithm converges even if the number of transmitted
streams is set to a smaller value compared to the DoF of the network. It can be seen
that by sacrificing the number of streams that are sent over the network the powers
needed for transmission of certain number of streams can be reduced. The reason is
that reducing the number of transmitted streams decreases the amount of leakage
interference, thus, lower powers would be sufficient for successful communication.
5.4. Test-bed Implementation
123
Mutual Information [bits/channel use]
7
Algorithm
Algorithm
Algorithm
Algorithm
6
3,
2,
3,
3,
I11
I12
I11
I12
5
4
3
2
1
0
0
10
20
30
40
50
60
70
80
90
100
Number of training slots
Figure 5.9: The mutual information of the lth stream l ∈ {1, 2} of the sourcedestination pair S1 − D1 (I1l ) versus the number of iterations. The mutual information values of the other source-destination pairs similarly converge to the corresponding transmission rates.
5.4
Test-bed Implementation
Wireless test-beds (e.g. the ones based on USRP or WARP hardware) are powerful tools for the experimental verifications of novel interference management algorithms. In recent years, several experimental validations of the algorithms inspired by the interference alignment concept have been reported in the literature [APH10,GRS+ 11,ZM12,MSS+ 12,MAH+ 13,Zet14,MFZS14,MFZ+ 14]. We implemented the proposed algorithm in the previous sections on KTH four-multi testbed and verified its performance in realtime measurements [MFZS14]. KTH fourmulti is a USRP-based wireless communication test-bed consisting of three stationary terminals and three movable terminals, where each terminal is equipped with
two antennas [Zet, ZM12]. The channel state feedback links are provided through
separate Ethernet links. A three-level synchronization scheme is also applied to
synchronize the signals in the time and frequency level.
We performed measurements in an indoor office environment at KTH. The
measurements indicate at least 4 dB reduction in transmission power in 90% of
124
K-user MIMO Interference Networks: Transceiver Design and Power Control
20
Algorithm
Algorithm
Algorithm
Algorithm
18
16
2,
2,
3,
3,
p11
p21
p11
p21
pl1 /N0 [dB]
14
12
10
8
6
4
2
0
10
20
30
40
50
60
70
80
90
100
Number of training slots
Figure 5.10: The power of the lth stream (l ∈ {1, 2}) of source S1 (pl1 ) versus the
number of iterations. The computed power values of the other sources also show
similar convergence behavior.
the experiments and at the same time a better bit-error-rate (BER) performance
compared to the case where MaxSINR algorithm in [GCJ11] with no power control
was implemented. The power saving gains as high as 13 dB was also observed in
10% of the measurements.
The benefits of performing power control in the proposed transceiver design
and power control algorithm is in fact two-fold. By decreasing the transmission
power, while retaining the target SINR, not only less interference is received at each
destination, but also the distortion noise due to transceiver hardware impairments
(e.g. distortions due to the power amplifier nonlinearities) decreases.
5.5
Summary
In this chapter we have addressed transceiver design and power control for MIMO
interference networks to provide reliable communication at given rates. Each source
intends to send possibly multiple independent data streams where the number of
the streams is selected according to the DoF of the network. Each stream is encoded
with a fixed data rate while different streams transmitted by one source can possibly
5.5. Summary
125
20
first scenario, p11
first scenario, p12
first scenario, p13
second scenario, p11
second scenario, p12
second scenario, p13
15
p1k /N0 [dB]
10
5
0
−5
−10
0
10
20
30
40
50
60
70
80
90
100
Number of training slots
Figure 5.11: Computed p1k /N0 versus the number of training slots when each terminal has four antennas. In the first scenario each terminal sends two streams, and
in the second scenario each terminal sends only a single stream.
have different rates. We have proposed two distributed iterative algorithms which
require only local CSI at each terminal. The transmitter beamforming matrices and
the receiver filtering matrices are designed such that the SINR corresponding to
each received data stream can be maximized. In addition, power control is performed to assign the minimum required power to each data stream while keeping
the mutual information of the corresponding stream larger than the transmission
rate. We have investigated the convergence behaviour of the power control scheme,
and provided a condition that needs to be fulfilled to have feasible solutions. Numerical performance evaluations confirmed the convergence behavior and that the
proposed algorithms required substantially lower powers compared to the conventional orthogonal transmission schemes. Experimental verifications of the proposed
algorithm on KTH wireless test-bed confirmed a considerable performance gain
compared to the transceiver design algorithms without power control.
126
5.A
K-user MIMO Interference Networks: Transceiver Design and Power Control
The Proof of Theorem 5.2.1
Since it is assumed that pmax → ∞, we need to show that there is a power vector
p satisfying p DFp + n if and only if λmax (DF) < 1. The proof is inspired by
the analysis in [HC00a] addressing the feasibility of power control in uplink transmission of the cellular networks. For K > 2, the matrix DF is a regular matrix (a
stochastic matrix in which all the entries of some power of the matrix are positive
is called regular matrix) because all the elements of (DF)2 are positive. Therefore,
the Perron-Frobenius Theorem [PSS05] guarantees the existence of a positive eigenvalue λmax (DF) and the corresponding positive right and left eigenvectors pr and
pl which satisfy λmax (DF)pr = DFpr and λmax (DF)pTl = pTl DF, respectively.
To prove the necessary condition, assuming p1 ∈ PH we have
p1 ∈ PH
(a)
⇒
p1 DFp1 + n
⇒
p1 − DFp1 ≻ 0
(b)
(c)
⇒
pTl (p1 − DFp1 ) > 0
⇒
(1 − λmax (DF))pTl p1 > 0
⇒
λmax (DF) < 1,
(d)
(e)
(5.39)
where (a) follows from the definition in (5.26); (b) follows from the fact that n ≻ 0;
(c) follows from the positivity of the left eigenvector (pl ≻ 0); (d) follows from
the characteristic of eigenvectors and (e) follows from the fact that pl ≻ 0 and
p1 ≻ 0, and consequently pTl p1 > 0. To prove the sufficient condition, assuming
λmax (DF) < 1 we have
(a)
λmax (DF)pr = DFpr ⇒ p2 = DFp2 + n + a(1 − λmax (DF))DFpr − n
(b)
⇒ p2 DFp2 + n
(c)
⇒ p2 ∈ PH
⇒ PH 6= ∅,
(5.40)
where (a) holds if we set p2 as p2 = aλmax (DF)pr ; (b) holds if we set a =
n(i,1)
maxi (1−λmax (DF))(DF)(i,:)p
where n(i, 1) is the ith element of n and (DF)(i, :) is
r
the ith row of DF, then since λmax (DF) < 1 we have a(1 − λmax (DF))DFpr − n 0;
(c) follows from the definition in (5.26) and the fact that pmax → ∞.
5.B
The Proof of Theorem 5.2.2
According to Theorem 2 in [Yat95], if there is at least one positive vector p (p ≻ 0)
that satisfies p I′ (p) and the function I′ (p) is a standard interference function,
then for any initial power vector p[0] the sequence generated by the recursive equation p[i] = I′ (p[i − 1]) converges to a unique fixed-point p∗ . A function I′ (p) is
5.B. The Proof of Theorem 5.2.2
127
called a standard interference function if for all p, p′ 0, it satisfies the following
conditions [Yat95]:
1) Positivity: I′ (p) ≻ 0;
2) Monotonicity: I′ (p) I′ (p′ ), ∀p p′ ;
3) Scalability: αI′ (p) ≻ I′ (αp), ∀α > 1.
First, we prove that the function I(p) introduced in (5.18) is a standard interference function by showing that it satisfies the above three conditions. To simplify
this verification, we rewrite Ikl (p) as
Ikl (p) = L · ϕlk (p) + N0 ,
(5.41)
where L =
Rl
2 k −1
|(ulk )∗ Hkk vlk |2
> 0 is a constant.
1) To check the positivity condition we have:
(a)
Ikl (p) = L · ϕlk (p) + N0 ≥ LN0 > 0,
(5.42)
where (a) follows from the fact that ϕlk (p) ≥ 0. Therefore, the positivity condition
is satisfied.
2) To verify the monotonicity condition, we assume p p′ :
2
2
∗
∗
(a)
l p p′ ⇒ plj ulk Hkj vlj ≥ p′ j ulk Hkj vlj , ∀l, k, j
(b)
⇒ ϕlk (p) ≥ ϕlk (p′ ), ∀l, k
(c)
⇒ Ikl (p) ≥ Ikl (p′ ), ∀k, l
⇒ I(p) I(p′ ),
(5.43)
2
where (a) follows from the fact that (ulk )∗ Hkj vlj ≥ 0; (b) follows from the definition of ϕlk (p) in (5.14); (c) follows from the definition in (5.17) and the fact that
l ∗
(u ) Hkk vl 2 > 0. Therefore, the monotonicity condition is satisfied.
k
k
3) To check the scalability condition we assume α > 1, then we have
Ikl (αp) = L · ϕlk (αp) + N0
= L · αϕlk (p) + N0
(5.44)
< αL · ϕlk (p) + N0 = αIlk (p).
This condition is valid for any k ∈ {1, 2, ..., K}. Thus, the scalability condition
is also satisfied. Now, we can conclude that I(p) given in (5.18) is a standard
interference function. Based on Theorem 7 in [Yat95], if I(p) is a standard inh
iT
terference function, then ˆI(p) = Iˆ1 (p), ..., Iˆd1 (p), ..., Iˆ1 (p), ..., IˆdK (p) , where
1
1
K
K
128
K-user MIMO Interference Networks: Transceiver Design and Power Control
Iˆkl (p) = min{Ikl (p), pmax }, is also a standard interference function. According to
Theorem 2 in [Yat95], if the problem is feasible (PH 6= ∅), for any initial power
vector p[0] the iterative algorithm in (5.27) converges to a unique fixed-point p∗ .
Lemma 1 in [Yat95] implies that this fixed-point is the solution with the minimum
required transmission power for each source.
Chapter 6
Multi-cell Interference Networks:
Pilot-assisted Opportunistic User Scheduling
M
ULTI-CELL interference networks are mathematical models for cellular
communication systems. These networks consist of multiple cells each of
which has one base station that communicates to multiple mobile terminals. The transmitted signals of each base station may interfere to the reception
at the mobile terminals within neighboring cells. The number of mobile terminals
in cellular systems has been rapidly increasing, and it has been predicted that
ultra-dense networks is one of likely scenarios in the next generations of wireless
systems (e.g. 5G). The impact of increasing the number of mobile terminals on
future networks is twofold: in one hand, coordinated transmission becomes even
more important for dealing with interference; and on the other hand, acquiring CSI
at mobile terminals and base stations becomes even more difficult to handle and
may cause a huge overhead to the system. Therefore, low-complexity coordinated
transmission based on low-overhead CSI acquisition schemes must be developed. In
order to obtain CSI and to perform data transmissions, we propose a pilot-assisted
opportunistic user scheduling (PAOUS) scheme. The proposed scheme consists of
a low-complexity channel training mechanism to acquire local CSI at mobile terminals, and a one-bit feedback scheme to enable scheduling at base stations. We
compute the achievable rate region for the proposed scheme and characterize the
achievable DoF region.
The structure of this chapter is as follows. We present the multi-cell interference
network in Section 6.1. In Section 6.2, we propose the pilot-assisted opportunistic
user scheduling scheme. The achievable rate region and the achievable degrees of
freedom region are characterized in Section 6.3. We provide numerical evaluations
of the proposed scheme in Section 6.3. Finally, Section 6.4 summarizes this chapter.
129
130
Multi-cell Interference Networks: Pilot-assisted Opportunistic User Scheduling
Figure 6.1: Schematic representation of different phases of the PAOUS scheme: (a)
orthogonal pilot transmission; (b) feedback transmission and user scheduling; and
(c) data transmission phase.
6.1. Multi-cell Interference Network
6.1
131
Multi-cell Interference Network
Consider a downlink transmission in a cellular network of K neighbouring cells.
In each cell, there is one single-antenna base station serving N single-antenna mobile terminals. The base station in the kth cell (k ∈ {1, . . . , K}) is denoted as
BSk , and the mobile terminals are shown as MSkj (j ∈ {1, . . . , N }). Each base
station intends to transmit independent messages to mobile terminals in the associated cell. The channel gain between BSk and MSpj (p ∈ {1, . . . , K}) is denoted
as hpj,k . We consider the block-fading channel model with coherence time T , where
channel gains are constant within one fading block, i.e. hpj,k (nT + t) = hpj,k (nT )
(t ∈ {1, . . . , T − 1}, n ∈ {1, 2, ...}), and change to independent values across subsequent blocks. Channel gains have zero-mean complex Gaussian distribution, i.e.
hpj,k ∼ CN (0, 1), and are mutually independent across different users and cells. We
assume that no a priori CSI is available at mobile terminals and base stations.
6.2
Pilot-assisted Opportunistic User Scheduling Scheme
The proposed PAOUS scheme at each fading block is conducted in three subsequent phases: pilot transmission, feedback transmission and user scheduling, and
data transmission phase as shown in Figure 6.1. Within each fading block, αT
time slots are allocated to pilot transmission phase where each base station sends
Tτ = αT /K pilot symbols. The remaining Td = (1 − α)T time slots are left for data
transmission phase. The parameter α (0 < α < 1) is the channel sharing factor.
A larger α implies that more channel uses are allocated to pilot transmission. The
transmission power of pilot symbols is Pτ and the one for data symbols is Pd . These
can be different in general as follows
Pd
Pτ
=
=
βP,
K ((1 − (1 − α) β)/α) P
(6.1)
(6.2)
where P is the transmission power of each base station and β (0 ≤ β ≤ 1/(1 − α))
is a power allocation factor. This power allocation follows the energy conservation
law
αT Pτ /K + (1 − α)T Pd = T P.
(6.3)
Clearly, there is a trade-off between the power allocation for pilot transmission
and the one for data transmission. In the following, we explain the aforementioned
phases in more details.
6.2.1
Pilot Transmission Phase
We consider a pilot-assisted channel training scheme to acquire an estimation of
local CSI at each mobile terminal, i.e. channel gains between base stations and
132
Multi-cell Interference Networks: Pilot-assisted Opportunistic User Scheduling
the corresponding mobile terminal. Channel training is performed in an orthogonal
fashion in which the training period is divided into K equal training slots each
of which has the duration of Tτ = αT /K. Each base station transmits Tτ pilot
symbols during one training slot and remains silent during other time slots as shown
in Figure 6.2. Then, each mobile terminal in the network estimates the gain of the
corresponding link between the active base station and itself. Consider transmission
at the nth fading block. The base station BSk (k ∈ {1, . . . , K}) sends Tτ known
pilot symbols as follows
p
Xτk (t) = Pτ , t ∈ Tnk ,
(6.4)
where Tnk = {nT + (k − 1)Tτ + 1, . . . , nT + kTτ }. Consequently, the received signals at mobile terminal MSpj (p ∈ {1, . . . , K}, j ∈ {1, . . . , N }) is
p
p
Yτ,j
(t) = Pτ hpj,k (nT ) + Zjp (t), t ∈ Tnk ,
where the receiver noise Zjp (t) has Gaussian distribution, i.e. Zjp (t) ∼ CN (0, N0 ).
The mobile terminal performs an MMSE estimation of the channel gain hpj,k (nT )
as follows
˜ p (nT ) =
h
j,k
Pτ
N0 + Tτ Pτ
nTX
+kTτ
p
Yτ,j
(t).
(6.5)
t=nT +(k−1)Tτ +1
The following equation holds
hpj,k (nT ) = ˜
hpj,k (nT ) + εpj,k (nT ),
(6.6)
where εpj,k (nT ) denotes the corresponding channel estimation error. The random
˜ p (nT ) are independent zero-mean Gaussian distributed
variables εpj,k (nT ) and h
j,k
with variances σε2 and 1 − σε2 , respectively, where
σε2 =
1
.
1 + Tτ Pτ /N0
(6.7)
At the end of the training phase, mobile terminal MSpj obtains the estimation
˜ p (nT ) (k ∈ {1, ..., K}). Then, this noisy estimation of
of local channel gains, i.e. h
j,k
CSI is used to compute the feedback signal as described in the next part.
6.2.2
Feedback Transmission and User Selection Phase
Each mobile terminal computes a measure based on the estimated strength of interference links and locally makes a decision whether this measure is below a certain
threshold. Then, it sends a one-bit feedback signal to the associated base station.
Specifically, in the nth fading block, MSpj computes δjp (nT ) that is defined as follows
δjp (nT )
K
X
2
p
˜
,
hj,i (nT ) .
i=1,i6=p
(6.8)
6.2. Pilot-assisted Opportunistic User Scheduling Scheme
KTτ = αT
133
Td = (1 − α)T
Tτ
BS1
BS2
BSK
Figure 6.2: Transmitted symbols by base stations BSk (k ∈ {1, ..., K}) within one
fading block. The crosshatched red slot, the plain green slot, and the blue angle
lined slots denote no transmission, pilot symbols, and data symbols, respectively.
Next, it sends the one-bit feedback signal fjp (nT ) defined as
(
1
δjp (nT ) ≤ ǫth
p
fj (nT ) ,
,
0
δjp (nT ) > ǫth
(6.9)
where ǫth (ǫth > 0) is a design parameter. The feedback channels are orthogonal to
each other and they are assumed to be error-free.
The base station BSk collects feedback signals from all mobile terminals within
the corresponding cell, i.e. fjk (nT ) (j ∈ {1, ..., N }). A mobile terminal is called a
candidate mobile terminal to be scheduled if the corresponding feedback signal is
one. We define the set of candidate mobile terminals in the kth cell as follows
Ak , ifik (nT ) = 1, i ∈ {1, ..., N } .
(6.10)
The number of candidate mobile terminals in the kth cell is ρk = |Ak |, where
|A| is the cardinality of the set A. If ρk 6= 0, then BSk schedules a randomly
selected mobile terminal from the set of the candidate mobile terminals Ak . The
selected mobile terminal is denoted as MSkαk . Otherwise, no mobile terminal will
be scheduled. The random scheduling ensures that all mobile terminals will be
scheduled with the same probability. This implies that the proposed scheme indeed
offers fairness.
6.2.3
Data Transmission Phase
There are N buffers at each base station, and each of them stores messages that
should be sent to a specific mobile terminal. In the data transmission phase, each
base station communicates to the scheduled mobile terminal in the associated cell.
Messages are encoded according to the multiplexed coding scheme similar to the
134
Multi-cell Interference Networks: Pilot-assisted Opportunistic User Scheduling
one proposed in [GV97]. Corresponding to each mobile terminal, there are multiple
codebooks each associated with a specific channel state. For a given channel state,
BSk (k ∈ {1, 2, . . . , K}) selects
message mkαkoindependently with a uniform distrin
′
˜
˜ k > 0 is the code rate, and
bution from the set M = 1, 2, . . . , 2N Td Rk , where R
N ′ is the number of fading blocks that spans one codeword. Then, it encodes the
′
Td
k
message mkαk to a length N ′ Td codeword {Xd,α
(i)}N
i=1 . Moreover, the codewords
k
must satisfy a power constraint
h
2 i
k
< Pd .
(6.11)
E Xd,α
k
In fading block n, BSk sends
n
o(n+1)T
k
Xd,α
(i)
k
i=nT +KTτ +1
during Td data transmission
time slots. All base stations transmit simultaneously at the same frequency band.
Consequently, the channel output at MSkαk is
k
Yd,α
(i)
k
k
(i)+
= hkαk ,k (nT )Xd,α
k
K
X
l
(i) + Z k (i),
hkαk ,l (nT )Xd,α
k
l=1,l6=k
i = nT + KTτ + 1, . . . , (n + 1)T
(6.12)
where Z k (i) ∼ CN (0, 1). The mobile terminal collects all N ′ received signals, decodes the received codeword and estimates the transmitted message.
6.3
Achievable Rate Region
In this section, we study the achievable rate region of the PAOUS scheme.
Theorem 6.3.1. The pair BSk − MSki can achieve the following rate
"
!#
2
h P
β ˜
Tτ
k
k
γi Eh˜ log 1+
,
Ri = 1−K
T
1 + β (Kσε2 + ǫth ) P
(6.13)
where
1
,
(6.14)
1 + KTτ ((1 − (1 − α) β) /α) P/N0
!
N
1
ǫth
γik =
1− 1−F
N
σε2
γ K − 1, x2
,
(6.15)
F (x) ,
Γ(K − 1)
R
˜ ∼ CN 0, 1 − σ 2 . The function Γ(z) , ∞ tz−1 e−t dt is the Gamma function,
and h
0
R x z−1ε −t
and γ(z, x) , 0 t e dt is the lower incomplete Gamma function.
σε2
=
6.3. Achievable Rate Region
135
Proof. See Appendix 6.A.
Corollary 6.3.1. The achievable sum-rate of the network RΣ ,
PK PN
k=1
i=1
Rik is
"
!#
2
h P
β ˜
Tτ
RΣ =K 1−K
γEh˜ log 1+
,
T
1 + β (Kσε2 + ǫth ) P
where
N
ǫth
γ =1− 1−F
.
σε2
(6.16)
This result can be used to compute the achievable DoF of the network.
6.3.1
Achievable Total Degrees of Freedom
The achievable total DoF is characterized in the following theorem.
Theorem 6.3.2. In a multi-cell network with K base stations and coherence time
T , the achievable total DoF is
dΣ = Kopt 1 −
Kopt
T
,
(6.17)
where
Kopt = min
T
2,K
,
(6.18)
if the number of mobile terminals (N ) scales proportional to SNR.
Proof. We set ǫth = 1/P , and N ∝ P . Then, the achievable total DoF can be
computed as dΣ = limP →∞ RΣ / log P , where RΣ is given in Corollary 6.3.1. Using
the dominated convergence theorem [MW12], it can be shown that this limit is equal
to K (1 − KTτ /T ). We select Tτ = 1 to maximize the achievable total DoF. It can
be shown that when K > T2 , the number of active base stations that maximizes the
total DoF is K ′ = T2 .
To maximize the total DoF, or equivalently the network throughput at highSNR regime, in large networks (K > T /2), Theorem 6.3.2 recommends that turn
on only a subset of base stations, and perform the proposed PAOUS scheme within
the cells with active base station. Since the network is symmetric, a random base
station selection works. In addition, this theorem crystalizes the dependency of the
optimum number of active base stations on the channel coherence time.
136
Multi-cell Interference Networks: Pilot-assisted Opportunistic User Scheduling
45
uT
40
bC
rS
Sum-rate [bits/channel use]
35
N
N
N
N
=3
=5
= 10
= 100
rS
rS
rS
bC
rS
30
bC
rS
25
bC
rS
bC
20
uT
uT
bC
rS
bC
uT
uT
bC
rS
uT
uT
uT
rS
5
uT
bC
bC
rS
10
uT
bC
rS
uT
bC
rS
15
uT
bC
rS
uT
bC
uT
0
10
20
30
40
50
60
70
SNR [dB]
Figure 6.3: Sum-rate versus power for different number of mobile terminals in each
cell (N ).
6.3.2
Numerical Evaluation
In this section, we numerically evaluate the performance of the PAOUS scheme
in a three-cell network (K = 3). Figure 6.3 shows the achievable sum-rate versus
SNR for different number of mobile terminals in each cell (N ). It can be seen that
the sum-rate increases as N increases. The reason is that as the number of mobile
terminals increases, it is more likely that the set of candidate mobile terminals in
(6.10) be nonempty and mobile terminals be scheduled.
Figure 6.4 illustrates the achievable sum-rate as a function of threshold ǫth defined in (6.9) for different SNR values. It can be observed that, for a given SNR, a
specific ǫth maximizes sum-rate. The optimum ǫth decays as SNR increases. Increasing ǫth , on one hand increases the probability that a mobile terminal is scheduled
in each cell, but on the other hand, the corresponding mutual information decays
as a consequence of a larger interference. By increasing SNR interference becomes
dominant and ǫth should be reduced in order to limit the level of interference and
increase the achievable rate.
Figure 6.5 shows the achievable sum-rate versus β for different values of N .
It can be observed that for each value of N a specific β maximizes the sum-rate.
6.4. Summary
137
25
SNR = 10
SNR = 20
SNR = 30
SNR = 40
Sum-rate [bits/channel use]
20
dB
dB
dB
dB
15
10
5
0
−60
−50
−40
−30
−20
−10
0
10
20
ǫth [dB]
Figure 6.4: The achievable sum-rate versus ǫth for different SNR values.
The optimum value of β increases as N increases. This implies that when there
are large mobile terminals in the network more power should be allocated to data
transmission instead of pilot transmission.
6.4
Summary
In this chapter, we have investigated a typical scenario in 5G communication systems, where a large number of mobile terminals in a multi-cell network have to be
served efficiently (e.g. with low training and feedback overhead), when no a priori
CSI is available neither at mobile terminals nor at base stations. We proposed the
PAOUS scheme, and we have shown that the proposed scheme is well-suited for such
scenarios: it offers low-computational complexity, and requires only a one-bit feedback signal from mobile terminals to their respective base stations. Furthermore, we
computed the achievable rate region for the proposed scheme. We have illustrated
that the achievable sum-rate scales as the number of mobile terminals in each cell
increases. Our results reveal that in a multi-cell network with K base stations,
given that the number of mobile terminals properly scales with the SNR, the sum
degrees of freedom Kopt (1 − Kopt /T ) is achievable, where Kopt = min {K, T /2} is
the optimum number of the base stations that need to be activated in the network.
138
Multi-cell Interference Networks: Pilot-assisted Opportunistic User Scheduling
12
Sum-rate [bits/channel use]
10
8
6
4
N
N
N
N
2
0
0
0.2
0.4
=5
= 10
= 50
= 100
0.6
0.8
1
1.2
1.4
β
Figure 6.5: Sum-rate versus β for different number of users in each cell (N ).
Finally, performance evaluations confirm that the proposed opportunistic transmission scheme can exploit the multi-user diversity to mitigate interference even
with single-antenna terminals at mobile terminals and base stations, and thereby
enhance the achievable sum-rate.
6.A. The Proof of Theorem 6.3.1
6.A
139
The Proof of Theorem 6.3.1
Assuming that MSkαk is scheduled, the mutual information between BSk and the
selected mobile terminal MSkαk can be lower bounded as
=
=
=
(a)
≥
k
k ˜ k
˜k
hαk ,1 , . . . , h
I Xd,α
; Yd,α
αk ,K
k
k
k
k
k
k
k
˜k
˜k
˜
˜
hαk ,1 , . . . , h
h Xd,α
αk ,K − h Xd,αk hαk ,1 , . . . , hαk ,K , Yd,αk
k
k
k
k
k
k
˜k
ˆ d,α
˜
h Xd,α
,
.
.
.
,
h
,
Y
h
−
h
X
−
X
α
,K
d,α
α
,1
d,α
k
k
k
k
k
k
k
k
k
k
k
˜
ˆ
˜
log (2πePd ) − h Xd,αk−X
d,αk hαk ,1 , . . . , hαk ,K , Yd,αk
log (2πePd ) − log 2πeσ 2 ,
(6.19)
k
˜k , . . . , h
˜k
ˆk , f h
where X
is a function of the received signal and
αk ,1
αk ,K , Yd,αk
d,αk
k
ˆk
the estimated local CSI, and σ 2 is the variance of Xd,α
−
X
d,αk . In this equation
k
(a) follows from the fact that the entropy of a random variable with a given variance
is upper bounded with the entropy of a Gaussian distributed random variable. We
ˆ k to be the MMSE estimate of X k as follows
select X
d,αk
d,αk
ˆk
X
d
=
=
∗ i
h
˜ k
˜k
ˆk Y k
E X
h
,
.
.
.
,
h
α
,1
αk ,K
d,αk
d
k
k
∗ h
i Yd,α
k
˜ k
k
k
k
˜
E Yd,αk Yd,αk hαk ,1 , . . . , h
αk ,K
∗
k
˜k
h
Yd,α
Pd
αk ,k
k
.
2
1 + (Kσε + ǫth ) Pd
(6.20)
Therefore, the variance σ 2 in (6.19) is
σ2
=
(a)
=
=
i
∗ ˜k , . . . , h
˜k
ˆ k h
ˆ k Xk − X
Xdk − X
αk ,1
αk ,K
d
d
d
i
h
∗ ˜k
ˆ dk ˜
hkαk ,1 , . . . , h
E Xdk Xdk − X
αk ,K
E
h
1+
P
d 2
h˜ kα ,k Pd
,
(6.21)
k
1+(Kσε2 +ǫth )Pd
where (a) follows the orthogonality principle. Substituting the computed variance
in (6.19), the lower bound on the mutual information can be computed. In addition,
the probability that the mobile terminal MSki is scheduled is γik = N1 γ k , where γ k is
the probability that one mobile terminal is scheduled in the kth cell. The probability
140
Multi-cell Interference Networks: Pilot-assisted Opportunistic User Scheduling
γ k can be computed as
(
k
γ = Pr
= 1−
N
[
k
δi < ǫth
i=1
N
Y
i=1
Pr
δik
)
= 1 − Pr
(N
\
i=1
δik
> ǫth
N
(a)
ǫth
> ǫth = 1− 1−F
,
σε2
δk
)
(6.22)
where (a) follows the fact that the random variable σi2 has Chi-squared distribution
ε
with degrees of freedom 2(K − 1). The corresponding cumulative density function
(CDF) is
γ K − 1, x2
,
(6.23)
F (x) =
Γ(K − 1)
R∞
Rx
where Γ(z) = 0 tz−1 e−t dt is the Gamma function, and γ(z, x) = 0 tz−1 e−t dt is
the lower incomplete Gamma function. Substituting Pτ = K ((1 − (1 − α) β) /α) P
in (6.7), σε2 given in (6.14) can be computed.
Chapter 7
Conclusion
7.1
Concluding Remarks
This thesis addresses coordinated transmission schemes in wireless interference networks and their performance limits. The focus has been put mainly on issues regarding channel training, CSI feedback, adaptive transmission, transceiver design,
and user scheduling. These issues have been investigated in different wireless communication scenarios, summarized as follows.
Fixed-rate transmission in two-user SISO interference networks has first been
studied. This scenario can model several practical wireless networks. One obvious
example is the real-time controlling signal delivery in machine-to-machine communication systems. To effectively minimize implementation complexity, point-topoint codes are employed at each user. Depending on whether the two sources are
activated orthogonally and the decoding strategy at the destinations, five different transmission schemes have been considered. The power control problem for
each scheme has been thoroughly analyzed. The inner and outer bounds of the
ǫ-outage achievable rate region of these schemes have also been computed. The
results have shown that the transmission strategy should be decided based on the
relative statistical strength of interference links and direct links to enhance the
system performance.
The thesis has also considered K-user SISO interference networks with the timevarying fading environment. Due to the fact that in practice normally no CSI is
a priori available at terminals (transmitters and receivers), a unified framework
has been developed to assign radio resources for channel training and data transmission. Based on the framework, a pilot-assisted ergodic interference alignment
scheme has been proposed to conduct coordinated transmission. Via achievable
rate region analysis, it has been revealed that with coherence time T , the sum DoF
Kopt (1 − Kopt /T )/2 is achievable when the number of active users is selected to be
Kopt = min{T /2, K}. Thus, in large networks (K > T /2), it is recommended to
first schedule a subset of T /2 users, then perform channel training and coordinated
transmission (in this case, performing ergodic interference alignment) within the set
141
142
Conclusion
of the scheduled users. Moreover, the optimum way of allocating power to channel
training and data transmission has been found. The observations have implied that
to increase the achievable rate, when the network is large or the channel coherence
time is small, more power should be allocated to the channel training instead of
the data transmission. The above results have revealed the inherent performance
limits of wireless interference networks due to the intrinsic requirements for the
radio resources dedicated for channel training. They can provide intuition helpful
for the design of coordinated transmission schemes over interference networks with
time-varying channels when no CSI is a priori available at terminals.
In these K-user SISO interference networks, the case that only quantized CSI
is available at sources has also been investigated. The coordinated transmission
design has been established again based on the ergodic interference alignment concept and focused on two problems. An iterative power control algorithm has been
proposed to find the smallest power consumption to guarantee a fixed-rate transmission. In addition, a rate adaptation scheme has been presented to maximize
network throughput. For the latter case, performance in delay-sensitive systems
has also been studied. The above results have shown that with proper transmission
design, applying the ergodic interference alignment scheme can still provide performance enhancements over conventional orthogonal transmission strategies, even if
only quantized CSI with limited resolution is available at the sources. If the total
number of feedback bits is fixed, different feedback bit allocation approaches would
have diverse impacts on system performance. Specifically, when the network is operating in noise-limited regime, more bits should be assigned to the direct links to
more accurately conduct rate adaptation (or power control). On the other hand, in
interference-dominant scenarios, more accurate quantization should be provided to
interference links to realize better interference alignment. Furthermore, when acceptable delay is limited, the accuracy of channel quantization and the number of
users within the network may affect the system performance in a way that is different from the case in systems without strict delay limits. These results can be used
to properly choose transmission strategies and select design parameters according
to different system requirements.
Joint transceiver design and power control for K-user MIMO interference networks has been addressed to provide reliable communication at given rates. Equipped
with multiple antennas, potentially multiple independent data streams with different data rates can be delivered between each user pair. Two distributed iterative
algorithms, which require only local CSI at each terminal, have been proposed.
Their performance advantages over conventional orthogonal transmission schemes
in terms of power consumption have been confirmed by both computer simulations
and practical hardware test-bed implementations.
A typical scenario in the next generation of communication systems (i.e. 5G
systems) has been studied, where a large number of users in a multi-cell network
have to be served efficiently (e.g. with low training and feedback overhead), when
no a priori CSI is available at mobile terminals and base stations. A pilot-assisted
opportunistic user scheduling (PAOUS) scheme is proposed. It has been shown that
7.2. Future Work
143
the proposed scheme is well-suited for such scenarios: it offers low-computational
complexity, and requires only a one-bit feedback signal from each mobile terminal
to its respective base station. The achievable rate region for the proposed scheme
is computed. It has been illustrated that the proposed scheme takes advantage of
the crowd of mobile terminals as the achievable sum-rate scales by increasing the
number of mobile terminals in each cell. The results reveal that in a multi-cell
network with B base stations, given that the number of mobile terminals properly
scale with the SNR,
the sum degrees of freedom Bopt (1 − Bopt /T ) is achievable,
where Bopt = min B, T2 is the optimum number of the base stations that need
to be activated in the network. Finally, performance evaluations confirm that the
proposed opportunistic transmission scheme can exploit multiple users in the network to mitigate interference even with single-antenna terminals and no a priori CSI
available at mobile terminals and base stations, and thereby enhance the achievable
sum-rate.
7.2
Future Work
There are several possible directions to extend the results of this thesis to support
the development of future wireless systems. In this part, we highlight some of these
extensions for further study:
Self-organizing Coordinated Networks
By self-organizing coordinated networks we mean the systems in which multiple terminals without any a priori knowledge about the propagation environment establish
coordination in order to conduct information transmissions. In these networks, part
of the radio resources can be allocated for establishing coordination, and the rest
can be used for coordinated transmissions. Several communication scenarios including ad-hoc networks, wireless sensor networks with random deployment of the
sensors, and machine-to-machine communication networks in non-stationary environments are examples of such networks. The theoretical framework which has been
developed in Chapter 4 and Chapter 6 for the design and analysis of coordinated
transmission when no a priori CSI is available at terminals can be applied to design
and to analyze transmission schemes for these networks.
Multi-layer Coding for Coordinated Transmission
In Chapter 4 and Chapter 6 we have considered systems in which noisy CSI is available at destinations. As we have shown, this degrades the performance of decoders.
Developing coding schemes that are more robust to such imperfections is of high
importance. To address this issue, a multi-layer coding technique can be designed
for fading interference networks in which only noisy CSI is available. Specifically,
the information bits can be encoded using a multi-layer coding scheme, an interference alignment scheme or a user scheduling can be applied to partially remove the
144
Conclusion
interference, and successive decoding can be applied at receivers to decode different layers of encoded signal. An initial attempt for developing multi-layer coding
technique for fading multiple-access channel with noisy CSI is reported in [AP14],
where it has been shown that a larger rate region can be achieved compared to the
single-layer coding scheme.
Synchronization in Interference Networks
The synchronization of terminals is essential for coordinated data transmission
and coherent signal detection. In this thesis, we have assumed that terminals are
perfectly synchronized. To realize this assumption, low-complexity synchronization
schemes for the schemes proposed in this thesis should be developed. The performance limits of such synchronization schemes should be quantified. In addition,
similar to channel training and feedback transmission, part of radio resources must
be allocated for performing synchronization. Therefore, the optimum resource allocation for synchronization is also an important problem to be investigated.
Channel Training and Feedback Design for Two-user Networks
In Chapter 3, we have considered transmission schemes based on point-to-point
coding when perfect global CSI is available at terminals. Design of low complexity
channel training and channel state feedback schemes for these transmission schemes
is a possible extension.
Data-aided channel Training for Interference Networks
In Chapter 4 and Chapter 6, we have studied a pilot-based channel training for
interference networks. We have shown that allocating radio resources for channel
training is a limiting factor of the investigated coordinated transmission schemes.
It is an interesting problem to design more efficient channel training to enhance
the performance of these schemes. For example, in packet-based communication
systems, data packets have headers which are highly protected with error correcting
codes. Terminals first decode the headers, and then proceed with decoding the
payload data. Therefore, if the headers are successfully decoded, then they can be
used as pilots for channel estimation purpose. This idea can be further extended
to communication systems in which a high-rate data source is multiplexed with a
low-rate data source. If the low-rate data is decoded successfully, then it can be
used as a pilot for channel estimation and decoding of the high-rate data. The early
results on this channel training scheme has been presented in [YT03]. This approach
can be investigated to further enhance the channel training schemes devised in this
thesis.
7.2. Future Work
145
Distributed Transceiver Design and Power Control for WLAN
In Chapter 5, we proposed algorithms for distributed transceiver design and power
control for MIMO systems. The proposed algorithms can provide multi-stream
transmissions by each user at desired rates. The test-bed implementation of these
algorithms confirmed that they can provide efficient and reliable communication in
indoor environments [MFZS14, MFZ+ 14]. There have been recent developments in
the standardizations of high-rate multi-stream data transmission schemes for wireless local area networks (WLANs) (such as IEEE 802.11ac standard). The aforementioned algorithms can be further investigated as possible solutions for multi-stream
fixed-rate transmission in WLAN.
Adaptive Channel Training and Channel State Feedback
The real wireless channels can be modeled as temporally correlated channels in
which the current channel realization is dependent on the previous channel realizations. This temporal correlation can be exploited in order to reduce the radio
resources allocated for channel training and channel state feedback. For single-user
MIMO communication systems, adaptive feedback schemes have been studied (see
e.g. [KLC11]), and channel prediction algorithms are proposed (see e.g [WE06]).
These solutions exploit the temporal correlations of the channel to reduce the overhead of channel training and feedback. For interference networks, developing such
schemes is a more involved problem, however, a considerable performance gain can
be predicted to be achieved.
Coordinated Cluster Design
The theoretical results in Chapter 4 and Chapter 6 for the performance limits of
interference networks can be used for the design of future clustered coordinated
networks. These networks are composed of multiple clusters, where within each
cluster multiple users perform coordinated transmission. The results developed in
these two chapters can be used to find the optimum size of the clusters and perform
radio resource allocation between the clusters.
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