Resource Allocation and Precoding in Cognitive Radio and Relay Networks
by
Hua Mu
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
December 8, 2012
Keywords: Cognitive radio, Relay network, Precoding, Interference alignment, Multiple
input multiple output (MIMO)
Copyright 2012 by Hua Mu
Approved by
Jitendra K. Tugnait, James B. Davis Professor of Electrical and Computer Engineering
Soo-Young Lee, Professor of Electrical and Computer Engineering
Shiwen Mao, McWane Associate Professor of Electrical and Computer Engineering
Abstract
In this dissertation, two promising technologies which improve the spectrum efficiency
in wireless networks are investigated: cognitive radio (CR) and relay technology. Cognitive
radio, as a promising technology to dynamically access spectrum resources, has drawn great
research attention recently. It provides a way to further improve the spectrum efficiency by
allowing unlicensed radio devices to learn from the radio environment and adapt their trans-
mit and receive parameters. This dissertation addresses the design of unlicensed networks,
including their transmission scheme, resource allocation and precoding/beamforming design
when multiple antennas are deployed.
Another topic in this dissertation is that of relay networks. By introducing relay stations
into the network, multiple benefits can be obtained, such as extended network coverage,
improved throughput and higher spectrum efficiency. Also, a relay network makes it possible
to enable two-way (even multi-way) transmission among multiple users within the network.
In this dissertation, precoding design in multiuser relay networks is discussed. Also, networks
based on combined cognitive radio and relay technologies are considered to leverage higher
performance, in terms of spectrum efficiency and network throughput.
This dissertation is organized into six chapters. Chapter 1 introduces the background
of cognitive radio and cooperative relay networks. In Chapter 2, a soft-decision spectrum
sensing concept is proposed and based on which, a joint spectrum sensing, access and power
allocation problem is addressed for multi-band cognitive radio networks by means of convex
optimization. Chapters 3 and 4 deal with the precoding design in multi-user cognitive relay
networks. Chapter 3 considers the multi-way transmission among multiple users and adopts
minimum mean square error (MMSE) as the design objective while Chapter 4 considers a
two-way relay network anda joint signal and interference alignment algorithmis proposed. In
ii
Chapter 5, a signal groupbased interference alignment algorithmis proposed fora generalized
MIMO Y channel where each user sends messages to all the other users. In Chapter 6,
conclusions are drawn and possible future research avenues are highlighted.
Some interesting ideas about how to improve the spectrum efficiency in wireless net-
works have been proposed and analyzed in this dissertation. The proposed soft-decision
spectrum sensing method allows more flexibility in designing the radio access strategies
in cognitive radio systems and achieves significantly higher throughput compared with a
traditional hard decision spectrum sensing based algorithm. Furthermore, the proposed
precoding/beamforming algorithms in the latter part of this dissertation enable concurrent
transmission of multiple users within the same frequency band, which can significantly reduce
or completely remove the inter-user interference. These technologies make it possible to uti-
lize the limited radio resources more efficiently and therefore can support the ever-increasing
demand arising from various wireless devices and applications.
iii
Acknowledgments
First of all, my sincere appreciation goes to my advisor Prof. Jitendra K. Tugnait for
his persistent support and inspiring guidance through my doctoral study. His broad and
deep expertise in the area of communication and signal processing and professional attitude
towards research has been very enlightening and encouraging for me. Without his kind help
and support, the fulfilment of this dissertation won?t be possible.
I would like to thank my dissertation committee members, Prof. Shiwen Mao and Prof.
Soo-Young Lee for their valuable advice on my research work and dissertation. Special
thanks go to Prof. Chris Rodger for being the university reader of my dissertation.
I also want to thank all my friends. Their friendship has made the past three year a
pleasant journey for me. I have been very luck to have them.
Last but not least, I would like to express my appreciation to my parents and grand-
mothers for their financial investment in my education as well as their never-ending love,
caring and support.
iv
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Cognitive Radio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Spectrum Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Resource Allocation in Cognitive Radio . . . . . . . . . . . . . . . . . 5
1.1.3 Beamforming/Precoding in Cognitive Radio . . . . . . . . . . . . . . 6
1.2 Cooperative Communication and Relay Networks . . . . . . . . . . . . . . . 6
1.2.1 Multi-user MIMO Relay Network . . . . . . . . . . . . . . . . . . . . 7
1.3 Overview of this Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Resource Allocation in Cognitive Radio Network . . . . . . . . . . . . . . . . . . 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . 14
2.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Spectrum Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Optimization Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Alternative Formulation with Individual Interference Constraints . . . . . . . 25
2.5 Heuristic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Implementation issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6.1 Obtaining primary user related prior knowledge . . . . . . . . . . . . 30
v
2.6.2 Effect of Imperfect Channel Estimation . . . . . . . . . . . . . . . . . 31
2.7 Comparison with Hard Decision based Spectrum Sensing . . . . . . . . . . . 32
2.8 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Precoding Design in Multi-User Cognitive Relay Network: a MMSE Approach . 46
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . 49
3.3 Iterative Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.1 Update Decoding Matrices . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.2 Update Precoding Matrices at the Secondary Transmitters . . . . . . 56
3.3.3 Updating Precoding Matrix at Relay Station . . . . . . . . . . . . . . 59
3.3.4 Algorithm Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.5 Effects of Allowing Interference to Primary Users . . . . . . . . . . . 61
3.4 Non-iterative Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.1 Matrix Distance based Secondary User Precoding/Decoding Matrices
Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.2 MSE based Relay Precoding Matrix Design . . . . . . . . . . . . . . 68
3.4.3 Distributed Implementation . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.4 Effects of Allowing Interference to Primary Users . . . . . . . . . . . 70
3.5 Robust Algorithm Design with Imperfect CSI . . . . . . . . . . . . . . . . . 70
3.5.1 Robust SU Precoding/Decoding Matrices Design . . . . . . . . . . . 71
3.5.2 Robust Relay Precoding Design . . . . . . . . . . . . . . . . . . . . . 73
3.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4 Precoding Design in Multi-Pair Two-Way Cognitive Relay Networks . . . . . . . 83
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . 86
vi
4.3 Matrix Distance based Precoding Design . . . . . . . . . . . . . . . . . . . . 90
4.3.1 Secondary User Precoding/Decoding Matrices Initialization . . . . . . 90
4.3.2 Iterative Relay Precoding and Secondary User Precoding/Decoding
Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.3 Convergence of Proposed Iterative Method . . . . . . . . . . . . . . . 97
4.3.4 Refining Decoding Matrices to Decode Each Datastream . . . . . . . 99
4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5 Beamforming for Generalized MIMO Y Channels . . . . . . . . . . . . . . . . . 105
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Signal Group Based Alignment Algorithm . . . . . . . . . . . . . . . . . . . 112
5.3.1 User Beamforming in MAC Phase . . . . . . . . . . . . . . . . . . . 112
5.3.2 Relay Receive Combining Vectors Design in MAC Phase . . . . . . . 116
5.3.3 User Receive Combining Design in BC Phase . . . . . . . . . . . . . . 117
5.3.4 Relay Beamforming Design in BC Phase . . . . . . . . . . . . . . . . 119
5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A Convexity of (2.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
B Proof of Proposition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
C Proof of (3.19) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
D Proof: Problem P3 in Chapter 3 is convex . . . . . . . . . . . . . . . . . . . . . 139
vii
E Proof: Problem P5 in Chapter 3 is convex . . . . . . . . . . . . . . . . . . . . . 140
F Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
viii
List of Figures
2.1 Sum transmission rate of secondary network versus sum received SNR (2.73) at
secondary receivers when 3% of the primary network?s bandwidth is interfered
with by secondary BS?s transmission and the received SNR of PU?s signal is -
10dB. [The curves labeled ?optimal,? ?Actual sum rate: soft decision,? and ?Alg.
2 with power control? are all quite close to each other toward the top of the fig.] 37
2.2 Sum transmission rate of secondary network versus sum received SNR (2.73) at
secondary receivers when 3% of the primary network?s bandwidth is interfered
with by secondary BS?s transmission and the received SNR of PU?s signal is -
20dB. [The curves labeled ?optimal? and ?Alg. 2 with power control? are all quite
close to eachother in the top half of the figure.] . . . . . . . . . . . . . . . . . . 38
2.3 Actual total interference versus sum received SNR (2.73) at secondary receivers
when 3% of the primary network?s bandwidth is interfered with by secondary
BS?s transmission and the received SNR of PU?s signal is -20dB . . . . . . . . . 39
2.4 Actual total interference versus sum received SNR (2.73) at secondary receivers
when 3% of the primary network?s bandwidth is interfered with by secondary
BS?s transmission and the received SNR of PU?s signal is -10dB. . . . . . . . . . 40
2.5 Actual total interference versus design bound on total interference to primary
network when sum received SNR (2.73) at secondary receivers is 15dB and SNR
of PU?s signal is -10dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
ix
2.6 Sum transmission rate of secondary network versus design bound on total inter-
ference to primary network when sum received SNR (2.73) at secondary receivers
is 15dB and SNR of PU?s signal is -10dB. . . . . . . . . . . . . . . . . . . . . . 42
2.7 Imperfect CSI: Sum transmission rate of secondary network versus sum received
SNR (2.73) at secondary receivers when up to 3% of the primary network?s band-
width is interfered with by secondary BS?s transmission and the received SNR of
PU?s signal is -10dB. Channel estimation error variance ?2h is set to be either 2%
or 10% of the true channel power gain of its corresponding channel (see (2.54)). 42
2.8 Imperfect CSI: Actual total interference versus sum received SNR (2.73) at sec-
ondary receivers when up to 3% of the primary network?s bandwidth is interfered
with by secondary BS?s transmission and the received SNR of PU?s signal is -
10dB. Channel estimation error variance ?2h is set to be either 2% or 10% of the
true channel power gain of its corresponding channel (see (2.54)). . . . . . . . . 43
2.9 Imperfect CSI: Sum transmission rate of secondary network versus sum received
SNR (2.73) at secondary receivers when up to 3% of the primary network?s band-
width is interfered with by secondary BS?s transmission and the received SNR of
PU?s signal is -20dB. Channel estimation error variance ?2h is set to be either 2%
or 10% of the true channel power gain of its corresponding channel (see (2.54)). 43
2.10 Imperfect CSI: Actual total interference versus sum received SNR (2.73) at sec-
ondary receivers when up to 3% of the primary network?s bandwidth is interfered
with by secondary BS?s transmission and the received SNR of PU?s signal is -
20dB. Channel estimation error variance ?2h is set to be either 2% or 10% of the
true channel power gain of its corresponding channel (see (2.54)). . . . . . . . . 44
x
2.11 Individual PU interference constraint: Sum transmission rate of secondary net-
work versus sum received SNR (2.73) at secondary receivers when up to 3% of the
primary network?s total bandwidth and up to 5% of individual subchannel band-
widths (total 16 subchannels) are interfered with by secondary BS?s transmission
and the received SNR of PU?s signal is -10dB. . . . . . . . . . . . . . . . . . . . 44
2.12 Individual PU interference constraint: Actual total interference versus sum re-
ceived SNR (2.73) at secondary receivers when up to 3% of the primary network?s
total bandwidth and up to 5% of individual subchannel bandwidths (total 16 sub-
channels) are interfered with by secondary BS?s transmission and the received
SNR of PU?s signal is -10dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 System set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Sum rate (3.91) vs SNR per secondary user 10log10(Ptot,i/?2n): K = 4, Ptot,i is
the same for i = 1,??? ,K, M = 4, N = 10, d = 2, Mp = Jp = 1, Ptot,r = KPtot,i,
? = 10. The label ?Iterative, Itot = 2?2n? refers to the approach of Sec. 3.3.5 and
the label ?Iterative, Itot = 0? refers to the approach summarized in Sec. 3.3.4. . 77
3.3 Sum MSE vs SNR per secondary user 10log10(Ptot,i/?2n): K = 4, Ptot,i is the same
for i = 1,??? ,K, M = 4, N = 10, d = 2, Mp = Jp = 1, Ptot,r = KPtot,i, ? = 10.
The label ?Iterative, Itot = 2?2n? refers to the approach of Sec. 3.3.5 and the label
?Iterative, Itot = 0? refers to the approach summarized in Sec. 3.3.4. . . . . . . . 78
3.4 SumMSE using non-iterativeapproachvs SNR per secondary user 10log10(Ptot,i/?2n):
K = 4, Ptot,i is the same for i = 1,??? ,K, M = 4, N = 10, d = 2, Mp = Jp = 1,
Ptot,r = KPtot,i, variable ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5 Sum MSE vs SNR per secondary user 10log10(Ptot,i/?2n): K = 4, Ptot,i is the same
for i = 1,??? ,K, M = 3 or 4, N = 10, d = 1 or 2, Mp = Jp = 2, Ptot,r = KPtot,i,
? = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
xi
3.6 Sum MSE vs SNR per secondary user 10log10(Ptot,i/?2n): K = 4, Ptot,i is the same
for i = 1,??? ,K, M = 3, N = 8, 9 or 12, d = 2, Mp = Jp = 1, Ptot,r = KPtot,i,
? = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.7 Sum MSE vs iteration (for a single run): 10log10(Ptot,i/?2n) = 5dB, K = 4, Ptot,i is
the same for i = 1,??? ,K, M = 4, N = 10, d = 2, Mp = Jp = 1, Ptot,r = KPtot,i.
Recall that noise variances have been normalized to one. . . . . . . . . . . . . . 81
3.8 Sum MSE vs SNR per secondary user 10log10(Ptot,i/?2n): K = 4, Ptot,i is the same
for i = 1,??? ,K, M = 4, N = 10, d = 2, Mp = Jp = 2, Ptot,r = KPtot,i, ? = 10,
receiver noise variance ?2n = 1, design interference bound Itot = 2?2n. . . . . . . . 82
3.9 Average interference power to primary receivers vs SNR per secondary user
10log10(Ptot,i/?2n): K = 4, Ptot,i is the same for i = 1,??? ,K, M = 4, N = 10,
d = 2, Mp = Jp = 2, Ptot,r = KPtot,i, ? = 10, receiver noise variance ?2n = 1,
design interference bound Itot = 2?2n. . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1 System model with2KSUs, onesecondary relay andoneprimarysource-destination
pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Sum rate (4.27) vs. transmit power per secondary user . . . . . . . . . . . . . . 101
4.3 Sum rate (4.27) vs. transmit power per secondary user . . . . . . . . . . . . . . 102
4.4 Sum rate (4.27) vs. transmit power per secondary user . . . . . . . . . . . . . . 103
5.1 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2 Sum rate vs SNR log2
parenleftBig P
?2n
parenrightBig
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
xii
List of Tables
5.1 Required minimum number of antennas for the signal group based alignment
algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
xiii
Chapter 1
Introduction
Cognitive radio is a concept aimed at tackling the spectrum scarcity problem by permit-
ting unlicensed users (secondary users) to access the licensed spectrum as long as interference
to the licensed users (primary users) is ?tolerable?. This dissertation addresses some design
and analysis problems in cognitive radio, with emphasis on spectrum sensing, joint resource
allocation and interference management in multi-user cognitive radio networks. The aim is
to improve the performance of secondary networks, such as throughput, bit error rate (BER)
and mean square error (MSE) of received signal, while keeping the interference to primary
users (PU) within a tolerable range.
The concept of cooperative communication and the deployment of relays in the net-
work can significantly increase throughput and extend network coverage. Various network
topologies, transmission schemes and techniques have been proposed recently which explore
the benefits of relay networks. Among these, two-way relay networks have attracted great
research interest for its ability of facilitating the information exchange of two partners. The
two-way relaying concept has also been extended to support multi-pair users and multi-way
transmission. This dissertation address some key problems in two-way relay networks. Also,
this work extends to multi-user two-way relay networks and also supports multi-way trans-
mission of multiple users. A focus in this dissertation is on precoding/beamforming design
at transmitter (both relay and end users) when multiple antennas are deployed. Precod-
ing/beamforming is performed from multiple aspects including mean square error (MSE),
zero-forcing (ZF) and interference alignment. Also, the effect of imperfect channel state
information (CSI), mainly due to the time varying nature of wireless channel, is considered
and a robust design is proposed to deal with this problem.
1
This chapter provides background for the problems investigated in this dissertation and
also introduces the motivations for our work.
1.1 Cognitive Radio
With the demand for more bandwidth from widespread wireless applications growing
rapidly, spectrum scarcity becomes a major problem which makes further improvement of
wireless network?s performance difficult. In order to overcome the limitation of traditional
spectrum regulation principle which assigns wireless spectrum to licensed users on a long-
term basis, the cognitive radio concept has been proposed which allows more flexibility in
spectrum usage and therefore increases spectrum efficiency [31], [32], [33]. This concept
is motivated by the fact that licensed users access their spectrum non-continuously and
thus leave spectrum holes (non occupied spectrum within a certain time) available to other
users/applications. Federal Communications Commission (FCC) defines CR as [33]: ?Cog-
nitive Radio: A radio or system that senses its operational electromagnetic environment and
can dynamically and autonomously adjust its radio operating parameters to modify system
operation, such as maximize throughput, mitigate interference, facilitate interoperability, ac-
cess secondary markets.? As an intelligent wireless communication system, cognitive radio
should be be aware of its environment, such as the spectrum occupancy by licensed users and
should adapt its operating parameters including access probability, transmit power, modu-
lation strategies, etc to make sure that licensed users? network (primary network) can still
function well.
Generally, cognitive radio system?s deployment doesn?t need to modify the primary net-
work and can operate transparently to the licensed users. In such cognitive radio system,
unlicensed users can access the spectrum when it is not occupied by licensed users (under-
lay scheme) or the unlicensed users can dynamically adjust their transmit power such that
the interference to the primary users does not exceed a certain level (overlay scheme) [30].
However some recent research work indicates that by allowing some cooperation and/or
2
information exchange between primary and secondary users (SU), secondary network?s per-
formance can be further improved without increasing the interference to licensed users. Such
system brings together heterogeneous operators and services, various network topologies and
radio access technologies and includes the joint management of several networks and recon-
figuration of wireless devices such as base stations and terminals. For example, by allowing
the secondary network to access the primary channel state information (CSI), secondary
precoding/beamforming can be performed at secondary transmitters to completely remove
the interference to the primary network by using the secondary-to-primary CSI.
During the past decade, many efforts have been made to design and develop effective
cognitive radio technologies. For obtaining knowledge of cognitive radio?s operational and ge-
ographical environment, spectrum sensing, geolocation, etc. have received significant research
attention, while for dynamically and autonomously adjusting its operational parameters and
protocols, many resource management techniques such as dynamically optimizing transmit
power and access probabilities of multiple frequency bands as well as beamforming/precoding
schemes have been proposed.
1.1.1 Spectrum Sensing
In cognitive radio, spectrum sensing is the essential task to determine the ?spectrum
holes? which can be used by unlicensed users. The term ?spectrum holes? refers to sub-
bands of radio spectrum which are not occupied by licensed users within a certain time
and location. Therefore spectrum sensing involves three dimensions: frequency, time and
location.
Spectrum sensing can be categorized into three types: energy detection [3], [4] matched
filter detection [5] and cyclostationary feature detection [5], [6]. While the latter two
can provide more sensing accuracy, energy detection is the most widely used method for its
computational simplicity and least requirement on prior knowledge.
3
Frequently, the spectrum sensing problem is formulated as a binary hypothesis testing
problem and secondary users make hard decisions about primary users? behavior (PU is
present or not). Then a spectrum access strategy is implemented based on the sensing
decision. However, no spectrum sensing technique can detect spectrum holes perfectly. So
secondary users? transmissions will always cause collisions with primary users with some
nonzero probability. In the binary test based spectrum sensing, informationloss is introduced
when transforming the continuous sensing statistics into a binary detection decision. When
this decision is made separately from other cognitive radio operating parameters, such as
spectrum access probabilities (or binary access decisions), transmit power, etc, the secondary
users? performance is usually not optimized. Also, in the scenario where secondary users
access the channels with some probabilities, it is not reasonable to make the access decisions
based on a binary sensing result. Motivated by these, we propose a soft-decision spectrum
sensing concept which allows more design flexibility and significantly improves the system
throughput.
The accuracy of signal user spectrum sensing is always compromised by multipath fading
and shadowing. Deep fading and shadowing can make the primary users? signal at the sensing
node very weak and therefore cause mis-detection of PU?s presence. Cooperative spectrum
sensing is proposed to overcome this problem and also to increase sensing speed. Cooperative
sensing can be performed in either centralized or distributed manner [5].
In centralized sensing, a central controller collects sensing information from all or some
of the secondary users, makes detection decision and then broadcasts this decision to all
secondary users to regulate the secondary traffic. A control channel usually exists between
central controller and secondary users to allow information reporting and decision distri-
bution. The subset of secondary users which will report their sensing information can be
selected based on their credibility which is usually determined by their channel conditions
from primary transmitter. Many decision fusion rules have been developed, including hard
information combining, such as AND, OR and M-out-of-N methods, and soft information
4
combining like equal gain combining (EGC), selection combining (SC) and switch and stay
combining (SSC). Usually soft information combining provides lower mis-detection and false
alarm probabilities but it requires more information shared by participating secondary users.
In distributed sensing, secondary users can share their sensing information with each
other but they make spectrum detection and access decisions independently. Therefore the
network structure is simplified since no central controller or control channel is needed. In
this case, coordinating protocol for information sharing decision distribution among sec-
ondary users needs to be designed wisely to avoid collision and reduce protocol overhead and
complexity.
1.1.2 Resource Allocation in Cognitive Radio
Cognitive radio introduces several challenges to network resource allocation algorithms.
Wireless spectrum is now shared between primary and secondary networks. To secondary
network, this resource may only be partially available during certain time interval. Resource
allocation of secondary network, generally in terms of channel and power allocation, should
consider the interference they cause to the primary network.
When resource allocation is performed based upon the hard decisions of channel avail-
abilities from the spectrum sensing stage, it is similar to traditional resource allocation
optimization problems with more constraints on interference to primary users. Some of the
recent works extend the resource optimization problem to include both spectrum sensing,
access and power allocation. Ref. [26] jointly optimizes the sensing thresholds over a wide-
band spectrum in both single cognitive radio and cooperative cognitive radio scenarios. Ref.
[17] jointly optimizes sensing thresholds and transmit power in a multiband system, and Ref.
[22] considers a similar problem in orthogonal frequency division multiplexing (OFDM) sce-
nario. However, given the fact that the statistics collected in the spectrum sensing stage
only provides a probability of channel?s occupancy, it is preferable to optimize resource al-
location problems based on these channel availability probabilities. Therefore soft sensing
5
concept is incorporated into resource allocation stage. Ref. [19] optimized spectrum access
probabilities based on quantized sensing statistics aiming at maximizing secondary user?s
service rate while maintaining primary user?s queue stable. In [16] an adaptive power and
rate allocation algorithm is developed for a single band cognitive radio system. Ref. [27]
considers the optimal power control problem in single band cognitive radio based on soft de-
cision spectrum sensing. In this dissertation, a soft-decision spectrum sensing based resource
allocation algorithm is proposed to jointly optimize channel access and transmit power.
1.1.3 Beamforming/Precoding in Cognitive Radio
In underlay cognitive radio networks, primary users and secondary users transmit con-
currently over the same spectrum band. One way to avoid interference to licensed PUs is
to employ multiple antennas at SU transmitters and design proper precoding matrix. When
secondary-to-primary CSI can be perfectly obtained, interference to PU can be completely
removed by choosing beamforming vectors that align in the null space of the secondary-
to-primary channel matrix. This also restricts the number of antennas at the secondary
transmitters. When CSI can not be perfectly obtained due to the time-varying nature of
wireless channels or quantization errors, or when number of transmit antennas at SUs are
not large enough, a compromise can be made by allowing some residual interference at the
PU receivers.
1.2 Cooperative Communication and Relay Networks
In cooperative communication, users exploit the broadcast nature of wireless channels to
help others? transmission [12], [13]. Sometimes, dedicated relay stations are deployed to help
forward users? information in order to increase multiplexing and/or diversity order, extend
network coverage and achieve higher throughput, etc.
6
In a typical two-hop relay network operating in a half-duplex mode (this is a widely
used case since it is practical to assume that each user can not transmit and receive simul-
taneously), the transmission is completed within two time slots. In the first time slot relay
receives signals from all transmitting users while in the secondary time slot the relay pro-
cesses and forwards its received signal to users. Two commonly employed relay techniques
are decode-and-forward (DF) and amply-and-forward (AF). These two techniques refer to
different signal processing methods at the relay node. In the DF scheme the relay decodes
its received information from the first time slot, re-encodes it and transmits it to desired
users in the second time slot. While in the AF scheme, the relay simply amplifies its received
signal according to its power budget in the single antenna case or designs a precoding matrix
to be multiplied with its received signal vector and then transmits this new signal vector
to receivers in the multiple antennas case. The key difference between these two schemes is
whether relay decodes the information it receives or not. The advantage of the AF scheme
is that it is simpler for the relay since no decoding is required. Also, from the security point
of view, the relay will not gain access to users? information since no decoding is performed,
therefore the privacy of users? data is preserved.
Relay deployment also enables two-way and multi-way transmission in two-user and
multi-user networks since users can exchange information via the help of the relay node. Net-
work coding concept [58] is usually combined with the relay techniques to achieve multiplex-
ing gain due to the fact that each user knows its own signal and can perform self-interference
cancellation. For each user, its desired signal and its ?self? signal can be network coded
together at the relay station which reduces the number of independent datastreams in the
network and in turn increases the multiplexing gain.
1.2.1 Multi-user MIMO Relay Network
MIMO is known as an advanced technique to improve data throughput without increas-
ing bandwidth or power [9?11]. In a multi-user relay network, concurrent transmission of
7
multiple users can be supported by deploying multiple antennas at the relay station or by
deploying multiple relays which form a virtual multiple antenna node. A multi-user network
is usually interference limited. However a proper design of precoding/decoding matrices at
transmitters/receivers with multiple antennas can mitigate the negative effects of inter-user
interference and therefore enable concurrent transmission of multiple users [14,48]. Next we
discuss three types of beamforming/precoding methods in a multi-user MIMO relay network:
minimum mean square error (MMSE), zero-forcing (ZF) and interference alignment.
One way to design precoding/decoding matrices is using mean square error (MSE) of
estimated signal as the design criterion. This method considers both inter-user interference
and noise as errors and tries to minimize them. MSE has been widely utilized in MIMO relay
networks. When both relay and end users employ multiple antennas, joint optimization of
precoding/decoding matrices at all nodes seems impossible since the problem is non-convex.
So suboptimal iterative methods are often utilized. In a two-user two-way relay network,
Ref. [14] designs optimal beamforming matrix at the relay node and derives capacity region.
Ref. [13] proposes an iterative source and relay precoding algorithm for two-user two-way
MIMO relay system. [8], [10] considers precoder design in a system where one pair of
users are assisted by multiple relays. While in a two-user two-way relay system inter-user
interference can be completely removed, this kind of interference can significantly degrade the
performance of a multi-user two-way relay system. For a multi-user two-way relay system,
Ref. [6] designs MIMO relay beamforming matrix based on zero-forcing and minimum MSE
criteria.
Zero-forcing is another method to eliminate the harmful effect of inter-user interference.
By performing ZF, interference is removed completely by forcing interference and desired
signal orthogonal subspaces at each receiver. For zero-forcing relay systems, Ref. [55]
proposes distributed beamforming for a multi-pair two-way relay system where each node
has a single antenna. The inter-pair interference is canceled out by designing relay weights
of multiple signal antenna relays. In [1], MIMO technique is employed so that inter-pair
8
interference is nulled out. In [57], a coordinated eigen beamforming method is proposed to
optimize the throughput given that the inter-pair interference has already been eliminated.
Interference alignment is a recently emerged technique for the interference-limited net-
work which can dramatically increase the system?s degree of freedom (DOF) and therefore
supports multi-user transmission [8]. This technique shows that interference is not a pro-
hibitive factor in multi-user networks anymore. The basic idea is to align all interference
from different interferers into the same subspace therefore more dimensions are left to useful
signals. Many alignment schemes have been proposed for various networks including multi-
user networks, cellular networks, MIMO X-channel, relay networks and Y channel, etc. For
interference alignment in MIMO Y channel, feasible schemes have been proposed for both
3-user [63] and multi-user scenarios [7]. By incorporating the network coding concept,
signals that can be network coded together at the relay node are aligned into the same direc-
tion. Therefore a higher DOF is achieved and equivalently, the number of required antennas
are reduced. In this dissertation a signal group based alignment method is proposed for a
generalized MIMO Y channel, which supports the concurrent transmission of K users to
convey K(K ?1) datastreams for each other. Our scheme significantly reduces the number
at users at the cost that more antennas are deployed at the relay node.
1.3 Overview of this Dissertation
The remainder of this dissertation is organized as follows.
In Chapter 2, a soft-decision spectrum sensing concept is introduced. Instead of making
binary hard decisions regarding the channel state, the channel availability probabilities are
directly utilized in the optimization problem including spectrum access and power allocation.
The benefits of this proposed algorithm is illustrated via simulations. Also some practical
implementation issues are discussed to leverage design tradeoffs for practical implementa-
tions.
9
Chapter 3 through Chapter 5 address the precoding/beamforming design problems for
two-way andmulti-way transmission inrelaynetworks. The aimofthese precoding/beamforming
methods are to alleviate the harmful effects of inter-user interference and therefore support
concurrent transmissions of multiple users utilizing MMSE criterion and interference align-
ment methods. Chapters 3 and 4 deal with a cognitive radio system where a multi-user
relay network is the secondary network which coexists with a primary network. In this kind
of system setup, the design of secondary precoding schemes address not only the inter-user
interference problem but also the interference avoidance from the secondary network to the
primary network. More specially, Chapter 3 investigates the precoding design in a multi-
way multi-user MIMO relay secondary network based on the MMSE criterion. Both iterative
and low-complexity non-iterative algorithms are presented. Also, the effects of channel un-
certainty is taken into consideration and a robust non-iterative algorithm is proposed. In
Chapter 4, a joint signal and interference alignment precoding method is proposed for multi-
user two-way relay systems. This method can be viewed as a generalization of zero-forcing
precoding and provides better performance in the low-to-medium SNR regime. This method
incurs low computational complexity while provides high design flexibility.
Chapter 5 considers a generalized MIMO Y channel. This channel includes multiple
users and each user intends to transmit to all the other users. By the help of a relay node
and the deployment of multiple antennas, the concurrent transmissions of multiple users is
supported with no inter-user interference. A novel signal group based interference alignment
method is proposed which only requires a minimum number of antennas at each user and
a relatively small number of antennas at the relay. Although this chapter considers a non
cognitive radio system model, the extension to cognitive radio network is straightforward
provided that the channel state information between primary and secondary networks is
available.
In Chapter 6, conclusions are provided and future research directions are presented.
10
Chapter 2
Resource Allocation in Cognitive Radio Network
The proliferation of wireless applications is requiring more and more radio resources.
The cognitive radio concept has been proposed to address the spectrum scarcity by relaxing
the traditional fixed spectrum regulation and allowing unlicensed users to access spectrum
provided they do not degrade licensed users?s quality of service. Various techniques have
been proposed to ensure that licensed and unlicensed users can share the radio resource.
The design of cognitive radio networks is mainly focused on the design of the secondary
network since the goal is to increase spectrum efficiency without modifying the operation of
the primary network. The most important design criteria for cognitive radio networks can
be summarized into two aspects. Firstly, interference to primary network should be avoided.
This is usually achieved by spectrum sensing, power control and beamforming, etc. Secondly,
the utilization of resources should be optimized to achieve a better performance for the CR
network. This requires an optimized design of spectrum access strategies, power allocation
and network scheduling protocol, etc. In this chapter, the design of a multiband cognitive
radio network is presented in which a soft-decision spectrum sensing method is proposed and
spectrum access and power allocation are jointly optimized.
This chapter is organized as follows. In Sec. 2.1, an introduction to the background and
related work of resource allocation in CR networks is presented. In Sec. 2.2 we describe the
system model, underlying assumptions, optimization objective function and constraints in
detail. In Sec. 2.3 we provide a solution to the proposed convex optimization problem via a
dual convex optimization formulation. The interference constraint used in this formulation
is the total interference to the primary network. An alternative formulation is presented
in Sec. 2.4 where additionally interference to individual PUs is also constrained. In order
11
to reduce computational complexity of the optimal solution, two heuristic algorithms are
proposed in Sec. 2.5 for the primary network interference formulation. In all our problem
formulations knowledge of the CSI (channel state information) for all links (SUs to secondary
network base station, and PUs to SUs) is assumed to be available. In Sec. 2.6 some practi-
cal/implementation issues involving CSI acquisition and imperfect CSI are briefly discussed.
In Sec. 2.7 a hard decision spectrum sensing scheme is presented for comparison purpose
following [17]. Numerical results are offered in Sec. 2.8 to illustrate the proposed soft sensing
based algorithms and to compare them with the hard sensing approach of Sec. 2.7. Con-
cluding remarks are given in Sec. 2.9 and some technical details for the solution in Sec. 2.3
are provided in the Appendix.
2.1 Introduction
Spectrum sharing is central to cognitive radio which permits unlicensed users (secondary
users) to access the licensed spectrum as long as the interference (instantaneous power) to
licensed users (primary users) is tolerable. There are two popular spectrum sharing schemes.
The first one (spectrum underlay) is to allow primary and secondary users access to the
same channel simultaneously while constraining the transmitted power of secondary users so
that it can be treated as background noise at primary users without exceeding the primary
users noise floor. In the second scheme (spectrum overlay) secondary users need to detect
spectrum holes and then access spectrum white space in an nonintrusive way [30]. In this
chapter, we will focus on the second scheme.
In this chapter we propose a joint spectrum sensing, access and power allocation al-
gorithm for spectrum overlay schemes where spectrum sensing is of great importance since
this is how secondary users find available spectrum holes which could be used. Improving
secondary network?s performance while keeping the interference to primary network accept-
able (tolerable) is an important tradeoff in cognitive radio networks. In traditional spectrum
sensing, secondary users make hard decisions about primary users? behavior (PU is present
12
or not). As discussed in [24] and [25] (also [16] and [27]), significant performance improve-
ment can be expected if one employs a soft-decision sensing technique where one uses the
continuous-valued sensing statistics directly (explicitly) in joint optimization of secondary
network?s parameters. On the other hand, a well-designed power allocation strategy is an-
other effective way to further improve secondary users? throughput.
Past work on joint power allocation and spectrum access using hard-decision spectrum
sensing includes [19], [26], [17], [22] and [29], and that using soft-sensing includes [16] and [27].
In [16] an adaptive power and rate allocation algorithm is developed for a single band cogni-
tive radio system. Ref. [27] considers optimal power control in single band cognitive radios
based on soft decision spectrum sensing. Both [16] and [27] allow secondary users to trans-
mit simultaneously with primary user, which means they are dealing with spectrum underlay
system and access probability is not considered. Note that [16] and [27] define interference
to primary users to be the average received power at primary users from the secondary
users? transmissions. This is different from the ?overlay? scenario where the interference is
usually defined to be the probability of collisions or percentage of bandwidth corrupted by
collisions [26]. Few papers consider power adaptation in overlay systems because under the
same spectrum access decision, adapting power does not change the collision probability.
Therefore, in such a case, spectrum sense-access can be separated from power allocation.
However, joint optimization of spectrum sensing, spectrum access and power adaptation can
achieve better sum throughput of secondary users while keeping the interference to primary
users under certain tolerable level.
Notation: We will use the variable h with various subscripts to denote the flat-fading
channel gain from various transmitters (primary or secondary users) to various receivers
(PUs or SUs).
hk gain of subchannel k from the secondary BS to the SU using subchannel k during the
transmission phase (after sensing has been done).
13
hpk gain of subchannel k from the PU using subchannel k to the SU also using subchannel
k during the transmission phase (after sensing has been done).
hpi,k gain of subchannel k from the PU using subchannel k to the SU sensing subchannel k
during the sensing phase.
In the following the CSI for all links (i.e. hk, hpk and hpi,k) is assumed to be available to all
nodes of the secondary network (if needed). We will use 1{?} to denote an indicator function
which takes value 1 when {?} is true and 0 when {?} is false. (?)+ := max{0,?}. If random
variable X is Gaussian with mean ? and variance ?2, we write X ?N(?,?2).
2.2 System Model and Problem Formulation
2.2.1 System Model
Consider a cognitive radio scenario where secondary users can potentially access K
orthogonal subchannels. One of the secondary users is designated as cognitive radio base
station (BS); we will refer to this node as BS. [Alternatively, the BS instead of being just one
of many users in the network, could be a single secondary user which transmits to multiple
users.] There are also N secondary users in the system. The secondary BS determines
spectrum access probabilities and allocates resources to other secondary users while both
secondary BS and users participate in spectrum sensing. We consider a downlink scenario
for the secondary network where the BS transmits to the secondary users and the association
between secondary users and subchannels are preassigned. Assume that the subchannels
experience block fading, which means the channel gains of these subchannels remain constant
during one time slot. Also assume the aggregated primary users? behavior over each channel
follows a block static pattern, i.e. during each time slot, primary users will access subchannel
k, (k = 1,2,...,K), with probability PH1k and remain idle with probability PH0k = 1 ?
PH1k where H1 denotes the fact that primary user is transmitting while H0 indicates the
alternative. Primary users? behavior over the K subchannels are assumed to be independent.
14
2.2.2 Spectrum Sensing
Spectrum sensing is performed at the beginning of each time slot. Secondary BS and
secondary users (or a subset of secondary users) sense the K subchannels and collect suf-
ficient sensing metrics for each channel. Any sensing technique can be used, such as the
commonly used energy detector [20,23]. Let Ak, k = 1,2,...,K, denote the sensing user
subset for subchannel k where Ak ? {0,1,...,N} with m = 1,2,??? ,N indexing one of N
secondary users and m = 0 indexing secondary BS. Let Qik,i ? Ak,k = 1,2,...,K, be the
sensing statistic for subchannel k from secondary user i. The conditional probability density
function of Qik conditioned on primary users? behavior (transmitting or silent) are assumed
to be known and denoted as f(Qik|H0k) and f(Qik|H1k). Let Qk = [Qik,i ?Ak] be the sens-
ing metric vector for subchannel k. With the knowledge of Qk?s distribution and primary
users? access/idle probabilities PH1k / PH0k, the probabilities of H0 and H1 for subchannel k
conditioned on Qk is given by
P(Hjk|Qk) = f(Qk|Hjk)PHjkf(Q
k|H0k)PH0k +f(Qk|H1k)PH1k
, j ?{0,1}, k = 1,2,...,K, (2.1)
where f(Qk|Hjk) = producttexti?Ak f(Qik|Hjk), j ? {0,1}. An example of Qik?s for energy detectors
and resultant probability densities may be found in Sec. 2.8.
2.2.3 Problem Formulation
After the sensing phase, the secondary BS will jointly optimize spectrum access and
power allocation. Assume that during each time slot, channel conditions (flat fading gains)
of secondary links (assumed to be flat fading), denoted ash = [h1,h2,...,hK], can be obtained
using channel estimation techniques. Let s = [s1,s2,...,sK] and p = [p1,p2,...,pK] denote
the access probabilities and allocated powers, respectively, over the K subchannels, that is,
sk is the probability with which the secondary BS accesses the k-th subchannel (recall we
are considering a downlink scenario for the secondary network). We allow sk ? [0,1] because
15
we believe that this will increase secondary network?s throughput, otherwise the optimal
value of sk will end up being either 0 or 1. Our goal is to maximize the expectation of sum
throughput (instantaneous capacity) of secondary network while keeping the interference to
primary users under some bound ?, under a transmit power constraint.
Let Wk represent the bandwidth of the k-th subchannel and N0 represent the (one-
sided) power spectrum density of white noise at secondary receivers. The transmission rate
of secondary BS over subchannel k consists of two parts. One part is the transmission rate
when primary user is not transmitting over this channel and the other part is the transmission
rate when primary user is also transmitting. Let hpk denote the gain of subchannel k from PU
transmitter to secondary receiver and let ppuk be the PU?s transmitted power in subchannel
k. Then the maximum transmission rate (instantaneous capacity) over subchannel k is given
by
R(pk,sk) = skP(H0k|Qk)Wk log2
parenleftBigg
1 + pk|hk|
2
N0Wk
parenrightBigg
+skP(H1k|Qk)Wk log2
parenleftBigg
1 + pk|hk|
2
N0Wk +ppuk|hpk|2
parenrightBigg
(2.2)
where the first term on the right-side of (2.2) is the instantaneous capacity when PU is not
transmitting while the second term is the instantaneous capacity when PU is transmitting,
ppuk|hpk|2 representing the interference power at the secondary receiver from PU transmitter
over subchannel k, and P(H0k|Qk) and P(H1k|Qk) are the posterior probability of PU not
transmitting and transmitting, respectively, in the k-th subchannel conditioned on the sub-
channel sensing statistic Qk. Interference over subchannel k occurs when the secondary BS?s
decision is to transmit over this subchannel and the PU is also transmitting over the same
subchannel. Defining the overall interference to be the average fraction of bandwidth cor-
rupted by secondary BS?s transmissions, we have the interference expression for subchannel
16
k as
I(sk) = skP(H1k|Qk)Wk. (2.3)
Introduce variables
tk = skpk, uk = |hk|2/(N0Wk), vk = |hk|2/(N0Wk +ppuk|hpk|2), (2.4)
and let t = [t1,t2,...,tK]. Then we have pk = tk/sk. Then the optimization problem under
consideration can be formulated as a convex optimization problem [18,21]:
P1 : maxt,s
Ksummationdisplay
k=1
R(sk,tk) (2.5)
s.t.
Ksummationdisplay
k=1
I(sk) ? ? (2.6)
Ksummationdisplay
k=1
tk ? Ptot (2.7)
0 ? sk ? 1, tk ? 0, k = 1,2,??? ,K, (2.8)
where
R(sk,tk) = skP(H0k|Qk)Wk log2
parenleftbigg
1 + tkuks
k
parenrightbigg
+skP(H1k|Qk)Wk log2
parenleftbigg
1 + tkvks
k
parenrightbigg
. (2.9)
In the above problem, (2.6) is the total interference constraint with a pre-specified bound ?,
and (2.7) is the total transmit power constraint with upper limit Ptot. As discussed in [24]
and [25], direct maximization of (2.5) w.r.t. p and s does not lead to a convex problem since
the constraint summationtextKk=1 tk = summationtextKk=1 skpk ? Ptot is not convex in p and s. The objective function
(2.5) in problem P1 is a concave function of variables t,s (a proof is given in Appendix A)
and all the constraints are affine in the variables t and s. Also, it is obvious that the feasible
set of t and s are non-empty since we can always find non negative t and s such that all the
17
constraints are satisfied. Therefore this is a convex problem [18, Sec. 4.2.1] and the global
optimal solution can be obtained. This solution is discussed next.
2.3 Optimization Solution
We now discuss solution to the convex problem P1. We will solve it via a dual formula-
tion. Since objective function (2.5) is concave, all constraints are affine, and the feasible set
is non-empty, Slater?s condition is satisfied and there is no duality gap [18, Sec. 5.2.3].
Let ? and ? be the dual variables associated with constraints (2.6) and (2.7). The
Lagrangian of problem P2 can be expressed as
J(?,?,s,t) =
Ksummationdisplay
k=1
skP(H0k|Qk)Wk log2
parenleftbigg
1 + tkuks
k
parenrightbigg
+skP(H1k|Qk)Wk log2
parenleftbigg
1 + tkvks
k
parenrightbigg
+?
parenleftBigg
??
Ksummationdisplay
k=1
skP(H1k|Qk)Wk
parenrightBigg
+?
parenleftBigg
Ptot ?
Ksummationdisplay
k=1
tk
parenrightBigg
(2.10)
= ??+?Ptot +
Ksummationdisplay
k=1
Jk(?,?,sk,tk) (2.11)
where
Jk(?,?,sk,tk) = skWkP(H0k|Qk)log2
parenleftbigg
1+ tkuks
k
parenrightbigg
+skWkP(H1k|Qk)log2
parenleftbigg
1 + tkvks
k
parenrightbigg
??skP(H1k|Qk)Wk ??tk. (2.12)
Let?denotethedomainof problemP1, therefore, ? = ?Kk=1?k where ?k = {tk ? 0, 0 ? sk ? 1}.
Then the Lagrange dual function can be expressed as
g(?,?) = max
(s,t)??
J(?,?,s,t) = ??+?Ptot +
Ksummationdisplay
k=1
max
(sk,tk)??k
Jk(?,?,sk,tk) (2.13)
18
and the dual optimization problem is
P2 : min
?,?
g(?,?) (2.14)
s.t. ? ? 0, ? ? 0. (2.15)
From (4.26) we have
g(?,?) = ??+?Ptot +
Ksummationdisplay
k=1
gk(?,?) (2.16)
where
gk(?,?) = maxs
k,tk
Jk(?,?,sk,tk). (2.17)
First we calculate g(?,?) for fixed ? ? 0 and ? ? 0 and then solve problem P2. From
(2.16), we observe that calculation of g(?,?) can be decomposed into solving for gk(?,?)
for each subchannel k. Note that Jk(.) is concave in tk for tk ? max{?sk/uk,?sk/vk} =
?sk/uk and second order differentiable. Also, we have ?Jk(?,?,sk,tk)?t
k
|tk=?sk/uk = ? > 0 and
?Jk(?,?,sk,tk)
?tk |tk=? = ?? < 0. So there exists exactly one t
?
k ??sk/uk with
?Jk(?,?,sk,tk)
?tk |tk=t?k =
0which maximizes Jk(?,?,sk,tk). To solve forgk(?,?), we takepartial derivative ofJk(?,?,sk,tk)
with respect to tk and set ?Jk(?,?,sk,tk)?t
k
= 0 to obtain two solutions
?tk1 = sk
?
??bk ?
radicalBig
b2k ?4ukvkck
2ukvk
?
? (2.18)
and
?tk2 = sk
parenleftBigg d
k
2ukvk
parenrightBigg
(2.19)
19
where
bk := uk +vk ? Wkukvk?ln2 , (2.20)
ck := 1? (P(H0k|Qk)uk +P(H1k|Qk)vk)Wk?ln2 , k = 1,2,...,K, (2.21)
dk := ?bk +
radicalBig
b2k ?4ukvkck. (2.22)
If t?k = ?tk1, then since ?tk2 ? ?tk1 ? ?sk/uk and ?Jk(.)?t
k
|tk=?tk
2
= 0, it follows that ?tk2 is
another point which maximizes Jk(.). This contradicts the fact that exactly one t?k ??sk/uk
maximizes Jk(.). Hence, ?tk2 is the only possible value to maximize Jk(.); therefore, the value
of tk ? ?k that maximizes Jk for a fixed sk ? [0,1] is given by
t?k = sk
parenleftBigg d
k
2ukvk
parenrightBigg+
(2.23)
where (?)+ := max{0,?}. Since tk = skpk, (2.23) allows us to obtain the optimal power
allocation for subchannel k as
p?k =
parenleftBigg d
k
2ukvk
parenrightBigg+
(2.24)
even when sk = 0. Then we have
Jk(?,?,sk,t?k) = sk
braceleftBigg
P(H0k|Qk)Wk
bracketleftBigg
log2
parenleftBigg
1 + dk2v
k
parenrightBiggbracketrightBigg+
+P(H1k|Qk)Wk
bracketleftBigg
log2
parenleftBigg
1 + dk2u
k
parenrightBiggbracketrightBigg+
??
parenleftBigg d
k
2ukvk
parenrightBigg+
??P(H1k|Qk)Wk
bracerightBigg
. (2.25)
Define
?k := P(H0k|Qk)Wk
bracketleftBigg
log2(1 + dk2v
k
)
bracketrightBigg+
+P(H1k|Qk)Wk
bracketleftBigg
log2(1 + dk2u
k
)
bracketrightBigg+
??
parenleftBigg d
k
2ukvk
parenrightBigg+
(2.26)
20
Then maximizing Jk(?,?,sk,t?k) w.r.t. sk ? ?k leads to
s?k =
?
???
???
???
???
1 if ?k ??P(H1k|Qk)Wk > 0
0 if ?k ??P(H1k|Qk)Wk < 0
? [0,1] if ?k ??P(H1k|Qk)Wk = 0.
(2.27)
Therefore,
gk(?,?) = Jk(?,?,s?k,t?k) = (?k ??P(H1k|Qk)Wk)+ . (2.28)
We optimize problem P2 (2.14)-(2.15) with respect to ? first. Define
ak := P(H1k|Qk)Wk, (2.29)
which leads to gk(?,?) = (?k ??ak)+. Let
g(?) = min
??0
g(?,?) = min
??0
braceleftBigg Ksummationdisplay
k=1
gk(?,?)+??
bracerightBigg
+?Ptot (2.30)
= min
??0
?
?
?
?
Ksummationdisplay
k=1
ak
?
parenleftBigg?
k
ak ??
parenrightBigg+
+?
?
?
?+?Ptot. (2.31)
Proposition 2.1 : Sort ?k/ak in a non-increasing order of magnitude and denote the
sorted sequence as {?d1, ?d2,..., ?dK}, and also re-order ak in the same order and denote the
re-ordered sequence as {?ak, k = 1,2,??? ,K}. Then (2.31) is minimized w.r.t. ? ? 0 for
?? =
?
???
???
?dl? if 1?summationtextl??1k=1 ?ak
? ? 0 and 1?
summationtextl?
k=1
?ak
? < 0 for some l
? ? K
0 if 1?summationtextKk=1 ?ak? ? 0
(2.32)
where, by definition, summationtext0k=1 ?ak? = 0.
Proof : See Appendix B. square
21
Then we have
g(?) = g(??,?) = ?Ptot +??? +
l?summationdisplay
k=1
(?ak ?dk ??ak??)+ (2.33)
where l? = K for the second case in (2.32). Since g(?,?) is a convex function, g(?) =
min??0 g(?,?) is also convex. Therefore, min??0 g(?) can be obtained via a line search.
When we search for ?, the lower bound can be set as ?l = 0. As to the upperbound, we
need to make sure that BS will allocate a positive power to at least one subchannel. From
(2.24), we choose ?u = max{(P(H0k|Qk)uk +P(H1k|Qk)vk)Wk/ln2,k = 1,2,...,K}.
After we obtain the optimal dual variables?? and ??, we can recover the optimal primary
variables according to (2.24) and (2.27). Note that the optimal solution sk will take a value
between 0 and 1 only when ?k ???P(H1k|Qk)Wk = 0. From Proposition 2.1, it follows that
?i? ? ?P(H1i? |Qi?)Wi? = 0 where i? corresponds to l? pertaining to re-ordered ?k/ak in
(2.32). However, we may have ?i/ai = ?i?/ai? for some i negationslash= i?. Define the sets
M = {i| ?ia
i
= ?i?a
i?
, i = 1,2,...K} (2.34)
and
Mc = {i| ?ia
i
negationslash= ?i?a
i?
, i = 1,2,...K}. (2.35)
Then for any k ? M, we have s?k ? [0,1]. If |M| = 1, then as in [24] (see discussion therein
after Proposition 2.1), we have only one sub-channel, i.e., the i?-th sub-channel whose access
probability s?i? ? [0,1], and we pick the largest si? satisfying all the constraints and then set
p?k =
parenleftBig d
k
2ukvk
parenrightBig+
for s?k > 0, and p?k = 0 for s?k = 0. When |M| > 1, the optimal solution should
satisfy the complementary slackness conditions [24, (29),(30)] since strong duality holds. If
?? = 0, then constraint (2.6) is never active. Therefore, we can access all subchannels with
probability 1. Also, from (2.20)-(2.22) and (2.24), we notice that ?? > 0, otherwise as long
as we access some subchannel, the total transmitted power will tend to infinity. For ?? > 0
22
and ?? > 0, constraints (2.6) and (2.7) hold true with equality (this is [24, (31)]). For
i ?Mc, s?i are determined using (2.27). We now address how to compute the optimal value
for si ? [0,1], i ?M, which satisfy (2.6) and (2.7) with equality. Define the set
D =
braceleftbigg
?s
vextendsinglevextendsingle
vextendsinglevextendsingle(?s,?t) = arg max
(s,t)??
J(??,??,s,t),
Ksummationdisplay
k=1
?skP(H1k|Qk)Wk = ?
bracerightbigg
; (2.36)
recall also from (4.26) that max(s,t)?? J(??,??,s,t) = g(??,??). Let ?pk = p?k as in (2.24)
with ? = ?? and ? = ??. Then for ?s ? D, the corresponding ?t are determined as ?tk = ?sk?pk,
k = 1,2,...,K, and ??g(??,??)|?s = Ptot ? summationtextKk=1 ?sk?pk is a subgradient of g(?,?) w.r.t. ?
evaluated at (??,??). We must have one of these subgradients w.r.t. ? to be 0 since (??,??)
denotes the optimal solution which minimizes g(?,?). Let
s1 = argmaxs?D
braceleftBigg
Ptot ?
Ksummationdisplay
k=1
sk?pk
bracerightBigg
(2.37)
s2 = argmins?D
braceleftBigg
Ptot ?
Ksummationdisplay
k=1
sk?pk
bracerightBigg
. (2.38)
Then we have ??g(??,??)|s1 ? 0 and ??g(??,??)|s2 ? 0. Therefore, there exists a scalar ?,
0 ? ? ? 1, such that for s? = s1 + ?(s2 ?s1), we have s? ? D and ??g(??,??)|s? = 0. Let
t?k = s?k?pk, k = 1,2,...,K. Since (s?,t?) is a solution to argmax(s,t)?? J(??,??,s,t), primary
feasible, and satisfies (2.6) and (2.7) with equality, it is an optimal solution of problem P2.
Let p?k = ?pk for s?k > 0 and let p?k = 0 for s?k = 0. This yields optimal primary variables
s? = s? and p?.
Remark 1. In the preceding developments, we have provided an algorithm to find an
optimal solution for p and s. This optimal solution is not necessarily unique even though
the solution for t is unique as stated in (2.23). Note that t?k in (2.23) is a function of yet
to be determined sk. For example, consider a special case in which all secondary links have
the same channel gain and the channel available probabilities over all sub-channels are the
same. By (2.23), this would lead to identical t?k/sk (for sk negationslash= 0), k = 1,2,??? ,K. After
23
the optimal dual variables ?? and ?? are obtained, the optimal power allocation for each
sub-channel will be as in (2.24), identical for k = 1,2,??? ,K. Then the remaining task is
to determine the optimal channel access probabilities s?k. If ?? = 0, then it implies that the
interference constraint is not active, therefore the secondary BS can access all sub-channels
with probability 1. For the case where ?? > 0, (2.6) should be satisfied with equality.
Since p?k are the same for all sub-channels, all s ? D (D is defined in (2.36)) should satisfy
summationtextK
k=1 skp
?
k = Ptot. Therefore, any s ? D is an optimal solution. Thus, p
?
k is unique whereas
s?k may not be. square
Actual throughput (instantaneous capacity) of the secondary network is then given by
Ksummationdisplay
k=1
1H0ks?kWk log2(1 +p?kuk) + 1H1ks?kWk log2(1 +p?kvk) (2.39)
where 1{?} is an indicator function which takes value 1 when {?} is true and 0 when {?} is
false. Actual interference to primary users caused by secondary BS?s transmission is given
by
Ksummationdisplay
k=1
1H1ks?kWk. (2.40)
Remark 2. We notice from (2.27) that most channel access probabilities turn out to be
0 or 1. However, this is still different from hard decision spectrum sensing where one makes
decisions solely based on spectrum sensing statistics. In our proposed algorithm, channel
access decisions are make jointly with power allocation based on channel availabilities as well
as secondary channel conditions. Eqn. (2.27) shows that even when two channels have exactly
the same busy and idle probabilities P(H0k|Qk) and P(H1k|Qk), the one with better channel
condition (uk and vk defined in (2.4)) is more likely to be accessed by the secondary BS. Also,
from (2.24) we observe that the optimal power allocation is a function of the continuous-
valued sensing statistics Qk for k = 1,2,...,K, which is different from hard spectrum sensing
24
where all that matters is whether the sufficient sensing metric is larger than a threshold or
not. square
This optimal algorithm is summarized as follows:
ALGORITHM OPT
1) Initialize ?l,?u and set ? = (?l +?u)/2.
2) While (??g(?,?) negationslash= 0)
Calculate ?? given the current ? according to Proposition 2.1.
if |M| = 1, calculated p?k and s?k as in (2.24) and (2.27).
if |M| > 1, find s1 and s2 as in (2.37) and (2.38) and let s? = s? = s1 +?(s2?s1),
where ? is chosen such that (2.6) and (2.7) are satisfied with equality.
If (??g(?,?) = Ptot ?summationtextKk=1 s?kp?k) > 0, set ?u = ?; otherwise set ?l = ?.
End while
3) Set p?k =
parenleftBig d
k
2ukvk
parenrightBig+
for s?k > 0, and p?k = 0 for s?k = 0.
2.4 Alternative Formulation with Individual Interference Constraints
In Sec. 2.2.3 we define the interference to primary network (see (2.6)) to be the total
fraction of bandwidth that is corrupted by secondary network?s transmission. Here we con-
sider the case where the interference over each sub-channel is also required to be bounded.
This is realistic when a primary user may occupy no more than one sub-channel simultane-
ously. In this case we need to guarantee that no single primary user is interfered with too
much. Let ?k be the interference bound for sub-channel k. Then the soft-decision spectrum
sensing based optimization problem with an individual interference constraint is formulated
25
as follows:
P3 : maxt,s
Ksummationdisplay
k=1
R(sk,tk) (2.41)
s.t.
Ksummationdisplay
k=1
I(sk) ? ? (2.42)
I(sk) ? ?k, k = 1,2,...,K (2.43)
Ksummationdisplay
k=1
tk ? Ptot (2.44)
0 ? sk ? 1, tk ? 0, k = 1,2,??? ,K. (2.45)
In order for (2.42) and (2.43) to make sense, we assume that summationtextKk=1 ?k > ? else (2.42) is
redundant.
From (2.3), we notice that the individual interference constraint (2.43) is actually an
upper bound of each channel access probability. Let ??k = min{1, ?kP(H
1k|Qk)Wk
}. Then problem
P3 becomes
P4 : maxt,s
Ksummationdisplay
k=1
R(sk,tk) (2.46)
s.t.
Ksummationdisplay
k=1
I(sk) ? ? (2.47)
Ksummationdisplay
k=1
tk ? Ptot (2.48)
0 ? sk ? ??k, tk ? 0, k = 1,2,??? ,K. (2.49)
Similar to problem P1, this is also a convex problem which can be solved by a Langrangian
dual formulation. Let ? and ? be the dual variables associated with constraints (2.47) and
(2.48) respectively. Then for a given ? and ? the optimal power allocation has a same
26
expression as in (2.24). The optimal channel access probabilities become
s?k =
?
???
???
???
???
??k if ?k ??P(H1k|Qk)Wk > 0
0 if ?k ??P(H1k|Qk)Wk < 0
? [0, ??k] if ?k ??P(H1k|Qk)Wk = 0,
(2.50)
where ?k is as in (2.26). Then corresponding to (2.28), we have
gk(?,?) = Jk(?,?,s?k,t?k) = ??k (?k ??P(H1k|Qk)Wk)+ . (2.51)
Define ??k = ??k?k and ?ak = ??kP(H1k|Qk)Wk. Then (2.51) becomes gk(?,?) = (??k ? ?ak)+.
The solution for ?? follows from Proposition 2.1 by substituting ?k and ak with ??k and ?ak,
respectively. And ?? can be found by a line search as in the non-individual interference
constraint case discussed earlier in this section.
2.5 Heuristic Algorithms
We notice that in Sec. 2.3 the computational complexity of the optimal solution is
relatively high because in order to find the optimal dual variable ?, a line search is employed
and within each iteration of this search, sorting metric ?k, k = 1,2,...,K is required to
obtain subchannel access probabilities s, which has a complexity of O(K logK). In order to
reduce the computational complexity, we propose two heuristic algorithms. Both first solve
for the access probabilities sk, k = 1,2,...,K, and then allocate power. The details of these
algorithms are provided next.
ALGORITHM 1
1) Allocate equal power pk = Ptot/K to each subchannel.
27
2) Calculate the rate of each subchannel as R(pk,sk). Sort the rate vector in non-
increasing order of magnitude and re-order P(H1k|Qk)Wk accordingly.
3) For k = 1 : K
Set s?k = 1 if summationtextki=1 P(H1i|Qi)Wi ? ?,
otherwise set s?k = ??
summationtextk?1
i=1 P(H1i|Qi)Wi
P(H1k|Qk)Wk , and if k < K, break after setting
s?j = 0 for j = k + 1,...,K.
end for.
4) Using s?k, k = 1,2,...K, obtained from Step 3 and following a similar way as in Sec.
2.3, optimal power allocation is given as follows: p?k =
parenleftBig d
k
2ukvk
parenrightBig+
if s?k > 0, and p?k = 0
otherwise. The parameter ? is selected to satisfy summationtextKk=1 s?kp?k = Ptot; from (2.21)-(2.22),
dk is a function of ck which, in turn, is a function of ?.
The sum rate of Algorithm 1 is
Ksummationdisplay
k=1
s?kWk [P(H0k|Qk)log2(1 +p?kuk) +P(H1k|Qk)log2(1+p?kvk)]
and the actual rate is calculated as in (2.39). In Algorithm 1, the access probabilities s?k,
k = 1,2,...,K, are determined based on equal power allocation. The subchannels with larger
expected transmission rate are more likely to used by the secondary BS.
In Algorithm 1, when the secondary BS chooses subchannels, each subchannel?s proba-
bility of interference is not taken into consideration. To remedy this drawback, we propose
Algorithm 2.
ALGORITHM 2
1) Allocate power pk = Ptot/K to each subchannel.
28
2) Calculate the rate-to-interference ratio for each subchannel as R(pk,sk)/I(sk). Sort
the rate-to-interference ratio vector in non-increasing order of magnitude, and re-order
P(H1k|Qk)Wk accordingly.
3) Repeat Step 3 of Algorithm 1.
4) Repeat Step 4 of Algorithm 1 using the results of the Step 3 of Algorithm 2.
The sum rate of Algorithm 2 is
Ksummationdisplay
k=1
s?kWk [P(H0k|Qk)log2(1 +p?kuk) +P(H1k|Qk)log2(1+p?kvk)]
and the actual rate is calculated as in (2.39).
The complexity of these two heuristic algorithms is comprised of two parts. The first
part comes from sorting rate or rate-to-interference ratio, which is O(K logK), and the
second part is from allocating power using the pre-determined channel access probabilities.
This can be solved using a line search over ? to satisfy the transmit power constraint. A
(crude) comparison based on computer simulation run time of these two heuristic algorithms
as well as the optimal algorithm is presented in Sec. 2.8.
2.6 Implementation issues
In this section we discuss some implementation issues that arise in our algorithms.
In the proposed soft-decision cooperative sensing based algorithms one requires knowledge
of conditional distribution of sensing statistics f(Qik|Hjk), i ? Ak, k = 1,2,...,K, j ?
{0,1}, interference power from primary users? transmission at secondary users ppuk|hpk|2,k =
1,2,...,K and the CSI hk between secondary BS and users. In this section we briefly discuss
possible solutions to how this knowledge can be obtained. We assume that the CSI among
the secondary links in the secondary network can be acquired as in [17] (and others) ?by
29
a feedback link from receiver to transmitter, or just exploiting channel reciprocity when
transmitter and receiver transmit over the same band.? [Note that a feedback link is also
needed to transmit the sensing metric to the secondary BS.] Knowledge of this CSI is also
critical to the hard decision approaches of [26] and [17], as it is needed to calculate the
needed instantaneous capacity. The CSI information among various nodes is also required
in [16,27]. The nature of prior knowledge needed regarding the CSI between SUs and PU
(hpk and hpi,k in our notation) depends upon the nature of the spectrum sensing metric. This
aspect is discussed next.
2.6.1 Obtaining primary user related prior knowledge
Although our algorithms are open to the selection of sensing techniques, suppose that we
use the energy detector (sole choice in [26] and [17], and also used in our simulations in Sec.
2.8). For energy detector, the sensing metric Qik follows the following normal distributions
under both hypotheses [26]:
f(Qik|H0k) ?N
parenleftBig
M?20,2M?40
parenrightBig
(2.52)
f(Qik|H1k) ?N
parenleftBig
M(?20 +ppuk|hpi,k|2),2M?20(?20 + 2ppuk|hpi,k|2)
parenrightBig
(2.53)
where N(?,?2) denotes a Gaussian distribution with mean ? and variance ?2, M is the
number of signal samples collected in the sensing phase, ?20 is the noise power and hpi,k is
the channel gain between the primary user and the secondary user i over subchannel k. In
this case, as noted in [26] (see also [20]), one only needs estimates of ?20 and ?20 +ppuk|hpi,k|2,
representing received signal power under no PU transmission and PU transmission, respec-
tively. These entities can be estimated a priori during the (confirmed) periods the PU is
known to be inactive or active which would allow computation of the probabilities in (2.52)
and (2.53).
30
For more general sensing metric Qik, one may need knowledge of hpi,k which would
require cooperation between the primary and secondary networks in the sense that PU?s
may need to transmit training sequences to enable CSI acquisition.
2.6.2 Effect of Imperfect Channel Estimation
Due to the time-varying nature of the wireless channel and noise, error is always present
in the CSI estimation results. This will result in performance degradation for both primary
and secondary network. An analysis of this aspect is outside the scope of this chapter
but we have investigated this aspect to some extent via simulations in Sec. 2.8 where we
have evaluated the imperfect CSI effect on the system performance by modeling the channel
estimation errors as zero-mean Gaussian random variables. For a single source-destination
pair, let h and ?h denote the true and estimated channel coefficient (scalar, flat fading),
respectively, and let e be the channel estimation error. Then we have (Nc denotes complex
Gaussian distribution)
?h = h+e, e ? Nc(0,?2h) (2.54)
(In Sec. 2.8 we have used ?2h = a|h|2 for a = 0.02 or 0.10.) Due to these channel estimation
errors, secondary network?s throughput will be affected. Also interference to primary users
may exceed the design interference bound since the calculation of channel a posteriori prob-
abilities is related to the channel gain between primary transmitter and secondary spectrum
sensor. The simulation results show that the throughput of secondary network is not sen-
sitive to channel estimation error, however, the interference to primary network can exceed
the interference bound.
Another observation is when the cost of sending CSI of secondary links and sensing
metrics of sub-channels back to BS is too expensive and becomes a major implementation
obstacle, we can quantize the CSI between secondary-links into fewer bits since our algo-
rithms are more robust to this kind of error than to the errors in the sensing metric. In
31
general, the cost of feeding back the required variables is likely to be ?small.? In the feed-
back link from the i-th SU to the secondary BS, one needs to send the estimated CSI hk and
the sensing metric Qik (with its associated distribution parameters) for the k-th subchannel,
once every time slot. For the energy detector, the sensing metric Qik is the received energy
(a real number) and the associated parameters are the two means and two variances under
hypotheses H0k and H1k (all real numbers), whereas the CSI hk is complex-valued. Thus
six scalars (one complex and five reals) are required to be sent over the feedback link once
every time slot. This is to be contrasted with the number of data samples (symbols) per slot
in each subchannel during direct transmission. In our simulations we used M = 50 signal
samples (symbols) per time slot for sensing. Assuming that we have ?150 symbols in the
information transmission phase (total ?200 symbols per slot), it is seen that the feedback
overhead is just 6 samples per slot versus ?200 samples in direct transmission, leading to
commensurate bandwidth requirements. A detailed comparison would depend upon actual
implementation including number of bits per sample.
2.7 Comparison with Hard Decision based Spectrum Sensing
For comparison, we also consider hard decision spectrum sensing and power adaptation.
An approach is available in [17], which in turn extends the approach of [26]. Both [17]
and [26] consider energy detectors for spectrum sensing; the case of more general detectors
is yet unsolved. Therefore, we will assume that energy detector is used and let Qik denote
the sensing statistic (received energy) at secondary user i over subchannel k. Then for each
subchannel k, the sensing metric is given by
Tk = summationdisplay
i?Ak
Qik. (2.55)
Assume Qik follows a normal distribution N(?(j)i,k,?(j)2i,k ) under Hjk, j ?{0,1}. Then we have
Tk ?N(summationtexti?Ak ?(j)i,k,summationtexti?Ak ?(j)2i,k ) under hypothesis Hjk, j ?{0,1}.
32
Let ?sk be the access action to subchannel k. Then in hard spectrum sensing based on
the test threshold ?, we have the hard decision
?s?k =
??
??
??
1 if Tk ? ?
0 if Tk < ?;
(2.56)
therefore, optimizing ?sk, k = 1,2,...,K, is actually optimizing ?. [Note that under the
null hypothesis of only noise at any sensing receiver, the sensing statistic Qik is identically
distributed over every sensor i and every subchannel k if the number of samples used at each
sensor are identical and the noise variance is identical in each subchannel and receiver. This
allows for consideration of identical threshold ? for each subchannel. This is the case for
the numerical example considered in Sec. VI.] Hence joint optimization of spectrum access
parameter ?sk, k = 1,2,...,K and power allocation pk, k = 1,2,...,K, can be formulated as
follows:
P5 : maxp,? ?R(?,p) (2.57)
s.t.
Ksummationdisplay
k=1
PH1k (1?Pdk(?))Wk ? ? (2.58)
Ksummationdisplay
k=1
PH0k (1?Pfa(?))pk +PH1k (1?Pdk(?))pk ? Ptot (2.59)
pk ? 0, k = 1,2,...,K, (2.60)
where
?R(?,p) = Ksummationdisplay
k=1
PH0kWk (1?Pfa(?))log2(1 +pkuk)
+PH1kWk (1?Pdk(?))log2(1 +pkvk) (2.61)
and Pfa (the same for all subchannels) and Pdk denote the probability of false alarm and
detection, respectively, for subchannel k. Notice that Pfa and Pdk are functions of ? rather
than sensing statistics Qk. While ?sk will still be determined by Qk from (2.56), the optimal
power p?k will be a function of only the subchannel state and ?. This problem set-up is
33
patterned after [17] (modified to accommodate our interference bound constraint), and the
optimal solution to this optimization problem can be obtained by following [17]. [Note
that [17] extends the results of [26] to include power allocation.]
For a fixed threshold ?, the value of Pfa(?) and Pdk(?) are fixed, where ?(j)i,k, ?(j)i,k,
i ? Ak, k = 1,2,...,K, j ? {0,1} are known. Define the feasible region of ? as F =
braceleftBig
?
vextendsinglevextendsingle
vextendsingle summationtextKk=1 PH1k(1?Pdk(?))Wk ? ?
bracerightBig
. Then for any ?0 ?F, optimal power allocation can be
obtained by solving
P6 : maxp ?R(?,p) (2.62)
s.t.
Ksummationdisplay
k=1
PH0k (1?Pfa(?0))pk +PH1k (1?Pdk(?0))pk ? Ptot (2.63)
pk ? 0, k = 1,2,...,K. (2.64)
Similar to Sec. 2.3, the optimal solution is as follows:
p?k =
?
???bk +
radicalBig?
b2 ?4?ek?ck
2?ek
?
?
+
, (2.65)
where
?ek = ukvk??(PH0k (1?Pfa(?0)) +PH1k (1?Pdk(?0)))ln2 (2.66)
?bk = ??[PH
0k (1?Pfa(?0)) +PH1k (1?Pdk(?0))](uk +vk)ln2
?[PH0kWk (1?Pfa(?0)) +PH1kWk (1?Pdk(?0))]ukvk (2.67)
?ck = ??[PH0k (1?Pfa(?0)) +PH1k (1?Pdk(?0))]ln2
?PH0kWk (1?Pfa(?0))uk ?PH1kWk (1?Pdk(?0))vk, (2.68)
34
and ?? is chosen to satisfy summationtextKk=1 PH0k (1?Pfa(?0))p?k + PH1k (1?Pdk(?0))p?k = Ptot. The
optimal sensing threshold ?? can be obtained by solving
P7 : max? ?R(?,p?) (2.69)
s.t. ? ? F. (2.70)
Problem P7 can be solved by a line search over ?.
Thus, in each time slot, the optimal spectrum access actions are given by s?k = 1 if
Tk ? ?? and s?k = 0 if Tk < ??, for k = 1,2,...,K. The optimal power allocation is given
by (2.65). The actual transmission rate (bound) we obtain in each time slot using this hard
spectrum sensing method is given by
?Ra = Ksummationdisplay
k=1
1H0ks?kWk log2(1 +p?kuk) + 1H1ks?kWk log2(1 +p?kvk). (2.71)
The interference to primary user is
Ksummationdisplay
k=1
1H1ks?kWk. (2.72)
2.8 Simulation Examples
Now we provide numerical results using the algorithms proposed in Secs. 2.3 and 2.5.
and compare them with hard sensing algorithm in Sec. 2.7. Secondary users associated with
K = 16 subchannels are located within a circular ring area with radii between 200 m to 1 km,
while the secondary BS is located at the center. The path-loss exponent for large-scale fading
is set to be 3.5. The secondary BS transmits to secondary users through K subchannels; recall
that, in this chapter, we are considering a downlink scenario for the secondary network. We
assume that the subchannels between the secondary BS and secondary users experience flat
Rayleigh fading. The channel fading coefficients remain constant during each time slot and
change independently from slot to slot. The ?sum? received SNR of K secondary receivers
35
is defined to be
sum received SNR = Ptot
parenleftBigg 1
K
Ksummationdisplay
k=1
E{|hk|2}
N0Wk
parenrightBigg
(2.73)
where E{|hk|2} is calculated according to the kth subchannel?s path-loss. The randomness
of hk, k = 1,2,...,K is caused by Rayleigh fading.
We assume that the secondary BS employs energy detector as in [26]; (note that our
algorithms are open to the selection of sensing techniques). Assume the primary users?
transmit power ppuk is normalized to 1 in all K subchannels. Then in each time slot, the
sensing statistics Qik for i ? Ak and k = 1,2,...,K, given primary users? behavior on the
k-th subchannel, follows a normal distribution given by (2.52) under H0k, and under H1k we
have
f(Qik|H1k) ?N
parenleftBig
M(?20 +|hpi,k|2), 2M?20(?20 + 2|hpi,k|2)
parenrightBig
(2.74)
Considering that sharing each secondary user?s sensing statistics Qik with BS can be ex-
pensive, we assume only 4 out of 16 users will transmit their Qik, k = 1,2...,K, to the
BS.
In simulations, we use M = 50, and (?normalized?) Wk = 1 and N0 = 1 leading to
?20 = WkN0 = 1. The subchannel power gain gpi,k := |hpi,k|2 from transmitting primary user
to secondary user i over subchannel k follows an exponential distribution with mean ?gpi,k.
The received SNR of PU?s signals, averaged over all secondary users? and BS?s receivers, is
given by 1K summationtextKk=1 1|A
k|
summationtext
i?Ak ppuk?gpi,k/?20. Finally, we assume that the primary users occupy
each subchannel with probability 0.5 during each time slot. All simulation results are based
on 1000 runs.
Figs. 2.1 and 2.2 show the sum transmission rate vs sum received SNR (? Ptot) for the
secondary network of 16 subchannels/secondary users for two different PU SNR?s when the
(total) PU interference bound ? is set to be 3% (by which we mean that ? = 0.03summationtextKk=1 Wk
in (2.6)); we will also refer to this as 0.03 fraction of bandwidth). Also, to better evaluate
36
the performance of our soft-decision sensing based algorithm, the actual throughput using
both soft and hard spectrum sensing is also presented in the following figures. Here the
actual throughput is calculated according to (2.39). It is seen from Figs. 2.1 and 2.2 that
the actual rate of the optimal algorithm agrees well with the theoretical value. Also, the soft
decision algorithm significantly outperforms the hard decision algorithm in both high PU
SNR (Fig. 2.2) and low PU SNR (Fig. 2.1) scenarios, and the heuristic Algorithm 2 (with
power control) has a near optimal performance. Even when the PU?s SNR is low (-20dB),
which implies that sensing is less accurate, the soft-decision sensing based optimal algorithm
still outperforms the hard sensing scheme.
10 12 14 16 18 200.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Sum Received SNR (dB)
Sum Rate (bps/Hz)
Interference bound 3%; Ave PU SNR=?10dB
 
 
Actual sum rate: soft decision
Actual sum rate: hard decision
Optimal
Alg.1 equal power
Alg.1 with power control
Alg. 2 equal power
Alg. 2 with power control
Figure 2.1: Sum transmission rate of secondary network versus sum received SNR (2.73)
at secondary receivers when 3% of the primary network?s bandwidth is interfered with by
secondary BS?s transmission and the received SNR of PU?s signal is -10dB. [The curves
labeled ?optimal,? ?Actual sum rate: soft decision,? and ?Alg. 2 with power control? are all
quite close to each other toward the top of the fig.]
Figs. 2.3 and 2.4 show the actual fraction of bandwidth which is interfered with by
secondary transmissions, when the bound ? is set to be 3% (0.03). Both soft and hard-sensing
algorithms have interferences close to the bound of 3%. We observe that the soft decision
algorithm causes a slightly higher interference than hard decision algorithm. However, given
37
10 12 14 16 18 20
0.2
0.3
0.4
0.5
0.6
0.7
Sum Received SNR (dB)
Sum Rate (bps/Hz)
Interference bound 0.03; Ave. PU SNR= ?20dB
 
 Actual sum rate: soft decision
Actual sum rate: hard decision
Optimal
Alg. 1 equal power
Alg. 1 with power control
Alg. 2 equal power
Alg. 2 with power control
Figure 2.2: Sum transmission rate of secondary network versus sum received SNR (2.73)
at secondary receivers when 3% of the primary network?s bandwidth is interfered with by
secondary BS?s transmission and the received SNR of PU?s signal is -20dB. [The curves
labeled ?optimal? and ?Alg. 2 with power control? are all quite close to eachother in the top
half of the figure.]
the fact that soft decision algorithm has a much higher feasible transmission rate than
hard decision algorithm, we can still conclude that using soft decision spectrum sensing can
significantly improve the performance of the secondary user network.
Fig. 2.5 shows that the the actual interference is close to the interference bound when
the SNR of PU?s signal is -10dB and the total received SNR of secondary users is 15dB.
From Fig. 2.6 we can observe that the actual rate of soft decision optimal algorithm is close
to the the optimal value and is significantly higher than hard decision algorithm. Also, the
heuristic Algorithm 2 has a near optimal performance.
Figs. 2.7 and 2.8 show the imperfect CSI effect where the channel estimation error
variance is set to be either 2% or 10% of the true channel power gain of its corresponding
channel (see (2.54) and discussion after it) and the PU SNR at secondary sensor is ?10dB.
It is observed that the secondary network throughput is not sensitive to channel estimation
error. However, the interference to primary users exceeds the 3% bound, marginally for
38
10 12 14 16 18 200
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Sum Received SNR (dB)
Actual Interference (fraction of bandwidth)
Interference bound 0.03; Ave PU SNR=?20dB
 
 
Soft decision
Hard decision
Figure 2.3: Actual total interference versus sum received SNR (2.73) at secondary receivers
when 3% of the primary network?s bandwidth is interfered with by secondary BS?s transmis-
sion and the received SNR of PU?s signal is -20dB
?2h = 0.02|h|2 but more significantly for ?2h = 0.10|h|2. This is because the primary signal
power at the spectrum sensor of secondary network is used in calculation of the a posteriori
channel probabilities. Figs. 2.9 and 2.10 show the results corresponding to Figs. 2.7 and 2.8
except that now the PU SNR at the secondary sensor is ?20dB. Compared to Figs. 2.7
and 2.8, now the effect of imperfect CSI on interference to PU network is much less severe.
[We have also investigated the case where with noise variance normalized to one, we used
?2h = 0.02 or 0.1 (i.e. it is not a function of |h|2); there was no significant difference from the
results shown in Figs. 2.7-2.10, and the results are not shown.]
Figs. 2.11 and 2.12 show the results when we use individual interference constraints
as discussed in Sec. 2.4. The total PU interference bound ? was set to be 3% while the
individual PU interference bound ?k was set to 5% (i.e. ?k = 0.05Wk in (2.43)) for each
of the 16 subchannels (k = 1,2,??? ,16). It is seen that under additional constraints, the
performance (sum rate) is poorer but not by much while the total interference is reduced.
39
10 12 14 16 18 200
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Sum Received SNR (dB)
Actual Interference (fraction of bandwidth)
Interference bound 3%; Ave PU SNR=?10dB
 
 
Soft decision
Hard decision
Figure 2.4: Actual total interference versus sum received SNR (2.73) at secondary receivers
when 3% of the primary network?s bandwidth is interfered with by secondary BS?s transmis-
sion and the received SNR of PU?s signal is -10dB.
Finally, the CPU times (secs) for 1000 runs on a computer with Intel Pentium(R) Dual-
Core CPU E5200@2.5GHz processor running under Windows 7 (professional) were 2.9267,
0.2083 and 0.1606 for the optimal algorithm, heuristic Algorithms 1 and 2, respectively, for
the results shown in Fig. 2.1.
2.9 Conclusions
We investigated joint optimization of cooperative spectrum sensing, channel access and
power allocation in an overlay multi-band cognitive radio network. Instead of making tradi-
tional hard binary decisions, a soft-decision cooperative spectrum sensing concept using the
continuous-valued sensing test statistics was considered to maximize the secondary users?
sum throughput while keeping the interference to primary users under a specified thresh-
old. The problem was shown to be a convex optimization problem and the Lagrangian
dual method was employed to obtain the optimal solution. We also provided an alternative
formulation where additionally interference to individual PUs was also constrained. Two
40
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Design interference bound
Actual Interference (fraction of bandwidth)
Ave PU SNR=?10dB; sum received SNR=15dB
 
 
Soft decision
Hard decision
Figure 2.5: Actual total interference versus design bound on total interference to primary
network when sum received SNR (2.73) at secondary receivers is 15dB and SNR of PU?s
signal is -10dB.
heuristic algorithms were also proposed to reduce the computational complexity. Simulation
results showed that our soft sensing based algorithm significantly outperforms traditional
hard decision sensing algorithms and one of the proposed heuristic algorithms (Algorithm 2
with power control) achieves a near optimal performance. Practical implementation issues
are also discussed, regarding obtaining channel state information of both secondary links
and primary to secondary links. The performance of our proposed optimal algorithm under
imperfect CSI was illustrated via simulations.
41
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Design interference bound
Sum Rate (bps/Hz)
Ave PU SNR=?10dB; sum received SNR=15dB
 
 
Actual sum rate: soft decision
Actual sum rate: hard decision
Optimal
Alg. 1 equal power
Alg.1 with power control
Alg. 2 equal power
Alg. 2 with power control
Figure 2.6: Sum transmission rate of secondary network versus design bound on total inter-
ference to primary network when sum received SNR (2.73) at secondary receivers is 15dB
and SNR of PU?s signal is -10dB.
0 5 10 15 200.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Sum Received SNR (dB)
Sum rate (bps/Hz)
Interference bound 3%; Ave PU SNR = ?10dB
 
 
Perfect CSI, soft decision ? optimal
Imperfect CSI, ?h2 = 0.02, soft decision
Imperfect CSI, ?h2 = 0.1, soft decision
Figure 2.7: Imperfect CSI: Sum transmission rate of secondary network versus sum received
SNR (2.73) at secondary receivers when up to 3% of the primary network?s bandwidth is
interfered with by secondary BS?s transmission and the received SNR of PU?s signal is -10dB.
Channel estimation error variance ?2h is set to be either 2% or 10% of the true channel power
gain of its corresponding channel (see (2.54)).
42
0 5 10 15 200
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Sum Received SNR (dB)
Actual Interference (fraction of bandwidth)
Interference bound 3%; Ave PU SNR = ?10dB
 
 
Perfect CSI, soft decision
Imperfect CSI, ?h2 = 0.02, soft decision
Imperfect CSI, ?h2 = 0.1, soft decision
Figure 2.8: Imperfect CSI: Actual total interference versus sum received SNR (2.73) at
secondary receivers when up to 3% of the primary network?s bandwidth is interfered with
by secondary BS?s transmission and the received SNR of PU?s signal is -10dB. Channel
estimation error variance ?2h is set to be either 2% or 10% of the true channel power gain of
its corresponding channel (see (2.54)).
0 5 10 15 200.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Sum Received SNR (dB)
Sum rate (bps/Hz)
Interference bound 3%; Ave PU SNR = ?20dB
 
 
Imperfect CSI, ?h2 = 0.1, soft decision
Perfect CSI, soft decision ? optimal
Imperfect CSI, ?h2 = 0.02, soft decision
Figure 2.9: Imperfect CSI: Sum transmission rate of secondary network versus sum received
SNR (2.73) at secondary receivers when up to 3% of the primary network?s bandwidth is
interfered with by secondary BS?s transmission and the received SNR of PU?s signal is -20dB.
Channel estimation error variance ?2h is set to be either 2% or 10% of the true channel power
gain of its corresponding channel (see (2.54)).
43
0 5 10 15 200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Sum Received SNR (dB)
Actual Interference (fraction of bandwidth)
Interference bound 3%; Ave PU SNR = ?20dB
 
 
Imperfect CSI, ?h2 = 0.1, soft decision
Perfect CSI, soft decision
Imperfect CSI, ?h2 = 0.02, soft decision
Figure 2.10: Imperfect CSI: Actual total interference versus sum received SNR (2.73) at
secondary receivers when up to 3% of the primary network?s bandwidth is interfered with
by secondary BS?s transmission and the received SNR of PU?s signal is -20dB. Channel
estimation error variance ?2h is set to be either 2% or 10% of the true channel power gain of
its corresponding channel (see (2.54)).
0 5 10 15 200
0.2
0.4
0.6
0.8
1
1.2
1.4
Sum Received SNR (dB)
Sum rate (bps/Hz)
Interference bound 3%; Ave PU SNR = ?10dB
 
 
Total interference constraint
Total and individual interference constraint
Figure 2.11: Individual PU interference constraint: Sum transmission rate of secondary
network versus sum received SNR (2.73) at secondary receivers when up to 3% of the primary
network?s total bandwidth and up to 5% of individual subchannel bandwidths (total 16
subchannels) are interfered with by secondary BS?s transmission and the received SNR of
PU?s signal is -10dB.
44
0 5 10 15 200
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Sum Received SNR (dB)
Actual Interference (fraction of bandwidth)
Interference bound 3%; Ave PU SNR = ?10dB
 
 
Total interference constraint
Total and individual interference constraint
Figure 2.12: Individual PU interference constraint: Actual total interference versus sum
received SNR (2.73) at secondary receivers when up to 3% of the primary network?s total
bandwidth and up to 5% of individual subchannel bandwidths (total 16 subchannels) are
interfered with by secondary BS?s transmission and the received SNR of PU?s signal is -10dB.
45
Chapter 3
Precoding Design in Multi-User Cognitive Relay Network: a MMSE Approach
In this chapter, the cognitive radio network employs a relay station to facilitate the
transmissions of multiple users. These secondary users intend to transmit data streams to
multiple partners, therefore form a multi-way transmission, which is a generalization of multi-
user two-way relay network. In designing this cognitive radio network with multiple users,
how to manage interference is of significant importance since interference not only exists
between primaryandcognitive radio networks, but alsoamong multiple cognitive users. With
the help of multiple antennas at transmitters and receivers, proper precoding/beamforming
design can cancel inter-network (between primary and secondary network) interference and
inter-user interference. In this chapter, an MSE-based joint source and relay precoding
algorithm is proposed for a multi-user multi-way cognitive radio network.
This chapter is organized as follows. Sec. 3.1 introduces the background of precoding
design in relay and cognitive radio networks. In Sec. 3.2 we introduce the system model
and formulate the optimization problem. Then this problem is decomposed into convex
subproblems and solved iteratively in Sec. 3.3. To reduce the complexity, we also propose
an non-iterative algorithm in Sec. 3.4. In Sec. 3.5, a robust design is proposed to consider
the situation when only imperfect CSI is available, based on our non-iterative algorithm.
Simulation results are presented in Sec. 3.6 to illustrate the effectiveness of our algorithms.
Finally, conclusions are drawn in Sec. 3.7 and some technical details are provided in the
Appendix.
46
3.1 Introduction
In cognitive radio network, when SUs concurrently access the spectrum with licensed pri-
mary users, how to mitigate interference to PUs becomes a crucial design issue of secondary
network. By employing multiple antennas at the secondary transmitters, interference to PUs
can be potentially nulled out by designing precoders provided the channel state information
(CSI) between the relevant source-destination pair is known. On the other hand, multi-
ple input multiple output (MIMO) technology can further improve spectrum efficiency by
providing multiplexing gain. However in multi-user MIMO systems, the performance is usu-
ally limited by inter-user interference which can be mitigated by system precoding/decoding
designs. In traditional multi-user MIMO systems, many interference alignment (IA) algo-
rithms have been proposed which focus on eliminating (or minimizing) the interference. For
example, in [34] inter-user interference is minimized by adjusting transmitting and receiving
?directions?. A similar transmitting and receiving directions based iterative algorithm has
been proposed in [35] which also considers the received signal?s directions. In this chapter
we adopt mean square error (MSE) of the decoded signal as the design criterion rather than
residual interference because by minimizing the MSE, desired signal?s strength is also taken
into consideration. In [36], a duality based robust transceiver design method is proposed for
multi-user MIMO system using MMSE (minimum MSE) as the design criterion, while in [37]
optimal receivers are designed using the MMSE criterion for a downlink multi-user MIMO
system to maximize the weighted sum rate.
Considerable research has been done regarding the transceiver design in CR MIMO sys-
tems. A rate balanced transceiver design for multiuser cognitive system is studied in [38]
where fairness among users is considered. In [39], a linear precoding method was proposed
for CR multiuser downlink MIMO system based on MMSE. For systems with knowledge of
the CSI at the transmitter, a nonlinear transceiver design problem in a multi-tier MIMO
cognitive radio network has been investigated in [53] while a linear transceiver design in a
downlink cognitive MIMO system has been proposed in [54]. All these previous works deal
47
with one-way transmission. For a network where multiple users have multiple data streams
for each other, a relay station could be employed to enable multi-user two-way transmission.
Moreover, introduction of the relay node can further improve spectrum efficiency. Therefore,
in this chapter, we employ an amplify-and-forward relay node in the secondary network to
support multi-user transmission. For non-cognitive radio relay systems, optimal beamform-
ing matrix at the relay node has been designed in [40] where capacity region for a two-user
two-way system has also been derived. An iterative source and relay precoding algorithm
for two-user two-way MIMO relay system has been proposed in [41] using the MMSE crite-
rion. Precoder design in a system where one pair of users are assisted by multiple relays has
been considered in [42,43]. While in two-user two-way relay system inter-user interference
can be completely removed, this kind of interference can significantly degrade the perfor-
mance of multi-user two-way relay system. For multi-user two-way relay systems, design of
MIMO relay beamforming matrix based on zero-forcing (ZF) and MMSE criteria has been
investigated in [44].
In this chapter, we develop a precoding and decoding matrices design algorithm in
a multi-user multi-way relay MIMO system for cognitive radio using MSE as our design
criteria. Due to the non-convexity of this problem, an iterative algorithm is proposed to
alternately optimize precoding matrices at secondary transmitters and the relay station, and
decoding matrices at secondary receivers. Our object is to minimize the sum MSE of all users
under transmit power constraints at each secondary transmitter as well as the relay station,
while interference to the primary user is nulled out by proper precoding matrices design. A
matrix distance based non-iterative algorithm is also proposed to reduce the computational
complexity.
We also consider the effect of imperfect CSI. In practical system, exact CSI may be
difficult to obtain due to the time-varying nature of channels and the cost of information
feedback from the receiver to transmitter. Therefore designing a robust precoder which is
insensitive to CSI errors maybe a more practical choice. A robust relay precoder design of a
48
two-hop amplify-and-forward(AF) MIMO system with multiple relays under imperfect CSI
has been proposed in [45]. Joint relay precoder and destination receive filters designs in
a nonregenerative MIMO relay network have been investigated in [46] under two models of
CSI errors: stochastic error and norm-bounded error. For two-way relay systems, transceiver
design for a three node two-way relay system under imperfect CSI has been considered
in [47]. In this chapter, we model the errors in channel coefficients as additive Gaussian
random variables which have zero mean and known variance. With this uncertainty in the
CSI, the expected sum MSE over channel coefficient errors becomes our objective function.
Also, interference to the PUs can not be eliminated completely. Therefore the constraint on
interference to PU now becomes keeping the mean interference power less than a threshold.
Notation: We use bold lower-case and upper-case letters to denote vectors and matrices,
respectively, I is the identity matrix, vec{.} is the vectorization operator, ? is the Kronecker
product, and(.)T,(.)?,(.)? and(.)?1 denote thetranspose, conjugate transpose, conjugateand
inverse of a matrix, respectively. We use bardbl.bardbl and bardbl.bardblF to denote the 2-norm and Frobenius
norm, respectively, tr(.) and rank(.) are the trace and rank of a matrix, R{.} indicates the
real part of a complex number, E(.) is the expectation operation, {A}mn denotes the mn-th
element of matrix A, andCm?n is the space of m by n complex matrices. ?Subject to?, ?with
respect to? and ?quality of service? are abbreviated as s.t., w.r.t. and QoS, respectively.
3.2 System Model and Problem Formulation
Consider a cognitive radio network consisting of K secondary users and one non-
regenerative relay station. The secondary users intend to send information to each other
through the help of the relay node. Each secondary user is equipped with M antennas while
the relay station has N antennas. A two time slot half-duplex transmission scheme is em-
ployed here. In the first time slot all of the K users who have information to send transmit
to the relay node while in the second time slot relay sends a linear combination of its received
signal from the first time slot, and K secondary users listen and decode their desired signals,
49
as shown in Fig. 3.1. This secondary network coexists with a primary source-destination
pair within a single band. Assume primary transmitter and receiver are equipped with Jp
and Mp antennas respectively. We assume that the network operates in a flat-fading envi-
ronment. We assume that the signals transmitted from different SUs and PUs, and noise at
all receivers, are all mutually statistically independent. Suppose that the i-th SU transmits
CR: Data stream
CR: First time slot transmission
CR: Second time slot transmission
CR: i-th Secondary user
CR: Relay station
Primary transmission
uni0053uni0052
uni0053uni0065uni0063uni006Funi006Euni0064uni0061uni0072uni0079uni0020uni004Euni0065uni0074uni0077uni006Funi0072uni006B
uni0050uni0074
uni0050uni0072
uni0050uni0072uni0069uni006Duni0061uni0072uni0079uni0020uni004Euni0065uni0074uni0077uni006Funi0072uni006B
uni0031
uni002E
uni002E
uni004D
uni0031
uni002E
uni002E
uni004D
uni0031
uni002E
uni002E
uni004D
uni0031
uni002E
uni002E
uni004D
uni0031
uni002E
uni002E
uni004D
uni0053uni0052
uni0031
uni002E
uni002E
uni004E
uni0031
uni002E
uni002E
uni002E
uni004Auni0070
uni0031
uni002E
uni002E
uni002E
uni004Duni0070
uni0053uni0034
uni0053uni0033
uni0053uni0032
uni0053uni0031
uni0053uni0035
uni0053uni0069
Figure 3.1: System set-up
di independent data streams. Then we will denote the i?th SU?s transmitted signal vector
as si = [si1,si2,...,sidi]T, i = 1,2,...K. Let ?si = [?si1,?si2,...,?si?di]T, i = 1,2,...K, denote the
50
desired signal vector at i-th SU (acting as receiver), where ?di is the number of desired data
streams of user i. [Note that ?di is not necessarily equal to di but summationtextKi=1 ?di = summationtextKi=1 di.] For
each secondary user i, the data streams in ?si can be from multiple secondary transmitters.
Therefore this is a multi-user multi-way transmission scheme [49?51]. Let
?s = [?sT1 ,?sT2 ,...,?sTK]T and s = [sT1 ,sT2 ,...,sTK]T. (3.1)
Then we have the relationship between transmitted signal and desired signal as
?s = Es (3.2)
where E is a permutation matrix.
Assume that the transmitted signal s satisfies E{ss?} = ?2sI. Let Hir ? CN?M and
Hri ? CM?N, i = 1,2,...,K denote the channel coefficient matrices from user i to relay
station and from relay to user i. Also let Hip ? CMp?M, Hpr ? CN?Jp, Hrp ? CMp?N and
Hpi ?CM?Jp, i = 1,2,...,K denote the channel coefficient matrices from secondary user i to
primary receiver, from primary transmitter to relay, from relay to primary receiver and from
primary transmitter to secondary user i, respectively. Assume that these channel coefficients
remain constant during one time slot and are known to the secondary network.
In the first time slot, let Ti ?CM?di be the precoding matrix at the i-th secondary user.
Then the transmitted signal from the i-th user is xi = Tisi. Secondary users? transmission
will cause interference to primary receiver, which is given by
i1 =
Ksummationdisplay
i=1
HipTisi . (3.3)
Let xp ?CJp?1 denote the signal vector sent from primary transmitter during the first time
slot withE{xpx?p} = ?2pI, and let nr denote complex white Gaussian noise at the relay station
with zero mean and covariance E{nrn?r} = ?2nI. Then the received signal at the relay station
51
is given by
yr =
Ksummationdisplay
i=1
HirTisi +Hprxp +nr. (3.4)
In the second time slot, the relay station transmits a linear combination of its received signal
yr, using a precoding matrix Tr ?CN?N :
xr = Tryr =
Ksummationdisplay
i=1
TrHirTisi +TrHprxp +Trnr. (3.5)
The interference to primary receiver caused by relay?s transmission is
i2 = Hrpxr = HrpTr
parenleftBigg Ksummationdisplay
i=1
HirTisi +Hprxp +nr
parenrightBigg
. (3.6)
Let ni denote the complex white Gaussian noise at secondary user i with zero mean and
covariance E{nin?i} = ?2nI and let ?xp ? CJp?1 denote the signal vector sent by the primary
transmitter during the second time slot with E{?xp?x?p} = ?2pI. Then the received signal at
i-th secondary user is given by
yi = Hri
?
?
Ksummationdisplay
j=1
TrHjrTjsj +TrHprxp +Trnr
?
?+Hpi?xp +ni (3.7)
Each secondary user i can subtract its own signal from yi. Then the self-interference free
received signal at user i is
?yi = Hri
?
?
Ksummationdisplay
j=1,jnegationslash=i
TrHjrTjsj +TrHprxp +Trnr
?
?+Hpi?xp +ni. (3.8)
To ensure that the secondary network?s transmission does not interfere with the primary
network, we need i1 = i2 = 0 for every possible si, i = 1,2,..,K. This can be satisfied by
52
letting
HipTi = 0, i = 1,2,...,K, and Hrp Tr = 0. (3.9)
Denote the null space matrices of Hip, i = 1,2,...,K and Hrp as H?ip, i = 1,2,...,K and H?rp,
respectively, i.e. the columns of H?ip span the null space of Hip, and similarly for H?rp. Since
Hip, i = 1,2,...,K and Hrp are random channel coefficient matrices, they have full rank
almost surely. Therefore rank{H?ip} = M ?Mp and rank{H?rp} = N ?Mp. With matrices
Pi,i = 1,2,...,K and Pr denoting arbitrary (M ?Mp)?di and (N ?Mp)?N matrices, to
satisfy (3.9) we choose
Ti = H?ipPi, i = 1,2,...,K, and Tr = H?rpPr. (3.10)
We can obtain rank{Ti} = min{M ? Mp,di}, i = 1,2,...,K by choosing full rank Pi;
similarly rank{Tr} = N ?Mp for a full rank Pr. At user i, the number of antennas should
at least satisfy M > Mp in order to null interference to the primary network (otherwise Ti is
null). Similarly, the number of relay antennas should satisfy N > Mp. As long as these two
conditions are meet, H?ip,i = 1,2,...,K and H?rp are non-empty and the secondary user and
relay precoders can be designed over Pi,i = 1,2,...,K and Pr. Thus our design parameters
become Pi, i = 1,2,...,K and Pr instead of Ti, i = 1,2,...,K and Tr, respectively.
Note that since user i wants to transmit di independent datastreams, to mitigate inter-
datastream interference (particularly when transmit power is high), intuitively it is desirable
to chooserank{Ti}? di to achieve di linearly independent (preferablyorthogonal)directions,
which leads to M?Mp ? di. Similarly, since the relay receives and transmits all datastreams
from K secondary users, to mitigate the inter-datastream interference it is desirable to have
number of relay antennas N such that N ?Mp ? summationtextKi=1 di. We do note that the conditions
M?Mp ? di and N?Mp ?summationtextKi=1 di are preferred but not essential in the iterative algorithms
53
presented in this chapter. As noted in Sec. 3.4.1, one needs M?Mp ? di for the non-iterative
algorithm discussed therein.
Let ?si denote the estimated signal at SU i, given by the decoding matrix Ri at secondary
user i operating upon the self-interference free received signal ?yi at SU i, as specified in (3.8).
Then we have
?si = Ri?yi = RiHri
?
?
Ksummationdisplay
j=1,jnegationslash=i
TrHjrTjsj +TrHprxp +Trnr
?
?+RiHpi?xp +Rini. (3.11)
Then the estimated signal at all K users is given by
?s = [?sT1 ,?sT2 ,...,?sTK]T. (3.12)
We assume that all the signals from the primary/secondary users and the noise are all
independent from each other. Let Ei denote a submatrix of the permutation matrix E such
that Eis is the desired signal at receiver i. The sum MSE at all K secondary users can be
expressed as
E{bardbl ?s??s bardbl22} = E{bardbl ?s?Es bardbl22} =
Ksummationdisplay
i=1
E{bardbl ?si ?Eis bardbl22}. (3.13)
Our goal is to minimize the sum MSE under a transmit power constraint for all transmitters
while ensuring that the interference to primary receiver is zero.
Let Ptot,i,i = 1,2,...,K and Ptot,r be the maximum transmit power at secondary user i
and relay node, respectively. Then using (3.13), the optimization problem under considera-
tion can be expressed as
P1 : minPr,Pi,Ri,i=1,2,...,KsummationtextKi=1E{bardbl ?si ?Eis bardbl22} (3.14)
s.t. E{tr{Tisis?iT?i}}? Ptot,i, i = 1,2,...,K, (3.15)
E{tr{xrx?r}}? Ptot,r. (3.16)
54
The objective function is not jointly convex in Pr,Pi and Ri. therefore, problem P1 is not
a convex optimization problem.
3.3 Iterative Algorithm
Due to the non-convexity of problem P1, optimizing Pr, Pi and Ri, i = 1,2,...,K
simultaneously to get the global optimal may require exhaustive numerical search whose
complexity is likely to be too high. Therefore an iterative method is proposed to alternately
optimize precoding matrices at secondary users, precoding matrix at relay station, and de-
coding matrices at the secondary receivers; each of these three subproblems is convex (shown
in the Appendix).
3.3.1 Update Decoding Matrices
For givenTi, i = 1,2,...,K andTr, the optimization problem P1 with respect to receiver
matrices Ri, i = 1,2,...,K can be decomposed into K independent optimization problems
as
P2 : min
Ri
E{bardbl ?si ?Eis bardbl22}, i = 1,2,...,K. (3.17)
DefineAi = HriTr[H1rT1,...,H(i?1)rTi?1,0,H(i+1)rTi+1,...,HKrTK], Bi = [HriTrHpr,Hpi],
Ci = [HriTr,I] ?ni = [nTr ,nTi ]T and ?xp = [xTp ,?xTp ]T. Then using (3.11) the estimated signal
?si can be expressed as
?si = RiHriTr
parenleftBigsummationtextK
j=1,jnegationslash=iHjrTjsj +Hprxp +nr
parenrightBig
+RiHpi?xp +Rini
= Ri (Ais+Bi?xp +Ci?ni). (3.18)
Problem P2 is an unconstrained least-squares problem which is obviously convex having
a well-known solution. Therefore the minimum MSE decoder at the i-th SU which solves
55
problem P2 can be expressed as (details are provided in Appendix C):
R?i = ?2sEiA?i
parenleftBig
?2sAiA?i +?2pBiB?i +?2nCiC?i
parenrightBig?1
. (3.19)
3.3.2 Update Precoding Matrices at the Secondary Transmitters
Let ?Ei denote the submatrix of the permutation matrix E comprising its columns
summationtexti?1
j=1 dj + 1 through
summationtexti
j=1 dj. Then ?Eisi denotes the desired signal at all K secondary
users from the i-th secondary user?s transmission. Consequently we can rewrite Es as
Es = [?E1, ?E2,..., ?EK][sT1 ,sT2 ,...,sTK]T. (3.20)
With the optimal decoder R?i obtained from Sec. 3.3.1, set Ri = R?i and define
Di =
bracketleftBig
(R1Hr1)T,...,(Ri?1Hr(i?1))T,0,(Ri+1Hr(i+1))T,...,(RKHrK)T
bracketrightBigT
TrHirH?ip . (3.21)
Let ?s(i) denote the estimated signal at all K secondary users from i-th secondary user?s
transmission, after self-interference cancellation. Then using (3.21), we have ?s(i) = DiPisi.
Since si,i = 1,2,...,K are independent, optimizing (3.14) w.r.t. Pi, i = 1,2,...,K, reduces
to the problem
P3 : minPi,i=1,2,...,K EsummationtextKi=1 bardbl (DiPi ? ?Ei)si bardbl22 (3.22)
s.t. E{tr{H?ipPisis?iP?iH??ip }} ? Ptot,i, i = 1,2,...,K, (3.23)
E{tr{xrx?r}} ? Ptot,r . (3.24)
56
This is a convex problem, as shown in Appendix B. Let P =
bracketleftBig
P1, P2, ??? ,PK
bracketrightBig
. The
Lagrangian for problem P3 is given by
L(?,?,P) = summationtextKi=1
braceleftbigg
Li(?i,?,Pi)??iPtot,i
bracerightbigg
+?tr{Tr(HprH?pr?2p +?2nI)T?r}??Ptot,r (3.25)
where ?i and ? are the dual variables for constraints (3.23) and (3.24), respectively, ? =
[?1,...,?K] and
Li(?i,?,Pi) = tr
braceleftbigg
(DiPi ? ?Ei)(DiPi ? ?Ei)??2s +?iH?ipPiP?iH??ip ?2s
+?TrHirH?ipPiP?iH??ip H?irT?r?2s
bracerightbigg
. (3.26)
The Lagrangian dual function is expressed as
g(?,?) = min
P
L(?,?,P) =
Lsummationdisplay
i=1
gi(?i,?)+?tr{Tr(HprH?pr?2p +?2nI)T?r}??Ptot,r, (3.27)
where gi(?i,?) = minPi {Li(?i,?,Pi)??iPtot,i}. The dual optimization problem is given by
P4 : max?,? g(?,?) (3.28)
s.t. ? followsequal 0, ? ? 0. (3.29)
To calculate gi(?i,?), take derivative of Li(?i,?,Pi) with respect to P?i and set it to zero to
obtain the optimal Pi given ?i and ?, as
P?i(?i,?) =
parenleftBig
D?iDi +?iH??ip H?ip +?H??ip H?irT?rTrHirH?ip
parenrightBig?1
D?i ?Ei .
In solving the dual problem P4 over the two dual variables, we employ a two-loop algorithm.
In the inner loop, optimal ? is solved for a fixed value of ?, so we denote the optimal ? as
57
??(?). In the outer loop, ?? is found using the value of ??(?). The inner loop problem can
be expressed as g??(?) = max?g(?,?), s.t. ? followsequal 0 and the outer loop problem can then be
expressed as max? g??(?)s.t. ? ? 0. The dual problem is convex, therefore both the outer
and inner loop problems are convex.
Define
V1i(??i(?),?) = ?2sH?ipP?i(??i(?),?)P??i (??i(?),?)H??ip . (3.30)
Then for the inner loop, for a fixed value of ?, the optimal value ??i(?) of ?i(?),i = 1,2,...,K
should satisfy (complementary-slackness condition [18]) ??i(?)(tr{V1i(??i(?),?)}?Ptot,i) =
0. Therefore we pick ??i(?) = 0 if tr{V1i(0,?)} ? Ptot,i and ??i(?) > 0 if tr{V1i(0,?)} >
Ptot,i. For the ??i(?) > 0 case, we can find ??i(?) by solving the following problem
max?i g(?i,?) = max?i
braceleftBig
Li(?i,?,P?i(?i,?))??iPtot,i
bracerightBig
(3.31)
s.t. ?i > 0, i = 1,2,...,K (3.32)
using a line search over ?i. We want to find optimal ??i(?) such that tr{V1i(??i(?),?)} =
Ptot,i. Set the lower bound of ?i as ?i,l = 0 since by complementary-slackness tr{V1i(0,?)} >
Ptot,i for the ??i(?) > 0 case. Then we increase the value of ?i to find its upper bound ?i,u
such that tr{V1i(?i,u(?),?)} < Ptot,i. A bisection method can then be employed to find the
optimal ??i(?). Finally we solve for optimal ?? in the outer loop. Define
Z = Tr
parenleftbigg Ksummationdisplay
i=1
HirV1i(??i(0),0)H?ir +?2pHprH?pr +?2nI
parenrightbigg
T?r.
The optimal ?? should satisfy (complementary-slackness condition [18]) ?? = 0 if tr{Z} <
Ptot,r and ?? > 0 if tr{Z} ? Ptot,r For the ?? > 0 case, ?? can be found by solving the
following problem using a line search (a bisection method can be employed similar to the
58
one we used to solve the inner dual problem):
max? g??(?) s.t. ? > 0
.
3.3.3 Updating Precoding Matrix at Relay Station
Define the matrices ?H1, ?H2, ?Hp, E(i)l , E(i)r and R and vector n
?H1 = [(R1Hr1)T,(R2Hr2)T,...,(RKHrK)T]T, (3.33)
?H2 = [H1rT1,H2rT2,...,HKrTK], (3.34)
?Hp = [(R1Hp1)T,(R2Hp2)T,...,(RKHpK)T]T, (3.35)
E(i)l = diag{0?d1??d1,...,0?di?1??di?1,I?di??di0?di+1??di+1,...,0?dK??dK}, (3.36)
E(i)r = diag{0d1?d1,...,0di?1?di?1,Idi?di,0di+1?di+1,...,0dK?dK}, (3.37)
R = diag{R1,R2,...,RK}, n = [nT1 ,nT2 ,...,nTK]T. (3.38)
Then from (3.11), (3.12) and (3.33)-(3.37), the estimated signal vector at secondary receivers
after self-interference cancellation can be expressed as
?s = ?H1Tr ?H2s?
Ksummationdisplay
i=1
E(i)l ?H1Tr ?H2E(i)r s+ ?H1TrHprxp + ?H1Trnr + ?Hp?xp +Rn (3.39)
where E(i)l ?H1Tr ?H2E(i)r s is the received signal part at secondary receiver i from itself (user
i). Define
V3 = ?H2 ?H?2?2s +HprH?pr?2p +?2nI and V2(Pr) = H?rpPrV3P?rH??rp . (3.40)
59
For given Ti and Ri, i = 1,2,...,K, minimizing the sum MSE w.r.t. Pr can then be formu-
lated as
P5: minPr E{bardbl ?s?Es bardbl22} (3.41)
s.t. tr{V2(Pr)}? Ptot,r. (3.42)
This is a convex problem as shown in Appendix C. The Lagrangian of problem P5 is
L(?,Pr) = Etr{(?s?Es)(?s?Es)?}+ tr{?V2(Pr)}??Ptot,r. (3.43)
The Lagrangian dual function is g(?) = minPr L(?,Pr). To calculate the dual function,
set derivative of L(?,Pr) w.r.t. P?r to zero. Then we can obtain the optimal value of Pr for
a fixed value of ? as
vec{P?r(?)} =
braceleftbigg
VT3 ?[H??rp ?H?1 ?H1H?rp]?
Ksummationdisplay
i=1
[?H2E(i)r ?H?2]T ?[H??rp ?H?1E(i)l ?H1H?rp]?2s
+?VT3 ?[H??rpH?rp]
bracerightbigg?1
?vec
braceleftBigg
H??rp ?H?1E?H?2?2s ?
Ksummationdisplay
i=1
H??rp ?H?1E(i)l EE(i)r ?H?2?2s
bracerightBigg
. (3.44)
The optimal value ?? of the dual variable ? satisfies the complementary-slackness con-
dition
?? (tr{V2(P?r(??))}?Ptot,r) = 0 and we should have ?? = 0 if tr{V2(P?r(0))} < Ptot,r and
?? > 0 if tr{V2(P?r(0))} ? Ptot,r. The dual problem is always convex, so the optimal dual
variable ?? in the case ?? > 0 can be obtained by solving the following dual problem using
a line search (a bisection method can be used here):
max
?
g(?) s.t. ? > 0,
where g(?) = L(?,Pr(?)).
60
3.3.4 Algorithm Summary
The iterative algorithm is summarized as follows:
1) Initialize Pi (i = 1,2,...,K) and Pr such that tr{H?ipPiP?iH??ip } = Ptot,i/?2s and
Etr{H?rpPryry?rP?rH??rp} = Ptot,r. [One way to accomplish this is to randomly and
independently choose each element in Pi and Pr, and then scale the respective ma-
trices to satisfy the aforementioned power constraints. An alternative (used in our
simulations) is to use the results of the non-iterative algorithm proposed in Sec. 3.4 for
initialization; this algorithm needs no initialization.]
2) Repeat (iterate):
? Update Ri, i = 1,2,...,K using (3.19) for fixed Pi, i = 1,2,...,K and Pr. We use
the latest available estimates of Pi and Pr.
? Update Pi, i = 1,2,...,K by solving problem P3 for fixed Ri, i = 1,2,...,K and
Pr. We use the latest available estimates of Ri and Pr.
? Update Pr by solving problem P5 for fixed Ri, i = 1,2,...,K and Pi, i =
1,2,...,K. We use the latest available estimates of Ri and Pi.
Continue until sum MSE converges.
Since all subproblems have exactly the same objective function (sum MSE) and each sub-
problem is convex, the objective function is decreasing in each subproblem, hence in each
iteration. The sum MSE is lower bounded by zero. As a decreasing sequence that is lower
bounded always converges, the proposed iterative algorithm is convergent to a local optimum.
3.3.5 Effects of Allowing Interference to Primary Users
In this section we investigate the case when the interference from secondary users to
primary network is not completely removed (nulled). This may improve the secondary
network?s performance because the constraints on interference to PU are relaxed from zero
61
to some nonzero upperbound, namely Itot (the same for the two time slots). The optimization
problem becomes
P6 : minTr,Ti,Ri,i=1,2,...,K E{bardbl ?s?Es bardbl22} (3.45)
s.t. E{tr{Tisis?iT?i}}? Ptot,i, i = 1,2,...,K, (3.46)
E{tr{xrx?r}} ? Ptot,r, (3.47)
summationtextK
i=1E{tr{HipTisis
?
iT
?
iH
?
ip}} ? Itot, (3.48)
E{tr{Hrpxrx?rH?rp}} ? Itot. (3.49)
As in the previous sections, P6 is not a convex problem, therefore we decompose it into three
convex subproblems and solve them iteratively.
First consider the design of decoding matrices Ri given Ti,1 ? i ? K and Tr. Since
the design of Ri does not affect interference to the primary network, it is exactly the same
as in Sec. 3.3.1.
Similar to (3.21), define
?Di = bracketleftBig(R1Hr1)T,...,(Ri?1Hri?1)T,0,(Ri+1Hri+1)T,...,(RKHrK)TbracketrightBigT TrHir.
For user precoding design, given Ri, 1 ? i ? K, and Tr, the problem is formulated as
follows:
P7 : minTi,i=1,2,...,K EsummationtextKi=k bardbl (?DiTi ? ?Ei)si bardbl22 (3.50)
s.t. (3.46)?(3.49) are satisfied. (3.51)
Following a similar argument as in Appendix B, we can show that problem P7 is a convex
optimization problem and the optimal solution for Ti,1 ? i ? K can be obtained. Define
V4 = Tr(HprH?pr?2p +?2nI)T?r and V5 = HrpV4H?rp. (3.52)
62
With P = [P1, P2, ??? ,PK,], the Lagrangian for problem P7 can be expressed as
L(?,?,?,?,P) =
Ksummationdisplay
i=1
braceleftbigg
Li(?i,?,?,?,Pi)??iPtot,i
bracerightbigg
+?tr{V4}+?tr{V5}??Ptot,r ??Itot ??Itot (3.53)
where ?i, ?, ? and ? are the dual variables, ? = [?1,...,?K] and
Li(?i,?,?,?,Ti) = tr
braceleftbigg
(?DiTi ? ?Ei)(?DiTi ? ?Ei)??2s +?iTiT?i?2s +?HipTiT?iH?ip?2s
+?TrHirTiT?iH?irT?r?2s +?HrpTrHirTiT?iH?irT?rH?rp?2s
bracerightbigg
. (3.54)
WithT = [T1, T2, ??? ,TK,], theLagrangiandual functiong(?,?,?,?) = minTL(?,?,?,?,T)
is expressed as
g(?,?,?,?) =
Lsummationdisplay
i=1
gi(?i,?,?,?)+?tr{V4}+?tr{V5}??Ptot,r ??Itot ??Itot, (3.55)
where gi(?i,?,?,?) = minTi {Li(?i,?,?,?Ti)??iPtot,i}. The dual optimization problem is
max
?,?,?,?
g(?,?,?,?)
s.t. ? followsequal 0, ? ? 0, ? ? 0, ? ? 0.
To calculate gi(?i,?,?,?), take derivative of Li(?i,?,?,?,Ti) with respect to T?i and set it
to zero to obtain the optimal Ti given ?i,?,? and ?, as
T?i(?i,?,?,?) =
parenleftbigg
?D?i ?Di +?iI+?H?irT?rTrHir +?H?ipHip +?H?irT?rH?rpHrpTrHir
parenrightbigg?1
?D?i ?Ei.
In solving the dual problem, there are four dual variables. A two-loop algorithm, similar
to the one that solves P4 is employed here. In the inner loop, optimal ? is solved for fixed
values of ?,? and ?, and the optimal ? is denoted by ??(?,?,?). In the outer loop, ??,
63
?? and ?? are jointly found using the value of ??(?,?,?) by a subgradient method. The
inner loop problem can be expressed as g??(?,?,?) = max?g(?,?,?,?), s.t. ? followsequal 0 and the
outer loop problem can then be expressed as max?,?,? g??(?,?,?) s.t. ?,?,? ? 0. The dual
problem is convex, therefore both the outer and inner loop problems are convex. The inner
loop problem is similar as the one in problem P4, therefore the steps are omitted due to
space limitations. For the outer loop, a subgradient method is employed to solve for the
optimal ??, ?? and ??. The subgradient of g??(?,?,?) w.r.t. these three dual variables are
given by
??g??(?,?,?) = tr{V4 +summationtextKi=1 TrHirTiT?iH?irT?r?2s}?Ptot,r (3.56)
??g??(?,?,?) = summationtextKi=1 tr{HipTiT?iH?ip?2s}?Itot (3.57)
??g??(?,?,?) = tr{V5 +summationtextKi=1 HrpTrHirTiT?iH?irT?rH?rp?2s}?Itot. (3.58)
For designing the precoder at the relay for given Ri and Ti, 1 ? i ? K, the optimization
problem P6 can be reformulated as follows where ?H2 and V3 are as in (3.34) and (3.40):
P8: minTr Ebardbl?s?Esbardbl2 (3.59)
s.t. tr
braceleftBig
TrV3T?r
bracerightBig
? Ptot,r (3.60)
tr
braceleftBig
HrpTrV3T?rH?rp
bracerightBig
? Itot . (3.61)
The Lagrangian of problem P8 can be expressed as
L(?,?,Tr) = Etr{(?s?Es)(?s?Es)?}+?tr{TrV3T?r}
+?tr
braceleftBig
HrpTrV3T?rH?rp
bracerightBig
??Ptot,r ??Itot. (3.62)
64
By setting ?L(?,?,Tr,?)?T?
r
= 0 and using (3.33)-(3.37), we obtain the optimal T?r satisfying
vec{T?r} =
braceleftbigg
VT3 ?[?H?1 ?H1]?summationtextKi=1[?H2E(i)r H?2]T ?[?H?1E(i)l ?H1]?2s +?VT3 ?I
+?V3 ?
parenleftBig
H?rpHrp
parenrightBigbracerightbigg?1
?vec
braceleftBig?
H?1E?H?2 ?summationtextKi=1 ?H?1E(i)l EE(i)r ?H?2
bracerightBig
?2s. (3.63)
To obtain the optimal dual variables ?? and ??, a subgradient method can be employed. The
subgradients of the Lagrangian dual function g(?,?) = L(?,?,T?r,?) w.r.t. these two dual
variables are given by
??L(?,?,T?r) = tr
braceleftBig
T?rV3T??r
bracerightBig
?Ptot,r (3.64)
??L(?,?,T?r) = tr
braceleftBig
HrpT?rV3T??r H?rp
bracerightBig
?Itot. (3.65)
Compared to the zero PU interference approach of Secs. 3.3.1-3.3.3, in the current case
we have higher computational complexity since in solving for Ti (compared with solving for
Pi in Sec. 3.3.2), the outer loop has three dual variables instead of one variable in Sec. 3.3.2,
therefore, a subgradient method is employed in the current case instead of a line search as
in Sec. 3.3.2. For solving Tr (compared with solving for Pr), there are two dual variables
instead of one, hence again, a subgradient method is used instead of a line search.
3.4 Non-iterative Algorithm
In this section we propose a non-iterative algorithm in order to reduce the computational
complexity. We want to design an algorithm such that each secondary user can design its
precoding and decoding matrices based on local channel matrices. Then the precoding
matrix at the relay station is designed to minimize the sum MSE of all secondary users.
This algorithm is based on a matrix (pseudo-)distance defined following [34] (see also [35]).
Let U be an n?m (m ? n) (complex) matrix with orthonormal columns and let A be an
arbitrary n?p complex matrix. The orthogonal projection of A onto the subspace spanned
65
by the columns of U is given by ?A = UU?A. Define the distance between A and its
orthogonal projection ?A as
bardblA,Ubardbldis = bardblA?UU?AbardblF. (3.66)
If the columns of A lie in the subspace spanned by U, then bardblA,Ubardbldis = 0. If none of the
columns of A lie in the subspace spanned by U, then bardblA,Ubardbldis = bardblAbardblF.
3.4.1 Matrix Distance based Secondary User Precoding/Decoding Matrices De-
sign
Precoding matrix Ti at the i-th secondary user is designed to minimize its distance
to H?ir so that ?more information? can be transmitted to the relay station. (If a symbol b
is transmitted along a (column) direction p1, then by the Cauchy-Schwartz inequality, the
received bit ?b = rT1p1 has maximum magnitude if and only if p1 and r?1 are in the same
one-dimensional subspace. This motivates picking columns of the precoding matrix so that
they span the same subspace as the columns of H?ir.) We also need to make sure that
secondary users? transmission will not degrade primary user?s QoS; therefore, Ti = H?ipPi
should be satisfied. We assume a priori that Pi has orthonormal columns, i.e. P?iPi = I;
this is suboptimal but leads to analytical tractability. Another reason for this assumption
is that each column corresponds to one datastream?s transmit beamforming vector. For a
user that is transmitting multiple datastreams, it is reasonable to make these beamforming
vectors orthogonal to each other to mitigate inter-datastream interference. The requirement
P?iPi = I also implies that the power scaling of each datastream is set to be the same; power
allocation and fairness among multiple datastreams is outside the scope of this chapter. A
scaler ?i is introduced to satisfy the transmit power constraint of user i. Let Ti = ?iH?ipPi.
Then the relay receives the signal HirTisi = ?iHirH?ipPi from the i-th secondary user. Since
H?ip is ?fixed,? we make it a part of Hir for the design of Pi and define an equivalent channel
66
matrix as Gir = HirH?ip. Then the precoding design at user i is formulated as
P9: P?i = arg min
P?iPi=I
bardblG?ir,Pibardbl2dis = arg min
P?iPi=I
bardblG?ir ?PiP?iG?irbardbl2F. (3.67)
In problem P9, the objective function represents the signal leakage that falls out of the
subspace spanned by the rows of equivalent channel Gir. By simplifying (4.7), we obtain
P?i = argmaxP?
iPi=I
tr{P?iG?irGirPi}. Since Pi ? C(M?Mp)?di has orthonormal columns, the
columns of P?i should be the di dominant eigenvectors of matrix G?irGir and each column?s
2-norm is normalized to 1. This is possible only if M ? MP ? di; thus, this condition is
essential for the non-iterative algorithm whereas for the iterative algorithm (as discussed in
Sec. 3.2) we require only M?Mp > 0. To satisfy the transmit power constraint, the optimal
value of ?i is chosen such that Etr{Tisis?iT?i} = Ptot,i, leading to
??i =
radicalbigg
Ptot,i/tr
braceleftBig
?2sH?ipP?iP??i H??ip
bracerightBig
, i = 1,2,...,K. (3.68)
For the secondary receiver i, the decoding matrix design could follow the same method
as for the precoder design, i.e. minimizing the matrix distance of decoding matrix Ri and
corresponding channel matrix H?ri (equivalently the distance between R?i and Hri). However,
interference from the primary user should also be taken into consideration, which means
that we should also minimize the matrix distance between Hpi and R??i where R?i denotes
a matrix whose columns are orthonormal basis for the complement of the subspace spanned
by Ri. As for the design of Pi, for analytical tractability we assume a priori that RiR?i = I.
Then the decoder design of i-th secondary user is formulated as
P10: R?i = argminRiR?
i=I
bardblHri,R?ibardbl2dis +?bardblHpi,R??i bardbl2dis
= argminRiR?
i=I
bardblHri ?R?iRiHribardbl2F +?bardblHpi ?R??i R?i Hpibardbl2F , (3.69)
67
where ? is a non-negative scalar weight. In problem P10, the first term in the objective
function is the signal leakage that falls out of the signal subspace spanned by rows of Ri while
the second term represents the interference leakage which falls out the interference subspace
spanned by rows of R?i . The weight ? is chosen empirically to strike a balance between
minimizing the interference leakage and the minimizing of signal leakage. By simplifying
(3.69) we have R?i = argminRiR?
i=I
tr{Ri(?HpiH?pi ?HriH?ri)R?i}. Since Ri ? C?di?M, the
columns of R??i should be the ?di least dominant eigenvectors of (?HpiH?pi ?HriH?ri) whose
2-norms are normalized to 1.
3.4.2 MSE based Relay Precoding Matrix Design
Given Ri and Ti for i = 1,2,...,K, the precoding matrix design at the relay node is
based on an MSE criterion. Let the estimated received signal at secondary receivers be
?s = ??s. We introduce the scaling parameter ? because the decoding matrices at secondary
users are designed to consist of orthonormal rows. Therefore a scalar ? is necessary to adjust
the scaling of the received signal. Then using (3.40), the MSE based relay precoding design
problem is formulated as
P11: minP
r,?
Ebardbl ??s?Es bardbl2 s.t. tr{V2(Pr)} ? Ptot,r. (3.70)
We now show that the optimal solution of P11 always satisfies the constraint with equality.
Assume that ?Pr,?? are the optimal solution of P11 and tr
braceleftBig
V2(?Pr)
bracerightBig
< Ptot,r. We introduce
a positive scalar ?, let ?Pr = ??Pr, and choose ? such that tr
braceleftBig
V2(?Pr)
bracerightBig
= Ptot,r is satisfied.
It is obvious that ? > 1. Let ?? = ??/?; therefore we have ???Pr = ???Pr. Use ?Pr,?? and ?Pr,??
to evaluate the objective function values of P11 and denote them by ?O and ?O, respectively.
Using (3.39) it can be shown that
?O? ?O = (|??|2 ?|??|2)tr
braceleftbigg
(?Hp ?H?p?2p +RR??2n)
bracerightbigg
> 0.
68
This shows that ?Pr,?? could not be the optimal solution of P11. Therefore, we will use the
equality constraint tr{V2(Pr)} = Ptot,r instead of the inequality constraint in P11. The
Lagrangian of problem P11 is
L(?,Pr,?) = Etr{(??s?Es)(??s?Es)?}+?tr{V2(Pr)}??Ptot,r . (3.71)
By setting ?L(?,Pr,?)?Pr = 0 and ?L(?,Pr,?)?? = 0, and using constraint (3.70), we have
??
??2 =
?2nsummationtextKi=1 ?di +?2ptr{?Hp ?H?p}
Ptot,r , (3.72)
vec{P?r} =
braceleftbigg
VT3 ?[H??rp ?H?1 ?H1H?rp]?summationtextKi=1[?H2E(i)r ?H?2]T ?[H??rp ?H?1E(i)l ?H1H?rp]?2s
+ ????2VT3 ?[H??rpH?rp]
bracerightbigg?1
?vec
braceleftbigg
H??rp ?H?1E?H?2 ?summationtextKi=1 H??rp ?H?1E(i)l EE(i)r ?H?2
bracerightbigg
?2s???1. (3.73)
Let P?r = ?P?r???1. Then ?? =
radicalBigg
tr
braceleftbigg
V2(?P?r)
bracerightbigg
/Ptot,r. Thus, first solve (3.72) and use it
in (3.73) to obtain ?P?r. Next using (3.40), the calculated value of ?P?r and the expression
?? =
radicalBigg
tr
braceleftbigg
V2(?P?r)
bracerightbigg
/Ptot,r, find ?? to complete the solution.
3.4.3 Distributed Implementation
We now present a distributed implementation of the non-iterative algorithm. Firstly,
assume that only local channel coefficient matrices are available at each secondary user. For
example, the i-th secondary user only has the knowledge of Hir, Hri, Hip and Hpi, and the
relay node only has the knowledge of Hir, Hri, i = 1,2,...,K, Hrp and Hpr. Also assume
that there exist control channels between relay and secondary users in order to facilitate
exchanges of design precoding and decoding matrices. The proposed distributed algorithm
is as follows:
69
1) Each secondary user i designs its own precoding and decoding matrices Pi and Ri by
solving problems P9 and P10, using only the local channel information. Then it feeds
back R?i and T?i to the relay station.
2) The relay station designs its precoding matrix by solving problem P11, using its local
channel information and R?i, T?i, i = 1,2,...,K.
3) The relay station sends its precoding matrix to the secondary users which will be used
by them to perform self-interference cancellation.
The benefits of this distributed implementation are: firstly, there is no need to assign a
central node and collect global channel information. Secondly, computation is broken down
into small parts and performed at all secondary nodes.
3.4.4 Effects of Allowing Interference to Primary Users
When a small amount of interference to the primary network is allowed instead of
complete interference cancellation, the problem formulation and optimization solution will
be similar to the robust algorithm in the next section while the only modification required
will be setting all the variances in CSI to be zero. Due to space limitations, the problem
formulation and solution is omitted.
3.5 Robust Algorithm Design with Imperfect CSI
In this section, we will take into account some practical design concerns and try to
design a more practical and robust precoding algorithm. Due to the time-varying nature
of wireless channels, the channel state information may already be out-dated when used in
the precoder design. The feedback of channel information may also introduce some errors
to the channel estimation results. Therefore, we propose a robust precoder design in this
section by modeling the channel estimation errors as random errors with known variance,
and considering its effect in the optimization problem. On the other hand, computational
70
complexity is also another practical concern, which means that a non-iterative algorithm
may be more cost-efficient for implementation. Also we want to design an algorithm that
can be implemented in a distributed manner so that no central controller is needed and the
computation can be distributed to all secondary users as well as the relay. Therefore, we will
propose a robust precoder using the matrix-distance based non-iterative algorithm proposed
in Sec. 3.4.
Assume that the CSI of all links is subject to stochastic errors. Channel matrices Hir,
Hri, Hip, Hpi, Hpr and Hrp are used to denote actual channel coefficient matrices among
various source-destination pair nodes, as in the earlier sections. Then the estimated channel
coefficients is the actual channel plus some random errors given by
?H? = H? +??, (3.74)
where ? ? {ir,ri,ip,pi,pr,rp} and the matrix ?? models the estimation error in channel
coefficient matrix H?. The various components of ?? are mutually independent and we have
E{??} = 0, and E{[vec??][vec??]?} = ?2?I. (3.75)
3.5.1 Robust SU Precoding/Decoding Matrices Design
Due to imperfect CSI of link from the i-th secondary transmitter to the primary receiver,
interference to the primary receiver can not be canceled completely. Therefore the design of
Ti should not only consider its distance to H?ir but also its distance to H?ip. Let Ti = ?i?Ti,
where ?i is introduced to scale Ti so that the transmit power constraint can be satisfied and
the interfering power to primary receivers can be bounded. We first state the MSE and all
constraints in terms of unknown (true) H? and then use (3.74) to express them in terms of
71
?H? and ?2?. The precoding design at secondary user i is formulated as:
P12 : ?T?i = argmin?T?
i ?Ti=I
E
braceleftBig
bardblH?ir, ?Tibardbl2dis +?bardblH?ip, ?T?i bardbl2dis
bracerightBig
= argmin?T?
i ?Ti=I
tr
braceleftbigg
?T?iparenleftBig?H?ip ?Hip? +?2ipMp?I? ?H?ir ?Hir ??2irNIparenrightBig?Ti
bracerightbigg
(3.76)
where ? is a non-negative scalar weight which can be empirically chosen and as in the design
of Pi in Sec. IV, ?Ti ? CM?di is assumed to have orthonormal columns. Therefore, the
columns of ?Ti should be the di least dominant eigenvectors of matrix
parenleftBig?
H?ip ?Hip?+?2ipMp?I?
?H?ir ?H?ir??2irNIparenrightBigwhose 2-norms are normalized to 1. After ?T?i is determined, we need to make
sure that the total transmit power constraint at secondary user i is satisfied. Besides, we
also need to make sure the interference caused by the i-th secondary user to primary receiver
is strictly less than a threshold. In order to design a distributed algorithm we choose this
interference threshold as Itot/K for each secondary user so that they can determine their
own transmitting parameters ?T?i and ??i . Therefore the following two constraints should be
satisfied.
tr
braceleftbigg
T?iT??i ?2s
bracerightbigg
? Ptot,i and tr
braceleftbiggparenleftBig
?H?ip ?Hip +?2ipMpIparenrightBigT?iT??i ?2s
bracerightbigg
? Itot/K. (3.77)
This leads to
??i = min
braceleftbiggradicalbigg
Ptot,i/tr
braceleftBig?
T?i ?T??i ?2s
bracerightBig
,
radicalbigg
(Itot/K)/tr
braceleftBigparenleftBig?
H?ip ?Hip +?2ipMpI
parenrightBig?
T?i ?T??i ?2s
bracerightBigbracerightbigg
. (3.78)
The robust design of Ri is similar to that for problem P10: Ri is assumed to have
orthonormal rows. However, uncertainty in Hri and Hpi also needs to be taken into account.
This leads to the optimization problem
P13: R?i = argminRiR?
i=I
E
braceleftBig
bardblHri,R?ibardbl2dis +?bardblHpi,R??i bardbl2dis
bracerightBig
= argminRiR?
i=I
braceleftBig
Ri
parenleftBig
? ?Hpi ?H?pi +??2piJpI? ?Hri ?H?ri ??2riNI
parenrightBig
R?i
bracerightBig
. (3.79)
72
Since Ri ? C?di?M, the columns of R??i should be the ?di least dominant eigenvectors of
parenleftBig
??Hpi ?H?pi +??2piJpI? ?Hri ?H?ri ??2riNI
parenrightBig
whose 2-norms are normalized to 1.
3.5.2 Robust Relay Precoding Design
In this section we design a robust precoder at the relay station using R?i and T?i obtained
in Sec. V-A, based on the MMSE criterion. Define the estimated channels as
?H1 = [(R?1 ?Hr1)T,(R?2 ?Hr2)T,...,(R?K ?HrK)T]T, (3.80)
?H2 = [?H1rT?1, ?H2rT?2,..., ?HKrT?K], (3.81)
?Hp = [(R?1 ?Hp1)T,(R?2 ?Hp2)T,...,(R?K ?HpK)T]T. (3.82)
Since interference to the primary users can not be eliminated completely due to the channel
uncertainty, we now impose a constraint on the interfering power at the primary receiver.
Define
V6 = Tr
parenleftBig?
H2 ?H?2?2s +HprH?pr?2p +?2nI
parenrightBig
T?r. (3.83)
Then this optimization problem is formulated as follows:
P14: minTr,? Ebardbl??s?Esbardbl2 (3.84)
s.t. Etr{V6} ? Ptot,r and Etr
braceleftBig
HrpV6H?rp
bracerightBig
? Itot. (3.85)
The Lagrangian of problem P14 can be expressed as:
L(?,?,Tr,?) = Etr{(??s?Es)(??s?Es)?}+?Etr{V6}
+?Etr
braceleftBig
HrpV6H?rp
bracerightBig
??Ptot,r ??Itot. (3.86)
73
By setting ?L(?,?,Tr,?)?T?
r
= 0 and substituting true channels in terms of estimated channels and
estimation errors, we obtain
vec{T?r} = vec{?Tr??1} =
braceleftbigg
[?H2 ?H?2 +e2I]T ?[?H?1 ?H1 +e1I]?2s
?summationtextKi=1[?H2E(i)r ?H?2 +?2irtr{T?iT??i }I]T ?[?H?1E(i)l ?H1 +?2ritr{R??i R?i}I]?2s
+[?Hpr ?H?pr +?2prJpI]T ?[?H?1 ?H1 +e1I]?2p +I?[?H?1 ?H1 +e1I]?2n
+ ??2[?H2 ?H?2?2s +e2?2sI+ ?Hpr ?H?pr?2p +?2prJp?2pI+?2nI]T ?I ??2[?H2 ?H?2?2s +e2?2sI+ ?Hpr ?H?pr?2p
+?2prJp?2pI+?2nI]?
parenleftBig?
H?rp ?Hrp +?2rpNI
parenrightBigbracerightbigg?1
vec{?H?1E?H?2 ?summationtextKi=1 ?H?1E(i)l EE(i)r ?H?2}?2s???1, (3.87)
where
e1 =
Ksummationdisplay
i=1
?2ritr{R??i R?i}, e2 =
Ksummationdisplay
i=1
?2irtr{T?iT??i }. (3.88)
Now we need the optimal value of ?,? and ?, or equivalently ??2, ??2 and ? to obtain the
optimal precoder T?r. We have the following Proposition.
Proposition 3.1 : The optimal value of ??2 and ??2 should satisfy
?
?2Ptot,r +
?
?2Itot = tr
braceleftbigg
?Hp ?H?p?2p
bracerightbigg
+?2pJp
Ksummationdisplay
i=1
?di?2ip + Ksummationdisplay
i=1
?di?2n. (3.89)
Proof : See Appendix F. square
Variables ?,? and ?2 should be nonnegative. Therefore finding the optimal value of ??2 and
?
?2 can be performed using a line search over one variable, say,
?
?2, since for a given value of
?
?2, the other one
?
?2 is determined according to Proposition 1. A bisection method over
?
?2
in the region ??2 > 0 can be performed to find the optimal value of ??2. Within this bisection
method, for a fixed value of ??2, ??2 is obtained using Proposition 1, Tr is given by (3.87) and
74
? is chosen to satisfy the two constraints of P11:
?? = argmax?
??
?tr
braceleftbigg
?T?rparenleftBig?H2 ?H?2?2s +e2?2sI+ ?Hpr ?H?pr?2p +?2prJp?2pI+?2nIparenrightBig?T??r
bracerightbigg
/Ptot,r,
tr
braceleftbiggparenleftBig
?H?rp ?Hrp +?2rpNIparenrightBig?T?rparenleftBig?H2 ?H?2?2s +e2?2sI+ ?Hpr ?H?pr?2p +?2prJp?2pI+?2nIparenrightBig?T??r
bracerightbigg
/Itot
??
?.(3.90)
(Note however that, from (3.87), when the values of ??2 and ??2 are given, the value of ? will
not change the value of the objective function or of the Lagrangian of P14). This robust
algorithm can also be implemented in a distributed manner, as in Sec. 3.4.3.
3.6 Simulation Results
We consider a secondary network with K = 4 secondary users and one relay station.
There is one primary transmitter-receiver pair within the same channel band. Assume that
all channel links experience flat Rayleigh fading. The noise power ?2n at every receiver, the
PU transmit power ?2p and the mean channel power gain of all links are all normalized to
one. Furthermore we also normalize the SU information sequence power ?2s to one, with
desired transmit power, hence the desired receiver signal-to-noise ratio (SNR), achieved by
scaling the precoder Pi at individual SU; for the case where there is no precoder, ?2s is scaled
to achieve the desired transmit power. Each secondary user has M antennas and relay
is equipped with N antennas. The primary transmitter has Jp antennas and the primary
receiver has Mp antennas. Each secondary user has d data streams to transmit and it wants
to receive d data streams. All simulation results are based on 100 Monte Carlo runs. For
illustrating the performance of various algorithms, in addition to the sum MSE, we also use
sum rate (instantaneous capacity) of the secondary network as a performance metric. Let
D = summationtextKk=1 dk. Then the sum rate is defined as
Rate =
Dsummationdisplay
i=1
log2
?
??1 + |
braceleftBig?
H1Tr ?H2
bracerightBig
i,j(i) |
2
iNi
?
?? (3.91)
75
where iNi is noise and interference power given by
iNi =
vextendsinglevextendsingle
vextendsinglevextendsingle
braceleftbiggparenleftBigg
?H1Tr ?H2 ? Ksummationdisplay
i=1
E(i)l ?H1Tr ?H2E(i)r
parenrightBiggparenleftBigg
?H1Tr ?H2 ? Ksummationdisplay
i=1
E(i)l ?H1Tr ?H2E(i)r
parenrightBigg?
?2s
?H1TrHprH?prT?r ?H?1?2p + ?H1TrT?r ?H?1?2n+RR??2n+HpH?p?2p
bracerightbigg
i,i
vextendsinglevextendsingle
vextendsinglevextendsingle2?|braceleftBig?H1Tr ?H2bracerightBig
i,j(i) |
2 (3.92)
and {A}m,n denotes the mn?th element of matrix A and j(i) := {l|{E}i,l = 1}.
Figs. 3.2 and 3.3 show the sum rate (3.91) and the sum MSE, respectively, of our
proposed iterative and non-iterative algorithms together with that of three other cases, as
a function of secondary user SNR. [The parameter ? in the non-iterative algorithm was
chosen to be 10.] The results of the non-iterative algorithm were used for initializing the
proposed iterative algorithm. It is seen from Figs. 3.2 and 3.3 that the sum rates and
the sum MSEs, respectively, of our proposed algorithms are significantly higher and lower,
respectively, than that of the schemes with no precoding. In the non-iterative algorithm,
performing self-information cancellation provides noticeable improvement in the sum rate
and the sum MSE. However, this comes at a cost that the precoding matrix of relay node
needs to be forwarded to all secondary users. In the no source and relay precoding scheme,
only decoding matrices Ri,i = 1,2,...,K are designed according to problem P2 while in the
no source precoding algorithm, Ri,i = 1,2,...,K and Tr are designed according to problems
P2 and P5 and no iteration is performed. It is also seen from Figs. 3.2 and 3.3 that allowing
some interference to the PU networks improves the performance of the proposed iterative
schemes but the improvement is not ?substantial?; in the remaining simulation examples we
omit this case. Fig. 3.4 shows the effect of varying values of the scalar weight ? used in the
proposed non-iterative algorithm (see (3.69)). It is seen that over a wide range of values (?
ranging from 10 through 100) the sum-MSE performance is essentially unchanged.
Fig. 3.5 shows the sum MSE vs the SNR per secondary user (relay?s transmit power is
K times the secondary user?s power) under several different secondary network setups. It is
observed that increasing the number M of SU antennas for the same number of datastreams
76
0 2 4 6 8 10 12 14 16 18 200
10
20
30
40
50
60
70
SNR per secondary user (dB)
Sum rate (bps/Hz)
 
 
Iterative, Itot = 2?n2
Iterative, Itot = 0
Non?iterative
Non?iterative without SI cancelation
No precoding at source
No precoding at source and relay
Figure 3.2: Sum rate (3.91) vs SNR per secondary user 10log10(Ptot,i/?2n): K = 4, Ptot,i is
the same for i = 1,??? ,K, M = 4, N = 10, d = 2, Mp = Jp = 1, Ptot,r = KPtot,i, ? = 10.
The label ?Iterative, Itot = 2?2n? refers to the approach of Sec. 3.3.5 and the label ?Iterative,
Itot = 0? refers to the approach summarized in Sec. 3.3.4.
d significantly improves the sum MSE. It was noted in Sec. 3.2 that one needs M?Mp > 0 in
order to null interference to the primary network, and M?Mp ? d was desirable to mitigate
inter-datastream interference. In Fig. 3.5 with Mp = 2, the case M = 3, d = 2 represents
M ?Mp = 1 > 0 but < d = 2 whereas all other cases satisfy M ?Mp ? d. It is seen that
the performance for the case M = 3, d = 2 is much worse than all other cases shown in Fig.
3.5. In Fig. 3.6 it is observed that increasing the number of relay antennas can decrease the
sum MSE for a fixed number of SU antennas and datastreams per secondary user, and the
gap between the iterative and non-iterative algorithms tends to be smaller as the number of
antennas at the relay station increases. Fig. 3.7 shows the convergence of sum MSE in our
iterative algorithms for a single run.
Figs. 3.8 and 3.9 compare the performance of our proposed robust algorithm with the
?non-robust? non-iterative algorithm of Sec. 3.4. In each Monte Carlo run various channel
gains generated according to Rayleigh fading were further perturbed with zero-mean complex
77
0 2 4 6 8 10 12 14 16 18 2010
?2
10?1
100
101
SNR per secondary user (dB)
Sum MSE
 
 
Iterative, Itot = 2?n2
Iterative, Itot = 0
Non?iterative
Non?iterative without SI cancelation
No precoding at source
No precoding at source and relay
Figure 3.3: Sum MSE vs SNR per secondary user 10log10(Ptot,i/?2n): K = 4, Ptot,i is the
same for i = 1,??? ,K, M = 4, N = 10, d = 2, Mp = Jp = 1, Ptot,r = KPtot,i, ? = 10.
The label ?Iterative, Itot = 2?2n? refers to the approach of Sec. 3.3.5 and the label ?Iterative,
Itot = 0? refers to the approach summarized in Sec. 3.3.4.
Gaussian noise (error). The curves labeled ?non-iterative ??? perfect CSI? are based on
perfect knowledge of CSI (error-free unperturbed CSI) whereas in the other cases the noisy
channel CSI was used in the algorithm. We assume all the channel links have the same level of
estimation errors, which means ?2? = ?2e, ???{ir,ri,ip,pi,pr,rp} (see (3.74) and (3.75)). In
the interference constraints (see (3.77) and (3.85)), we set Itot = 2?2n = 2. It is seen in Fig. 3.8
that when the channel estimation errors are relatively small (?2e = 0.01), the robust design
achieves a performance improvement (lower sum MSE) over the non-robust non-iterative
algorithm in the low transmit power (equivalently low SNR) regime. At higher SNRs, the
channel errors are dominant over the noise power leading to ?plateauing? of the sum MSE
curve with increasing SNR. When the channel estimation errors are high (?2e = 0.1), the
robust design achieves a performance improvement (lower sum MSE) over the non-robust
non-iterative algorithm over the entire transmit power range. Now the channel errors seem
to dominate the noise power at all SNRs, showing little improvement with increasing SNR.
78
0 2 4 6 8 10 12 14 16 18 20
10?1
100
SNR per secondary user (dB)
Sum MSE
 
 
? = 1
? = 4
? = 10
? = 20
? = 50
? = 100
Figure 3.4: Sum MSE using non-iterative approach vs SNR per secondary user
10log10(Ptot,i/?2n): K = 4, Ptot,i is the same for i = 1,??? ,K, M = 4, N = 10, d = 2,
Mp = Jp = 1, Ptot,r = KPtot,i, variable ?
From Fig. 3.9 we also see that although the sum MSE of the robust algorithm in Fig. 3.8 in
the high SNR regime and lower channel estimation errors case of ?2e = 0.01 is slightly higher
than that for the non-iterative algorithm, the interference power to the primary network is
well constrained (see Fig. 3.9), while the interference power of the non-robust algorithm has
increased drastically and significantly exceeds the bound Itot = 2. By design, when perfect
CSI is available, the non-robust algorithm can null out the interference to the primary
network; this is seen in Fig. 3.9.
3.7 Conclusions
We investigated joint design of precoders and decoders in a multiuser multi-way relay
system in cognitive radio networks, which operates concurrently with a primary network
within the same frequency band. The design objective was to minimize the sum MSE of
all secondary users under a transmit power constraint for each transmitting node while
keeping the interference to primary receiver to be zero, assuming complete knowledge of the
79
0 2 4 6 8 10 12 14 16 18 20
10?2
10?1
100
SNR per secondary user (dB)
sum MSE
 
 
M=4, d=1, iterative
M=4, d=1, non? iterative
M=4, d=2, iterative
M=4, d=2,non? iterative
M=3, d=1, iterative
M=3, d=1, non?iterative
M=3, d=2, iterative
Figure 3.5: Sum MSE vs SNR per secondary user 10log10(Ptot,i/?2n): K = 4, Ptot,i is the
same for i = 1,??? ,K, M = 3 or 4, N = 10, d = 1 or 2, Mp = Jp = 2, Ptot,r = KPtot,i,
? = 10
channel state information (CSI) was available. We considered iterative optimization as well
as a non-iterative approach, which can be implemented in a distributed manner. A channel
error-aware robust matrix distance based algorithm was also proposed to address the case
of imperfect CSI knowledge. The efficacy of our proposed algorithms was illustrated via
computer simulations. Significant gains in the sum rate of the secondary network can be
obtained by proper design of precoders and decoders in the multiuser multi-way relay system
compared to the case of no precoders, while mitigating interference to the primary system.
80
0 2 4 6 8 10 12 14 16 18 2010
?2
10?1
100
101
SNR per secondary user (dB)
sum MSE
 
 
N=12, iterative
N=12, non?iterative
N=9, iterative
N=9, non?iterative
N=8, iterative
N=8, non?iterative
Figure 3.6: Sum MSE vs SNR per secondary user 10log10(Ptot,i/?2n): K = 4, Ptot,i is the
same for i = 1,??? ,K, M = 3, N = 8, 9 or 12, d = 2, Mp = Jp = 1, Ptot,r = KPtot,i, ? = 10.
0 50 100 150 200 250 3000.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Iteration
MSE
 
 
Iterative, Itot = 0
Iterative, Itot = 2?n2
Figure 3.7: Sum MSE vs iteration (for a single run): 10log10(Ptot,i/?2n) = 5dB, K = 4, Ptot,i
is the same for i = 1,??? ,K, M = 4, N = 10, d = 2, Mp = Jp = 1, Ptot,r = KPtot,i. Recall
that noise variances have been normalized to one.
81
0 2 4 6 8 10 12 14 16 18 2010
?1
100
101
SNR per secondary user (dB)
sum MSE
 
 
robust design, ?e2=0.01, N=10
non?iterative, ?e2=0.01, N=10
non?iterative, ?e2=0.01/0.1, perfect CSI, N=10
robust design, ?e2=0.1, N=10
non?iterative, ?e2=0.1, N=10
robust design, ?e2=0.01, N=12
robust design, ?e2=0.1, N=12
Figure 3.8: Sum MSE vs SNR per secondary user 10log10(Ptot,i/?2n): K = 4, Ptot,i is the
same for i = 1,??? ,K, M = 4, N = 10, d = 2, Mp = Jp = 2, Ptot,r = KPtot,i, ? = 10,
receiver noise variance ?2n = 1, design interference bound Itot = 2?2n.
0 2 4 6 8 10 12 14 16 18 2010
?1
100
101
102
SNR per secondary user (dB)
total interference to PU
 
 
robust design, ?e2=0.01
non?iterative, ?e2=0.01
robust design, ?e2=0.1
non?iterative, ?e2=0.1
design interference bound
Figure 3.9: Average interference power to primary receivers vs SNR per secondary user
10log10(Ptot,i/?2n): K = 4, Ptot,i is the same for i = 1,??? ,K, M = 4, N = 10, d = 2,
Mp = Jp = 2, Ptot,r = KPtot,i, ? = 10, receiver noise variance ?2n = 1, design interference
bound Itot = 2?2n.
82
Chapter 4
Precoding Design in Multi-Pair Two-Way Cognitive Relay Networks
Two-way relay network is an effective way to support the data exchange of two users
within the same user pair. Traditional two-way relay system contains two users who want to
share information and one relay node. Given the fact that each user has perfect knowledge
of its own transmitted signal, the information from one user can be coded with its desired
signal, called network coding, and the recovery of its desired signal is achieved by subtracting
its own signal, which is usually called self-interference cancellation. Suppose both the end
nodes and the relay operate in a half-duplex mode, the information exchange of two users
can be completed by a two-time slot transmission scheme. In the first time slot, also called
as multiple access (MAC) phase, two end users transmit to the relay simultaneously, while
in the second time slot, also called as broadcasting (BC) phase, the relay broadcasts its
received and coded signals to the two end users. Similar to one-way relaying, the relay node
can either perform decode-and-forward or amplify-and-forward [15,52]. The advantage of
AF is its simplicity in relay design and implementation since the relay only amplifies its
received signal instead of decoding it as in DF scheme.
With the deployment of multiple antennas, one relay node may support the data ex-
change of multiple users pairs since multiple antennas provide extra spacial dimensions which
can accommodate multiple network coded datastreams. The MIMO techniques can signif-
icantly improve the system throughput. However in multi-pair two-way relay networks,
interference between user pairs can dramatically degrade the system?s performance. Trans-
mit and relay precoding is an efficient way to explore the multiplexing gain provided by
multiple antennas and diminish the harmful effects of inter-pair interference.
83
In this chapter, we propose a joint source and relay precoding method for a multi-pair
two-way relay cognitive network. In this network, multiple secondary user pairs exchange
information with the help of a relay node. They coexist with a primary source-destination
pair within the same frequency band and both the secondary nodes and the primary node
transmit concurrently. With the perfect CSI available, a matrix-distance based precod-
ing/decoding scheme is proposed which align both the signal and inter-pair interference so
that an improved performance under certain condition is achieved, compared with the MSE-
based method as in Chapter 3. The iteration between precoding/decoding at the users and
at the relay can be performed to further improve the system performance at a cost of higher
complexity.
This chapter is organizedasfollows. Background of precoding in cognitive radio andtwo-
way relay networks and related works are presented in Sec. 4.1. System model is introduced
in Sec. 4.2. A matrix distance based precoding method is proposed in Sec. 4.3. In Sec. 4.4
simulation results show the performance of the proposed precoding design. Conclusions are
drawn in Sec. 4.5.
4.1 Introduction
Cognitive radio is a promising technology to increase spectrum efficiency. The unli-
censed users can access licensed spectrum either opportunistically or concurrently. When
secondary users concurrently access the spectrum with licensed primary users, how to miti-
gate interference to primary users becomes a crucial design issue of secondary network. The
employment of multiple antennas at secondary transmitters makes it possible to cancel out
the interference to primary network by aligning the transmit signal direction given complete
channel state information. Therefore in multi-user cognitive radio system, both the inter-
secondary user interference and the secondary-primary interference should be mitigated or
at lease minimized by proper transceiver design. In [38] a rate balanced transceiver was pro-
posed in multiuser cognitive radio network. In [39], a linear precoding method was proposed
84
for CR multiuser downlink MIMO system based on minimum mean square error (MMSE)
criterion. For systems with imperfect CSI, [53] studied a nonlinear transceiver design prob-
lem in a multi-tier MIMO cognitive radio network while [54] proposed a linear transceiver
design in a downlink cognitive MIMO system.
Relay based two-way transmission schemes provide significant performance improvement
such as higher power and spectrum efficiency and extended coverage (compared to the case
when there is no relay). By performing self-interference cancellation, each user can obtain its
desired information. However, when there are multiple user pairs, the inter-pair interference
can dramatically reduce the system performance. Therefore how to eliminate interference or
alleviate the negative effects of this interference is crucial in system design. Two common
methods to perform interference cancellation are based on MMSE and zero-forcing (ZF)
criteria. By adopting the MMSE criterion, in [48], the precoders at relay and sources are
designed to minimize the sum MSE of received signals at all users. For zero-forcing system
design, [55] proposed a distributed beamforming for a multi-pair two-way relay system where
each node has a single antenna. The inter-pair interference is canceled out by designing relay
weights of multiple signal antenna relays. In [44] the relay beamforming matrix was designed
based on ZF and MMSE criteria for a multi-user MIMO relay system. By employing MIMO
techniques, many interference alignment methods have been proposed to null out the inter-
pair interference therefore enabling multi-pair transmission [1,34]. In [35], a joint signal
and interference alignment algorithm is proposed. In [57], a coordinated eigen beamforming
method is proposed to optimize the throughput given that the inter-pair interference has
already been eliminated.
In this chapter, a multi-pair two-way relay system with a single relay in a cognitive
radio network is considered. Secondary end users and the relay are equipped with multiple
antennas. Unlike Chapter 3 in which the relay precoder is designed using a MMSE criterion
for both iterative and non-iterative algorithms, in this chapter we propose an interference
alignment (IA)-like relay precoder to align both the interference and desired signal directions
85
while ensuring no interference is caused to the primary network. By allowing some residual
inter-pair interference, the system performance can be further improved compared with ZF-
based design in which the inter-pair interference is completely removed in the low SNR
regime. Also this is a more general algorithm which can be employed whether the number
of antennas at the relay station is large enough to permit zero-forcing or not.
Notation: We use bold lower-case and upper-case letters to denote vectors and matrices
respectively. In denotes the n ? n identity matrix. (.)T and (.)? denote the transpose and
conjugate transpose of a matrix respectively. bardbl.bardbl and bardbl.bardblF are 2-norm and Frobenius norm
respectively. Cn?m denotes the space of n?m complex matrices. {A}m,n denotes the mn?th
element of matrix A.
4.2 System Model and Problem Formulation
Consider a cognitive radio network consisting of 2K secondary users and one relay
station. This cognitive radio network coexists with a primary source-destination pair within
a single band. Suppose secondary user k and k + 1, (k < 2K and k is odd) form a user
pair who want to send information to each other through the help of the relay node. Each
secondary user is equipped with M antennas while the relay station has N antennas and the
primary transmitter and receiver have J antennas. A two-time-slot half-duplex transmission
scheme is used to support the two-way transmission of K secondary user pairs. In the first
time slot all of the 2K users transmit to the relay while in the second time slot relay sends a
linear combination of its received signal fromthe first time slot, and 2K secondary users listen
and decode their desired signals, showed in Fig. 4.1. Let Hir ? CN?M and Hri ? CM?N,
i = 1,2,...,K denote the channel coefficient matrices from user i to relay station and from
relay to user i. Also Hip ?CJ?M, Hpr ?CN?J, Hrp ?CJ?N and Hpi ?CM?J, i = 1,2,...,K
denote the channel coefficient matrices from secondary user i to primary receiver, from
primary transmitter to relay, from relay to primary receiver and from primary transmitter
86
to secondary user i, respectively. It is assumed that these channel matrices remain constant
during the two-time slot transmission and are available (known) to the secondary network.
Assume each secondary user has d independent data streams to transmit. Let si be
the d?1 information vector from i-th secondary user with zero mean and E{sis?i} = ?2sId.
Ti ?CM?d is the precoding matrix of user i. Then in the first time slot, transmitted signal
from user i is xi = Tisi. During the first time slot, the received signal yr at secondary relay
node and interference i1 to primary receiver are given by
i1 =
2Ksummationdisplay
i=1
HipTisi
yr =
2Ksummationdisplay
i=1
HirTisi +Hprxp +nr,
wherexp istheJ?1 signal vector transmitted by primaryuser withzero mean andE{xpx?p} =
?2pIJ and nr is the N?1 complex white Gaussian noise vector at relay with nr ?N(0,?2nIN).
In the second time slot, relay transmits a linear combination of its received signal, using a
precoding matrix Tr. The transmitted signal is denoted as xr = Tryr
The interference i2 to primary receiver and received signal yi at 2K secondary users
during the second time slot are expressed as follows:
i2 = HrpTr
parenleftbigg 2Ksummationdisplay
i=1
HirTisi +Hprxp +nr
parenrightbigg
(4.1)
yi = Hri
parenleftbigg 2Ksummationdisplay
j=1
TrHjrTjsj +TrHprxp +Trnr
parenrightbigg
+Hpi?xp +ni, i = 1,2,...,2K, (4.2)
?xp is the J?1 signal vector transmitted by primary user in second time slot, with zero mean
and E{?xp?x?p} = ?2pIJ and ni is the M ?1complex white Gaussian noise at secondary user i
which follows N(0,?2nIM).
87
Suppose each user i uses a decoding matrix Ri to estimate its received signal ?si. Then
we have
?si = Riyi = Ri
braceleftbigg
Hri
parenleftbigg 2Ksummationdisplay
j=1
TrHjrTjsj +TrHprxp +Trnr
parenrightbigg
+Hpi?xp +ni
bracerightbigg
. (4.3)
As in other interference alignment schemes, we assume noise is negligible compared to inter-
ference from other users. So we rewrite (4.3) as
?si = Ri
braceleftbigg
Hri
parenleftbigg 2Ksummationdisplay
j=1
TrHjrTjsj +TrHprxp
parenrightbigg
+Hpi?xp
bracerightbigg
.
One crucial requirement for cognitive radio design is to cause no harm to primary network.
This means i1 = i2 = 0 should be satisfied for any possible transmit signal vectors si,i =
1,2,...,K. Therefore we enforce the following constraints on precoders Ti and Tr:
HipTi = 0, i = 1,2,...,2K, (4.4)
HrpTr = 0. (4.5)
Let H?ip and H?rp be the null space matrices of Hip and Hrp respectively. Then letting
Ti = H?ipPi and Tr = H?rpPr satisfies (4.4) and (4.5). The precoding parameters that are
to be designed now become Pi, i = 1,2,...,K and Pr.
Lemma 1 : In order to cause no interference to primary network, the number of
antennas at secondary users and relay should satisfy M > J, N > J. The maximum
multiplexing order that each SU can employ should not exceed M ?J, i.e. d ? M ?J.
Proof: This follows directly from the fact that channel matrices Hip,Hrp have full rank
almost surely. For the channel between secondary user i and primary receiver, rank{Hip} =
min{M,J}. If M ? J, then the dimension of Hip?s null space will be empty, which makes it
impossible to cancel the interference to primary network completely. So it is required that
M > J, which leads to rank{Hip} = J, rank{H?ip} = M ? J and H?ip is a M ? (M ? J)
88
PUtPUr
R
SU 1
SU 2K-1
SU 2
SU 2K
..
..
..
..
..
..
....
Figure 4.1: System model with 2K SUs, one secondary relay and one primary source-
destination pair
matrix. Then rank{Ti} ? M ? J which means the multiplexing order user i can employ
should not exceed M ?J. For a similar reason, the requirement on relay antenna number is
N > J. H?rp is an N ?(N ?J) matrix and its rank is N ?J. square
In this chapter, we assume secondary users employ the maximum multiplexing order, i.e.
d = M ?J.
For notational simplicity, define the equivalent channel matrices ?Hir = HirH?ip, ?Hri =
HriH?rp, for i = 1,2,...,K. Since it is guaranteed that secondary users? transmission won?t
cause any interference to primary network, we can concentrate on designing precoding and
decoding matrices to alleviate the negative effects of inter-user interference and improve
secondary network?s performance. From the above discussion, the precoding matrices design
at the relay and SU i now becomes the task of designing Pi ?C(M?J)?d and Pr ?C(N?J)?N.
89
4.3 Matrix Distance based Precoding Design
In this section, we present a matrix distance based precoding design at secondary users
as well as the relay station. Unlike most IA schemes aiming to remove inter-user interference
completely, we propose a matrices distance based algorithm in which the direction of both the
desired signal and inter-user interference are jointly considered. A scalar weight is introduced
to strike a balance between the ?desired signal? and ?interference?. It will be shown that
by allowing a small residual inter-user interference, it is possible to increase the strength of
desired signal; therefore better performance can be achieved.
The distance between two matrices A and B, each with orthonormal columns, is defined
as [35] (the motivation of this definition was discussed in detail in Sec. 3.4)
bardblA,Bbardbldis = bardblA?BB?AbardblF. (4.6)
The design goal is to align the signal direction as close to its desired receiver as possible while
trying to keep it away from the direction to other receivers so that inter-user interference
is mitigated. Firstly we initialize the precoding and decoding matrices Ti and Ri at each
user i using a matrix distance criterion. Then the relay precoding matrix is expressed as
a cascade of 3 parts including a receive combining matrix, power amplifying matrix and
transmit beamforming matrix. The two relay beamforming matrices are then designed by
jointly considering the inter-user interference and desired signal strength. Finally another
SU decoding matrix ?Ri at user i is cascaded with Ri in order to decode each datastream
for user i, i.e. removing the inter-datastream interference. The transmit precoder design at
SUs and relay guarantees that no interference is caused to the primary network.
4.3.1 Secondary User Precoding/Decoding Matrices Initialization
Before designing precoding matrix Pr at relay node, precoding matrix Ti and decoding
matrix Ri at i-th SU?s are initialized. Ti is designed to minimize its distance to the channel
90
matrix Hir so ?more information? can be transmitted to relay station. Besides, to make sure
that primary user?s transmission is not interfered with by secondary network, Ti = H?ipPi
is imposed. Without loss of generality, we assume Pi has orthonormal columns. A scaler ?i
is introduced to satisfy the transmit power constraint of user i; therefore Ti is of the form
?iH?ipPi. The equivalent channel matrix for Pi is ?Hir. Then the precoding design at user i
is formulated as:
Pi = argminP?
iPi=Id
bardbl?H?ir,Pibardbl2dis
= argminP?
iPi=Id
bardbl?H?ir ?PiP?i ?H?irbardbl2F. (4.7)
By simplifying (4.7), we obtain
Pi = arg max
P?iPi=Id
tr{P?i ?H?ir ?HirPi}. (4.8)
Since Pi ?C(M?J)?d has orthonormal columns, the columns of Pi should be the d dominant
eigenvectors of matrix ?H?ir ?Hir and each column?s 2-norm is normalized to 1. Let P?i be the
optimized solution of 4.7. ?i is chosen to satisfy the total transmit power constraint Ptot,i at
user i. So ?i =
radicalBig
Ptot,i/tr{?2sH?ipP?iP??i H??ip }, i = 1,2,...,K.
The decoding matrices initialization at SU i is to minimize the distance of decoding
matrix Ri and corresponding channel matrix Hri. The interference from primary user should
also be considered, which means that we should let the signal from primary user lie closer
to the orthogonal complement of the subspace spanned by Ri. Let R?i be a matrix whose
rows are orthonormal basis for the complement of the subspace spanned by Ri?s rows. It is
easy to show that R??i R?i = Id ?R?iRi. Then the design of decoder Ri is formulated as
Ri = argminRiR?
i=Id
braceleftbigg
?bardblHri,R?ibardbl2dis +bardblHpi,R??i bardbl2dis
bracerightbigg
= argminRiR?
i=Id
braceleftbigg
?bardblHri ?R?iRiHribardbl2F +bardblHpi ?R??i R?i Hpibardbl2F
bracerightbigg
, (4.9)
91
where ? is a scalar weight to strike a balance between signal from secondary relay and
interference from primary user?s transmission. By simplifying (4.9) we have
Ri = arg min
RiR?i=Id
tr{Ri(HpiH?pi ??HriH?ri)R?i}. (4.10)
Since Ri ? Cd?M, the rows of Ri should be the d least dominant eigenvectors of (HpiH?pi ?
?HriH?ri) whose 2-norms are normalized to 1. R?i is used to denote the solution of (4.9).
4.3.2 Iterative Relay Precoding and Secondary User Precoding/Decoding De-
sign
After Ti and Ri are determined, we present a joint transmit and receive beamform-
ing design at the relay station. Its precoding matrix Pr is designed to have the following
structure:
Pr = F?G, (4.11)
where G ? CKd?N is the receiving combining matrix, F ? C(N?J)?Kd is the transmitting
beamforming matrix, and ? is a Kd ? Kd diagonal matrix. It is the amplifying matrix
of Kd datastreams after network coding. Let ?si be the desired signal vector at secondary
user i and ?s = [?sT1 ?sT2 ... ?sT2K]T be the desired signal at all 2K users. Partition G =
[GT1 GT2 ... GTK]T and F = [F1 F2 ... FK]. Also denote ? = diag{?1 ?2,...,?K}.
Then Gi ? Cd?N, Fi ? C(N?J)?d and ?i ? Cd?d are the receive beamforming matrix,
transmit beamforming matrix and amplifying matrix of data-streams of the i-th SU pair i.e.
users 2i?1 and 2i.
Let s and ?s be the transmitted and received signal vector from all 2K secondary users.
Also define x = [sT xTp ]T to be the transmitted signal in the first time slot including both
signals from primary and secondary network. Then x is a (2Kd + J) ? 1 vector and the
92
received signal of all 2K secondary users can be expressed as
?s = H2F?GH1x+R?xp, (4.12)
where H1 and H2 are the equivalent channel matrices of the two hops, defined as
H1 = [?H1rP1, ?H2rP2, ..., ?H2KrP2K, Hpr], (4.13)
H2 = [(R1Hr1)T, (R2Hr2)T, ..., (R2KHr2K)T]TH?rp. (4.14)
Only the first term in (4.12) is related to relay precoding matrix Pr. The second term is
the received interference from primary network?s transmission in the second time slot. It
is up to the receive filters Ri, i = 1,2,...,2K to mitigate this interference. Therefore in
this subsection, we only focus on the first term. Let s(R) = H2F?GH1x be the relay-related
signal part. Since we need to cancel the interference to primary users, the equivalent channels
from SU i to relay and from relay to SU i do not have reciprocity anymore. Therefore the
receive and transmit beamformers of relay should be designed differently. Firstly we design
the i-th SU pair?s receive beamforming vector Gi.
For the i-th SU pair, only the signal from users 2i?1 and 2i are desired or considered
self-interference which can be completely removed by self-interference cancellation. Signals
transmitted by all other users are considered interference and should be mitigated by pre-
coding design at the relay node. Define ?H1i as the interference channel matrix associated
with i-th user pair. Then ?H1i is formed by removing (2i?2)d+1 : 2id-th columns from H1.
Let B1i be the (2i?2)d + 1 : 2id-th column of matrix H1, then B1i is considered channel
matrix associated with useful signals. The goal is to receive ?more? of the useful signal and
remove as much as possible the harmful interference. Then receive beamforming matrix Gi
93
is designed according to the following matrix-distance criterion:
Gi = argminGiG?
i=Id
braceleftbigg
?bardblB1i,G?ibardbl2dis +bardbl?H1i,G??i bardbl2dis
bracerightbigg
= argminGiG?
i=Id
braceleftbigg
?bardblB1i ?G?iGiB1ibardbl2F +bardbl?H1i ?G??i G?i ?H1ibardbl2F
bracerightbigg
(4.15)
where G?i is a matrix with its rows being the orthonormal basis for the orthogonal com-
plement of the subspace spanned by Gi?s rows and ? is a scalar weight to strike a balance
between the desired signal and interference. It will be shown later that by setting ? = 0,
this matrix-distance based beamforming design becomes zero-forcing beamforming; therefore
zero-forcing is a special case of our method. By simplifying (4.15) and using the fact that
G??i G?i = Id ?G?iGi, we get
Gi = argminGiG?
i=Id
tr
braceleftbigg
Gi[?H1i ?H?1i ??B1iB?1i]G?i
bracerightbigg
. (4.16)
Since Gi has d rows, its rows should be the d least eigenvectors of the matrix ?H1i ?H?1i ?
?B1iB?1i, with its 2-norm normalized to 1. The solution of (4.15) is denoted as G?i.
The design of transmit beamforming matrix Fi of i-th user pair?s data-stream follows
in a similar way. Let ?H2i be the interference channel matrix for i-th user pair. It is formed
by deleting (2i ? 2)d + 1 : 2id-th rows from H2. The channel matrix associated with the
useful datastream of user pair i is the (2i?2)d+1 : 2id-th rows of H2, which is denoted as
B2i. Using ? as a scalar weight to strike a balance between aligning the desired signal and
mitigating inter-user interference, the relay transmit beamforming matrix for i-th SU pair is
chosen to be
Fi = argminF?
iFi=Id
braceleftbigg
?bardblB?2i,Fibardbl2dis +bardbl?H?2i,F?i bardbl2dis
bracerightbigg
= argminF?
iFi=Id
braceleftbigg
?bardblB?2i ?FiF?iB?2ibardbl2F +bardbl?H?2i ?F?i F??i ?H??2i bardbl2F
bracerightbigg
, (4.17)
94
Here F?i is a matrix whose columns are orthonormal basis of the orthogonal complement of
the subspace spanned by Fi?s columns. By simplifying (4.17) and using the fact F?i F??i =
Id ?FiF?i, the transmit beamforming matrix Fi is
Fi = arg min
F?iFi=Id
tr
braceleftbigg
F?i
bracketleftBig?
H?2i ?H2i ??B?2iB2i
bracketrightBig
Fi
bracerightbigg
. (4.18)
Fi?s columns are chosen to be the d least dominant eigenvectors of matrix ?H?2i ?H2i??B?2iB2i
with each column?s 2-norm normalized to 1. The solution is denoted as F?i. The amplifying
matrix ? is chosen to be an identity matrix since the power amplifying to different datas-
treams is not the focus in this chapter. A scalar ?r is introduced to satisfy the transmit power
constraint for relay node. Let Ptot,r be the maximum transmit power for relay node, then?r =
radicalBig
Ptot,r/tr{?2sH?rpF??G?H1diag{?2sI2Kd ?2pIJ}H?1G????F??H??rp +?2nH?rpF??G?G????F??H??rp}
and the precoding matrix at relay station is ?rH?rpF?G.
Now we show that zero-forcing beamforming is a special case of our proposed matrix-
distance based IA-like precoding design. By setting ? = 0 in (4.15), rows of receive beam-
forming matrix Gi should be chosen as the least dominant eigenvector of matrix ?H1i ?H?1i. In
the case when ?H1i?s row null space has at least d dimension, Gi should satisfy Gi ?H1i = 0.
For transmit beamforming vector Fi, by setting ? = 0 in (4.17), columns of Fi are the d least
dominant eigenvectors of ?H?2i ?H2i. Therefore when ?H2i?s null space has at least dimension d,
Fi should satisfy ?H2iFi = 0. Therefore, when zero-forcing is possible, by setting ? = 0, our
proposed precoder becomes the zero-forcing precoder.
Lemma 2 : By performing relay precoding, the number of antennas at relay station
should satisfy N ? (2K ? 1)d + J to perform zero-forcing and cancel out all inter-user
interference (including the interference between primary and secondary network).
Proof: We already know that for each Gi and Fi, to satisfy the zero-forcing criteria G?i ?H1i =
0 and ?H2iFi = 0, we need the dimension of ?H1i?s column null space and ?H2i?s row null space
to be at least d. Since ?H1i is an N ?(2K ?2)d+J matrix who has full rank almost surely,
95
rank{?H1i} = min{N,(2K?2)d+J}? N?d should be satisfied such that its row null space?s
dimension N ?rank{?H1i} ? d. Therefore, we need N ? (2K ?1)d+J. Similarly, for ?H2i,
we need rank{?H2i} = min{N?J,(2K?2)d} ? N?J?d. Therefore, (2K?2)d ? N?J?d
should be satisfied, which also leads to N ? (2K ?1)d+J. square
After the relay precoding matrix is designed, end users can adjust their precoding and
decodicng matrices to further improve the performance. Since the relay precoding matrix
is decomposed into three parts including receive combining matrix G, transmit beamform-
ing matrix F, and amplifying matrix ? for Kd network coded datastreams, the two-way
transmission can be decomposed into two one-way transmission, namely SU-to-relay and
relay-to-SU. Therefore, the secondary user precoding can be designed only based on G while
the secondary user decoding matrix is designed based on F.
For the transmit precoding design at secondary user k,k ? {1,2,...,2K}, let Ak be
the equivalent channel matrix of its signal, and ?Ak be the channel matrix associated with
interference. Since at relay, the signal of two users? are network coded together, then k-th
user?s desired signal is indexed as ?k/2?-th datastream at the relay station. Ak and ?Ak are
defined as the follows
Ak = G?k/2?HkrH?kp , ?Ak = ?G?k/2?HkrH?kp, (4.19)
where
?G?k/2? = [GT1 ,...,GT?k/2??1,GT?k/2?+1,...,GTK]T.
The design of k-th user?s precoding matrix Pk follows a similar way as the design of relay
transmit precoding matrix Fi.
Pk = argminP?
kPk=Id
braceleftBig
?bardblAk,Pkbardbl2dis +bardbl?Ak,P?k bardbl2dis
bracerightBig
= argminP?
kPk=Id
tr
braceleftbigg
P?k
bracketleftbigg
?A?k ?Ak ??A?kAk
bracketrightbigg
Pk
bracerightbigg
(4.20)
96
Then the columns of Pk should be the d least dominant eigenvectors of matrix ?A?k ?Ak ?
?A?kAk.
For k-th user?s receive decoding matrix Rk design, we define
Ck = HrkH?rpF?k/2? , ?Ck = [HrkH?rp?F?k/2?,Hpi]. (4.21)
where
?F?k/2? = [?F1,..., ?F?k/2??1, ?F?k/2?+1,..., ?FK].
The design of Ri is formulated as
Rk = argminR
kR
?
k=Id
braceleftBig
?bardblCk,R?kbardbl2dis +bardbl?Ck,R??k bardbl2dis
bracerightBig
= argminR
kR
?
k=Id
tr
braceleftbigg
Rk[?Ck ?C?k ??CkC?k]R?k
bracerightbigg
. (4.22)
Therefore the rows of Rk should be the d least dominant eigenvectors of matrix ?Ck ?C?k ?
?CkC?k. The detailed procedure of how to perform iteration is summarized in Algo. 1.
4.3.3 Convergence of Proposed Iterative Method
In this two time slot transmission scheme, the design of precoding/decoding matrices is
divided into two parts. The first part is the precoding matrices design at the users and the
decoding matrix G design at the relay in the first time slot. The second part is to design the
relay precoding matrix F and user decoding matrices in the second time slot. The iteration
is performed for these two parts in parallel. In the next we show that for each part, our
algorithm converges.
97
For the design in the first time slot. The two objectives of the two problems (4.16) and
(4.20) are denoted as Oi and ?Ok respectively. We have
Oi =
Ksummationdisplay
k=1,knegationslash=2i?1,knegationslash=2i
tr
braceleftBig
Gi(?HkrPk)(?HkrPk)?G?i
bracerightBig
+ tr
braceleftBig
GiHprH?prG?i
bracerightBig
??tr
braceleftBig
Gi(?H2i?1,rP2i?1)(?H2i?1,rP2i?1)?G?i
bracerightBig
??tr
braceleftBig
Gi(?H2i,rP2i)(?H2i,rP2i)?G?i
bracerightBig
?Ok = Ksummationdisplay
i=1,inegationslash=?k/2?
tr
braceleftBig
P?k(Gi ?Hkr)?(Gi ?Hkr)Pk
bracerightBig
??tr
braceleftBig
P?k(G?k/2? ?Hkr)?(G?k/2? ?Hkr)Pk
bracerightBig
(4.23)
It can be shown that
Ksummationdisplay
i=1
Oi =
2Ksummationdisplay
k=1
?Ok + Ksummationdisplay
i=1
tr
braceleftBig
GiHprH?prG?i
bracerightBig
The user precoding design in the first time slot can be expressed as
min
P?kPk=I,k=1,2...2K
?Ok ? min
P?kPk=I,k=1,2...2K
?Ok + Ksummationdisplay
i=1
tr
braceleftBig
GiHprH?prG?i
bracerightBig
(4.24)
The equality holds because summationtextKi=1 tr
braceleftBig
GiHprH?prG?i
bracerightBig
is not dependent on value of Pk therefore
the two problems in (4.24) will lead to the same solution. This problem can be decomposed
into 2K subproblems as in (4.20) and these subproblems can be solved in parallel and
independent of each other.
On the other hand, the relay decoding matrix G is designed by solving the following
problem
min
GG?=I
Ksummationdisplay
i=1
Oi, (4.25)
which can be decomposed into K subproblems as in (4.16). These subproblems can also be
solved in parallel and independent of each other.
The iteration for the first time slot precoding/decoding design is performed by alter-
nately solving problem (4.24) and (4.25). We have shown that both problems are to minimize
98
the same objective. Since this objective is lower bounded by zero, we can conclude that the
iteration converges.
The convergence of the iteration in the second time slot, i.e. the iteration between the
optimization of relay precoding matrix F and user receive decoding matrices Ri, can be
proved in a similar way. They try to minimize the same objective and the objective is lower
bounded by zero.
4.3.4 Refining Decoding Matrices to Decode Each Datastream
In the previous subsection, we designed relay precoding which tries to mitigate inter-pair
interference. After performing self-interference cancellation, the multiple data-streams for
each SU are still ?mixed up? together which leaves to the decoding filter at user i to solve.
Since Ri has already been designed, a secondary decoding matrix ?Ri ? Cd?d is designed to
be cascaded with Ri to distinguish among the d different data-streams for user i. Therefore
the decoding matrix at user i is ?RiRi. Express the received signal ?si as
?si = ?Ri ?Hixt,
where ?Hi is the equivalent channel matrix from both primary and secondary transmitters to
secondary user i, ?Hi = Ri[HriTrH1 Hpi], and xt = [s xp ?xp] is the transmitted signal
from all secondary users in the first time slot and from primary user in both time slots.
Denote j-th row of ?Ri by ?rij which is the receive combining vector of j-th datastream at
user i. It wants to receive the j-th datastream sent by user i?1 if i is even and user i+1 if
i is odd. Let l(i,j) = j +id if i is odd while l(i,j) = j +(i?2)d if i is even. Let hij be the
channel vector associated with j-th desired signal at user i; then hij is the l(i,j)-th column of
?Hi. Also, by performing self-interference cancellation, the signal sent by i-th secondary user
can be completely removed from j-th estimated signal at user i. Let ?Hij, which is formed
by deleting l(i,j)-th and (i?1)d + 1 : id-th columns from ?Hi, be the interference channel
99
matrix. ?ril can be chosen using the matrix-distance criterion:
?rij = argmin?rij?r?
ij=1
braceleftbigg
?bardbl?hij,?r?ijbardbl2dis +bardbl?Hij,?r??ij bardbl2dis
bracerightbigg
. (4.26)
So ?rij should be chosen as the least dominant eigenvector of ?H1i ?H?1i???hil?h?1i, with its 2-norm
normalized to 1.
It can be shown that when N ? (2K ? 1)d + J, the inter-user interference can be
completely removed. In ?Hij only the columns associated with i-th user?s desired datastreams
(not including j-th) are non zero. So we have rank{?Hij} = d ? 1 if zero-forcing precoder
at relay is used. The rank of ?Hij?s column null space is 1 and by setting ? = 0 in (4.26),
?rij ?Hij = 0, which means the interference among different datastreams of the same user can
also be removed completely.
The proposed iterative algorithm which optimize the precoding/decoding matrices at
the secondary users and the precoding matrix at the relay node is summarized as follows.
ALGORITHM 4.1
1) Each user initialize Pi and Ri as described in Sec. 4.3.1.
2) Repeat (iterate):
? Design G,F and ? based on most current Pi and Ri,i = 1,2,...,K as described
in Sec. 4.3.2.
? Update Pi and Ri based on G and F as described in Sec. 4.3.2.
Continue until ending iteration criterion is meet.
3) Refine each Ri to recover d datastreams wanted by each user as in Sec. 4.3.4.
100
One simple way to end the iteration process is to set a maximum iteration number and
algorithm ends when this max iteration number is reached.
4.4 Simulation Results
In this section we provide simulation results to show the effectiveness of our algo-
rithm. Consider a secondary network consisting of K = 2 SU pairs (total 4 SUs) and
one non-regenerative relay node, coexisting with one source-destination primary user pair.
All channels experience flat Rayleigh fading and the channel gains remain constant during
the two time slots transmission. Assume the expectation of all channel power gains is 1 and
?2s = ?2p = ?2n = 1. Also assume the number of antennas at secondary users is M = 3 and
relay has N antennas. The number of data-streams d=2. The primary user is equipped with
a single antenna J = 1. We evaluate the sum rate of K secondary users of our proposed
algorithm. Also by comparing it with MSE-based algorithm in Chapter 3 and zero-forcing,
we illustrate the performance of this proposed matrix distance based alignment method in
both high and low SNR regime with different number of antennas at the relay.
0 5 10 15 20 25 30 35 400
2
4
6
8
10
12
14
16
18
20
Transmit power per secondary user (dB)
Sum rate (bps/Hz)
 
 
Iterative IA, N=7, ? = 0.01
MSE, N=7
Iterative IA, N=6, ? = 0.01
MSE, N=6
Figure 4.2: Sum rate (4.27) vs. transmit power per secondary user
101
Define the sum rate of 2K SUs as
Rate =
2Ksummationdisplay
i=1
dsummationdisplay
j=1
log2
parenleftBigg
1 + |{
?Ri ?Hi}j,l(i,j) |2 ?2s
iNij
parenrightBigg
(4.27)
where iNij is noise and interference power for j-th datastream at user i given by
iNij = ?rij ?Hijdiag{?2sI(2K?1)d?1,?2pI2J}?H?ij?r?ij
+?rijRiHriTrT?rH?riR?i?r?ij?2n +?rijRiR?i?r?ij?2n
0 5 10 15 20 25 30 35 400
2
4
6
8
10
12
14
16
18
20
Transmit power per secondary user (dB)
Sum rate (bps/Hz)
 
 
zero?forcing, N=8
Iterative, IA, ? = 0.001, N=8
zero?forcing, N=7
Iterative, IA, ? = 0.05, N=7
Figure 4.3: Sum rate (4.27) vs. transmit power per secondary user
In Fig. 4.2 the sum rate of secondary network using matrix distance alignment algorithm
is compared with MSE-based algorithm (non-iterative) in Chapter 3 for N ? (2K?1)d+J
and N < (2K ? 1)d + J cases. It is observed that IA-like algorithm significantly outper-
forms MSE-based algorithm in medium to high SNR regime, because the IA-like algorithm
focuses on inter-pair interference and primary-to-secondary interference, which are the most
important problems when SNR is high and the MSE-based algorithm deals with both in-
terference and noise. Fig. 4.3 compares the proposed algorithm with zero-forcing, which is
102
0 5 10 15 20 25 30 35 400
1
2
3
4
5
6
7
8
Transmit power per secondary user (dB)
Sum rate (bps/Hz)
 
 
Iterative, IA, ? = 1, N=6
Iterative, IA, ? = 0.1, N=6
Iterative, IA, ? = 0.01, N=6
Figure 4.4: Sum rate (4.27) vs. transmit power per secondary user
achieved by setting ? = 0 for N ? (2K ? 1)d + J case. When N = (2K ? 1)d + J, i.e.
N = 7, the IA-like algorithm outperforms zero-forcing in low to medium SNR regime while
the zero-forcing achieves better performance in high SNR regime because interference among
users is a more important factor compared to desired signal strength when SNR is high. For
N > (2K ? 1)d + J, e.g. N = 8 case, the proposed algorithm achieves a higher sum rate
for the entire SNR regime because more degrees of freedom are provided in this case and
the proposed algorithm not only tries to cancel the interference but also employ the extra
degrees of freedom to maximize the desired signal strength. In Fig. 4.4 we show the effect
of different choices of the scaler weight ? on the system performance. A larger value of ?
means more weight on the signal strength over interference. It is observed that in the low
SNR regime, it is better to choose larger ? while in the higher SNR regime the negative
effect of interference is more severe, therefore smaller value of ? is more desirable. In all
simulations, the proposed matrix distance based alignment algorithm performs 10 iterations.
103
4.5 Conclusion
In this chapter we proposed a interference and signal alignment precoding algorithm
based on matrices distance for multi-pair two-way relay system in a cognitive radio scenario
where secondary users and relay are equipped with multiple antennas. This proposed al-
gorithm not only considers the negative effect of inter-pair interference but also takes into
consideration the desired signal strength. We showed that our algorithm is a generalization of
zero-forcing algorithm and can also work when zero-forcing is not possible. The performance
of our proposed algorithm was compared with zero-forcing and MSE-based algorithms, and
discussed for different SNR regimes and for both N ? (2K?1)d+J and N < (2K?1)d+J
cases.
104
Chapter 5
Beamforming for Generalized MIMO Y Channels
Relaying is a powerful technique to support the concurrent transmission of multiple
users, including single/multiple user pair two-way transmission, multi-way transmission, etc.
In this chapter, we consider a newly proposed MIMO Y channel. A typical Y channel consists
of 3 users transmitting to each other through the help of a relay node. In this system model,
each user transmits two independent messages to the other two users. A generalized Y
channel refers to the network consisting of K,K ? 3 users and a relay. Each user has
K?1 independent messages for all the other K?1 users. With the deployment of multiple
antennas at both end users and the relay, K(K ? 1) messages can be conveyed to their
desired receivers within two time slot (assume this network works in a half-duplex mode).
The beamforming design of such a network is usually based on signal space alignment and
network coding concept in recent literatures. However, this requires that the end users have
abundant antennas and it is up to the end users to align their signals so the two signals that
should be network coded together fall into the same direction at the relay. In this chapter,
we propose an alternative method where the end users just try to align their signals into
a smaller subspace at the relay but it is eventually the relay?s beamforming design which
eliminates the harmful effect of inter-pair interference.
This chapter is organized as follows. In Sec. 5.1, the background and related work of
MIMO Y channel is introduced. System model and main results are presented in Sec. 5.2.
A signal group based alignment method is proposed in Sec. 5.3. This method significantly
reduces the required antenna number at end users with a higher antenna number at the relay,
compared with the signal space alignment method. Simulation results are given in Sec. 5.4
to show the achieved DOF of our proposed method. Conclusions are drawn in Sec. 5.5.
105
5.1 Introduction
Bi-directional communications between two users can be facilitated by a relay station.
Using the fact that each user is fully aware of its own sent signal, the network coding concept
[58] is utilized in bi-directional communication systems to allow information exchange of two
users within two time slots, assuming all nodes in this system operate in a half-duplex
manner. In a typical two user bidirectional communication system, data transmission is
completed in two phases: multiple access channel (MAC) phase and broadcast channel
(BC) phase. During the MAC phase, two users transmit to the relay node simultaneously.
In the BC phase, the relay retransmits its received signal during the first time slot (with
or without processing) to the two users and by performing self-interference cancellation,
the two users can decode their desired signals. This bi-direction relay channel has been
extensively investigated by researchers during the recent years. Many extensions have also
been considered, such as multi-pair two-way relaying, multi-user multi-way transmission
[61] and the recently proposed Y channel [62], [63]. In these multi-user networks, system
performance is interference limited and hence how to avoid inter-user interference becomes
a crucial design issue. Many zero-forcing based beamforming designs have been proposed
in recent literature for multi-user two-way or multi-way transmission. In [44], multiple
users which are equipped with single antenna conduct two-way transmission via a MIMO
relay. The relay transceiver is optimized based on MMSE and zero-forcing criteria. Both co-
channel interference (CCI) and self-interference (SI) are eliminated by the proposed methods.
In [65], a projection based separation of multiple operators (ProBaSeMo) relay transmit
strategy is proposed for a mulit-pair two-way relaying channel, which is inspired by block
diagonalization (BD) and regularized block diagonalization (RBD). This algorithm provides
significant sharing gain [65].
The three user Y channel, as a multi-user multi-way transmission scheme has been draw-
ing increasing research attention [63,64]. In a typical Y channel, three users communicate
with each other with the help of one relay node. Each user has two independent messages,
106
each message is meant to be sent to one of the other two users. A signal space alignment
method is proposed in [63,64] for MIMO Y channel. In these papers, a K, K = 3, user
relay channel is considered where each user intends to send two independent messages to the
other two users via the help of an intermediate relay. This system model is a generalization
of two-way transmission and by the concept of network coding, two messages that are to
be exchanged by two users can be network coded together given the fact that each user is
aware of its own sent messages and can perform self-interference cancellation. The proposed
signal space alignment method requires that the two messages to be network coded together
be aligned into the same direction at the relay station in the multiple access (MAC) phase.
Therefore, the K(K ?1) messages only occupy K(K ?1)/2 dimensions at the relay, which
reduces the minimum required number of antennas at the relay node. This novel method is
extended to more general system models in [62] such as multi-user Y channel (also referred
to as generalized Y channel) in which there are K, K > 3, users and each user conveys
K ? 1 independent messages to all other K ? 1 users. In order to support the transmis-
sion of K(K ? 1) messages in this system, the number of antennas Mi at the i-th user
and the number of antennas N at the relay should satisfy Mi ? K ?1, N ? K(K?1)2 , and
mini,j,inegationslash=j{Mi +Mj} > N [62].
In thischapter, a signal groupbased alignment is proposed which requires fewer antennas
at the end users compared to [62]. This method is suitable for the scenarios where the
relay serves as a central station while the end users are small mobile devices such as in a
cellular network. The key idea is to assign signals from all users into several groups and the
beamforming vectors and receive combining vectors for the signals in each group are jointly
designed to align these signals into a smaller subspace. This algorithm uses network coding
concept and signal group alignment to support the multi-user MIMO Y channel free from
inter-user and inter-message interference.
Notations: In this chapter, bold upper and lower case letters are used to denote matri-
ces and vectors respectively. Superscripts (.)?, (.)T and (.)H represent complex conjugate,
107
transpose and complex conjugate transpose (Hermitian) operation, respectively, on a vec-
tor/matrix. E{.} denotes the expectation operation and tr{.} denotes the trace of a matrix.
The null space of a matrix is denoted as null{.} and the subspace spanned by the columns
of a matrix is denoted by span{.}.
5.2 System Model
Consider a generalized MIMO Y channel with K end users and one relay node. Each
user is equipped with M antennas while the relay node has N antennas. Each user has K?1
independent messages for the other K?1 users. This system model is illustrated in Fig. 5.1.
Let sij denote the data signal from user i to user j and vij ?CM?1 denote the beamforming
vector for signal sij. Then the precoding matrix at user i can be expressed as Ti = ?iVi
where Vi = [vi1,...,vi(i?1),vi(i+1),...,viK] ? CM?(K?1) and ?i is a scalar to scale the transmit
power. The signal vector sent by user i is ?iTisi, where si = [si1,...,si(i?1),si(i+1),...,siK]T
is the information vector from user i. Assume the signals sij, i negationslash= j, are independent from
each other with zero mean and variance ?2s.
A half-duplex transmission scheme is considered in this chapter. In the first time slot,
all K users send signals to the relay while in the second time slot, relay sends a linear
combination of its received signal to K users. Let Hir ?CN?M and Hri ?CM?N denote the
channel matrices from user i to the relay and from the relay to user i and assume the channel
state information (CSI) is completely known. Assume each element in the channel matrix
is drawn from i.i.d. (independent and identically distributed) complex Gaussian distribution
with zero mean and unit variance. The received signal at the relay node during the first
time slot (MAC phase) is expressed as
yr =
Ksummationdisplay
i=1
?iHirVisi +nr, (5.1)
108
uni0055uni0073uni0065uni0072uni0020uni004B
uni0052uni0065uni006Cuni0061uni0079
uni0055uni0073uni0065uni0072uni0020uni0033
uni0055uni0073uni0065uni0072uni0020uni0032
uni0055uni0073uni0065uni0072uni0020uni0031
uni0053a0
a1a2
uni0053a0
a3
uni2026uni0020
uni0053a0
a4
uni0053
a5
a6
a7uni0053
a5
a8uni002Euni002Euni002E
uni0053
a5
a9
uni0053a10a11a12uni0053
a10
a13uni2026uni0020uni0053
a10
a14
a10
a15
a11a16
uni0053a17
a18a19
uni0053a17
a20a19
uni002Euni002Euni002E
uni0053a17
a21
uni0053a22
a23a24
uni0053a25
a23
uni2026uni0020
uni0053a26
a23
uni0053a6
a5a7uni0053a8
a5uni002Euni002Euni002Euni0053
a9
a5
uni0053a27
a28a29
uni0053a30
a28a29
uni002Euni002Euni002E
uni0053a31
a28
uni0053a32a33a34uni0053
a35
a33uni2026uni0020uni0053
a36
a33
a37
a32
a38a33
uni002Euni0020uni0020uni0020
uni002Euni0020uni0020uni0020
uni002E
Figure 5.1: System model
where nr ?CN?1 is the additive white complex Gaussian noise at the relay with zero mean
and E
braceleftBig
nrnHr
bracerightBig
= ?2nI. In the second time slot (BC phase), the relay precodes its received
signal yr with a precoding matrix Tr. So the received signal at each user during the second
time slot is
yi = HriTr
Ksummationdisplay
k=1
?kHkrVksk +HriTrnr +ni, (5.2)
where ni ? CN?1 is the additive white complex Gaussian noise at user i with zero mean
and E
braceleftBig
ninHi
bracerightBig
= ?2nI. Each user i can perform self-interference calcellation upon the receive
of signal vector yi. Let ?yi be the signal vector at user i after self-interference cancellation.
Then we have
?yi = HriTr
Ksummationdisplay
k=1,knegationslash=i
?kHkrVksk +HriTrnr +ni.
109
Let Ri = [ri,1,??? ,ri,i?1,ri,i+1,...,ri,K]T ? C(K?1)?M be the receiver combining matrix at
user i while rTi,j ?C1?M, j negationslash= i, is the combining vector for sij. Then the estimated signal at
user i is
?si = Ri?yi = [?s1,?s2,i,...,?si?1,i,?si+1,i,...,?sK,i]T, (5.3)
with ?sj,i = rTi,j?yi, i negationslash= j, being the estimated signal at user i from user j. By the concept of
network coding, signal sij and sji can be network coded together at the relay since each user
can perform self-interference cancellation to decode its desired signal. Assume the inter-user
and inter-message interference are treated as noise. Define
aji = rTijHriTr
Ksummationdisplay
k=1,knegationslash=i,knegationslash=j
Ksummationdisplay
j?=1,j?negationslash=k
?kHkrvkj?skj?
+rTijHriTr
Ksummationdisplay
j?=1,j?negationslash=i,j?negationslash=j
?jHjrvjj?sjj?,
bji = rTijHriTrnr +rTijni,
cji = ?jrTijHriTrHjrvjisji,
where aji represent the inter-user and inter-message interference, bji is the overall noise and
cji is the desired signal at user i from user j. Then ?sji = aji + bji + cji. Let P denote the
user transmit power constraint, i.e.
?2iEtr
braceleftBig
(Visi)(Visi)H
bracerightBig
? P, 1 ? i ? K, (5.4)
Etr
braceleftBig
(Tryr)(Tryr)H
bracerightBig
? P. (5.5)
Then the achievable rate of message sij is calculated as [28]
Rji(P) = 12
braceleftBigg
log2
parenleftBigg
1 + E{|cji|
2}
E{|aji|2}+E{|bji|2}
parenrightBiggbracerightBigg
(5.6)
110
The 12 factor is due to the fact that this system operates in a half-duplex mode and K users?
transmission is completed in two time slots. The sum achievable degree of freedom (DOF)
of this K-user MIMO Y channel is [62]
?(K) = lim
P??
summationtextK
i=1
summationtextK
j=1,jnegationslash=iRji(P)
log2 P (5.7)
Also, define the normalized DOF as:
??(K) = ?(K)average antenna number per node. (5.8)
In recent research works, the beamforming vectors vij and vji are usually designed
to satisfy span{Hirvij} .= span{Hjrvji}, i.e., the subspaces spanned by Hirvij and Hjrvji
are the same. This means the signal vectors associated with sij and sji are aligned into
the same direction at the relay node such that K(K ?1) signals only occupy K(K ?1)/2
dimensions. The requirement on the number of antennas is 2M > N, i negationslash= j, M ? K ? 1
and N ? K(K ?1)/2. This requires the end users to be equipped with a large number of
antennas especially when K is large. However, under some circumstances, it is more realistic
that the relay node can have more antennas while the number of antennas of end users should
be made as small as possible. One example is cellular network where the base station (BS)
serves as the relay and the end users are mobile users. Therefore we propose a signal group
based beamforming design. The requirement on the number of antennas is summarized in
the following theorem:
Theorem 1 : To achieve DOF 12K(K?1), when K is even, the number of antennas at
the relay and the users should satisfy N ? (K ?1)2 and KM ? N +(K ?1), and when K
is odd, the number of antennas should satisfy N ? K(K ?2) and (K ?1)M ? N + 1. ?
In the next section, we will show how to construct the beamforming vectors such that
Theorem 1 is achieved. From Theorem 1, it is easy to deduce the following minimum antenna
requirement.
111
Corollary 1 : The minimum number of antennas to achieve DOF 12K(K?1) is M = K?1
and N = (K?1)2 when K is even, and M = K?1 and N = K(K?2) when K is odd. ?
5.3 Signal Group Based Alignment Algorithm
In this section, a beamforming method is proposed to establish Theorem 1. Firstly, we
present the a lemma which leads to the constructions of this beamforming method.
Lemma 1 : Let H1,H2,...,Hl ? CN?M be l matrices such that their elements are i.i.d.
and generated from some continuous random distribution. Let x1,x2,...,xl ? CM?1 be l
beamforming vectors for signal d1,...,dl. Let S = span{H1x1,...,Hlxl}. If dim{S} = l ?1,
then a weighted sum of any two scalar signals di and dj can be obtained from summationtextlk=1Hkxkdk.
?
Proof : Without loss of generality, suppose we want to recover a weighted sum of scalar
signals d1 and d2. Since dim{S} = l?1, then N ? l?1. Let ?X = [H3x3,H4x4,...,Hlxl] ?
CN?(l?2). Then dim{?X} ? l ? 2. Therefore, N > dim{?X} and there exists a vector
y? ? null{?XH}, i.e. yT ?X = 0. This leads to yT summationtextlk=1Hkxkdk = yTH1x1d1 + yTH2x2d2 =
w1d1 + w2d2. Additionally, since elements of Hi, 0 ? i ? l, are generated independently
from some continuous distribution, the probability that w1 = 0 or w2 = 0 is zero. square
5.3.1 User Beamforming in MAC Phase
In this section, the beamforming vectors design for each end user in the MAC phase is
presented. Based on Lemma 1, a beamforming method is proposed to align signals from l
users into a l ?1 subspace at the relay, such that the dimensions occupied by these signals
is reduced from l to l ?1 while the network coded signal, i.e. weighted sum of two signals
can still be recovered. The beamforming vectors construction for both MAC and BC phase
will help prove theorem 1.
Firstly, we consider the case when the number of users K is even. Assume the number
of antennas at end users satisfy M ? K ? 1 so that K ? 1 independent datastreams can
112
be sent simultaneously. The goal is to align K signals from K users (one signal from each
user) into a K ?1 subspace at the relay node. There are K(K ?1) signals from all the end
users. Therefore, these signals need to be assigned into K ?1 groups. We propose that the
grouping of all K(K ?1) signals should satisfy two criteria. Firstly, each group contains K
signals from K users and these signals are meant to be sent to K users. Secondly, the signals
in each group can form K/2 pairs and the two signals within each pair can be network coded
together, i.e. each pair contains si,j and sj,i, i negationslash= j. Then at the relay, K(K ? 1) signals
only occupy (K ?1)2 dimensional subspace. Since the relay has N antennas, N ? (K ?1)2
should be satisfied so that the relay can accommodate these signals. The relay combining
matrix is designed to recover the network coded signals, which will be presented later.
To group signals from all end users into K?1 groups, we employ a graph edge-coloring
method. The K user Y channel can be represented as a K-complete graph, which is a graph
with K vertices and any two vertices are connected by an edge. In this K-complete graph,
each vertex represents a user and the edge connecting any two users represents two signals
that these two users want to send to each other. The vertices are indexed as A1,A2,...,AK
and the edge are indexed as e12,e13,...,e1K,e23,...,eK(K?1) with eij, i < j, being the edge
connecting Ai and Aj. Then eij represents two signals sij and sji that will be network coded
together at the relay.
When K is even, there exists a K ?1 proper edge coloring for this K-complete graph,
which is also a 1-factorization of this graph. Each 1-factor contains K/2 edges and the edges
belonging to each 1-factor are colored by the same color. By the definition of 1-factor, the
K/2 edges in each 1-factor connect all K users, which means the signals in each group are
from all K users and are meant to be sent to all K users. Also, the K/2 edges are associated
with K signals which can be paired into K/2 pairs. There are exactly K ?1 such 1-factors
for this graph. Therefore, the signals associated with each 1-factor form a group. There are
K ?1 such groups.
113
After the grouping of K(K?1) signals is obtained, the beamforming vectors should be
designed such that the K signals in each group are aligned into a K ? 1 subspace at the
relay. Denote the groups as G1,G2,...,GK?1. Let pii(.) denote the pairing method of group
Gi, i.e. if signals sj?j and sjj? form a pair in group i, then pii(j) = j? and pii(j?) = j. Denote
the signals in group i as Gi = {s1?i(1),s2?i(2),...,sK?i(K)}. The design of beamforming vectors
for signals in each group i should satisfy
dim
braceleftBig
span{H1rv1,?i(1),H2rv2,?i(2),...,HKrvK,?i(K)}
bracerightBig
= K ?1, 1 ? i ? K ?1.
This can be achieved by aligning vectors [vT1,?i(1),vT2,?i(2),...,vTK,?i(K)]T into the null space of
matric [H1r,H2r,...,HKr], i.e.
[vT1,?i(1),vT2,?i(2),...,vTK,?i(K)]T ? null{[H1r,H2r,...,HKr]}.
Scalar ?i,i = 1,2,...,K is chosen such that the power constraint in (5.4) is satisfied with
equality. Since we have E{sisHi } = ?2sI, ?i can be expressed as:
?i =
radicaltpradicalvertex
radicalvertexradicalbt P
tr{ViVHi }?2s .
Since there are K ?1 groups, the null space of [H1r,H2r,...,HKr] should have dimension of
at least K ?1. Also [H1r,H2r,...,HKr] is a N ?MK matrix with full rank almost surely.
So the number of antennas should satisfy MK ?(K ?1) ? N. Recall that the number of
relay antennas should satisfy N ? (K ? 1)2, so M ? K ?1 is implicitly satisfied. This is
the requirement in Theorem 1.
When K is odd, it is not reasonable to group K signals into one group since these K
signals can not be paired (simply because of the fact that K is odd). However, K?1 signals
can be assigned into one group and (K ? 1)/2 pairs can be possibly obtained. Similarly
as the case when K is even, the network can be represented as a K-complete graph with
114
each vertex representing a user and each edge connecting two vertices representing these two
signals the two users want to send to each other. For a K-complete graph with K being odd,
there exists a K proper edge coloring. Let ci, i = 1,2,...,K, be the number of edges colored
by color i. We have ci ? (K ?1)/2 since there are K vertices in the graph and K is odd.
So from the fact that summationtextKi=1 ci = K(K?1)/2, we have ci = (K?1)/2, 1 ? i ? K. Therefore,
each color colors exactly (K ? 1)/2 edges, which connects K ? 1 vertices. We assign the
signals associated with the edges colored by the same color into one group, because the signal
in each group can be paired into (K ?1)/2 pairs. Each group contains signals from/to only
K?1 users. Since each vertex is incident to K?1 edges, these edges are colored with K?1
different colors. Therefore, each user?s signals belong to K ? 1 groups, which means each
group is missing 1 user?s signals. Since there are K groups and K users, each group is missing
a different user?s signals. Let i-th group be the group that is missing i-th user?s signals. Then
the i-th group can be defined as Gi = {s1,?i(1),s2,?i(2),...,si?1,?i(i?1),si+1,?i(i+1),sK,?i(K)}.
After the grouping of signals, user beamforming vectors are designed such that the K?1
signals in each group are aligned into a K?2 subspace at the relay. Since there are K signal
groups in total and the signals in each group are aligned into a K ?2 dimensional subspace
at the relay, the relay needs at least K(K ?2) antennas to accommodate all these signals
from end users, i.e. N ? K(K ?2). For Gi, the beamforming vectors should satisfy
dim
braceleftbigg
span {H1rv1,?i(1),H2rv2,?i(2),...,Hi?1,rvi?1,?i(i?1),Hi+1,rvi+1,?i(i+1),...,HKrvK,?i(K)}
bracerightbigg
= K ?2, 1 ? i ? K ?1. (5.9)
This requires that vectors [vT1,?i(1),vT2,?i(2),...,vTi?1,?i(i?1),vTi+1,?i(i+1),...,vTK,?i(K)]T are aligned
into the null space of matric ?Hi = [H1,H2,...,Hi?1,Hi+1,...,HK], i.e.
[vT1,?i(1),vT2,?i(2),...,vTi?1,?i(i?1),vTi+1,?i(i+1),...,vTK,?i(K)] ? null
braceleftBig?
Hi
bracerightBig
, i = 1,2,...,K. (5.10)
115
The scalar ?i is chosen as
?i =
radicaltpradicalvertex
radicalvertexradicalbt P
tr{ViVHi }?2s
so that power constraint (5.4) is satisfied with equality. Since all groups are missing different
users? signals, ?Hi, 1 ? i ? K, are all different. So the dimension of the null space of ?Hi,1 ?
i ? K should be at least one. The number of antennas should satisfy M(K ?1) ? N + 1.
This condition plus the requirement N ? K(K ?2) imply that M ? K ?1 is also satisfied.
This is the requirement in Theorem 1.
5.3.2 Relay Receive Combining Vectors Design in MAC Phase
Let the relay precoding matrix be Tr = ?PU where U ?CK(K?1)2 ?N andP ? CN?K(K?1)2
are the receive combining and beamforming matrices, respectively, of the K(K?1)/2 signal
pairs, and ? is a scalar to satisfy the power constraint at the relay as in (5.5). In this
section we consider the design of receive combining matrix U for the MAC phase. During
the MAC phase, relay receives K(K ? 1) signals and needs to recover K(K ? 1)/2 signal
pairs, i.e. sij + sji, 1 ? i,j,? K, i negationslash= j. Let uTij ? C1?N,i < j be the receive combining
vector at the relay for signal pair sij and sji. Then U = [u12,...,uij,...,uK?1,K]T. In order
to recover signal pair sij and sji without inter-user or inter-message interference, u?ij should
fall into the null space of the interference matrix ?UHij. The columns of ?Uij are Hi?rvi?j? with
{i?,j?} negationslash= {i,j} and {i?,j?} negationslash= {j,i}. So ?Uij is a N ? (K(K ? 1) ? 2) complex matrix. In
the next Lemma, it is proved that such non-zero vectors uij, i < j, 1 ? i,j ? K exist when
beamforming vectors vi,j, i < j, 1 ? i,j ? K are designed as in Sec. 5.3.1.
Lemma 2: If the beamforming vectors vi,j, i negationslash= j, 1 ? i,j ? K are designed as in Sec.
5.3.1, then for even K, rank{?Uij} ? (K ? 1)2 ? 1 for all signal pairs i,j, and for odd K,
rank{?Uij}? K(K ?2)?1 for all signal pairs i,j. ?
Proof : ?Uij is a matrix whose columns are associated with all signals from K users
except for sij and sji. So the rank of ?Uij is the dimensions that all signals other than sij,sji
occupy at the relay.
116
When K is even, there are K?1 signal groups and in each group, K signals are aligned
into a K ?1 subspace. Since sij and sji form a signal pair, they belong to the same group.
We name this group as Gt and use ?Gt to denote the group formed by removing sij and sji
from Gt. Then there are K?2 signals in ?Gt and they can occupy at most K?2 dimensions
at the relay. Therefore, the signals in group ?Gt and signals in all other groups can occupy
at most (K ?1)2 ?1 dimensions at the relay.
When K is odd, there are K groups while each group contains K ?1 signals. Let Gt
be the group that signals sij and sji belong to and ?Gt be a group formed by removing sij
and sji from group Gt. There are K ? 3 signals in group ?Gt so they can occupy at most
K ?3 dimensions at the relay. Therefore, the signals in ?Gt and all other groups can occupy
at most K(K ?2)?1 dimensions at the relay. square
When the number of antennas at the relay satisfy N ? (K ? 1)2 for even K and
N ? K(K ?2) for odd K, from Lemma 2 we always have rank{?Uij} < N. So there always
exists a non-zero vector ui,j such that u?ij ? null{?UHij}.
5.3.3 User Receive Combining Design in BC Phase
In the BC phase, relay transmits the signals from all K users using a linear precoding
matrix Tr. All users receive signals from the relay and try to recover K?1 desired messages
sent from all other K ?1 users. Similar as in the MAC phase, we consider the user receive
combining design in two cases: K is even and K is odd.
When user number K is even, the K(K?1) total desired signals are assigned to K?1
groups while each group contains K signals from/to K users. Furthermore, the K signals
in each group form K/2 pairs and the two signals in each pair are the signals two users
want to send to each other, i.e. sij and sji, i negationslash= j. The grouping of signals is the same as
in MAC phase, i.e. signals from all K users are assigned to groups G1,G2,...,GK?1 with
Gi = {s1?i(1),s2?i(2),...,sK?i(K)}. The receive combining vector at user i to recover signal sji
is denoted as rTij ? C1?M. If these receive combining vectors are chosen randomly, in BC
117
phase, the signals in each group will be received along K directions. We propose to design
the receive combining vectors such that the signals in each group are received along K ?1
directions, which leads to
dim
braceleftBig
span
braceleftBig
[HTr1r1,?i(1),HTr2r2,?i(2),...,HTrKrK,?i(K)]
bracerightBigbracerightBig
= K ?1, 1 ? i ? K. (5.11)
When the channels in MAC and BC phases have reciprocity, Hrk = HTkr. In order to make
equation (5.11) hold, with zi = [rT1?i(1),rT2?i(2),...,rTK?i(K)] ? C1?MK, vector zTi should fall
into the null space of the stack of the corresponding channel matrices as follows:
zTi ? null
braceleftBig
[HTr1,HTr2,...,HTrK]
bracerightBig
, 1 ? i ? K ?1. (5.12)
Since there are K ? 1 signal groups, there are K ? 1 stacked receive combining vectors
zT1 ,...,zTK?1 that should fall into the null space of matrix [HTr1,HTr2,...,HTrK]. The null space
of matrix [HTr1,HTr2,...,HTrK] should at least have K ?1 dimension, which puts a constraint
on the number of antennas: KM ? N+(K?1). Considering that N ? (K?1)2, M ? K?1
is implicitly satisfied.
When K is odd, the K(K ? 1) signals are assigned to K groups with each group
containing K ?1 signals using the same method as in the user beamforming design for the
MAC phase. The i-th group is the group that is missing the signals to and from i-th user,
denoted as Gi = {s1,?i(1),s2,?i(2),...,si?1,?i(i?1),si+1,?i(i+1),sK,?i(K)}. The design of receive
combining vectors allows to receive K ? 1 signals in each group along K ? 2 dimensions.
This is achieved by designing rij, i negationslash= j, 1 ? i,j,? K to satisfy the following condition:
dim
braceleftbigg
span
braceleftBig
[HTr1r1,?i(1),...,HTr,i?1ri?1,?i(i?1),HTr,i+1ri+1,?i(i+1),...,HTrKrK,?i(K)]
bracerightBigbracerightbigg
= K ?2, 1 ? i ? K. (5.13)
118
Let zi = [rT1,?i(1),rT2,?i(2),...,rTi?1,?i(i?1),rTi+1,?i(i+1),...,rTK,?i(K)] ? C1?M(K?1). The above con-
dition is meet by letting zTi fall into the null space of its corresponding channel matrix as
follows:
zTi ?null
braceleftBig
span{[Hr1,Hr2,...,Hr,i?1,Hr,i+1,...,HrK]}
bracerightBig
,1 ? i ? K. (5.14)
Since for each group, the channel matrix as in the right-side of the above equation is different,
it is required that the null space of the channel matrix should have dimension ? 1. Therefore,
a restriction on the number of antennas for the users and the relay is M(K ?1) ? N + 1.
Considering N ? K(K ?2) for odd K, M ? K ?1 is implicitly satisfied.
5.3.4 Relay Beamforming Design in BC Phase
During the MAC phase, the relay recovers K(K ?1)/2 network coded signals. In the
BC phase, they are sent to K end users. Let P = [p12,...,pij,...,pK?1,K], where pij ? CN?1,
i < j is the beamforming vector for signal pair sij and sji. This signal pair should be sent to
user i and user j and rij and rji are the associated receive combining vectors at the desired
receivers. To avoid inter-user and inter-message interference, the beamforming vectors at
the relay should be designed such that each signal pair can only be recovered by desired
receive combining vectors at desired users. Define the interference matrix for i,j signal pair
as ?Vij,i < j. The rows of ?Vij are rTi?j?Hri? with {i?,j?} negationslash= {i,j}, {i?,j?} negationslash= {j,i}. Therefore,
?Vij is a (K(K ?1)?2)?N complex matrix.
Lemma 3 : When K is even, rank{?Vij} ? (K ?1)2 ?1 for all signal pairs i,j. When
K is odd, rank{?Vij}? K(K ?2)?1 for all signal pairs i,j. ?
The proof of Lemma 3 is similar to Lemma 2. Let Gt be the signal group that contains
signals sij and sji, and ?Gt is the group formed by removing sij and sji from Gt. When
K is even, the receive-channel vectors rTi?j?Hri? associated with signals in group ?Gt span at
most K ? 2 dimensions. Since the receive-channel vectors associated with signals in each
119
group will occupy K ?1 dimensions, the rank of ?Vij is at most (K ?1)2 ?1. When K is
odd, the receive-channel vectors ri?j? ?Hi? associated with signals in group ?Gt span at most
K?3 dimensions, and the receive-channel vectors associated with signals in each other group
occupy K ?2 dimensions. Therefore, the rank of ?Vij is at most K(K ?2)?1.
The design of beamforming vector pij, i < j, 1 ? i,j ? K should make sure that it falls
into the null space of ?Vij.
pij ? null
braceleftBig?
Vij
bracerightBig
,i < j,1 ? i,j,? K.
In order for the non-zero vector pij to exist, we need N > rank{?Vij}. Then the number of
antennas at the relay should satisfy N ? (K?1)2 when K is even, and N ? K(K?2) when
K is odd. This is the same requirement as in designing the receive combining matrix U.
Finally, the scalar ? is chosen as
? =
radicaltpradicalvertex
radicalvertexradicalbt P
tr
braceleftBigsummationtextK
i=1 PUHirH
H
irU
HPH?2
s +PUU
HPH?2
n
bracerightBig
to satisfy the relay power constraint (5.5) with equality. So far, a beamforming method based
on signal group alignment has been proposed. This method completes the transmission of
K(K ? 1) datastreams in two time slots and requires the numbers of antennas to satisfy
MK ? N + K ? 1 and N ? (K ? 1)2 for even K and to satisfy (K ?1)M ? N + 1 and
N ? K(K ?2) for odd K. Thus, Theorem 1 is proved.
Remark 1: In the proposed algorithm, ?(K) = K(K ?1)/2 is achieved for this K user
MIMO Y channel in a half-duplex mode. The averaged DOF is given by
??(K) =
?
???
???
K(K?1)/2
(MK+N)/(K+1) ?
K2+K
4K?2 for even K
K(K?1)/2
(MK+K(K?2))/(K+1) ?
K2?1
4K?6 for odd K
(5.15)
120
K Sum DOF M N total number of antenna
3 3 2 3 9
4 6 9 3 21
5 10 15 4 35
6 15 25 5 55
Table 5.1: Required minimum number of antennas for the signal group based alignment
algorithm
We can see that the averaged DOF increases in the order of K/4. The table 5.1 shows the
minimum required antenna numbers as well as the achieved DOF of the proposed signal
group alignment.
5.4 Simulation Results
In this section, we show that our proposed signal group based alignment algorithm can
achieve the DOF K(K ? 1)/2 when the antenna number conditions are met as stated in
Theorem 1. In Fig. 5.2, the sum rate of all K users? datastreams are evaluated versus SNR.
Two system set-ups are considered to show the effectiveness of the proposed beamforming
algorithm for both odd and even number of users. In Fig. 5.2, K is the number of users,
M is the number of user antennas and N is the number of relay antennas. All K users and
the relay transmit with the same power. With all channel coefficients assumed to be i.i.d.
Gaussian random variable with zero mean and unit variance, SNR is defined as
SNR = log2
parenleftBiggP
?2n
parenrightBigg
.
It is seen from Fig. 2 that as the SNR increases, the slope of the K = 4 curve is 6 while the
slope of the K = 5 curve is 10. Both slopes equal K(K?1)/2, thus verifying our theoretical
results.
121
0 5 10 15 20 250
20
40
60
80
100
120
140
160
180
200
SNR
Sum rate (bpsHz)
 
 
K = 4, M = 3, N = 9
K = 5, M = 4, N = 15
Figure 5.2: Sum rate vs SNR log2
parenleftBigP
?2n
parenrightBig
5.5 Conclusion
In this chapter, we proposed a signal group based alignment method for a generalized
K user MIMO Y channel in which each user has K ? 1 independent datastreams for the
other K ?1 users. This multi-way transmission is facilitated by a relay station and operate
in a two time-slot half-duplex mode. The K(K ? 1) datastreams are assigned to g groups
(g = K when K is odd and g = K ?1 when K is even) using a graph theory method. Let l
denote the number of signals in each group. The beamforming design is to align these signals
in each group in each group into a l?1 dimensional subspace at the relay node during the
MAC phase and let the receive combing vectors associated with these signals receive along a
l?1 dimensional subspace. To achieve K(K?1)/2 DOF, minimum number of antennas (M
at each user and N at the relay) are specified in Theorem 1. In our proposed approach we
significantly decrease the minimum M at the expense of higher N for a given number of users
K., compared to an existing approach. For instance, for K = 8 and the same of number of
antennas at each user, [62] would require N ? K(K ?1)/2 = 28, and M ? K ?1 = 7 as
122
well as 2M > N ? 28, leading to M > 14 or M ? 15. On the other hand, by our Corollary
1, we require M ? K ?1 = 7 and N ? (K ?1)2 = 49.
123
Chapter 6
Conclusions and Future Work
6.1 Conclusions
In this dissertation, two kind of problems were addressed: 1) resource allocation in
cognitive radios, 2) precoding design in cognitive radio and relay networks. They are two
effective ways to improve spectrum efficiency and to allow coexistence of multiple mobile
devices which are share the same radio resources.
A joint spectrum sensing, access and power allocation algorithm was proposed in Chap-
ter 2 in a multi-channel cognitive radio environment. By introducing the soft-decision spec-
trum sensing concept into the optimization problem and utilizing channel availability proba-
bilities instead of binary channel state decisions, it was shown that the system performance in
terms of sum throughput is significantly improved while interference to the primary network
is kept under a specified threshold. The optimization problem was transformed into a con-
vex optimization problem and optimal solution was obtained. To reduce the computational
complexity, two heuristic algorithms were also proposed which solve for the access strategy
first and then allocate power. Simulation results showed these soft-decision spectrum sensing
based algorithms outperform hard-decision counterpart and one of the heuristic algorithm
achieves a near optimal performance with a much smaller complexity.
In Chapter 3, a multi-user multi-way relay network acting as a secondary network in
a cognitive radio environment was considered. With the deployment of a relay node and
multiple antennas equipped at both users and relays, the multi-way transmission was made
possible via proper predcoding/decoding design at users and relays while interference to the
primary users was completely or partially eliminated. A two time slot half-duplex trans-
mission scheme was adopted in this network. An iterative algorithm based on the MMSE
124
criterion was first proposed which optimized the precoding matrices at transmit users, pre-
coding matrix at the relay and decoding matrices design at receive users iteratively. It was
shown all the three sub-problems are convex and optimal solutions to the sub-problems were
presented. Then a non-iterative algorithm was also proposed to provide a low complexity
solution. This algorithm optimized the user precoding and decoding matrices based on a
matrix distance criterion and the relay precoding was designed to minimize sum MSE. Simu-
lation results showed that this non-iterative algorithm with reduced complexity only results
in a small performance degradation compared with the iterative algorithm. Furthermore,
considering the face that perfect CSI is sometimes hard to obtain due to the time-varying na-
ture of wireless channel, a robust precoding method was proposed based on the non-iterative
algorithm. In this robust algorithm, the channel uncertainty was modeled into some stochas-
tic error which follows a zero mean normal distribution. Simulation results showed that this
robust algorithm can keep the interference to primary network under control and improve
the performance of secondary network compared with the non-robust algorithm.
In Chapter 4, a joint signal and interference alignment precoding was proposed for a
multi-pair two-way relay network as a secondary network in a cognitive radio system. We
considered a system in which multiple pairs of users wish to transmit multiple datastreams to
each other and a relay station is deployed to facilitate the multi-pair two-way transmission.
In this system model, three types of interference are present: inter-user interference, inter-
datastream interference and primary-secondary network interference. To avoid the harmful
effects of the interference, precoding and decoding matrices at users and relay are designed
to avoid or completely remove all the interference. A matrix distance based algorithm was
proposed to jointly consider the desired signal strength and avoidance of interference. A non-
negative scalar weight was used to balance the signal alignment and interference alignment.
It was shown that the zero forcing design is a special case of this proposed algorithm. Also,
by appropriately changing the value of the scalar weight, this algorithm can adapt to both
high SNR and low SNR regimes.
125
A generalized Y channel consisting of K,K > 3, users and each user having K ? 1
messages to be sent to all other K ?1 users was considered in Chapter 5. A relay node was
used to enable this multi-way transmission of multiple users. Based on the network coding
concept, a signal group based alignment algorithm was proposed which aligns the signals
from all users into a smaller subspace at the relay such that fewer antennas are required for
the relay as well as the users. The proposed signal group alignment is suitable for a system
in which it is more practical to equip the relay node with more antennas and the end users
are small or mobile devices in which only a small number of antennas can be installed.
6.2 Future Work
The research work reported in this dissertation about resource allocation and precoding
design in cognitive radio and relay networks also suggests the following future research ideas.
Robust precoding design
In most of this dissertation we assumed perfect CSI, which may be difficult to obtain
in practical systems due to time varying nature of the wireless channel and the overhead
of transmitting CSI back to the transmitter or central control node. A promising research
topic is to develop algorithms that do not require the knowledge of CSI or are insensitive to
the CSI errors.
Precoding/beamforming design with security concerns
The precoding and beamforming algorithms proposed in this dissertation can dramat-
ically improve the system performance by mitigating the interference effects. The beam-
forming and receive combining design can also be an effective way to provide secure data
transmission from a physical layer point of view. The broadcast nature of wireless signals
makes the security issue a major concern in wireless network design. Therefore incorporating
security concerns into the precoding and beamforming design will be a promising research
topic.
126
Bibliography
[1] C. Leow, Z. Ding, K. Leung and D. Goeckel ?On the Study of Analogue Network Coding
for Multi-Pair, Bidirectional Relay Channels,? IEEE Trans. Wireless Comm., vol. 10,
pp. 670-681, Feb. 2011.
[2] ITU-R SM. 2152, ?Definitions of Software Defined Radio (SDR) and cognitive Radio
System (CRS)?, Sept. 2009.
[3] D. Cabric, S. M. Mishra and R. W. Brodersen, ?Implementation Issues in Spectrum
Sensing for Cognitive Radios,? in Proc. 38th Asilomar Conf. Signals, Systems, Com-
puters, Pacific Groe, CA, USA, Nov. 2004.
[4] D. Cabric, A. Tkachenko and R. W. Brodersen, ?Experimental study of spectrum
sensing based on energy detection and network cooperation,? Proceeding TAPAS ?06
Proceedings of the first international workshop on Technology and policy for accessing
spectrum, Boston, MA, USA, Aug. 2006.
[5] T. Yucek, H. Arslan, ?A survey of spectrum sensing algorithms for cogntivive radio
applications,? Communications Surveys & Tutorials, IEEE, vol. 11, pp. 116-130, First
Quater, 2009.
[6] P. D. Sutton, K. E. Nolan and L. E. Doyle, ?Cyclostationary signatures in practical
cognitive radio applications,? in Communications Surveys & Tutorials, IEEE, vol. 26,
pp. 13-24, Jan, 2008.
[7] K. Lee, N. Lee and I. Lee, ?Feasibility Conditions of Signal Space Alignment for Net-
work Coding on K-user MIMO Y channels,? in Proc. 2011 IEEE IEEE International
Conference on Communications (ICC 2011), Kyoto, Japan, June 2011.
[8] S. Jafar, ?Interference Alignment: A New Look at Signal Dimensions in a Communica-
tion Network?, Foundations and Trends in Communications and Information Theory,
Vol. 7, No. 1, pages: 1-136.
[9] J. Winters, ?On the capacity of radio communication systems with diversity in a
Rayleigh fading environment, ? IEEE J. Sel. Areas Commun. vol. SAC-5, no. 6, pp.
871-878, Jun. 1987.
[10] G. Foschini, ?Layered space-time architecture for wireless communication in fading en-
vironments when using multi-element antennas, ? Bell Labs Tech. J. pp. 4159, 1996.
127
[11] I. Telatar, ?Capacity of multi-antenna Gaussian channels? AT&T Bell Labs. Tech. Rep.
BL0112170-950615-07TM, Jun. 1995
[12] A. Sendonaris, E. Erkip, and B. Aazhang, ?User cooperation diversitypart I: System
description,? IEEE Trans. Commun., vol. 51, no. 11, pp. 19271938, Nov. 2003.
[13] A. Sendonaris, E. Erkip, and B. Aazhang, ?User cooperation diversity-part II: Imple-
mentation aspects and performance analysis,? IEEE Trans. Commun., vol. 51, no. 11,
pp. 19391948, Nov. 2003.
[14] H. Mu and J. Tugnait, ?Interference Alignment-Like Precoder Design in Multi-Pair
Two-Way Relay Cognitive Radio Networks?, Proc. 2012 IEEE Global Telecom. Conf.
(GLOBECOM), Anaheim, CA, Dec. 2012.
[15] D. Hu and S. Mao ?Cooperative relay in cognitive radio networks: Decode-and-forward
or amplify-and-forward??, Proc. 2010 IEEE Global Telecom. Conf. (GLOBECOM), Mi-
ami, FL, Dec. 2010.
[16] V. Asghari and S. Aissa, ?Adaptive rate and power transmission in spectrum-sensing
system,? IEEE Trans. Wireless Commun., vol. 9, no. 10, pp. 3272-3280, Oct. 2010.
[17] S. Barbarossa, S. Sardellitti and G. Scutari, ?Joint optimization of detection thresholds
and power allocation for opportunistic access in multicarrier cognitive radio networks,?
in Proc. 2009 3rd IEEE Intern. Workshop Comp. Adv. Multi-sensor Adaptive Proc.
(CAMSAP), Aruba, Dutch Antilles, Dec. 2009.
[18] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.
[19] A. El-Sherif, A. Sultan and K. Seddik, ?Soft sensing-based multiple access for cognitive
radio networks,? in Proc. 2010 IEEE Global Telecom. Conf. (GLOBECOM), Miami,
FL, Dec. 2010.
[20] R. Fantacci and A. Tani, ?Performance evaluation of a spectrum-sensing technique for
cognitive radio applications in B-VHF communication systems,? IEEE Trans. Vehicular
Tech., vol. 58, no. 4, pp. 1722-1730, May 2009.
[21] J. Huang, V.G. Subramanian, R. Agrawal and R.A. Berry, ?Downlink scheduling and
resource allocation for OFDM systems,? IEEE Trans. Wireless Commun., vol. 8, no. 1,
pp. 288-296, Jan. 2009.
[22] X. Huang and B. Beferull-Lozano, ?Joint optimization of detection and power allo-
cation for OFDM-based cognitive radios,? in Proc. 2010 IEEE Global Telecom. Conf.
(GLOBECOM), Miami, FL, Dec. 2010.
[23] J. Ma, G.Y. Li and B.H. Juang, ?Signal processing in cognitive radio,? Proc. IEEE, vol.
97, no. 5, pp. 805-823, May 2009.
[24] H. Mu and J.K. Tugnait, ?Soft spectrum sensing and power adaptation in multiband
cognitive radios,? in Proc. 2011 IEEE Global Telecom. Conf. (GLOBECOM), Houston,
TX, Dec. 5-9, 2011.
128
[25] H. Mu and J.K. Tugnait, ?Joint soft-decision cooperative spectrum sensing and power
control in multiband cognitive radios,? in Proc. IEEE 2012 Inter Conf. Commun. (ICC),
Ottawa, Canada, June 10-15, 2012.
[26] Z. Quan, S. Cui, A.H. Sayed and H.V. Poor, ?Optimal multiband joint detection for
spectrum sensing in cognitive radio networks,? IEEE Trans. Signal Proc., vol. 57, no.
3, pp. 1128-1139, March 2009.
[27] S. Srinivasa and S.A. Jafar, ?Soft sensing and optimal power control for cognitive radio,?
IEEE Trans. Wireless Commun., vol. 9, no. 12, pp. 3638-3645, Dec. 2010.
[28] D. Tse and P. Viswanath, ?Fundamentals of Wireless Communication,? Cambridge
University Press, 2005.
[29] X. Wang, ?Joint sensing-channel selection and power control for cognitive radio,? IEEE
Trans. Wireless Commun., vol. 10, no. 3, pp. 958-967, March 2011.
[30] Q. Zhao and B.M. Sadler, ?A survey of dynamic spectrum access,? IEEE Signal Proc.
Mag., vol. 24, no. 3, pp. 79-89, May 2007.
[31] F. Fitzek and M. Katz (Editors). ?Cognitive Wireless Networks?, Springer, Dordrecht,
The Netherlands, 2007.
[32] E. Hossain, D. Niyato and Z. Han. ? Dynamic Spectrum Access and Management in
Cognitive Radio Networks?, Cambridge Univ. Press, Cambridge, UK, 2009.
[33] Federal Communications Commission, ?Notice of proposed rule making and order: Fa-
cilitating opportunities for flexible, efficient, and reliable spectrum use employing cog-
nitive radio technologies,? ET Docket No. 03-108, Feb. 2005.
[34] S. Peters and R. Heath, Jr, ?Interference alignment via alternating minimization,? in
Proc. 2009 IEEE Intern. Conf. Acoustics, Speech, Signal Proc. (ICASSP 2009), Taipei,
Taiwan, Apr. 2009.
[35] K. Kumar and F. Xue, ?An iterative algorithm for joint signal and interference align-
ment,? in Proc. 2010 IEEE Intern. Symp. Info. Theory (ISIT 2010), Austin, TX, USA,
Jun. 2010.
[36] T. Bogale and L. Vandendorpe, ?Robust sum MSE optimization for downlink multiuser
MIMO systems with arbitrary power constraint: generalized duality approach,? IEEE
Trans. Signal Proc., vol. 60, pp. 1862-1875, April 2012.
[37] T. Bogale and L. Vandendorpe, ?Weighted sum rate optimization for downlink multiuser
MIMO coordinated base station systems: Centralized and distributed algorithms,?
IEEE Trans. Signal Proc., vol. 60, pp. 1876-1889, April 2012.
[38] K. Cumanan, J. Tang and S. Lambotharan, ?Rate balancing based linear transceiver de-
sign for multiuser MIMO system with multiple linear transmit covariance constraints,?
in Proc. 2011 IEEE Intern. Conf. Commun. (ICC2011), Kyoto, Japan, June 2011.
129
[39] K. Lee, H. Sung and I. Lee, ?Linear precoder designs for cognitive radio multiuser
MIMO downlink systems,? in Proc. 2011 IEEE Intern. Conf. Commun (ICC 2011),
Kyoto, Japan, June 2011.
[40] R. Zhang, Y. Liang, C. Chai and S. Cui, ?Optimal beamforming for two-way multi-
antenna relay channel with analogue network coding,? IEEE J. Sel. Areas Commun.,
vol. 27, pp. 699-712, Jun. 2009.
[41] R. Wang and M. Tao, ?Joint source and relay precoding designs for MIMO two-way
relaying based on MSE criterion,? IEEE Trans. Signal Proc., vol. 60, pp. 1352-1365,
Mar. 2012.
[42] K. Lee, H. Sung, E. Park and I. Lee, ?Joint optimization for one and two-way MIMO
AF multiple-relay systems,? IEEE Trans. Wireless Commun., vol. 9, pp. 3671-3681,
Dec. 2010.
[43] C. Li, L. Yang and W. Zhu, ?Two-way MIMO relay precoder design with channel state
information,? IEEE Trans. Commun., vol. 58, pp. 3358-3363, Dec. 2010.
[44] J. Joung and A.H. Sayed, ?Multiuser two-way amplify-and-forward relay processing and
power control methods for beamforming systems,? IEEE Trans. Signal Proc., vol. 58,
pp. 1833-1846, Mar. 2010.
[45] Y. Fu, L. Yang, W. Zhu and C. Liu, ?Optimum linear design of two-hop MIMO relay
networks with QoS requirements,? IEEE Trans. Signal Proc., vol. 59, pp. 5473-5484,
May 2011.
[46] P. Ubaidulla and A. Chockalingam, ?Relay precoder optimization in MIMO-relay net-
works with imperfect CSI,? IEEE Trans. Signal Proc., vol. 59, pp. 5473-5484, Nov.
2011.
[47] J. Zou, W. Liu, M. Ding, H. Luo and H. Yu, ?Transceiver design for AF MIMO two-way
relay systems with imperfect channel estimation,? in Proc. 2011 IEEE Global Telecom.
Conf. (GLOBECOM), Houston, TX, Dec. 2011.
[48] H. Mu and J.K. Tugnait, ?MSE-based source and relay precoder design for cognitive
radio multiuser two-way relay systems,? in Proc. 2012 IEEE Wireless Commun. Net-
working Conf. (WCNC 2012), pp. 742-747, Paris, France, April 1-4, 2012.
[49] A. Amah and A. Klein, ?Non-regenerative multi-antenna multi-group multi-way relay-
ing,? EURASIP J. Wireless Commun. Networking, 2011, 2011:29.
[50] D. Gunduz, A. Yener, A. Goldsmith and H.V. Poor,, ?Multi-way relay channel,? in
Proc. 2011 IEEE Intern. Symp. Info. Theory, Seoul, Korea, 2009.
[51] W. Long, T. Lv, H. Gao and Y. Lu, ?Interference alignment for multi-user multi-
way relaying X networks,? in Proc. 2012 IEEE 75th Veh. Tech. Conf. (VTC Spring),
Yokohama, Japan, May 6-9, 2012.
130
[52] B. Rankov and A. Wittneben ?Spectral efficient protocols for half-duplex fading relay
channels,? IEEE J. on Selected Areas in Comm. , Vol. 25, pp. 379-389, Feb. 2007.
[53] E. Gharavol, Y. Liang and K. Mouthaan, ?Collaborative Nonlinear Transceiver Opti-
mization in Multi-tier MIMO Cognitive Radio Networks with Deterministically Imper-
fect CSI,? in Proc. 2010 IEEE Global Telecom. Conf. (GLOBECOM), Miami, FL, Dec.
2010.
[54] X. Gong, A. Ishaque, G. Dartmann and G. Ascheid, ?MSE-based Stochastic Transceiver
Optimization in Downlink Cognitive Radio Networks,? in Proc. 2011 IEEE Wireless
Comm. and Networking Conf. (WCNC), Quintana-roo, Mexico, Mar. 2011.
[55] C. Wang, H. Chen, Q. Yin, A. Feng and A. Molisch,?Multi-User Two-Way Relay Net-
works with Distributed Beamforming,? IEEE Trans. Wireless Comm., vol. 10, pp. 3460-
3470, Oct. 2011.
[56] A. AmahandA. Klein, ?Pair-Aware Transceive Beamforming for Non-Regenative Multi-
User Two-Way Relaying,? in Proc. 2010 IEEE Intern. Conf. Acoustics, Speech, Signal
Proc. (ICASSP 2010), Dallas, TX, Mar. 2010.
[57] H. Xin, Y. Peng, C. Wang, Y. Yang and W. Wang, ?Coordinated Eigen Beamforming
for Multi-Pair MIMO Two-Way Relay Network,? in Proc. 2011 IEEE Global Telecom.
Conf. (GLOBECOM2011), Houston, TX, Dec. 2011.
[58] R. Ahlswede, N. Cai, S. Li, and R. Yeung, ?Nenwork Information Flow, ? IEEE Trans.
Information Theory, Vol. 46, pp 1204-1216, July, 2000.
[59] H. Du and T. Ratnarajah, ?Robust joint signal and interference alignment for MIMO
cognitive radio network,? in Proc. 2012 IEEE Wireless Commun. Netw. Conf. (WCNC
2012), Paris, France, April 2012.
[60] Z. Zhou and B. Vucetic, ?An iterative beamforming optimization algorithm for gener-
alized MIMO Y channels,? in Proc. 2012 IEEE Intern. Conf. Commun (ICC 2012),
Ottawa, Canada, Jun. 10-15, 2012.
[61] N. Lee and J. Chun, ?Signal space alignment for an encryption message and successive
network code decoding on the MIMO K-way relay channel,? in Proc. 2011 IEEE Intern.
Conf. Commun (ICC 2011), Kyoto, Japan, Jun. 5-9, 2012.
[62] K. Lee, N. Lee, and I. Lee, ?Achievable degrees of freedom on K-user Y channel,? IEEE
Trans. Wireless Comm., pp. 1210-1218, March 2012.
[63] N. Lee, J. Lim and J. Chun, ?Degrees of freedom on the MIMO Y channel: Signal
space alignment for network coding,? IEEE Trans. Information Theory, vol. 56, pp.
3332-3342, July 2010.
[64] N. Lee and J. Lim, ?A novel signaling for communication on MIMO Y Channel: Signal
space alignment for network coding? in Proc. IEEE Intern. Symp. Info. Thy. (ISIT 09),
Seoul, Korea, June 28 - July 3, 2009.
131
[65] J. Zhuang, F. Roemer and M. Haardt, ?Relay assisted physical resource sharing: pro-
jection based separation of multiple operators (ProBaSeMO) for two-way relaying with
MIMO amplify and forward relays,? IEEE Trans. Signal Processing, vol. 60, pp. 4834-
4836, Sept. 2012.
132
Appendices
133
Appendix A
Convexity of (2.5)
Here we will provide a proof of our claim that the objective function (2.5) in problem P1
is a concave function of variables t,s. Since a positive weighted sum of concave functions is
concave, the desired result follows if we can show that R0(s,t) defined in (A.1) is a concave
function of scalar variables s and t (0 ? s ? 1 and t ? 0: compare with (4.17)) where
R0(s,t) = asln
parenleftBigg
1 + bts
parenrightBigg
+csln
parenleftBigg
1 + dts
parenrightBigg
(A.1)
and a, b, c and d are nonnegative real scalars. A necessary and sufficient condition for R0(s,t)
to be concave is that its Hessian matrix is negative semi-definite. We have the following:
?R0(s,t)
?s = aln
parenleftBigg
1 + bts
parenrightBigg
+cln
parenleftBigg
1 + dts
parenrightBigg
? abts+bt ? cdts+dt, (A.2)
?R0(s,t)
?t =
abs
s+bt +
cds
s+dt, (A.3)
?2R0(s,t)
?2s = ?
ab2t2/s
(s +bt)2 ?
cd2t2/s
(s+dt)2, (A.4)
?2R0(s,t)
?2t = ?
ab2s
(s+bt)2 ?
cd2s
(s+dt)2 (A.5)
and
?2R0(s,t)
?t?s =
ab2t
(s+bt)2 +
cd2t
(s+dt)2. (A.6)
Thus the Hessian of R0(s,t) is given by
?2R0(s,t) =
?
??
?
?t2s
parenleftBig ab2
(s+bt)2 +
cd2
(s+dt)2
parenrightBig
t
parenleftBig ab2
(s+bt)2 +
cd2
(s+dt)2
parenrightBig
t
parenleftBig ab2
(s+bt)2 +
cd2
(s+dt)2
parenrightBig
?s
parenleftBig ab2
(s+bt)2 +
cd2
(s+dt)2
parenrightBig
?
??
?. (A.7)
134
For any real scalars x1 and x2, we have
[x1 x2]?2R0(s,t)[x1 x2]T = ?
parenleftBigg ab2
(s+bt)2 +
cd2
(s+dt)2
parenrightBiggparenleftBigg t
?sx1 ??sx2
parenrightBigg2
? 0, (A.8)
and therefore, R0(s,t) is concave.
135
Appendix B
Proof of Proposition 2.1
Here we provide a proof of Proposition 2.1. Firstly ?k can be rewritten as
?k = P(H0k|Qk)Wk
braceleftbigg
[log(z)+ ?(1? 1z)+]
bracerightbigg
where
z = P(H0k|Qk)Wk?k? .
It turns out that [log(z)]+ ? (1? 1z) ? 0 for any z > 0 so that ?k ? 0 (hence ?dk ? 0) ?k.
Define
f(?) =
Ksummationdisplay
k=1
?ak
?
parenleftBig?
dk ??
parenrightBig+
+?. (B.1)
Then
min
?
g(?,?) ? min
?
f(?). (B.2)
The derivative of f(?) with respect to ? is given by
df(?)
d? = 1?
isummationdisplay
k=1
?ak
? if
?di+1 ? ? ? ?di. (B.3)
Since ?ak > 0 ?k, 1?summationtextlk=1 ?ak? is decreasing in l; we take 1?summationtext0k=1 ?ak? := 1. We have
df(?)
d?
vextendsinglevextendsingle
vextendsingle?=0 = 1?
summationtextK
k=1 ?ak
? (B.4)
df(?)
d?
vextendsinglevextendsingle
vextendsingle?=? = 1. (B.5)
136
If 1?summationtextKk=1 ?ak? > 0, then we have df(?)d? > 0 for ? ? 0, so f(?) is an increasing function of ?.
Minimizing f(?) with respect to ? gives us the optimal solution ?? = 0. If 1?summationtextKk=1 ?ak? = 0,
we will have df(?)d? = 0 for some 0 < ? ? ?dmin = argmin1?k?K{?dk | ?dk > 0} and df(?)d? > 0 for
? > ?dmin, so ?? = 0 is still the optimal solution.
If 1 ?summationtextKk=1 ?ak? < 0, then we have df(?)d?
vextendsinglevextendsingle
vextendsingle?=0 < 0 and df(?)d?
vextendsinglevextendsingle
vextendsingle?=? > 0. Then the optimal
solution ?? should be such that either df(?)d?
vextendsinglevextendsingle
vextendsingle?=?? = 0, or df(?)d?
vextendsinglevextendsingle
vextendsingle?=???? < 0 and df(?)d?
vextendsinglevextendsingle
vextendsingle?=??+? > 0
for some ? > 0. This leads to ?? = ?dl? if 1?summationtextl??1k=1 ?ak? ? 0 and 1?summationtextl?k=1 ?ak? < 0.
137
Appendix C
Proof of (3.19)
Here we derive (3.19). For the i-th user, the objective function of problem P2 becomes
Ebardbl ?si ?Eis bardbl22= E bardbl (RiAi ?Ei)s+RiBi?xp +RiCi?ni bardbl22
= tr
braceleftbigg
Ri[AiA?i?2s +BiB?i?2p +CiC?i?2n]R?i ?EiA?iR?i?2s ?RiAiE?i?2s
bracerightbigg
. (C.1)
Since AiA?i?2s+BiB?i?2p+CiC?i?2n is a positive semi-definite matrix, (C.1) is a convex function
in Ri. Take derivative of (C.1) with respect to R?i and set it to zero to obtain the optimal
decoding matrix of i-th secondary user as R?i = ?2sEiA?i
parenleftBig
?2sAiA?i +?2pBiB?i +?2nCiC?i
parenrightBig?1
.
138
Appendix D
Proof: Problem P3 in Chapter 3 is convex
The objective function of P3 can be written as
Ksummationdisplay
i=1
tr
braceleftBig
P?iD?iDiPi ?E?iDiPi ?P?iD?iEi +EiE?i
bracerightBig
?2s.
It it obvious that D?iDi is a positive semi-definite matrix. Therefore the objective function
of P3 is convex in Pi. Using the fact that tr{AB} = tr{BA}, it is easy to show that the left
side of the power constraint (3.23) of P3 is also convex in Pi. For the power constraint (3.24),
it can be shown thatEtr{xrx?r} = tr{P?iH??ip H?irT?rTrHirH?ipPi?2s+Tr(HprH?pr?2p +?2nI)T?r}.
This is a convex function of Pi because H??ip H?irT?rTrHirH?ip is positive semi-definite. Hence
problem P3 is convex. square
139
Appendix E
Proof: Problem P5 in Chapter 3 is convex
Define G = ?HT2 ? (?H1H?rp) ?summationtextKi=1(?H2E(i)r )T ? (E(i)l ?H1H?rp) and L = HTpr ? (?H1H?rp).
Using (3.39) and after considerable manipulations, the objective function of problem P5 can
be expressed as
bardbl ?H1H?rpPr ?H2s?
Ksummationdisplay
i=1
E(i)l ?H1H?rpPr ?H2E(i)r s+?H1H?rpPrHprxp+?H1H?rpPrnr+?Hp?xp+Rn?Es bardbl22
= tr
braceleftBig
vec{Pr}?G?Gvec{Pr}?2s
bracerightBig
?tr
??
?E
?
parenleftBigg
?H?2H?rpP?rH?1 ? Ksummationdisplay
i=1
E(i)?r ?H?2H?rpP?r ?H?1E(i)?l
parenrightBigg
?2s
?
parenleftBigg
?H1H?rpPr ?H2? Ksummationdisplay
i=1
E(i)l ?H1Tr ?H2E(i)r
parenrightBigg
E?+RR??2n
?
?
?+tr{vec{Pr}
?L?Lvec{Pr}?2
p}. (E.1)
In (E.1) the first term is convex in Pr since G?G is positive semi-definite and the the
second term is linear in Pr. The third term in (E.1) is also convex in Pr since L?L is a
positive semi-definite matrix. Therefore the objective function, which is the sum of these
three convex terms, is also a convex function in Pr. Now we consider the constraint of
P5. Since (?H2 ?H?2?2s + HprH?pr?2p + ?2nI) is Hermitian, it can be expressed as QQ?. So the
constraint of P5 can be expressed as tr
braceleftBig
vec{Pr}?(QT ?H?rp)?(QT ?H?rp)vec{Pr}
bracerightBig
? Ptot,r.
Since (QT ?H?rp)?(QT ?H?rp) is positive semi-definite, this constraint is convex. Therefore
problem P5 is convex. square
140
Appendix F
Proof of Proposition 3.1
By setting ?L(?,?,Tr,?)?T?
r
= 0 and ?L(?,?,Tr,?)?? = 0, we obtain respectively
bracketleftbigg
?H?1 ?H1 +e1I
bracketrightbigg
Tr
bracketleftbigg
?H2 ?H?2 +e2I+ ?Hpr ?H?pr?2p + ?2prJp?2pI+?2nI
bracketrightbigg
?2s
?
Ksummationdisplay
i=1
bracketleftbigg
?H?1E(i)l ?H1 +?2ritr{R??i R?i}I
bracketrightbigg
Tr
bracketleftbigg
?H2E(i)r ?H?2 +?2irtr{T?iT??i I}
bracketrightbigg
?2s
+
bracketleftBigg ?
?2
parenleftBig?
H?rp ?Hrp +?2rpNI
parenrightBig
+ ??2I
bracketrightBigg
Tr
bracketleftbigg
?H2 ?H?2?2s +e2?2sI+ ?Hpr ?H?pr?2p +?2prJp?2pI+?2nI
bracketrightbigg
=
parenleftBigg
?H?1E?H?2 ? Ksummationdisplay
i=1
?H?1E(i)l EE(i)r ?H?2
parenrightBigg
?2s???1 (F.1)
and
?
braceleftbiggbracketleftbigg
?H?1 ?H1 +e1I
bracketrightbigg
Tr
bracketleftbigg
?H2 ?H?2 +e2I+ ?Hpr ?H?pr?2p +?2prJp?2pI+?2nI
bracketrightbigg
T?r?2s
?
Ksummationdisplay
i=1
bracketleftbigg
?H?1E(i)l ?H1 +?2ritr{R??i R?i}I
bracketrightbigg
Tr
bracketleftbigg
?H2E(i)r ?H?2 +?2irtr{T?iT??i I}
bracketrightbigg
T?r?2s
bracerightbigg
??tr
braceleftbigg
?Hp ?H?p?2p +?2pJp Ksummationdisplay
i=1
?di?2pi +RR??2n
bracerightbigg
= R
braceleftbigg
tr
braceleftBigbracketleftBig?
H?1E?H?2 ?
Ksummationdisplay
i=1
?H?1E(i)l EE(i)r ?H?2bracketrightBig?2sT?rbracerightBig
bracerightbigg
.
(F.2)
Right multiply (F.1) by T?r?. We observe that
braceleftBigbracketleftBig?
H?1E?H?2 ?summationtextKi=1 ?H?1E(i)l EE(i)r ?H?2
bracketrightBig
?2sT?r
bracerightBig
is
Hermitian. Then we take the trace of equation (F.1) right multiplied by T?r?. We observe
that its right-side is exactly the same as the right-side of (F.2). Then we have
tr
braceleftbiggbracketleftbigg
?
?2
parenleftBig?
H?rp ?Hrp +?2rpNI
parenrightBig
+ ??2I
bracketrightbigg
Tr
bracketleftbigg
?H2 ?H?2?2s +e2?2sI+ ?Hpr ?H?pr?2p +?2prJp?2pI+?2nI
bracketrightbigg
T?r
bracerightbigg
= tr
braceleftbigg
?Hp ?H?p?2p
bracerightbigg
+?2pJpsummationtextKi=1 ?di?2pi +summationtextKi=1 ?di?2n. (F.3)
141
Let Ic and Pc denote the interfering power to primary users and ?consumed? power, respec-
tively, given by
Pc = Tr
bracketleftbigg
?H2 ?H?2?2s +e2?2sI+ ?Hpr ?H?pr?2p +?2prJp?2pI+?2nI
bracketrightbigg
T?r, (F.4)
Ic =
parenleftBig?
H?rp ?Hrp +?2rpNI
parenrightBig
Tr
bracketleftbigg
?H2 ?H?2?2s +e2?2sI+ ?Hpr ?H?pr?2p +?2prJp?2pI+?2nI
bracketrightbigg
T?r. (F.5)
Then we can observe that the left-side of (F.3) is actually ??2Pc+ ??2Ic. When the optimal val-
ues of the primal and dual variables are achieved, the following conditions (complementary-
slackness) should be satisfied:
??
?2
braceleftbigg
Ic ?Itot
bracerightbigg
= 0, ?
?
?2
braceleftbigg
Pc ?Ptot,r
bracerightbigg
= 0. (F.6)
Therefore, whether the values of ?? and ?? are positive or zero, we will always have (3.89).
square
142