On Multiantenna Spectrum Sensing and Energy-Aware Resource Allocation in
Cognitive Radio Networks
by
Guangjie Huang
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
August 3rd, 2013
Keywords: Cognitive radio, Cyclostationarity, Spectrum sensing, Energy efficiency,
Resource allocation, Multiple input multiple output(MIMO)
Copyright 2013 by Guangjie Huang
Approved by
Jitendra K. Tugnait, James B. Davis Professor of Electrical and Computer Engineering
Prathima Agrawal, Ginn Distinguished Professor of Electrical and Computer Engineering
Shiwen Mao, McWane Associate Professor of Electrical and Computer Engineering
Olav Kallenberg, Professor of Mathematics
Abstract
This dissertation covers two main parts: 1) low complexity cyclostationarity based spec-
trum sensing in cognitive radio, and 2) energy efficient resource allocation in cognitive radio
networks. Cognitive radio allows unlicensed users (secondary users) to access the frequency
spectrum allocated to licensed users (primary users). Secondary users are allowed to ac-
cess the spectrum licensed to primary users in two ways: underlay and overlay. Under the
overlay mode, secondary users need to sense the channel first and then decide whether to
transmit based on the presence or absence of a primary signal. The underlay mode allows
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. The most common sensing
method is based on the energy distribution of the primary signal, however its performance
is degraded when noise variance is not accurately known. For cyclostationary primary sig-
nals such as Orthogonal Frequency Division Multiplexing (OFDM) and Gaussian filtered
Minimum Shift Keying (GMSK) signals, we exploit their statistical characteristics under
both certain and uncertain Gaussian noise and propose a low complexity cyclostationarity-
based spectrum sensing method for cognitive radio networks. Multiple receiving antennas
are utilized to improve sensing performance. In addition to sensing complexity, energy effi-
ciency has become a ?hot? research topic due to the trend of sustainable energy and green
wireless communication and networks. For mobile terminals with battery for power supply,
the energy consumption issues are more serious because development of battery technology
lags behind the development of wireless communication technologies. This dissertation also
ii
addresses energy efficient sensing and power allocation for multichannel overlay cognitive ra-
dio networks (CRN), energy efficient precoding for MIMO (multiple input multiple output)
assisted spectrum sharing CRN and MIMO cognitive multiple access (C-MAC) channel.
The dissertation is organized into eight chapters. Chapter 1 provides an introduction to
cognitive radio networks and related research issues. Existing spectrum sensing algorithms
are addressed too. It also briefly outlines the whole dissertation. Chapter 2 through Chap-
ter 4 are mainly concerned with low complexity cyclostationarity-based spectrum sensing
for cognitive radio networks. This sensing scheme is explicitly designed for cyclostationary
primary signals under white and colored Gaussian additive noise. Chapter 2 introduces the
cyclostationarity background and related spectrum sensing algorithm. The cyclostationary
characteristics of the OFDM and GMSK signals are illustrated as examples. The hypothesis
testing framework is briefly described and existing spectrum sensing tests based on cyclo-
stationarity are presented. In Chapter 3, a novel low complexity cyclostationarity-based
spectrum sensing approach under white Gaussian noise is proposed. The theoretical per-
formance and computational complexity advantages compared with other counterparts are
given in this chapter. The simulation examples support the theoretical analysis. Chapter 4
focuses on cyclostationarity-based spectrum sensing under colored Gaussian noise. Besides
sensing performance, energy efficiency related with sensing and transmission for either over-
lay or underlay cognitive radio networks is also studied in Chapters 5, 6 and 7. Chapter
5 focuses on energy-aware sensing and power allocation for multi-channel transmission of a
single secondary link in overlay cognitive radio networks. The optimization algorithms for
sensing duration, test threshold and transmission power allocation are described. Simula-
tion results are provided in support of the proposed algorithm. Chapters 6 and 7 address
energy efficient resource allocation in underlay cognitive radio networks under both transmit
power constraint and interference power constraint. Chapter 6 proposes an energy efficient
transmit covariance matrix (precode) to maximize energy efficiency for a single secondary
link in spectrum sharing underlay CRN. Multiple antennas are employed at the secondary
iii
transmitter and receiver. Chapter 7 further extends the work in Chapter 6 to propose an
energy efficient precoding for MIMO cognitive multiple access channels. Conclusions and
future work are stated in Chapter 8.
iv
Acknowledgments
Here, I would take the chance to thank all people who help me during last four years?
study at Auburn University.
First of all, I should express my deepest gratitude to my advisor Dr. Jitendra K. Tugnait
who is a well-recognized scholar on digital signal processing and communication system.
The dissertation would not be finished without his guidance. He always encourages me to
explore research topic independently, challenges my idea, and shapes my mathematical model
formulation with his broad knowledge and deep experience on digital signal processing. He
also donates precious time and effort to correct my writing with patience. His kindness,
brilliant ideas hint, excellent example provided, and instructions help me grow to be an
eligible scientific researcher with self-discipline.
I would like to thank committed members: Dr. Prathima Agrawal, Dr. Shiwen Mao and
Dr. Olav Kallenberg. I really appreciate their time and effort to read my PhD proposal and
dissertation. I should give special thanks to Dr. Kevin Phelps who serves as outside reader
for this dissertation.
To all the professors from whom I took courses, I own my gratitude to their instruction
and knowledge that help me to finish PhD courses. I would also like to acknowledge my
classmates and friends in Auburn. With their accompany, I could enjoy the past four years?
life at Auburn.
Last, but not least, I will always be grateful for my parents, my brothers and sisters for
their spiritual support. My mere thanks would not be sufficient to express my thanks for
them.
Thanks you all again.
v
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Cognitive Radio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The History of Cognitive radio . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Spectrum Opportunity Discovery . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Dynamic Spectrum Access Strategies . . . . . . . . . . . . . . . . . . 4
1.1.4 Resource Allocation and Precoding . . . . . . . . . . . . . . . . . . . 5
1.1.5 Cognitive Radio Network QoS Issue, Security Issue and Protocols . . 6
1.1.6 Cognitive Radio Network Standard and applications . . . . . . . . . . 6
1.2 Existing Spectrum Sensing Algorithms in Cognitive Radio . . . . . . . . . . 7
1.2.1 Energy Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Coherent Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Cyclostationary Detector . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Other Spectrum Sensing Algorithms . . . . . . . . . . . . . . . . . . 9
1.3 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Cyclostationarity Background and Corresponding Spectrum Sensing Tests . . . 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Cyclostationary Characteristics of OFDM and GMSK Signals . . . . . . . . 14
2.2.1 OFDM Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 GMSK Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
vi
2.3 Introduction to Statistical Hypothesis Testing . . . . . . . . . . . . . . . . . 18
2.4 Existing Cyclostationarity-based Spectrum Sensing Algorithms . . . . . . . . 19
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 On Cyclic Autocorrelation Based Spectrum Sensing for Cognitive Radio Systems
in Gaussian Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Cyclostationarity Background . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Multivariate Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Large Sample Statistics of Sample CAF . . . . . . . . . . . . . . . . . . . . . 29
3.3.1 Nonconjugate CAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.2 Conjugate CAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Test Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.1 Nonconjugate CAF: OFDM Signals . . . . . . . . . . . . . . . . . . . 34
3.4.2 Conjugate CAF: GMSK Signals . . . . . . . . . . . . . . . . . . . . . 36
3.4.3 Single Antenna Case: Nonconjugate CAF . . . . . . . . . . . . . . . 36
3.5 Detection Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5.1 Nonconjugate CAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5.2 Conjugate CAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.7.1 Example 1: Single Antenna Receiver . . . . . . . . . . . . . . . . . . 40
3.7.2 Example 2: Cyclostationary Interference . . . . . . . . . . . . . . . . 41
3.7.3 Example 3: Multi-Antenna Receiver, OFDM PU signal . . . . . . . . 43
3.7.4 Example 4: Multi-Antenna Receiver, GMSK PU signal . . . . . . . . 44
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Cyclostationarity Based Spectrum Sensing Under Uncertain Gaussian Noise . . 52
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
vii
4.2 Cyclostationarity Background . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.1 Multivariate Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Large Sample Statistics of Sample CAF . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 Nonconjugate CAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.2 Conjugate CAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Test Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.1 Nonconjugate CAF: OFDM Signals . . . . . . . . . . . . . . . . . . . 63
4.4.2 Conjugate CAF: GMSK Signals . . . . . . . . . . . . . . . . . . . . . 65
4.4.3 Single Antenna Case: Nonconjugate CAF . . . . . . . . . . . . . . . 65
4.5 Detection Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5.1 Nonconjugate CAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5.2 Conjugate CAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.6 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.7 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.7.1 Example 1: Analytical Threshold Selection, p = 1, K = 1, Colored
Gaussian Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7.2 Example 2: Single Antenna Receiver, Colored Gaussian Noise, K =1
or 4, OFDM PU signal (IEEE 802.11a/g) . . . . . . . . . . . . . . . 70
4.7.3 Example 3: Multi-antenna Receiver (p = 1,2,4,8), OFDM PU sig-
nal (IEEE 802.11a/g), Colored Gaussian Noise, Rayleigh flat-fading
channel, K =1,2,4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.7.4 Example 4: Single Antenna Receiver, Colored Gaussian Noise, K =1,
OFDM PU signal (WiMAX) . . . . . . . . . . . . . . . . . . . . . . 72
4.7.5 Example 5: Multi-Antenna Receiver (p = 4), GSM GMSK PU signal,
colored Gaussian Noise, K = 1 . . . . . . . . . . . . . . . . . . . . . . 73
4.7.6 Example 6: Computational Complexity . . . . . . . . . . . . . . . . . 73
4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
viii
5 Design of Sensing and Power Allocation Strategies for Energy Aware Multi-
Channel Cognitive Radio Networks . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.1 Primary network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2.2 Spectrum Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2.3 Spectrum Access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3.1 Throughput and Energy Consumption . . . . . . . . . . . . . . . . . 90
5.4 Iterative Algorithm for optimizing Sensing Parameters and Power Allocation 92
5.4.1 Power Allocation Optimization . . . . . . . . . . . . . . . . . . . . . 92
5.4.2 Optimization of Sensing Parameters Ts and ?m . . . . . . . . . . . . . 95
5.4.3 Iterative algorithm for power allocation and sensing parameters opti-
mization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.5.1 Utility function versus total power constraint . . . . . . . . . . . . . . 98
5.5.2 Energy Efficiency versus energy price ? . . . . . . . . . . . . . . . . . 99
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6 On Energy Efficient MIMO-Assisted Spectrum Sharing for Cognitive Radio Net-
works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . 105
6.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2.2 Energy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3 Solutions based on Concave Fractional Program and Related ParametricProb-
lem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
ix
6.4 Multichannel Energy Efficient Precoding . . . . . . . . . . . . . . . . . . . . 110
6.5 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.5.1 Energy Efficiency and Capacity Comparisons . . . . . . . . . . . . . . 113
6.5.2 Multichannel Energy Efficient MIMO Transmission . . . . . . . . . . 114
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7 On Precoding for Maximum Weighted Energy Efficiency of MIMO Cognitive
Multiple Access Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . 121
7.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3 Solutions: Concave Fractional Program and Related Parametrized Problem . 124
7.4 SDP-based Cyclic Coordinate Ascent for Problem P2 . . . . . . . . . . . . . 126
7.5 Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A Proof of Lemma 1 of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B Proof of Lemma 2 of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
C Fractional Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
x
List of Figures
2.1 Theoretical cyclic Autocorrelation Function Value . . . . . . . . . . . . . . . . . 16
3.1 Chapter 3 Example 1: Theoretical and simulation-based performance compari-
son of probability of detection Pd versus SNR for the proposed single-antenna
spectrum sensing under different observation lengths. . . . . . . . . . . . . . . . 46
3.2 Chapter 3 Example 1: Simulation-based comparison of Pd versus SNR with single
antenna under Rayleigh flat-fading. . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Chapter 3 Example 1: Number of symbols needed to achieve Pd = 0.9 versus
SNR with single antenna under non-random channel with constant gain. . . . . 47
3.4 Chapter 3 Example 1: Simulation-based comparison of Pd versus SNR with single
antenna under non-random channel with constant gain. . . . . . . . . . . . . . . 47
3.5 Chapter 3 Example 2: Simulation-based comparison of Pd versus SNR with single
antenna under non-random channel with constant gain and possible presence of
cyclostationary interference-Part I. . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6 Chapter 3 Example 2: Simulation-based comparison of Pd versus SNR with single
antenna under non-random channel with constant gain and possible presence of
cyclostationary interference-Part II. . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.7 Chapter 3 Example3: Simulation-based comparison of probability of detection
versus SNR with p = 4 antennas under Rayleigh flat-fading. . . . . . . . . . . . 50
xi
3.8 Chapter 3 Example 3: Performance analysis: theoretical and simulation-based
performance comparison of probability of detection Pd versus SNR for the pro-
posed multi-antenna spectrum sensing. . . . . . . . . . . . . . . . . . . . . . . . 50
3.9 Chapter 3 Example 4: Simulation-based comparison of probability of detection
versus SNR for GMSK signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.10 Chapter 3 Example 4: Simulation-based comparison of probability of detection
versus SNR for GMSK signal under Rayleigh flat-fading. . . . . . . . . . . . . . 51
4.1 Example 1. Analytical threshold selection performance: colored Gaussian noise,
M = 2000, 2000MonteCarloruns. Theapproachlabeled?Dandawate-Giannakis?
is used in [59, 67, 64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Example 2: Performance analysis: theoretical and simulation-based performance
comparison of probability of detection Pd versus SNR for the proposed single-
antenna spectrum sensing: OFDM signal, non-random channel with unit gain,
2000 runs, Pfa = 0.01, p = 1, K = 1 or 4, colored Gaussian noise, M = 4000 or
20000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Example 3: Simulation-based comparison of Pd versus SNR with variable number
of antennas: OFDM signal, M = 20000 (250 OFDM symbols), Rayleigh flat-
fading channel, 2000 runs, Pfa = 0.01, p = 1 in (a) and p = 2 in (b), K = 1, 2
or 4, colored Gaussian noise. The approach labeled ?DG/L? refers to the ones
based on [59, 64, 73] and that labeled ?prop? refers to our proposed detector. the
labels ??? and ??? refer to cycle frequency and lag, respectively. . . . . . . . . . 77
xii
4.4 Example 3: Simulation-based comparison of Pd versus SNR with variable number
of antennas: OFDM signal, M = 20000 (250 OFDM symbols), Rayleigh flat-
fading channel, 2000 runs, Pfa = 0.01, p = 4 in (a) and p = 8 in (b), K = 1, 2
or 4, colored Gaussian noise. The approach labeled ?DG/L? refers to the ones
based on [59, 64, 73] and that labeled ?Proposed? refers to our proposed detector. 78
4.5 Example 3: Simulation-based comparison of Pd versus SNR with four antennas:
OFDM signal, M = 20000 (250 OFDM symbols), Rayleigh flat-fading channel,
2000 runs, Pfa = 0.01. The approach labeled ?Axell-Larsson? is from [76] and
that labeled ?Proposed? refers to our proposed detector. . . . . . . . . . . . . . 79
4.6 Example 4: Simulation-based comparison of Pd versus SNR with single antenna
under non-random channel with constant gain: Pfa = 0.05, p = 1, K = 1, M =
81920, 2000 Monte Carlo runs, OFDM signal (number of subcarriers Nc=256,
cyclic prefix length Ncp = 64), colored Gaussian noise. The approach labeled
?Tani-Fantacci? is that of [71]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.7 Example 4: Simulation-based comparison of probability of detection versus SNR
for GMSK signal with 2 samples per symbol, with p = 4 antennas under Rayleigh
flat-fading, colored Gaussian noise and M = 540: Pfa = 0.01, 2000 runs. The
approach labeled ?Zhong et al? is from [73]. . . . . . . . . . . . . . . . . . . . . 80
4.8 Example 5: Computational complexity of different spectrum sensing algorithms
for varying sizes of observation length M, (L,?) = (65,1) when nonconjugate
CAFs are used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.1 Utility function versus the total power constraint,when the minimum required
detection probability is fixed at 0.9, energy price ? = 10. . . . . . . . . . . . . . 99
5.2 Utility function versus the total power constraint, when the minimum required
detection probability is fixed at 0.9, energy price ? = 150. . . . . . . . . . . . . 100
xiii
5.3 Normalized criteria versus the price of energy ?, when the total power constraint
is fixed at ?20dBW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.1 Chapter 6: Capacity comparison for energy efficient maximization and capacity
maximization scheme in MIMO-spectrum sharing CRN . . . . . . . . . . . . . . 114
6.2 Chapter 6: Energy efficiency comparison for energy efficient maximization and
capacity maximization scheme in MIMO-spectrum sharing CRN . . . . . . . . . 115
6.3 Chapter 6: Capacity comparison for energy efficient maximization and capacity
maximization scheme in multichannel MIMO-spectrum sharing CRN . . . . . . 116
6.4 Chapter 6: Energy efficiency comparison for energy efficient maximization and
capacity maximization scheme in multichannel MIMO-spectrum sharing CRN . 117
7.1 Chapter7: Capacity comparison of energy efficient scheme and spectral efficient
design for MIMO cognitive MAC channel . . . . . . . . . . . . . . . . . . . . . 133
7.2 Chapter7: Energy efficiency comparison of energy efficient scheme and spectral
efficient design for MIMO cognitive MAC channel . . . . . . . . . . . . . . . . . 133
xiv
List of Tables
3.1 Number of complex multiplications for M observation samples with white Gaus-
sian noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 Number of complex multiplications for M samples with colored Gaussian noise . 69
xv
Chapter 1
Introduction
1.1 Cognitive Radio
1.1.1 The History of Cognitive radio
The development of microelectronics has spurred the emergence of cheaper, higher speed
and quality of service (QoS) devices and applications, resulting in high demand for radio
spectrum. On the other hand, the frequency spectrum is a limited natural resource managed
by the government agencies in a static allocation mode. The rights of frequency band are sold
to licensed users with guaranteed minimum frequency interference between adjacent users.
While most of the spectrum resources has been allocated to licensed users, the current radio
spectrum allocation can not accommodate the dramatic emergence of new wireless commu-
nication application. However, according to [1] the utilization of current licensed spectrum
is inefficient from 15% to 85%, especially the televison broadcasting band. Thus the true
problem is the fixed frequency spectrum allocation rather than true shortage of frequency.
To alleviate the confliction between increasing needs for frequency and inefficient spectrum
usage, the concept of cognitive radio (CR) was first proposed by Mitola in 1999 [2]. It is an
alternative method to allow the unlicensed users to opportunistically use frequency spectrum
that is not temporally or spatially occupied by the licensed users. The potential of improving
spectrum resource utilization of CR triggered Federal Communications Commission?s (FC-
C?s) work on new spectrum policy [3, 4]. FCC allows an unlicensed fixed, personal/portable
device to share the TV bands, leading the begining of CR network deployment.
1
In academia, a large amount of work and effort has been put into CR research [2, 5, 56, 6].
The research ?hot? points include: spectrum opportunity discovery such as spectrum sens-
ing, dynamic spectrum access strategies and protocols, CR network routing and transport
protocols, security issue, standardization and future application [7, 8, 9, 10].
The government officials also began to publish related new spectrum policy to pave the
way for CR network applications. In the United States, based on the spectrum utilization
research [1, 3], an official rule to guide unlicensed transmission of CR devices in TV bands
was specified in [4]. In the United Kingdom Office of Communications (Ofcom) has proposed
regulation that CR devices can access digital TV channels without interfering with PU users,
jointly based on white space databases and spectrum sensing to exploit spectrum opportunity
[11]. The Conference of European Post & Telecommunications Administrations (CEPT) has
also specified the frequency band plan and technical requirement for CR devices to access
790-862 MHz licensed band [12].
The first industrial research on CR network was launched in the United States motivated
by FCC. In 2008, the first testing CR network applied in TV bands was field tested with
prototype CR devices from Adaptrum, I2R, Motorola and Philips [13]. The results showed
that the performance of spectrum sensing for TV signals and wireless microphone is limited
and even unreliable. Thus, FCC then recommended combination of spectrum sensing and
geolocation databases for white band discovery. Then in 2009 the first public CR network
was deployed and tested in Virginia [14] and finally led to a ?Smart City? CR network built
in North Carolina in 2010, which used CR devices from Spectrum Bridge, Microsoft and Dell
[15]. The development of CR techniques and its potential application in TV white band has
attracted increasing attention from the public and industry.
1.1.2 Spectrum Opportunity Discovery
The first and the most important step of CR is to detect spectrum opportunity that
is also named spectrum hole or white frequency space [8, 9]. Recently emerging bunch of
2
proposals for discovery of spectrum opportunity can be classified into three basic types: PU
signal detection, auxiliary beacons detection and geolocation databases of white spaces or
spectrum holes.
The aim of PU signal detection, also named spectrum sensing, is to detect PU signals
in the frequency band of interest. Spectrum sensing can be further categorized into PU
transmitter signal detection and receiver detection. The former includes matched-filtering
sensing, waveform-based sensing, cyclostationarity sensing and energy detector sensing [57]
[60] [16] [65], which would be explained in detail in Sec. 1.2. Since the PU receiver?s local
oscillator leaks power, a leakage detector is used to passively identify spectrum opportunity
[17].
Besides spectrum sensing, beacon transmitter is another method to detect spectrum
holes. The beacon system occupies a narrow part of TV band, which periodically transmit
message to secondary users to utilize idle PU channels. At the same time, strong ?Stop?
beacon message is designed to protect the primary users such as wireless microphone oper-
ations from interference due to secondary transmission [18]. When the PU?s location and
activities are known a priori such as TV bands usage, a geolocation database [19] can be
built. Secondary user with GPS-aids would visit the database, and then decide when and
which channel to access. CR users can also predict the spectrum opportunity by reasoning
and learning from previous channel usage [20].
Spectrum sensing performance such as false alarm probability is determined directly by
sensing duration. Long sensing time would result in better detection. On the other hand,
too much sensing time will reduce potential network throughput since spectrum opportu-
nity is limited. Thus, MAC layer assisted sensing scheduling is needed to balance sensing
performance and transmission throughput [21, 22]. The dynamics of wireless channel such
as fading can make a single sensor unable to identify PU signal, thus wasting opportunity
or interfering with PU signal. To overcome the disadvantage of local sensors, collaborative
sensing can exploit spatial diversity to enhance detection performance [22].
3
Besides MAC layer cooperative sensing utilizing spatial diversity of sensors, Ref. [23]
works on relay-based cooperation that can improve cooperative spectrum sensing perfor-
mance with relay diversity to overcome heavy shadowing and frequency-selective Rayleigh
fading.
1.1.3 Dynamic Spectrum Access Strategies
The spectrum opportunity discovery provides the state of PU channels. Given this state,
the secondary users has to decide when and how to access the channel. On the other hand,
the white channel may become busy due to PU?s activity and other SU?s occupancy. Thus
how to avoid interfering with PU signal and share spectrum with other secondary users, how
to utilize spectrum for transmission and how to switch channel when current channel is not
available any more, are some of the issues that a secondary user needs to face. According
to [24], dynamic spectrum access can be classified into three groups: dynamic exclusive use
model, open sharing model and hierarchical access model. The first model maintains the
basic idea of current static spectrum allocation policy: spectrum allocated can only be used
exclusively. For CR network, the secondary users try to share the spectrum with primary
user who holds the spectrum rights. As a result, the only way is to do spectrum trading
between PU and SU to achieve exclusive usage [25]. The primary use can sell or trade the
spectrum to secondary user for profit. In [26] the stable throughput of secondary link in a
cognitive radio network composed of single SU and single PU is studied. The SU queues and
forwards packets for the PU to enhance transmission rate. SU also benefits from relaying
PU?s transmission becasue more idle channel time is created for SU itself.
Another popular access model is the hierarchical one, where there is no trading between
SUs and PUs. It can be further classified into: spectrum underlay and spectrum overlay. In
the spectrum underlay, the secondary user can transmit simultaneously with PU [27], with
limited interference to PU signal. In the underlay model, the SU can not necessarily detect
4
the PU signal. Spectrum overlay is also called opportunistic spectrum access, that is sec-
ondary users can only transmit when PU signal is absent. Based on the network structure,
the opportunistic spectrum access is categorized into: centralized and distributed. For cen-
tralized access, there is a central entity that controls spectrum allocation. Thus scheduling
is a typical centralized access method [28]. For distributed spectrum access, special MAC
layer protocols are needed. In [29] access strategies are studied to maximize single SU?s
throughput with PU protection. In multi-SU scenario, [30] proposes MAC designed for CR
networks to guarantee co-existence of different SUs and PUs.
1.1.4 Resource Allocation and Precoding
In CR networks, the dynamics of spectrum resource induce competition among sec-
ondary users. At the same time different PU channels have different interference temper-
ature requirement and activity models. To guarantee maximum SU?s throughput, fairness
among SUs and PU signal protection, resource allocation should be based on cross-layer
consideration. To manage interference among secondary users and to primary users, power
control is needed for spectrum access and related channel allocation is also needed [31].
In underlay spectrum sharing cognitive radio networks, MIMO (multiple input multiple
output) technique is employed to enhance spectral efficiency [106, 107, 101, 100, 99], to
mitigate interference to PU receiver and to maximize energy efficiency using spatial diversity.
When multiple antennas are equipped at secondary users and channel state information
from SU transmitter to both SU receiver and PU receiver is available, optimization on
transmit covariance matrix (precoding) can be performed to maximize capacity or energy
efficiency. The spatial efficient or energy efficient precoding can be applied in single secondary
link, cognitive multiple access channel and/or cognitive broadcasting channel with multiple
transmit antennas.
5
1.1.5 Cognitive Radio Network QoS Issue, Security Issue and Protocols
CR networks need awareness of the surrounding wireless environment, cooperation or
interaction among secondary users, and adapt to network variations by learning and rea-
soning from current and previous observations. However, the erroneous spectrum sensing,
dynamically changing channel state, temporally and spatially discrete spectrum resources
would bring intolerant latency or traffic jitter for high layer applications with QoS require-
ment. As a result, new routing protocols and transport layer protocols are needed to adapt
spectrum dynamics and to ensure required latency, network throughput and capacity [32].
Ref. [33] studies a distributed routing protocol based on traditional on-demand routing for
ad hoc networks with multi-flow multi-frequency scheduling. An unified adaptive transport
protocol for heterogeneous fourth-generation cognitive radio networks was studied in [34].
Since there is generally lack of cooperation between primary users and secondary users
in CR networks, CR networks are more vulnerable than traditional wireless networks to
security attacks. There are two types of attack on spectrum sensing: primary user emulation
attack (PUEA) and spectrum sensing data falsification (SSDF). The work in [35] directly
compared the true PU transmitter?s location with estimated location of signal source to
identify fake PU. Ref. [36] studied SSDF attack and proposed a statistics-based filtering
method to mitigate false sensing report?s affect. To alleviate CR users violating spectrum
etiquettes, a monitoring and punishing mechanisms may be embedded into hardware in CR
devices [2].
1.1.6 Cognitive Radio Network Standard and applications
Aiming to resolve the spectrum scarcity problem and to improve spectrum utilization,
cognitive radio has triggered standardization from all levels such as International Telecom-
munication Union (ITU), Institute of Electrical and Electronics Engineers (IEEE), European
Association for Standardizing Information and Communication Systems (ECMA) and Eu-
ropean Telecommunications Standards Institute (ETSI) [9, 8]. ITU-R 1B has developed
6
definitions, usage scenarios of cognitive radio systems (CRS) and its relation with software
defined radio (SDR) [37]. IEEE 802 group and SCC41 group are developing standardization
of cognitive radio and have proposed CR standards 802.22 [38], 802.11af [39], SCC41 [40]
and 802.19. IEEE 802.22 wireless regional area network (WRAN) is the first international
CR standard. WRAN is a cellular network designed for rural area, including base station
(BS) and customer premises equipment (CPE). IEEE 802.11af is evolved from current Wi-Fi
[41], based on the FCC policy to allow personal and/or portable devices [4]. IEEE 801.19
aims to provide coexistence among independent unlicensed wireless network, such as 802.22,
802.16 on TV band. IEEE SCC41, formally named IEEE 1900, is a CR standard to regulate
higher-layer protocols than MAC and PHY for dynamic access network (DSA). ECMA pro-
poses the first standard ECMA 392 that personal/portable cognitive devices utilize white
space in TV bands [42]. The TC RSS working group of ETSI has proposed a technical report
on how cognitive radio network operate on UHF white space frequency bands, primary user
protection and system requirements [43].
In [10] a survey on emerging cognitive radio network application is studied. The first
future potential cognitive radio network is on TV bands in the United States. The secondary
CR network is MBANS. There is another trend that leads traditional power grid into smart
grid with cognitive ability. For the public?s benefits, CR can also be applied into public
safety networks, cellular networks and wireless medical networks.
1.2 Existing Spectrum Sensing Algorithms in Cognitive Radio
Thegoal ofcognitiveradioistoachievemaximum spectrumutilization whileinterference
to licensed users is kept to a minimum. To avoid interference to license users (also named
primary users), the unlicensed users (also named secondary users) should detect the presence
of primary users? signal before accessing the specified spectrum band. On the other hand,
the secondary users should cancel transmission immediately once they detect the signal of
primary users. As a result, advanced spectrum sensing techniques are need for application of
7
cognitive radios. For local or non-cooperative spectrum sensing, researchers have proposed
many solutions, including matched-filter sensing, waveform-based sensing, cyclostationarity
sensing and energy detector sensing [65, 60, 16].
1.2.1 Energy Detector
Energy detector is the most popular one due to its simplicity and no needs for primary
users? signal [44, 45, 46]. The energy detector compares the power metric of signal with a
threshold to decide the presence or absence of the primary signal. However, the test thresh-
old is related to power or variance of noise, which is not available in many scenarios. In [45],
the authors propose a power spectrum density (PSD) estimation method for noise through a
assessment window, which is assumed to be free of primary signal. Thus, it does not require
exact knowledge of noise. Ref. [47] improves the PSD estimation bias and variance in [45] by
replacing Welch windowed periodogram with multitaper estimator. The multi-taper method
is proved to decrease the bias and variance for PSD estimation in [46].
1.2.2 Coherent Detector
When the primary user signal is deterministic and known to the secondary users,
matched filter can be used for coherently spectrum sensing. Even only part of the sig-
nal structure is known, such as pilot aided channel estimation, we can still achieve coherent
detection[48].
1.2.3 Cyclostationary Detector
Some of man-made signals are cyclostationary due to signal structure, modulation,
codingorpilotaddedforchannelestimation. Forthesecyclostationarysignals, theirstatistics
such as autocorrelation function exhibit periodicity, which can be used for detection of
primary user signal under noise and interference background [59, 73, 49, 64, 68, 67, 71].
8
1.2.4 Other Spectrum Sensing Algorithms
Other spectrum sensing algorithms include covariance-based detection and wavelet-
based detection. When the received PU signal exhibits spatial correlation due to multi-
antenna receiver, oversampling or dispersive channel, we can utilize such correlation to dis-
tinguish PU signal from background noise[50]. Wavelet-based detection [51] has been used to
estimate the power spectrum density (PSD) of the PU signal with discrete changes between
adjacent sub frequency bands and smooth changes within each sub frequency band.
1.3 Overview of the Dissertation
In this dissertation, we investigate low complexity cyclostationarity based sensing and
energy efficient design for sensing and power allocation in cognitive radio networks. Based
on the cyclostationarity of the primary signal, we propose low complexity sensing for both
single antenna and multiple antennas assisted secondary users. For battery-supported mo-
bile secondary users, we study the effect of energy on sensing and transmission, and then an
energy efficient sensing and power allocation scheme is proposed for multichannel secondary
transmission in cognitive radio network. We further propose energy efficient resource alloca-
tion for either single secondary link or cognitive multiple access channel in spectrum sharing
underlay cognitive radio networks. The main contents of the dissertation are presented in
Chapter 2 through Chapter 7.
Chapter 2 through Chapter 4 focus on low complexity cyclostationarity based spectrum
sensing under both white and colored uncertain Gaussian noise, in which multiple receive
antennas are utilized to enhance sensing performance. This sensing scheme is explicitly
designed for the cognitive radio networks where primary users? signals exhibit cyclostation-
arity. To opportunistically access the spectrum allocated to the primary users, the secondary
users need to first sense the presence of cyclostationary signals from primary users to avoid
transmission conflict.
9
In Chapter 2, the cyclostationarity background and corresponding existing spectrum
sensing schemes are introduced. First, the cyclostationary characteristics of OFDM and
GMSK signals are derived theoretically and illustrated with simulations. Second, the con-
cept of statistical hypothesis testing is reviewed before introducing existing cyclostationary
sensing algorithms.
Chapter 3 presents the details of spectrum sensing of cyclostationary primary user (PU)
signals in white Gaussian noise with both single and multiple receiving antennas. A novel
algorithm based on looking for a cycle frequency at a particular time lag in the cyclic au-
tocorrelation function (CAF) of the noisy PU signal is proposed. We explicitly exploit the
knowledge that under the null hypothesis of PU signal absent, the measurements originate
from white Gaussian noise with possibly unknown variance. Our formulation allows us to
computationally simplify the spectrum sensing detector, obviating the need for estimating
an unwieldy covariance matrix needed in prior works. We consider both single and multiple
antenna receivers. Supporting simulation examples are provided and they verify our perfor-
mance analysis and also show that our approaches either outperform or are at least as good
as existing approaches while being computationally much cheaper.
The cyclostationarity-based spectrum sensing under uncertain (colored) Gaussian noise
is addressed in Chapter 4. Detection of cyclostationary primary user (PU) signals in colored
Gaussian noise for cognitive radio systems is considered based on looking for a cycle fre-
quency at a particular time lag in the cyclic autocorrelation function (CAF) of the noisy PU
signal. We explicitly exploit the knowledge under the null hypothesis of PU signal absent,
the measurements originate from colored Gaussian noise with possibly unknown correlation
function. We consider both single and multiple antenna receivers. A performance analysis
of the proposed detector is carried out. Supporting simulation examples are provided using
a OFDM PU signal and they show that our proposed approaches are computationally much
cheaper than the Dandawate-Giannakis and related approaches while having quite similar
detection performance for a given false alarm rate.
10
In Chapter 5, energy-aware design of sensing and power allocation strategies for multi-
channel cognitive radio networks is presented and analyzed. Frequency spectrum and energy
are two key resources of green cognitive radio networks with battery-powered wireless termi-
nals. The issues of how to utilize sparse frequency spectrum and limited energy resource pose
challenges to the design of sensing and power allocation strategies that affect both through-
put and energy consumption. In this dissertation, we first construct an utility function that
incorporates throughput as reward and energy consumption as cost for a time slotted multi-
channel cognitive radio network. An optimization problem to maximize the utility function
is formulated involving optimization of both sensing parameters (sensing duration, local test
threshold) and power allocation strategy. The problem is non-convex, however, we decouple
it into two separate convex problems and propose an iterative algorithm to obtain a sub-
optimal solution. The simulation results show that our iterative algorithm converges fast
and performs better than an ?only power allocation optimization? approach and an existing
approach that ignores energy efficiency.
In Chapter 6, energy efficient spectrum sharing in cognitive radio network is designed
and analyzed. We formulate the design of energy efficiency maximization for a single sec-
ondary link in an underlay spectrum sharing cognitive radio network under an SU (secondary
user) transmit-power constraint and an upperbound on the interference power at the PU (pri-
mary user). We propose energy efficient precoding (beamforming) for the SU transmitter
to maximize energy efficiency defined as the transmission rate to power ratio, when the
terminals in the system are equipped with multiple antennas. The underlying channels are
assumed to be known at the SU transmitter. The nonlinear optimization (fractional pro-
gramming) problem is transformed into a parametrized convex optimization problem and
the corresponding solution is discussed. Computer simulation examples are provided to il-
lustrate the proposed approach and to explore the trade-off between energy efficiency and
spectrum efficiency.
11
Based on energy efficient precoding for MIMO assisted single secondary link in Chapter
6, we further study weighted energy efficiency maximization for multiple-input multiple-
output cognitive multiple access channels under both secondary user transmit power con-
straint and primary user interference power constraint in Chapter 7. Energy efficiency is de-
fined as the ratio of weighted sum rate and energy consumption including both transmission
and circuit energy consumption. The nonlinear (fractional programming) energy efficiency
optimization problem is transformed into a series of parametrized convex optimization prob-
lems. A combination of bisection search method and cyclic coordinate ascent-based iterative
water-filling algorithm is proposed to solve the parametrized problem. Computer simula-
tion examples are provided to illustrate the proposed approach and to explore the trade-off
between energy efficiency and spectrum efficiency.
Conclusions and related future work are presented in Chapter 8.
12
Chapter 2
Cyclostationarity Background and Corresponding Spectrum Sensing Tests
2.1 Introduction
Due to the signal structure, modulation, pilot or carriers, man-made signal exhibits
periodicity in its mean, correlation function or spectral descriptors. Such signal sequences
can be called (wide-sense) cyclostationary processes. A discrete time zero-mean complex
cyclostationary process x(t) is characterized by its time-varying correlation function as [59,
67]
Rxx(t,t+?) = E{x(t)x?(t+?)}. (2.1)
The correlation function Rxx(t,t + ?) can be further expressed in Fourier series (FS) form
[59, 68] as follows
Rxx(t,t+?) =
summationdisplay
??A
R(?;?)ej2??t (2.2)
where R(?;?) is called the cyclic autocorrelation function (CAF) at cycle frequency ? ? A
given by (M is the observation length)
R(?;?) = lim
M??
1
M
Msummationdisplay
t=1
Rxx(t,t+?)e?j2??t (2.3)
and A is the set of cycle frequencies [59]
Adefines{?,0 ? ? < 1 and R(?;?) negationslash? 0}. (2.4)
13
For some cyclostationary processes, CAF varnishes to zero but conjugate CAF R(?)(?;?) is
large enough to be detected
R(?)(?;?) = lim
M??
1
M
Msummationdisplay
t=1
Rxx?(t,t+?)e?j2??t (2.5)
where Rxx?(t,t+?) = E{x(t)x(t+?)}.
Later we will consider vector random processes arising due to a multiantenna receiver
with p ? 1 antennas. Therefore, we will consider a p?1 discrete time zero-mean complex-
valued cyclostationary signal x(t) with time-varying autocorrelation function Rxx(t,t+?) :=
E{x(t)xH(t+?)}. The corresponding Fourier series decomposition and nonconjugate CAFs
of Rxx(t,t + ?) can be obtained according to (2.2) and (2.3). Similar comments apply to
conjugate CAFs. In the following section, we illustrate the cyclostationarity exhibited in
Orthogonal Frequency Division Multiplexing (OFDM) and Gaussian Minimum Shift Keying
(GMSK) signals.
Notation: Bij denotes the ijth element of the matrix B, xi denotes the ith element
of vector x, and I is the identity matrix. The superscripts ? and H denote the complex
conjugate and the Hermitian (conjugate transpose) operations, respectively. ?i,j denotes the
Kronecker delta: = 1 if i = j, 0 otherwise. Given a column vector x, diag{x} denotes a
square matrix with elements of x along its main diagonal and zeros everywhere else. For
a real Gaussian variable x with mean ? and variance ?2, its distribution is described as
x ? N(?,?2). For circularly symmetric complex Gaussian vector x with mean vector ? and
the covariance matrix ?, the probability distribution is denoted as x ? CN(?,?).
2.2 Cyclostationary Characteristics of OFDM and GMSK Signals
2.2.1 OFDM Signal
An OFDM signal exhibits nonconjugate cyclostationarity [68], which is suitable for
cyclostationarity-based spectrum sensing. In an OFDM system, the Phase Shift Keying
14
(PSK) or Quadrature Amplitude Modulation (QAM) modulated information symbols are
transmitted over multiple orthogonal carriers in parallel. A typical baseband OFDM signal
can be expressed as
s(t) =
radicalbigg 1
Nc
?summationdisplay
n=??
Nc?1summationdisplay
i=0
dn,iej2?i(t?nTs??)/TugR(t?nTs ??)e?j2?Nc?12 t/Tu (2.6)
where dn,i is the nth information symbol modulated on the ith sub-carrier, Nc is the number
of sub-carriers, and ? is the unknown symbol timing. Ts is the OFDM symbol length and
Ts = Tu +Tg, where Tu is the useful OFDM symbol length and Tg is the cyclic prefix length.
gR(t) is the rectangular impulse function of length Ts.
The time varying nonconjugate autocorrelation of the OFDM signal can be expressed
as [68, 67]
Rss(t,t+?) = ?
2
d
Nc
sin(?Nc?/Tu)
sin(??/Tu)
?summationdisplay
n=??
gR(t?nTs ??)g?R(t?nTs ??+?) (2.7)
where ?2d = E{dn,id?n,i}. It is easy to show that Rss(t,t+?) is periodic in t with a period Ts.
As a result, the OFDM signal exhibits nonconjugate cyclostationarity with a cyclic frequency
? = k/Ts = kfs, where k = 0,?1,?2,.... The cyclic autocorrelation function of OFDM
signal can be calculated as [68, 67]
R(?;?) =
?
???
???
?2d
Nc
sin(?Nc?/Tu)sin[?kfs(Ts??)]
?ksin(??/Tu) e
?j2?kfs?, ? = k/Ts,|?| < Ts,
0, otherwise.
(2.8)
Consequently, although the OFDM signal does not exhibit conjugate cyclostationarity,
its nonconjugate cyclostationarity with a basic cycle frequency equal to the symbol rate of
the OFDM system ? = 1/Ts is large enough for detection.
Assume OFDM signal is sampled at the Nyquist rate such that the sampling interval is
Ts
Nc+Ncp. For each OFDM symbol, there are Ncp samples for the cyclic prefix and Nc samples
15
for useful OFDM symbol. Assume only one cyclic frequency 1Ts and one time lag Tu is
considered. Due to Nyquist sampling, the corresponding cyclic frequency and time lag in
discrete time domain is ? = 1Nc+Ncp and ? = Nc respectively. Let ?s = ?d, of which ?2s is the
power of the OFDM signal. The theoretical cyclic autocorrelation function with parameters
?,? and ?s is calculated as [67]
R(?;?) = ??
2
s
? sin
parenleftbigg
? NcpN
c +Ncp
parenrightbigg
. (2.9)
Based on (2.8), we set Nc = 8, Ts/Tu = 5/4 and plot a 3D theoretical CAF figure in Fig. 2.1.
Figure 2.1: Theoretical CAF function with 8 subcarriers and normalized power
2.2.2 GMSK Signal
GMSK modulation is used in Global System for Mobile Communications (GSM) system.
It exhibits conjugate cyclostationarity [68]. GMSK can also be treated as a 2-level FSK
modulation with a modulation index h = 0.5. The complex envelope of a GMSK signal is
16
[67]
s(t) = exp
bracketleftBig
j2?h
?summationdisplay
n=??
dn
integraldisplay t
??
g(? ?nTs)d?
bracketrightBig
(2.10)
with the symbol sequence dn ? {?1,1}, symbol rate fs = 1/Ts and frequency impulse g(t)
given as
g(t) = 1T
s
rect( tT
s
)?pGauss(t) (2.11)
where rect(t) is rectangular pulse and pGauss(t) is a Gaussian impulse with the time band-
width product BTs (B is the bandwidth), which is equal to 0.3 for the GSM system. In
practice, the Gaussian impulse is cut to a length LTs with L ? 3. With L = 4, a GMSK
signal can be approximated [52] as
s(t) ?
?summationdisplay
n=??
znc0(t?nTs) (2.12)
with elementary impulse form c0(t) and the modulating symbol sequence zn [53] such that
zn = exp
bracketleftBig
j?h
nsummationdisplay
i=??
di
bracketrightBig
= exp[j?2(dn +
n?1summationdisplay
i=??
di)]
= jdnzn?1. (2.13)
Using this approximation, the conjugate time varying autocorrelation function of the signal
can be calculated as [68, 67]
Rss?(t,t+?) ?
+?summationdisplay
n=??
z2??(?1)nc0(t???nTs)?c0(t???nTs +?) (2.14)
where ? is the unknown symbol timing and z2?? ? {?1,1} is a constant determined by the
initial state of zn. It is easy to show that Rss?(t,t+?) is periodic with a period 2Ts. Hence,
GMSK exhibits conjugate cyclostationarity with cycle frequencies ak = k/(2Ts) = kfs/2 for
17
integer k. Then we obtain the conjugate cyclic autocorrelation function (CAF) as
Rss(?)(?,?) =
?
???
???
z2??
Ts
integraltext?
?? C0(?)C0(
k
2Ts ??)e
?j2?? k2Ts ej2??( k2Ts??)d?, ? = kf
s/2,k odd,
0, otherwise
(2.15)
where C0(?) is the Fourier transform of c0(t). Similar deduction can be applied to noncon-
jugate CAF functions with a basic cycle frequency ?f = 1/Ts = fs. Then we have [67]
R( kT
s
,?) = 1T
s
integraldisplay ?
??
e?j2?(?? kTs )?e?j2?? kTs Co(?)C?0(?? kT
s
)d? (2.16)
where k = 0,?1,?2,...
Since the function C0(?) of GMSK has narrow spectrum in the GSM system, the non-
conjugate CAF functions are totally suppressed for all cycle frequencies. On the other hand,
conjugate CAF function with cycle frequencies ? 12Ts is not suppressed, which can be used
for identification of GMSK signals.
2.3 Introduction to Statistical Hypothesis Testing
In this section, we briefly review binary hypothesis tests under the Neyman-Pearson
criterion [54]. The goal of detection is to identify a PU signal s(t) embedded in white
Gaussian noise n(t). For simplicity, we assume s(t) = 0 or 1 to represent absence or presence
ofPUsignals. Assumen(t) ? N(0,?2)andletx(t)denotesthereceivedsignal. Foranalytical
simplicity, the detection is based on only one sample of x(t), represented as x[0]. Then we
have to decide whether s(t) is present with noise n(t) or only n(t) is present, which can be
formulated as
H0 : x[0] = n[0],
H1 : x[0] = s[0]+n[0] (2.17)
18
where H0 represents the null hypothesis, H1 is the alternative hypothesis, s[0] = 1 and
n[0] ? N(0,?2). Then, the distribution of received signal sample x[0] would be
p(x|H0) = 1?2??2exp(? 12?2x[0]2),
p(x|H1) = 1?2??2exp(? 12?2(x[0]?1)2). (2.18)
Based on Neyman-Pearson (NP) detection rule, to maximize detection probability for
a given false alarm probability ?, we design the NP test to decide H1 if
L(x) = p(x|H1)p(x|H
0)
=
1?
2??2exp(?
1
2?2(x[0]?1)
2)
1?
2??2exp(?
1
2?2x[0]
2) > ? (2.19)
where the test threshold ? is given by
Pfa =
integraldisplay
x:L(x)>?
p(x|H0)dx = ? (2.20)
and L(x) is termed as the likelihood ratio function. As a result, the detection probability of
NP test is defined as
Pd =
integraldisplay
x:L(x)>?
p(x|H1)dx. (2.21)
2.4 Existing Cyclostationarity-based Spectrum Sensing Algorithms
Since white Gaussian noise is wide-sense stationary (without cyclostationarity), the
primary signal with cyclostationarity can be detected based on its nontrivial CAFs at some
cycle frequencies and time lag. In the following subsections, existing cyclostationarity based
spectrum sensing algorithms are reviewed.
19
Dandawate-Giannakis test [59, 67, 64]
First we introduce the test in [59]. Define
?c :=
?
?? Re?R11,x(?;?)
Im?R11,x(?;?)
?
??, F(2?f) := Msummationdisplay
t=1
x(t)x?(t+?)e?j2?ft, (2.22)
?S := 1
MLw
s=(Lw?1)/2summationdisplay
s=?(Lw?1)/2
W(s)F(?? sM)F(?? sM), (2.23)
?S? := 1
MLw
s=(Lw?1)/2summationdisplay
s=?(Lw?1)/2
W(s)F?(?? sM)F(? + sM), (2.24)
?? :=
?
?? Re
braceleftBig?
S+?S?
2
bracerightBig
Im
braceleftBig?
S??S?
2
bracerightBig
Im
braceleftBig?
S+?S?
2
bracerightBig
Re
braceleftBig?
S???S
2
bracerightBig
?
?? (2.25)
where W(s) is a spectral smoothing window of odd length Lw. The test of [59] is given by
T2c := M?cT ???1?c
H1
greaterlessequal
H0
? where T2c H0? ?22. (2.26)
A multi-antenna version of this test for conjugate CAFs may be found in [73]. A Kaiser
window function has been used for W(s) in [59, 67, 68, 64, 73]. It is given by
W(s) =
?
??
??
I0
parenleftBig
?
radicalBig
1?( sLw/2)2
parenrightBig
, ?Lw?12 ? s ? Lw+12
0, elsewhere
where I0 is zero-order modified Bessel function of the first kind and ? and Lw are parameters
to be selected.
It should be pointed out that the tests of [59, 67, 68, 64, 73] are not generalized likelihood
ratio tests (GLRTs), contrary to the claims made therein. Consider [59] and let c denote
the true value of ?c under H1 (it is zero under H0) and let ? denote the true value of ??. As
20
discussed in [64, Sec. II.B], the likelihood ratio (LR) for this problem (?c is the ?observation?
and f(?) denotes probability density function) is given by
? = f(?c|H1)f(?c|H
0)
= exp
parenleftbig?1
2M(?c?c)
T??1(?c?c)parenrightbig
expparenleftbig?12M?cT??1?cparenrightbig . (2.27)
The unknown under H0 is ? and the unknowns under H1 are c and ?. In GLRT, one replaces
the unknowns with their maximum likelihood (ML) estimates. Under H1 the ML estimate
of c is given by ?c [55, Sec. 3.2], making the numerator in (2.27) a constant (one). Under H0
a positive-definite estimate of ? does not exist; by [55, Sec. 3.2] the ML estimate of ? would
have been ?c?cT if we had rank(?c?cT)=2, but it is of rank one. In [64, Sec. II.B], ?? is used as
the ML estimate of ? which is not the case. It is more appropriate to view T2c in (2.26) as
an ad hoc test statistic resulting in a CFAR test; same remark applies to [67, 68, 64, 73] (all
based on [59]).
Jallon Test [63]
This test applies to OFDM signals only. We will state the relevant results of [63] in
the notation of our dissertation. Given an OFDM signal as discussed in Sec. 2.2.1, the test
statistic of [63, Sec. 3] is given by
?Jx(Nb) := 1
2Nb +1
k=Nbsummationdisplay
k=?Nb
vextendsinglevextendsingle?R
11,x(k?0;Nc)
vextendsinglevextendsingle2 H1greaterlessequal
H0
?J1, ?0 = 1N
c +Ncp
= 1N, (2.28)
with Nb picked to be the largest integer < NcNcp. In [63] one is testing the estimated CAF at a
single lag ? = Nc but at 2Nb+1 cycle frequencies in the set { kN, k = ?Nb,?Nb+1,??? ,Nb}
= { kN, k = N ? Nb,N ? Nb + 1,??? ,N ? 1,0,1,??? ,Nb}. Under H0, [63, Cor. 1] states
that asymptotically M?4
1
(2Nb + 1) ?Jx(Nb) ? ?2(2(2Nb + 1)) where ?21 = E{|x1(t)|2|H0}. This
is incorrect in that a factor of two is missing; correct asymptotic distribution under H0 is
given by 2M?4
1
(2Nb + 1) ?Jx(Nb) ? ?2(2(2Nb + 1)), the reason being that a complex zero-mean
Gaussian random variable Y with variance ?2 consists of independent real and imaginary
21
parts, each with zero-mean and variance ?2/2 whereas [63] seems to use variance ?2 for real
and imaginary parts of Y. Notice that in order to calculate the test threshold in (2.28), one
needs the knowledge of the noise variance ?21. Lack of this knowledge is not addressed in
[63], but is discussed in [62, Sec. 3] (in the context of spread signals) where one estimates ?21
and use it to normalize the test statistic as
?J2x(Nb) := 2M?
R211,x(0)(2Nb +1)
?Jx(Nb) H1greaterlessequal
H0
?J2, where ?J2x(Nb) H0? ?2(2(2Nb +1)). (2.29)
Note that the factor of two error in [63] regarding the test statistic distribution under H0 is
also present in [62]. If one considers only a single cycle frequency at ?0 in (2.29) (and set
Nb = 0), then (2.29) reduces to our proposed detector in Chapter. 3.
Tani-Fantacci Test [71]
This test applies to OFDM signals only. We will state the relevant results of [71] in
the notation of our dissertation. Given an OFDM signal as discussed in Sec. 2.2.1, the test
statistic of [71, Sec. IV.A] is given by
T? :=
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
?R?11(Nc)
?R?+S11 (Nc)
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsingle
H1
greaterlessequal
H0
?T, ? = 1N
c +Ncp
= 1N, (2.30)
where
?R?11(?) := 1
M
Msummationdisplay
n=1
parenleftBigg
1
L
n+L?1summationdisplay
k=n
x(k)x?(k +?)
parenrightBigg
e?j2??n (2.31)
and L is picked to be shorter than the OFDM symbol duration. One picks the (fundamental)
cycle frequency ? = ?0 = 1N (for sampled OFDM signal) and S is an arbitrary integer value
that does not belong to the set of cycle frequencies { kTs, k = 1,2,???} for continuous-time
OFDM signal. It is established in [71] that under H0, asymptotically T? follows a Cauchy
distribution.
22
2.5 Conclusions
In this chapter, the background of cyclostationarity and corresponding spectrum sens-
ing schemes based on it are introduced. We also briefly introduce the concept of bina-
ry hypothesis test. In following Chapter 3, the details of our proposed low complexity
cyclostationarity-based spectrum sensing with both single and multiple receiving antennas
are presented. Those existing cyclostationary spectrum sensing cited in this chapter are
treated as counterparts of our proposed algorithm in the simulation chapter under different
scenarios.
23
Chapter 3
On Cyclic Autocorrelation Based Spectrum Sensing for Cognitive Radio Systems in
Gaussian Noise
Detection of cyclostationary primary user (PU) signals in white Gaussian noise for cog-
nitive radio systems is considered based on looking for a cycle frequency at a particular time
lag in the cyclic autocorrelation function (CAF) of the noisy PU signal. We explicitly exploit
the knowledge that under the null hypothesis of PU signal absent, the measurements orig-
inate from white Gaussian noise with possibly unknown variance. Our formulation allows
us to computationally simplify the spectrum sensing detector, obviating the need for esti-
mating an unwieldy covariance matrix needed in prior works. We consider both single and
multiple antenna receivers. A performance analysis of the proposed detector is carried out.
Supporting simulation examples are provided using both OFDM and GMSK PU signals and
they verify our performance analysis and also show that our approaches either outperform
or are at least as good as existing approaches while being computationally much cheaper.
3.1 Introduction
Cognitive radio allows for usage of licensed frequency bands (rights held by primary
users (PUs)) by unlicensed users (secondary or cognitive users) when the licensed spectrum
bands are unoccupied (a function of time and location). Therefore, one of the first steps to be
accomplished by a cognitive user is spectrum sensing: analysis of the received electromagnetic
transmissions to search for unoccupied spectrum bands (spectrum holes). One of the ways to
accomplish this is energy detection/spectral analysis and consequent (statistical) hypothesis
testing [56, 57, 60, 65]. Based on the received signal x(t), the cognitive user has to decide
if the primary user (PU) is present or not. This may be formulated as a binary hypothesis
24
testing problem
x(t) =
?
??
??
n(t) : H0
s(t)+n(t) : H1
(3.1)
where H0 is the null hypothesis that cognitive user is receiving just noise n(t), and H1 is
the alternative that PU signal s(t) is also present. A popular approach is that of energy
detection; see [56, 57, 60, 65] and references therein. One designs a CFAR (constant false
alarm rate) test requiring prior knowledge of some of the signal statistics under H0. For
instance, n(t) is taken to be thermal noise with known variance.
Energy detection based approaches can not distinguish among different types of pri-
mary user or secondary user signals. Cyclostationarity detection allows classifying signals
exhibiting cyclostationarity at different cycle frequencies [64] and/or different time lags. As
noted in [67], one can use cyclostationarity detection for recognizing the individual air in-
terfaces, a task at which energy detector would fail. Existing cyclostationarity detectors for
cognitive radio systems reported in [67, 64, 73] are all based on [59] where one looks for a
cycle frequency at a particular time lag in the cyclic autocorrelation function (CAF) of the
(noisy) PU signal; extensions to multiple cycle frequencies and/or multiple lags may also be
found in some of these papers. Under the alternative hypothesis, CAF is nonzero whereas
under the null hypothesis it is zero. Signal characteristics under the null hypothesis are not
exploited except for noting that its CAF would be zero. Implementation of the approaches
of [59, 67, 64, 73] requires computation of the covariance matrix of estimated CAF?s real and
imaginary parts. [In general, this covariance matrix does not have an analytical expression
even when the PU signal is precisely known.] In practice, it is estimated from the data, is
computationally expensive and it needs a frequency smoothing window [59] to smooth cer-
tain cyclic periodograms in the frequency domain. In this chapter we explicitly exploit the
knowledge that under the null hypothesis of PU signal absent, the measurements originate
from white Gaussian noise with possibly unknown variance. Such a framework is explicitly
25
stated in [73, Eqns. (1),(2)], and although not explicitly stated in [67, 64], is extensively
employed in simulation examples presented therein.
Other existing approaches based on single antenna receivers include [62], [63], [66] and
[71]. In [62] detection of spread signals using the signal nonconjugate CAF at multiple
lags and multiple cycle frequencies is considered under white Gaussian noise with possibly
unknown variance. In [63] detection of orthogonal frequency division multiplex (OFDM)
signals using the signal nonconjugate CAF at a single time lag and multiple cycle frequencies
is considered under white Gaussian noise with known variance. In [66] several single antenna
cyclostationarity detectors under white Gaussian noise have been considered including a peak
detector [66, (12)] which utilizes the noisy signal CAF at multiple cycle frequencies and a
single time-lag. This detector of [66, (12)] is the same as that of [63]; both assume known
noise variance and both can be modified to handle unknown noise variance following [62].
(It should be noted that sensitivity of the detector in [66, (12)] to uncertainty in noise
variance was recognized in [66] but was not fixed as in [62], rather a different detector was
proposed.) Our proposed detector when specialized to single antenna is similar to that in
[62], [63] and [66, (12)], except that we use a single time lag and a single cycle frequency. As
noted in [64] (and others), cyclostationarity detection allows discrimination among signals
exhibiting cyclostationarity at different cycle frequencies and/or time lags. Use of multiple
cycle frequencies or time lags can reduce this discrimination capability where an interfering
cyclostationary signal may be detected as a PU cyclostationary signal. In our simulation
examples we will discuss such cases where the detector of [63] detects a certain interfering
signal as a PU signal whereas our proposed detector does not. Further discussion of [63] is
postponed to Sec. 2.4. In [71] a single cycle frequency and single time lag cyclostationarity
detector has been presentedfor OFDM signals under noisevariance uncertainty. It is different
from that in [62], [63], [66] and from our proposed detector; further details regarding [71] are
in Sec. 2.4.
26
The proposed detectors are useful in several practical scenarios such as detection of
OFDM signals in digital video broadcasting (DVB) standard DVB-T (under IEEE 802.22
Working Group proposals), as discussed in [63]. As noted in [64] OFDM is employed by many
of the current as well as future wireless communications systems including 3GPP Long Term
Evolution (LTE), IEEE 802.11a/g WLANs, DVB standards DVB-T and DVB-H, as well as
IEEE 802.16 and WiMax wireless metropolitan area networks. Finally, a GSM network as
the primary system and coexisting with an OFDM based secondary WLAN system has been
considered in [68]. Our conjugate CAF based detectors for detecting GMSK signals in the
presence of interfering OFDM signals apply to the application considered in [68].
Our formulation allows us to simplify the spectrum sensing detector, both conceptually
and computationally, and obviates the need for estimating the unwieldy covariance matrix
needed in [59, 67, 64, 73]. We consider both single and multiple antenna receivers. Under
single antenna receivers, our detector is similar to that in [62], [63] and [66, (12)], except
that we use a single time lag and a single cycle frequency which allows discrimination among
signals exhibiting cyclostationarity at different cycle frequencies and/or time lags, unlike
receivers of [62], [63] and [66, (12)] that utilize multiple cycle frequencies and/or multiple
time lags. Also unlike[62], [63], [66, (12)] and [71], we investigate multiple antenna receivers
where we exploit cyclic cross-correlations among different antennas. Exploitation of cyclic
cross-correlations among different antennas has also been considered in [73] for GMSK signals
where [73] extends the single-antenna approach of [59] inheriting its computational complex-
ity. A performance analysis of the proposed detectors is presented in Sec. V. Computational
complexity issues are discussed in Sec. VI. Supporting simulation examples are provided us-
ing OFDM or GMSK PU signals are presented in Sec. VII where we compare our detectors
with those of [59], [73], [63] and [71].
Notation: Bij denotes the ijth element of the matrix B, xi denotes the ith element
of vector x, and I is the identity matrix. The superscripts ? and H denote the complex
conjugate and the Hermitian (conjugate transpose) operations, respectively. ?i,j denotes
27
the Kronecker delta: = 1 if i = j, 0 otherwise. Given a column vector x, diag{x} de-
notes a square matrix with elements of x along its main diagonal and zeros everywhere
else. We use cov(x,y?) := E{xyH}? E{x}E{yH} and let cum4(x1,x2,x3,x4) denote the
joint 4th cumulant of random variables xi (i = 1,2,3,4). Note that cum4(x1,x2,x3,x4) =
E{x1x2x3x4}? E{x1x2}E{x3x4}? E{x1x3}E{x2x4}? E{x1x4}E{x2x3} when all xi?s are
zero-mean.
3.2 Cyclostationarity Background
A discrete-time zero-mean complex-valued cyclostationary signal x(t) is characterized
by a time-varying autocorrelation function Rxx(t,t + ?) := E{x(t)x?(t + ?)} which has a
Fourier series representation [59, 67]
Rxx(t,t+?) =
summationdisplay
??A
Rxx(?;?)ej2??t (3.2)
where given x(t) for t = 0,1,??? ,M?1, the cyclic autocorrelation function (CAF) Rxx(?;?)
at cycle frequency ? ? A is given by
Rxx(?;?) = lim
M??
1
M
M?1summationdisplay
t=0
Rxx(t,t+?)e?j2??t (3.3)
and A is the set of cycle frequencies [59]
A := {?|0 ? ? < 1, Rxx(?;?) negationslash? 0}. (3.4)
A conjugate CAF is defined as
Rxx(?)(?;?) = lim
M??
1
M
M?1summationdisplay
t=0
Rxx?(t,t+?)e?j2??t (3.5)
28
where Rxx?(t,t+?) := E{x(t)x(t+?)}. Ref. [59] considers conjugate CAFs whereas [67, 64]
consider both types of CAFs. For (wide-sense) stationary (WSS) signals, Rxx(?;?) = 0 as
well as Rxx(?)(?;?) = 0 for any ? negationslash= 0.
3.2.1 Multivariate Processes
Later we will consider vector random processes arising due to a multiantenna receiver
with p ? 1 antennas. Therefore, we will consider a p?1 discrete-time zero-mean complex-
valued cyclostationary signal x(t) with time-varying autocorrelation function Rxx(t,t+?) :=
E{x(t)xH(t+?)} which has a Fourier series representation [59, 67]
Rxx(t,t+?) =
summationdisplay
??A
Rxx(?;?)ej2??t (3.6)
where the nonconjugate cyclic autocorrelation function (CAF) Rxx(?;?) at cycle frequency
? ? A is given by
Rxx(?;?) = lim
M??
1
M
Msummationdisplay
t=1
Rxx(t,t+?)e?j2??t. (3.7)
Similar comments apply to conjugate CAFs.
3.3 Large Sample Statistics of Sample CAF
Our focus will be on the binary hypothesis testing problem involving p-dimensional
processes (p ? 1)
x(t) =
?
??
??
n(t) : H0
s(t)+n(t) : H1
(3.8)
where n(t) is zero-mean white Gaussian (as in [73]) and s(t) = summationtextLcl=0 h(l)spu(t ? l) where
spu(t) is a scalar cyclostationary signal (emitted by a primary user) with a known cycle
frequency ? and known lag ? such that Rspuspu(?;?) negationslash= 0, and p-column h(l) is the complex
channel impulse response. Our approach (similar to [59, 67, 64, 73]) is to test the estimated
CAF of the observations at a pair (?,?) to check whether it is nonzero. However, unlike [59,
29
67, 64, 73]), we explicitly exploit the known statistics of observations under H0 (namely, we
have white Gaussian noise with possibly unknown variances). In this section we investigate
large sample statistics of estimated CAF when we have Gaussian noise as observations. We
are interested only in nonzero cycle frequencies.
3.3.1 Nonconjugate CAF
These results are appropriate for OFDM signals.
Spatially and Temporally White Gaussian Noise
Consider a p? 1 WSS proper (circularly symmetric) complex-valued zero-mean white
Gaussiansequencex(t) = n(t)withcovariance?n = E{n(t)nH(t)} = diagbraceleftbig?21, ?22, ??? ,?2pbracerightbig.
Since x(t) is WSS, Rxx(t,t+?) = Rxx(?) = ?n??,0. Also, since x(t) is proper, E{x(t)xT(t+
?)} ? 0. Given an observation length of M samples, we estimate the (nonconjugate) CAF
Rxx(?;?) as
?Rxx(?;?) = 1
M
Msummationdisplay
t=1
x(t)xH(t+?)e?j2??t. (3.9)
It is easy to see that limM?? E{?Rxx(?;?)} = Rxx(?;?). Let (xi is the i-th component of
x)
?Rik,x(?;?) := 1
M
Msummationdisplay
t=1
xi(t)x?k(t+?)e?j2??t. (3.10)
Then
E{?Rik,x(?;?)} = Rik,x(?)
bracketleftBigg
1
M
Msummationdisplay
t=1
e?j2??t
bracketrightBigg
(3.11)
where Rik,x(?) := E{xi(t)x?k(t+?)}. We have
E{xi(t1)x?j(t1 +?1)e?j2??t1x?k(t2)xl(t2 +?2)ej2??t2}
= cum4parenleftbigxi(t1),x?j(t1 +?1),x?k(t2),xl(t2 +?2)parenrightbige?j2?(?t1??t2)
30
+Rij,x(?1)R?kl,x(?2)e?j2?(?t1??t2) +Rik,x(t2 ?t1)Rlj,x(t1 ?t2 ??2 +?1)e?j2?(?t1??t2).
(3.12)
For Gaussian sequences, the 4th cumulants are identically zero. Hence,
cov
parenleftBig?
Rij,x(?;?1), ?R?kl,x(?;?2)
parenrightBig
= E
braceleftBig?
Rij,x(?;?1)?R?kl,x(?;?2)
bracerightBig
?E
braceleftBig?
Rij,x(?;?1)}E{?R?kl,x(?;?2)
bracerightBig
= 1M2
Msummationdisplay
t1=1
Msummationdisplay
t2=1
bracketleftbigR
ik,x(t2 ?t1)Rlj,x(t1 ?t2 ??2 +?1)e?j2?(?t1??t2)
bracketrightbig=: A. (3.13)
Using the fact that x(t) is white Gaussian, it follows that
A = 1M2
Msummationdisplay
t=1
Rik,x(0)Rlj,x(?1 ??2)e?j2?(???)t = 1M2Rik,x(0)Rlj(?1 ??2)
bracketleftBigg Msummationdisplay
t=1
e?j2?(???)t
bracketrightBigg
.
(3.14)
We have
1
M
Msummationdisplay
t=1
e?j2?(???)t =
??
?
??
1 for ? = ? ? A
e?jpi(???)(M+1)
M
sin(?(???)M)
sin(?(???)) for ? negationslash= ?, ?,? ? A.
(3.15)
It then follows that
lim
M??
MA = Rik,x(0)Rlj,x(?1 ??2)??,?. (3.16)
But under H0, x(t) is spatially and temporally white, leading to
lim
M??
Mcov
parenleftBig?
Rij,x(?;?1), ?R?kl,x(?;?2)
parenrightBig
= Rii,n(0)Rjj,n(0)?i,k?j,l??1,?2??,? (3.17)
where Rii,n(0) = ?2i , i = 1,2,??? ,p.
Using (3.10) it follows that
?R?ik,x(?;?) = ?Rki,x(??;??) = ?Rki,x(1??;??). (3.18)
31
Therefore, from (3.17) and (3.18), we have
lim
M??
Mcov
parenleftBig?
Rij,x(?;?1), ?Rkl,x(?;?2)
parenrightBig
= lim
M??
Mcov
parenleftBig?
Rij,x(?;?1), ?R?lk,x(??;??2)
parenrightBig
= Rii,n(0)Rjj,n(0)?i,l?j,k??1,??2??,??. (3.19)
Also, by [59, 67, 64], asymptotically (as M ? ?), ?Rik,x(?;?) is a Gaussian random variable
(complex-valued but not necessarily circularly symmetric) and ?vectorized? matrix ?Rxx(?;?)
is a Gaussian random vector for any ? and ?.
White Gaussian Noise? Single Cycle Frequency and Single Lag
We will test the observations for the presence of cyclostationarity at a single cyclic
frequency ? and a single lag ?; extensions to multi-cycles and multi-lags are straightforward
but tedious, hence are not considered in this chapter. Note that for white Gaussian noise,
Rxx(?;?) = 0 for any ? negationslash= 0 and any ?. Setting ? = ? negationslash= 0 and ?1 = ?2 = ? ? 0 in (3.17)
and (3.19), we have
lim
M??
Mcov
parenleftBig?
Rij,x(?;?), ?R?kl,x(?;?)
parenrightBig
= Rii,n(0)Rjj,n(0)?i,k?j,l, (3.20)
lim
M??
Mcov
parenleftBig?
Rij,x(?;?), ?Rkl,x(?;?)
parenrightBig
= 0. (3.21)
Invoking asymptotic Gaussianity, it then follows that
lim
M??
?M ?R
ij,x(?;?) ? Nc (0,Rii,n(0)Rjj,n(0)) ?i,j (3.22)
where Nc(m,?) denotes a circularly symmetric (proper) complex Gaussian (vector) distri-
bution with mean m and covariance matrix ?. Furthermore, ?Rij,x(?;?) is asymptotically
independent of ?Rkl,x(?;?) if i negationslash= k or j negationslash= l.
32
3.3.2 Conjugate CAF
These results are appropriate for GMSK signals. Given an observation length of M
samples, we estimate the conjugate CAF Rxx(?)(?;?) := limM?? 1M summationtextMt=1 E{x(t)xT(t +
?)}e?j2??t as
?Rxx(?)(?;?) = 1
M
Msummationdisplay
t=1
x(t)xT(t+?)e?j2??t. (3.23)
Spatially and Temporally White Gaussian Noise? Single Cycle Frequency, Single
Lag
When n(t) is proper and x(t) = n(t), it follows easily that E{?Rxx(?)(?;?)} = 0 and
cov
parenleftBig?
R(?)ij,x(?;?), ?R(?)kl,x(?;?)
parenrightBig
= E{?R(?)ij,x(?;?)?R(?)kl,x(?;?)} = 0 (3.24)
where R(?)ik,x(?) := E{xi(t)xk(t+?)}. Mimicking Sec. 3.3.1 we have
cov
parenleftBig?
R(?)ij,x(?;?), ?R?(?)kl,x(?;?)
parenrightBig
= E{?R(?)ij,x(?;?)?R?(?)kl,x(?;?)}
= 1M2
Msummationdisplay
t1=1
Msummationdisplay
t2=1
[Rik,n(t2 ?t1)Rjl,n(t2 ?t1)+Ril,n(t2 ?t1 +?)Rjk,n(t2 ?t1 ??)]e?j2??(t1?t2)
= 1M2
Msummationdisplay
t=1
[Rik,n(0)Rjl,n(0)+Ril,n(0)Rjk,n(0)??,0]. (3.25)
Hence we have
lim
M??
Mcov
parenleftBig?
R(?)ij,x(?;?), ?R?(?)kl,x(?;?)
parenrightBig
=
??
????
???
??
2Rii,n(0)Rjj,n(0) if i = k = l = j and ? = 0
Rii,n(0)Rjj,n(0) if (i,j) = (k,l) but i negationslash= j, or (i,j) = (l,k) and ? = 0
0 otherwise .
(3.26)
33
Invoking asymptotic Gaussianity, it then follows that under H0
lim
M??
?M ?R
(?)ij,x(?;?) ? Nc (0,(1 +??,0?i,j)Rii,n(0)Rjj,n(0)) ?i,j. (3.27)
NotingthatsincelimM?? ?R(?)ij,x(?;?) = limM?? ?R(?)ji,x(?;?)and ?R(?)ij,x(?;0) = ?R(?)ji,x(?;0),
unlike nonconjugate CAFs, in case of conjugate CAFs, ?R(?)ij,x(?;?) is asymptotically inde-
pendent of ?R(?)kl,x(?;?) for i negationslash= k or j negationslash= l if i ? j and k ? l (or if i ? j and k ? l).
3.4 Test Statistics
Based on the large sample statistics of CAFs of white Gaussian noise discussed in Sec.
3.3, we now propose ad hoc CFAR (constant false alarm rate) detectors for detection of
nonzero nonconjugate as well as conjugate CAFs. They are ad hoc as they do not follow any
optimality criterion.
3.4.1 Nonconjugate CAF: OFDM Signals
Consider the binary hypothesis testing problem (3.8). The PU channel impulse response
and the noise variances are unknown. We do know that noise is zero-mean complex (proper)
Gaussian, spatially uncorrelated and temporally white. Also, spu(t) in (3.8) is a scalar
cyclostationary signal (emitted by PUs) with a known cycle frequency ? and known lag
? such that Rspuspu(?;?) negationslash= 0. The binary hypothesis testing problem in this case can be
formulated as
H0 : Rij,x(?;?) = 0 ?i,j = 1,2,??? ,p
H1 : Rij,x(?;?) negationslash? 0 for some i,j, where i,j = 1,2,??? ,p.
(3.28)
Under both hypotheses, we have limM?? E{?R(?;?)} = R(?;?). Under H0, by the results
of Sec. 3.3.1, ?Rij(?;?) has zero-mean ?i,j, and under H1, it has non-zero mean for some i,j;
therefore, |Rij(?;?)| > 0 for some i,j. Let ?r denote the p2 ?1 vector composed of elements
34
?Rij,x(?;?)/radicalbigRii,n(0)Rjj,n(0) for all distinct (i,j). Then by (3.22), we have limM???M?r H0?
Nc (0,I) leading to 2M?rH?r H0? ?22p2 where ?2n denotes the central chi-square distribution with
n degrees of freedom (dof). Since |Rij(?;?)| > 0 for some i,j, we have E{?rH?r|H1} >
E{?rH?r|H0}. Furthermore, since the true values Rii,n(0), (i = 1,2,??? ,p), are not available,
we replace them by ?Rii,x(0) and consider the following ad hoc test motivated by 2M?rH?r and
E{?rH?r|H1} > E{?rH?r|H0}:
T := 2M
psummationdisplay
i=1
psummationdisplay
j=1
|?Rij,x(?;?)|2
?Rii,x(0)?Rjj,x(0)
H1
greaterlessequal
H0
? (3.29)
where
?Rii,x(0) := 1
M
Msummationdisplay
t=1
|xi(t)|2 (3.30)
and the threshold ? is picked to achieve a pre-specified probability of false alarm Pfa =
P{T ? ?|H0}. By the results of Sec. 3.3.1, we can find (shown next) asymptotic distribution
ofT underH0 and design a CFAR test. Such CFAR tests are not optimal in any sense but do
yield a constant false-alarm rate, a desirable property, under model parameter uncertainties.
CFAR tests have been used in radar and other problems under parameter uncertainty [61,
Sec. 8.1].
We now turn to calculation of ?. It follows from (3.22) that asymptotically (as M ? ?),
2M |
?Rij,x(?;?)|2
Rii,x(0)Rjj,x(0)
H0? ?2
2. (3.31)
Eqn. (3.20) applied to white Gaussian noise also implies that as M ? ?, ?Rii,x(0) converges
in the mean-square sense, hence in probability (i.p.), to Rii(0) [70, Appendix B]. Together
with (3.31) this results implies that asymptotically [70, Lemma B.4]
2M |
?Rij,x(?;?)|2
?Rii,x(0)?Rjj,x(0)
H0? ?2
2. (3.32)
35
As noted in Sec. 3.3.1, any element of ?Rxx(?;?) is asymptotically independent of all other
elements, therefore, in (3.29) we have mutually independent random variables for distinct ij
pairs. Since there are p2 such terms in T , asymptotically
T H0? ?22p2. (3.33)
3.4.2 Conjugate CAF: GMSK Signals
We will take ? = 0 since that is where the GMSK CAF is the strongest [68]. Following
Secs. 3.3.2 and 3.4.1, we propose the test statistic
Tconj := 2M
psummationdisplay
i=1
psummationdisplay
j=i
|?R(?)ij,x(?;0)|2
(1+?i,j)?Rii,x(0)?Rjj,x(0)
H1
greaterlessequal
H0
?conj. (3.34)
Using the results of Sec. 3.3.2 and noting that there are (p2 + p)/2 terms in the double
summation in (3.34), it follows that
Tconj H0? ?2p2+p (3.35)
which allows us to calculate the test threshold ?conj corresponding to a specified Pfa.
3.4.3 Single Antenna Case: Nonconjugate CAF
Proposed Test
For p = 1 (single antenna), the proposed nonconjugate CAF test reduces to
2M|
?R11,x(?;?)|2
?R211,x(0)
H1
greaterlessequal
H0
?. (3.36)
36
3.5 Detection Probability
In this section we derive the asymptotic distribution of the proposed test statistics of Sec.
3.4 under H1 for a specified PU signal. We will assume a non-random channel; for random
channels, our results offer conditional detection probability conditioned on the channel.
3.5.1 Nonconjugate CAF
Under H1, we have Rij,x(?;?) negationslash= 0 for at least one pair (i,j). Let ?r denote the p2 ?1
vector composed of elements ?Rij,x(?;?) := ?Rij,x(?;?)/
radicalBig
?Rii,x(0)?Rjj,x(0) for all distinct (i,j),
with m1 := E{?r} composed of elements Rij,x(?;?)/
radicalBig
R(1)ii,x(0)R(1)jj,x(0) (for large M) where
R(1)ii,x(0) := E{xi(t) x?i(t)|H1}. Let ?1 := limM??(1/M)cov(?r,?r?). Then invoking the
asymptotic Gaussianity [59, 67, 64], under H1 and for low SNRs,
lim
M??
?M (?r?m
1) ? Nc (0,?1). (3.37)
[Note that limM?? Mcov
parenleftBig?
Rij,x(?;?), ?Rij,x(?;?)
parenrightBig
negationslash= 0 when the PU signal is present, hence
?Rij,x(?;?) is not proper, but is approximately proper for ?low? SNRs.] Then for large M,
2M?rH??11 ?r H1? ?22p2(?1), (3.38)
where ?2n(?1) denotes noncentral chi-square distribution with n degrees of freedom and non-
centrality parameter ?1, and ?1 = 2MmH1 ??11 m1. Under low SNR, ?1 ? I?2M?rH??11 ?r ?
T and ?1 ? 2MmH1 m1. Thus,
T H1? ?22p2(?1) (3.39)
where (assuming s(t) =summationtextLcl=0 h(l)spu(t?l))
?1 = 2M
psummationdisplay
i=1
psummationdisplay
j=1
|Rij,x(?;?)|2
R(1)ii,x(0)R(1)jj,x(0)
, R(1)ii,x(0) = ?2i +E{|si(t)|2|hi(l), 0 ? l ? Lc}. (3.40)
37
Therefore, given a specified false alarm probability Pfa, the threshold ? satisfies Pfa =
integraltext?
? fT (x)dx where T ? ?
2
2p2. The resultant probability of PU detection Pd is then Pd =
P{T ? ?|H1, h(l), 0 ? l ? Lc} =integraltext?? fT (y)dy where T ? ?22p2(?1).
3.5.2 Conjugate CAF
WewillfollowthesamelineofargumentsasinSec.3.5.1. UnderH1, wehaveR(?)ij,x(?;0)
negationslash= 0 for at least one pair (i,j). Let ?r(?) denote the p2+p2 ? 1 vector composed of ele-
ments ?R(?)ij,x(?;0) := ?R(?)ij,x(?;0)/
radicalBig
(1 +?i,j)?Rii,x(0)?Rjj,x(0) for 1 ? i ? j ? p, with
m2 := E{?r(?)}composed of elements R(?)ij(?;0)/
radicalBig
(1 +?i,j)R(1)ii (0)R(1)jj (0) (for large M). Let
limM?? (1/M)cov(?r(?),?r?(?)) =: ?2. Then invoking the asymptotic Gaussianity [59, 67, 64],
under H1 and for low SNRs,
lim
M??
?Mparenleftbig?r
(?) ?m2
parenrightbig? N
c (0,?2). (3.41)
[Note that, as in Sec. 3.5.1, limM?? Mcov
parenleftBig?
R(?)ij,x(?;0), ?R(?)ij,x(?;0)
parenrightBig
negationslash= 0 when the PU
signal is present, hence ?R(?)ij,x(?;0) is not proper, but is approximately proper for ?low?
SNRs.] Then for large M,
2M?rH(?)??11 ?r(?) H1? ?2p2+p(?2), (3.42)
where ?2 = 2MmH2 ??12 m2. Under low SNR, ?2 ? I ? 2M?rH(?)??12 ?r(?) ? Tconj and ?2 ?
2MmH2 m2. Thus,
Tconj H1? ?2p2+p(?2) (3.43)
where
?2 = 2M
psummationdisplay
i=1
psummationdisplay
j=i
vextendsinglevextendsingleR
(?)ij,x(?;0)
vextendsinglevextendsingle2
(1 +?i,j)R(1)ii,x(0)R(1)jj,x(0)
. (3.44)
3.6 Computational Complexity
Here we will compare computational complexity of our proposed tests with that of [59]
(used in the single antenna case (p = 1) for testing a single cycle frequency at a single
38
lag in [67] and extended to multiple cycle frequencies in [64]), [73] (extension of [59] to the
multiple antennas case (p > 1)), [63] and [71]. We will express computational complexity in
terms of number of complex multiplications needed to implement the various approaches; the
respective counts for various approaches are listed in Table 3.1. Given an observation sample
size of M samples, for computing the CAF at given ? and ?, we need 2M multiplications. To
compute the (stationary) autocorrelation at zero lag we need M multiplications. This yields a
count of 3M multiplications for the proposed test (3.36) for single antenna. For the proposed
multi-antenna detectors (3.29) or (3.34) with p antennas, we have to calculate p2 CAFs (cyclic
auto- and cross-correlations) and p stationary autocorrelation functions, yielding a count of
Mp(1 + 2p) complex multiplications for (3.29) and a count of Mp(p + 2) multiplications
for (3.34). For the test of [59] described in Sec. 2.4, one needs 2M multiplications to
calculate the CAF at given ? and ?, (M/2)log2(2M) multiplications for the FFT involved
(to compute the cyclic periodogram), and M + 4Lw multiplications for smoothing the two
cyclic periodograms. This yields the number of complex multiplications needed as 3M +
M
2 log2 M + 4Lw. The extension to p > 1 given in [73] requires p
2parenleftbig3M + M
2 log2 M +4Lw
parenrightbig
complex multiplications for nonconjugate CAF-based test and p2+p2 parenleftbig3M + M2 log2 M +4Lwparenrightbig
multiplications for conjugate CAF-based test. As seen by comparing (3.36) and (2.28), the
test (2.28) of [63] requires (2Nb + 1)2M + M multiplications for p = 1. Using the results of
[71, Sec. IV-C], the test (2.30) requires 3M complex multiplications.
3.7 Simulation Examples
We now present some computer simulation examples using OFDM and GMSK signals
to illustrate our detectors and to compare their performance and computational complexity
with some existing detectors. In all presented examples noise variance is assumed to be
unknown.
39
3.7.1 Example 1: Single Antenna Receiver
The primary signal is an OFDM signal with number of subcarriers Nc = 32 and cyclic
prefix of length Ncp = 8. The number of OFDM symbols were taken to be Nofdm = 100, 10
or 1, with the corresponding observation size of M = Nofdm(Nc + Ncp) = 4000, 400 or 40,
respectively. The subcarrier modulation was QPSK. The false alarm probability is 0.01 and
all results are averaged over 2000 Monte Carlo runs. For PU signal detection we picked the
(normalized) cycle frequency ? = 1/(Nc + Ncp) = 1/N and time lag ? = Nc. We used zero-
mean circularly symmetric complex-Gaussian additive white noise. The OFDM signal was
passed through either a constant channel with unit gain or a flat Rayleigh fading channel.
In Fig. 3.1 we compare the theoretical performance based on (3.33) and (3.39) with the
simulations-based performance for the proposed detector when number of antennas p = 1
and the channel is unit gain. The theoretical CAF for the OFDM signal was obtained from
[67] (for rectangular pulse shape) for use in computing the non-centrality parameter ?1. It is
seen that the agreement between theory and simulations is quite good for larger observation
samples. In Fig. 3.2 we compare Pd performance (as a function of SNR for Pfa = 0.01) of our
proposed approach with that of [59] (also used in [67, 64]) via simulations when number of
antennas p = 1 and the channel is flat Rayleigh fading channel. For the approach of [59] we
used Kaiser windows with parameters (?,Lw) = (1,65), (1,513) and (1,513) for M =40, 400
and 4000, respectively. It is seen that the proposed approach is superior to [59] for smaller
observation sizes. On the other hand, using the results of Table 3.1, we find that the [59]
needs 4.05, 4.15 and 3.17 times more complex multiplications than the proposed approach
for M =40, 400 and 4000, respectively. Notice that both the proposed test (3.36) and test
(2.26) of [59] use the same CAF but normalize it differently. While (3.36) explicitly uses
white Gaussian nature of noise to normalize using ?R211,x(0), (2.26) uses a ?generic? covariance
estimator ?? which needs far more parameters (spectral smoothing window, FFT in (2.22),
etc.) which likely makes the estimate ?? of higher variance than ?R211,x(0). This fact (likely
higher variance of ?? under both H0 and H1, as compared to the variance of ?R211,x(0)) would
40
seem to account for poorer performance of Dandawate-Giannakis test (2.26) as compared to
the proposed test (3.36) for ?smaller? observation samples M; for larger M the variances
of normalization factors become small enough to have negligible difference in performance.
However, the computational complexity advantage of test (3.36) over test (2.26) remains. In
Fig. 3.3 for a constant-gain channel and OFDM PU signal in white Gaussian noise, we show
the number of observation samples needed to achieve Pd = 0.9 versus SNR, with Pfa = 0.01.
It is seen that the proposed test is superior to [59] for SNR?0 dB and has same performance
for SNR< 0dB, while being computationally cheaper (as discussed earlier). In Fig. 3.4 we
compare Pd performance (as a function of SNR for Pfa = 0.01) of the proposed test (3.36)
with the Tani-Fantacci test (2.30) [71] for an AWGN channel when M = 2000 (we took
L = 38 < N = 40 in (2.31)). It is seen that the proposed test vastly outperforms the
Tani-Fantacci test while by Table 3.1, the computational complexity of the two approaches
is the same. Comparisons with the Jallon test (2.28) are deferred to Sec. 3.7.2.
3.7.2 Example 2: Cyclostationary Interference
First we consider a constant-gain channel and an OFDM PU signal (p = 1) as in
Example 1 (Nc = 32, Ncp = 8) but under three different scenarios: (i) white noise alone, (ii)
white noise and an OFDM interfering signal with Nc = 64, Ncp = 16, QPSK modulation
(called ?Type I? interference hereinafter), with interfering signal power equal to noise power
(0dB SNR for interfering signal), and (iii) white noise and an OFDM interfering signal with
Nc = 32, Ncp = 16, QPSK modulation (called ?Type II? interference hereinafter), with
interfering signal power 10dB below the noise power (?10dB SNR for interfering signal).
Fig. 3.5 shows the performance (based on 2000 runs) of the nonconjugate CAF detector
(3.36) and the test (2.28) of [63] under the three scenarios when M = 2000 and Pfa = 0.01.
With fixed noise power and interference power fixed relative to noise power, we varied the
PU signal power to achieve the desired PU SNR (it does not include interference signal
power). For test (2.28) we consider two cases: Nb = 0 (smallest value) and Nb = 3 (largest
41
Nb < NcNcp = 328 = 4 based on the PU signal parameters, the largest Nb recommended by
[63]). Under white noise alone, [63] with Nb = 3 significantly outperforms the proposed test
(about 3dB SNR advantage for achieving Pd = 0.9) since test (2.28) of [63] uses more PU
cycle frequencies. This advantage is preserved under Type I interference. However, under
Type II interference, test (2.28) of [63] fails in that it views interference as a PU signal (even
at SNR=?20dB Pd is significantly higher than the design Pfa of 0.01, unlike scenarios (i)
and (ii)): CAFs of both PU signal and Type II interference peak at ? = Nc = 32. While the
proposed test (3.36) tuned to ? = 1/40 and ? = 32 is able to reject Type II interference (it
only worsens the effective SNR), Jallon?s test (2.28) tuned to ? = 32 and cycle frequencies
k/40, k = ?3,?2,??? ,3 also responds to significant CAF of interference at ? = 32 and
zero cycle frequency. Note that we picked Type II interference SNR to be ?10dB; for 0dB
SNR Pd for test (2.28) is almost 1 even at low SNRs. As noted earlier, cyclostationarity
detection allows discrimination among signals exhibiting cyclostationarity at different cycle
frequencies and/or time lags. As in the case of Type II interference in this example, use of
multiple cycle frequencies or time lags can reduce this discrimination capability where an
interfering cyclostationary signal may be detected as a PU cyclostationary signal. Finally,
using the results of Table 3.1 for p = 1, note that Jallon?s detector needs 1 +(4/3)Nb times
more complex multiplications than the proposed approach for any observation sample size.
Thus [63] has five times higher computational complexity for Nb = 3 while it obtains a 3dB
SNR advantage for achieving Pd = 0.9 over our proposed detector under white noise. When
Nb = 0, Jallon?s and the proposed detectors have same computational complexity, but the
SNR advantage now is less than 0.3 dB.
Fig. 3.6 shows the performance of the nonconjugate CAF detector and the approach of
[59] in the presence of an interfering signal in addition to noise, for a constant gain channel.
The PU signal is an OFDM signal with Nc=64, Ncp = 16 and QPSK modulation, whereas
the interfering signal is also OFDM but with Nc=32, Ncp = 8 and QPSK modulation.
The nonconjugate detector looks for (normalized) ? = 1/80 at ? = 64. The SNR for the
42
interfering signal was fixed at 0dB (i.e., noise power equals interfering signal power). With
fixed noise power, we varied the PU signal power to achieve the desired PU SNR. Fig. 3.6
shows the Pd for Pfa = 0.01, M = 2000, p =1 antenna, 2000 runs, and varying PU SNR. We
also show the results when the interfering signal is absent. The nonconjugate CAF detector
is able to distinguish between the PU signal and the interfering signal (which, as noted in
[64] and others, an energy detector fails to do). There is about a 3dB loss in performance
compared to the case when the interference is absent, for both the proposed approach and
that of [59]; this is not surprising since interference plus noise power is 3dB higher than noise
power. Furthermore, the proposed approach is superior to [59] (about 2dB SNR advantage
for achieving Pd = 0.9).
3.7.3 Example 3: Multi-Antenna Receiver, OFDM PU signal
Now we consider p = 4 antennas and an OFDM signal as in Example 3.7.1. In Fig.
3.7 we compare Pd performance (as a function of SNR for Pfa = 0.01) of our proposed
approach with that of [73] (modified to handle nonconjugate CAFs, an extension of [59] to
multiple antenna receiver) via simulations when observation samples M = 4000 (100 OFDM
symbols) and the channel is flat Rayleigh-fading vector channel with independent compo-
nents (each Rayleigh fading), and spatially uncorrelated zero-mean circularly symmetric
complex-Gaussian additive white noise. For the approach of [73], we used Kaiser window
with parameters (?,Lw) = (1,513). We tested for ? = 1/40 and ? = 32. It is seen that the
performance of the proposed approach is about the same as that of [73]. Using the results
of Table 3.1 for nonconjugate CAFs, we find that the [73] needs 4.22 times more complex
multiplications than the proposed approach for M =4000; thus the proposed approach is to
be preferred when both performance and computational complexity are considered. In Fig.
3.8 we compare the theoretical performance based on (3.33) and (3.39) with the simulations-
based performance for the proposed detector when number of antennas p = 4, M = 4000,
Pfa = 0.01 and the channel is non-random with unit gain. It is seen that the agreement
43
between theory and simulations is good. Note that the asymptotic distribution (3.39) under
H1 is a low-SNR approximation which accounts for some difference in theory and simulations
at SNRs around -10 to -9 dB (?higher? end).
3.7.4 Example 4: Multi-Antenna Receiver, GMSK PU signal
Now we consider a GMSK PU signal with N=16 samples per symbol. We seek to detect
the PU signal using conjugate CAF at the normalized cycle frequency ? = 132 and lag ? = 0.
The channel was either constant gain or flat Rayleigh fading, with circularly symmetric
complex Gaussian additive noise. Figs. 3.9 and 3.10 show the Pd performance (as a function
of SNR for Pfa = 0.01) of our proposed approach and that of [73] for constant-gain and
Rayleigh fading channels, respectively, based on 10000 runs when number of antennas p = 4
and M=320. For the approach of [73], we used Kaiser window with parameters (?,Lw)
= (1,65). In both figures there is little difference between the two approaches. Using the
results of Table 3.1 for conjugate CAFs, we find that the [73] needs 3.32 times more complex
multiplications than the proposed approach for M =320; thus the proposed approach is to
be preferred when both performance and computational complexity are considered.
3.8 Conclusions
Detection of cyclostationary primary user (PU) signals in white Gaussian noise for cog-
nitive radio systems was considered based on looking for a cycle frequency at a particular
time lag in the cyclic autocorrelation function (CAF) of the noisy PU signal. We explicitly
exploit the knowledge that under the null hypothesis of PU signal absent, the measurements
originate from white Gaussian noise with possibly unknown variance. Our formulation al-
lows us to simplify the spectrum sensing detector, and obviates the need for estimating an
unwieldy covariance matrix needed in the Dandawate-Giannakis [59] and related approaches.
We considered both single and multiple antenna receivers and both nonconjugate and conju-
gate CAFs. A performance analysis of the proposed detectors was carried out. Supporting
44
simulation examples were presented to compare our detectors with those of [59], [73], [63]
and [71], and to demonstrate the capability to discriminate among signals exhibiting cyclo-
stationarity at different cycle frequencies and/or time lags. Our proposed approaches are
computationally cheaper than the Dandawate-Giannakis and related approaches [59], [73]
while having quite similar detection performance. Our proposed approach significantly out-
performs [71] while having same computational complexity. In white noise channels, the
single-antenna detector of [63] has a 3dB SNR advantage for achieving a detection probabil-
ity of 0.9 over our proposed detector but requires five times more complex multiplications.
Also, Jallon?s test [63] fails to reject a class of cyclostationary interfering signals whereas our
proposed detector is successful.
Table 3.1: Number of complex multiplications for M observation samples with white Gaus-
sian noise
Proposed [59, 73] Jallon [63] Tani-Fantacci [71]
p=1 3M 3M + M2 log2 M + 4Lw (2Nb + 1)2M + M 3M
p > 1: nonconjugate CAF Mp(1 + 2p) p2
parenleftBig
3M + M2 log2 M + 4Lw
parenrightBig
p > 1: conjugate CAF Mp(p + 2) p2+p2
parenleftBig
3M + M2 log2 M + 4Lw
parenrightBig
45
?20 ?15 ?10 ?5 0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Proposed; Pfa = 0.01; White Gaussian noise; 2000 runs; Nfft=32, Ncp=8; M=2000
M=40 ?
M=400 ?
M=4000 ?
Theory
Simulation
Figure 3.1: Example 1: Performance analysis: theoretical and simulation-based performance
comparison of probability of detection Pd versus SNR for the proposed single-antenna spec-
trum sensing under different observation lengths: OFDM signal (number of subcarriers
Nc=32, cyclic prefix length Ncp = 8), non-random channel with unit gain, Pfa = 0.01,
p = 1, 2000 Monte Carlo runs.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
?20 ?15 ?10 ?5 0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Pfa = 0.01; White Gaussian noise; Rayleigh Flat Fading; 2000 runs
M=40 ?
M=400 ?
M=4000 ?
Proposed
Dandawate?Giannakis
Figure 3.2: Example 1: Simulation-based comparison of Pd versus SNR with single antenna
under Rayleigh flat-fading: Pfa = 0.01, p = 1, 2000 Monte Carlo runs, OFDM signal (number
of subcarriers Nc=32, cyclic prefix length Ncp = 8). The approach labeled ?Dandawate-
Giannakis? is used in [59, 67, 64].
46
?15 ?10 ?5 0 5 10 1510
0
101
102
103
104
SNR (dB)
Number of OFDM symbols to P
d=0.9
Pfa = 0.01; Pd = 0.9; White Gaussian noise; 2000 (> 0 dB 10000) runs
Proposed
Dandawate?Giannakis
Figure 3.3: Example 1: Number of symbols needed to achieve Pd = 0.9 versus SNR with
single antenna under non-random channel with constant gain: p = 1, Pfa = 0.01, 2000 runs
for SNR?0dB and 10000 runs for SNR>0dB, OFDM signal (number of subcarriers Nc=32,
cyclic prefix length Ncp = 8). The approach labeled ?Dandawate-Giannakis? is used in
[59, 67, 64].
?20 ?15 ?10 ?5 0 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Pfa = 0.01; 2000 samples; 2000 runs, Single Antenna, AWGN
Tani?Fantacci
Proposed
Figure 3.4: Example 1: Simulation-based comparison of Pd versus SNR with single antenna
under non-random channel with constant gain: Pfa = 0.01, p = 1, M = 2000, 2000 Monte
Carlo runs, OFDM signal (number of subcarriers Nc=32, cyclic prefix length Ncp = 8). The
approach labeled ?Tani-Fantacci? is that of [71].
47
?20 ?17 ?14 ?11 ?8 ?5 ?2 1 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Pfa = 0.01; 2000 samples; 2000 runs, Single Antenna, AWGN, type I/II interfering OFDM
Proposed: white noise
Proposed: Type I int.
Proposed: Type II int.
Jallon: Nb=0, white noise
Jallon: Nb=0, Type I int.
Jallon: Nb=0,Type II int.
Jallon: Nb=3, white noise
Jallon: Nb=3, Type I int.
Jallon: Nb=3, Type II int.
Figure 3.5: Example 2: Simulation-based comparison of Pd versus SNR with single antenna
under non-random channel with constant gain and possible presence of cyclostationary in-
terference: Pfa = 0.01, p = 1, M = 2000, 2000 Monte Carlo runs, OFDM signal (number of
subcarriers Nc=32, cyclic prefix length Ncp = 8). The approach labeled ?Jallon? is that of
[63]. Type I interference is an OFDM signal with number of subcarriers Nc=64 and cyclic
prefix length Ncp = 16 whereas Type II interference is an OFDM signal with number of
subcarriers Nc=32 and cyclic prefix length Ncp = 16. The SNR for Type I interfering signal
was fixed at 0dB (i.e., noise power equals interfering signal power) whereas that for Type II
interfering signal was fixed at -10dB. With fixed noise power, we varied the PU signal power
to achieve the desired PU SNR (horizontal axis).
48
?20 ?15 ?10 ?5 0 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Pfa = 0.01; White Gaussian noise;contstant gain; intterference (0dB) + Noise (0dB); 2000 samples; 2000 runs
Dandawate?Giannakis: only noise
Dandawate?Giannakis: noise+interference
Proposed: only noise
Proposed: noise+interference
Figure 3.6: Example 2: Simulation-based comparison of Pd versus SNR with single antenna
under non-random channel with constant gain and possible presence of cyclostationary in-
terference: Pfa = 0.01, p = 1, M = 2000, 2000 Monte Carlo runs, OFDM signal (number
of subcarriers Nc=64, cyclic prefix length Ncp = 16). The approach labeled ?Dandawate-
Giannakis? is used in [59, 67, 64]. Interference is an OFDM signal with number of subcarriers
Nc=32 and cyclic prefix length Ncp = 8. The SNR for the interfering signal was fixed at 0dB
(i.e., noise power equals interfering signal power). With fixed noise power, we varied the PU
signal power to achieve the desired PU SNR (horizontal axis).
49
?25 ?20 ?15 ?10 ?5 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
4 antennas; Pfa = 0.01; 4000 samples: White Gaussian noise; Rayleigh Flat Fading; 2000 runs
Proposed
Zhong et al
Figure 3.7: Example 3: Simulation-based comparison of probability of detection versus SNR
with p = 4 antennas under Rayleigh flat-fading and M = 4000: Pfa = 0.01, 2000 Monte-
Carlo runs, OFDM signal (number of subcarriers Nc=32, cyclic prefix length Ncp = 8). The
approach labeled ?Zhong et al? is from [73].
?30 ?25 ?20 ?15 ?10 ?5 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
M=4000, 32 subcarriers, prefix length = 8, 4000 runs
Simulation
Theory
Figure 3.8: Example 3: Performance analysis: theoretical and simulation-based performance
comparison of probability of detection Pd versus SNR for the proposed multi-antenna spec-
trum sensing: OFDM signal (number of subcarriers Nc=32, cyclic prefix length Ncp = 8),
non-random channel with unit gain, Pfa = 0.01, p = 4, M = 4000, 4000 runs.
50
?25 ?20 ?15 ?10 ?5 0 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
pfa = 0.01; 320 samples; 10000 runs, Multi Antenna, constant gain, Conjugate GMSK signal
Proposed
Zhong et al
Figure 3.9: Example 4: Simulation-based comparison of probability of detection versus SNR
for GMSK signal with 16 samples per symbol, with p = 4 antennas under non-random
channel with constant gains and M = 320: Pfa = 0.01, 10000 runs. The approach labeled
?Zhong et al? is from [73].
?25 ?20 ?15 ?10 ?5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
pfa = 0.01; 320 samples; 10000 runs, Multi Antenna, Rayleigh fading, Conjugate GMSK signal
Proposed
Zhong et al
Figure 3.10: Example 4: Simulation-based comparison of probability of detection versus
SNR for GMSK signal with 16 samples per symbol, with p = 4 antennas under Rayleigh
flat-fading and M = 320: Pfa = 0.01, 10000 runs. The approach labeled ?Zhong et al? is
from [73].
51
Chapter 4
Cyclostationarity Based Spectrum Sensing Under Uncertain Gaussian Noise
Detection of cyclostationary primary user (PU) signals in colored Gaussian noise for
cognitive radio systems is considered based on looking for single or multiple cycle frequen-
cies at single or multiple time lags in the cyclic autocorrelation function (CAF) of the noisy
PU signal. We explicitly exploit the knowledge that under the null hypothesis of PU sig-
nal absent, the measurements originate from possible colored Gaussian noise with unknown
correlation function. Our formulation allows us to simplify the spectrum sensing detector
and obviates the need for estimating an unwieldy covariance matrix needed in some prior
works. We consider both single and multiple antenna receivers, and both nonconjugate and
conjugate CAFs. A performance analysis of the proposed detector is carried out. Supporting
simulation examples are provided to demonstrate the efficacy of the proposed approaches
and to compare them with some existing approaches. Our proposed approaches are compu-
tationally cheaper than the Dandawate-Giannakis and related approaches while having quite
similar detection performance for a given false alarm rate.
4.1 Introduction
Cognitive radio allows for usage of licensed frequency bands (rights held by primary
users (PUs)) by unlicensed users (secondary or cognitive users) when the licensed spectrum
bands are unoccupied (a function of time and location). Therefore, one of the first steps to be
accomplished by a cognitive user is spectrum sensing: analysis of the received electromagnetic
transmissions to search for unoccupied spectrum bands (spectrum holes). One of the ways to
accomplish this is energy detection/spectral analysis and consequent (statistical) hypothesis
testing [56, 57, 60, 65]. Based on the received signal x(k) (k denotes discrete-time), the
52
cognitive user has to decide if the PU is present or not. This may be formulated as a
binary hypothesis testing problem where under the null hypothesis H0, the cognitive user is
receiving just noise x(k) = n(k), and under the alternative H1, PU signal s(k) is also present
with x(k) = s(k)+n(k). A popular approach is that of energy detection; see [56, 57, 60, 65]
and references therein. One designs a CFAR (constant false alarm rate) test requiring prior
knowledge of some of the signal statistics under H0. For instance, n(k) is taken to be thermal
noise with known variance.
Detectors for orthogonal frequency division multiplex (OFDM) signals over additive
white Gaussian noise (AWGN) channels with possible unknown noise variance have been
considered in [75] using likelihood ratio tests. The formulation of [75] does not consider
colored noise or frequency-selective channels, unlike this chapter. Autocorrelation based
detection of OFDM signals is investigated in [58] for both AWGN and multipath channels
under the assumption of white noise. In [76] spectrum sensing of a second-order cyclosta-
tionary signal (such as OFDM) received at multiple antennas is investigated under white
Gaussian noise and flat fading. It is noted in [76, Sec. VII] that ?The covariance structure is
changed if the noise is colored, and the performance of the proposed detector will degrade.
This is a problem for future study.? We consider colored noise in this chapter. In a typical
system due to filtering at the front-end to exclude out-of-band signals (and other operations
such as sampling), the noise at the receiver is seldom (exactly) white. Furthermore, any
interfering OFDM signal (with signal parameters such as cycle frequencies, not of interest)
can be modeled as a Gaussian process (by invoking the central limit theorem for large enough
number of subcarriers [58]). Such signals propagating through multipath channels are then
colored Gaussian processes.
Energy detection based and quite a few other approaches can not distinguish among
different types of primary user or secondary user signals. Cyclostationarity detection allows
classifying signals exhibiting cyclostationarity at different cycle frequencies [64] and/or dif-
ferent time lags. As noted in [67], one can use cyclostationarity detection for recognizing
53
the individual air interfaces, a task at which energy detector would fail. Existing cyclosta-
tionarity detectors for cognitive radio systems reported in [67, 64, 73] are all based on [59]
where one looks for a cycle frequency at a particular time lag in the cyclic autocorrelation
function (CAF) of the (noisy) PU signal; extensions to multiple cycle frequencies and/or
multiple lags may also be found in some of these papers. Under the alternative hypothesis,
CAF is nonzero whereas under the null hypothesis it is zero. Signal characteristics under the
null hypothesis are not exploited except for noting that its CAF would be zero. Implemen-
tation of the approaches of [59, 67, 64, 73] requires computation of the covariance matrix
of estimated CAF?s real and imaginary parts. (In general, this covariance matrix does not
have an analytical expression even when the PU signal is precisely known.) In practice, it is
estimated from the data, is computationally expensive and it needs a frequency smoothing
window [59] to smooth certain cyclic periodograms in the frequency domain. In this chapter
we explicitly exploit the knowledge that under the null hypothesis of PU signal absent, the
measurements originate from possibly colored Gaussian noise with possibly unknown corre-
lation function. This framework subsumes white Gaussian noise. Such a framework (white
Gaussian noise) is explicitly stated in [73, Eqns. (1),(2)], and although not explicitly stated
in [67, 64], is extensively employed in simulation examples presented therein.
Other existing approaches based on single antenna receivers include [62], [63], [66] and
[71]. In [62] detection of spread signals using the signal nonconjugate CAF at multiple
lags and multiple cycle frequencies is considered under white Gaussian noise with possibly
unknown variance. In [63] detection of OFDM signals using the signal nonconjugate CAF
at a single time lag and multiple cycle frequencies is considered under white Gaussian noise
with known variance. In [66] several single antenna cyclostationarity detectors under white
Gaussian noise have been considered including a peak detector [66, (12)] which utilizes
the noisy signal CAF at multiple cycle frequencies and a single time-lag. Our proposed
detector when specialized to single antenna and white Gaussian noise is similar to that
in [62], [63] and [66, (12)], except that we exclude the use of zero cycle frequency, the
54
contribution of the stationary part of the signal. In [71] a single cycle frequency and single
time lag cyclostationarity detector has been presented for OFDM signals under noise variance
uncertainty. It is different from that in [62], [63], [66] and from our proposed detector; further
details regarding [71] are in Sec. 2.4.
Preliminary conference versions of this chapter have appeared in [74] and [72]. In [74]
cyclostaionarity detectors in white Gaussian noise with unknown variance based on single
cycle frequency and single time lag have been considered whereas in [72] cyclostaionarity
detectors in colored Gaussian noise with unknown correlation function based on single cycle
frequency and single time lag have been investigated. In this chapter we discuss detectors
based on multiple cyclic frequencies and multiple time lags in colored Gaussian noise. Also
we consider both nonconjugate and conjugate CAFs appropriate for OFDM and GMSK
signals, respectively, whereas both [74] and [72] are restricted to nonconjugate CAFs.
The proposed detectors are useful in several practical scenarios such as detection of
OFDM signals in digital video broadcasting (DVB) standard DVB-T (under IEEE 802.22
Working Group proposals), as discussed in [63]. As noted in [64] OFDM is employed by many
of the current as well as future wireless communications systems including 3GPP Long Term
Evolution (LTE) [69], IEEE 802.11a/g WLANs, DVB standards DVB-T and DVB-H, as well
as IEEE 802.16 and WiMAX wireless metropolitan area networks. Finally, a GSM network
as the primary system and coexisting with an OFDM based secondary WLAN system has
been considered in [68]. Our conjugate CAF based detectors for detecting GMSK signals in
the presence of interfering OFDM signals apply to the application considered in [68]. Note
that we exploit only the temporal and spatial (in the multiantenna case) properties of the
received signal, as in [59, 67, 64, 73], but not its spectral properties such as cyclic spectrum
(or power spectrum) in that we can only detect presence or absence of the PU signal in the
?full? PU band instead of distinguishing between occupancy or vacancy of various subbands
in the full band. This is a limitation that our approach shares with that of [59, 67, 64, 73].
55
Our formulation allows us to simplify the spectrum sensing detector, both conceptually
and computationally, and obviates the need for estimating the unwieldy covariance matrix
needed in [59, 67, 64, 73]. We consider both single and multiple antenna receivers. Under
single antenna receivers, our detector is similar to that in [62], [63] and [66, (12)] when
specialized to single antenna and white Gaussian noise. Also unlike[62], [63], [66, (12)]
and [71], we investigate multiple antenna receivers where we exploit cyclic cross-correlations
among different antennas. Exploitation of cyclic cross-correlations among different antennas
has also been considered in [73] for GMSK signals where [73] extends the single-antenna
approach of [59] inheriting its computational complexity.
As in [59, 67, 64, 73], our approach is to test the estimated CAF of the observations
at selected cycle frequencies and lags: their mean value is zero under colored Gaussian
noise and is nonzero when the desired PU signal is present. Since we do not assume any
knowledge about the structure of the colored noise (its variance or correlation function) or the
underlying channel (flat or frequency selective), an optimal detector based on the likelihood
ratio or generalized likelihood ratio test, does not appear to be possible. We devise tests
based on the large sample properties of the estimated CAF without requiring knowledge
of the noise or the channel structure. The rest of the chapter is organized as follows. In
Sec. II we present some background material pertaining to cyclostationary signals and their
CAF. In Sec. III we derive some large sample statistics of the sample CAFs of colored
Gaussian sequences which are then used in Sec. IV to propose novel detectors to detect
PU signals exhibiting nonconjugate or conjugate cyclostaionarity. The results of Secs. III
and IV are new. A performance analysis of the proposed detectors is presented in Sec. V.
Computational complexity issues are discussed in Sec. VI. Supporting simulation examples
are provided using OFDM or GMSK PU signals are presented in Sec. VII where we compare
our detectors with those of [59], [64], [73] and [71].
Notation: Variable Bij denotes the ijth element of the matrix B, xi denotes the
ith element of vector x, and I is the identity matrix. The superscripts ? and H denote the
56
complex conjugate and the Hermitian (conjugate transpose) operations, respectively. The
function ?i,j denotes the Kronecker delta function, i.e. ?i,j = 1 if i = j, 0 otherwise. Giv-
en a column vector x, diag{x} denotes a square matrix with elements of x along its main
diagonal and zeros everywhere else. We use cov(x,y?) := E{xyH}? E{x}E{yH} and let
cum4(x1,x2,x3,x4) denote the joint 4th cumulant of random variables xi (i = 1,2,3,4).
Note that cum4(x1,x2,x3,x4) = E{x1x2x3x4} ? E{x1x2}E{x3x4} ? E{x1x3}E{x2x4} ?
E{x1x4}E{x2x3} when all xi?s are zero-mean. The notation Nc(m,?) denotes a circularly
symmetric (proper) complex Gaussian (vector) distribution with mean m and covariance
matrix ?, and Nr(m,?) denotes the same for a real-valued Gaussian vector.
4.2 Cyclostationarity Background
A discrete-time zero-mean complex-valued cyclostationary signal x(k) is characterized
by a time-varying autocorrelation function Rxx(k,k + ?) := E{x(k)x?(k + ?)} which has a
Fourier series representation [59, 67]
Rxx(k,k +?) =
summationdisplay
??A
Rxx(?;?)ej2??k (4.1)
where given M samples of observations x(k) for k = 1,2,??? ,M, the cyclic autocorrelation
function (CAF) Rxx(?;?) at cycle frequency ? ? A is given by
Rxx(?;?) = lim
M??
1
M
Msummationdisplay
k=1
Rxx(k,k +?)e?j2??k (4.2)
and A is the set of cycle frequencies [59]
A := {?|0 ? ? < 1, Rxx(?;?) negationslash? 0}. (4.3)
57
A conjugate CAF is defined as
Rxx(?)(?;?) = lim
M??
1
M
Msummationdisplay
k=1
Rxx?(k,k +?)e?j2??k (4.4)
where Rxx?(k,k+?) := E{x(k)x(k+?)}. Ref. [59] considers conjugate CAFs whereas [67, 64]
consider both types of CAFs. For wide-sense stationary (WSS) signals, Rxx(?;?) = 0 as
well as Rxx(?)(?;?) = 0 for any ? negationslash= 0.
4.2.1 Multivariate Processes
Later we will consider vector random processes arising due to a multiantenna receiver
with p ? 1 antennas. Therefore, we will consider a p?1 discrete-time zero-mean complex-
valued cyclostationary signal x(k) with time-varying autocorrelation function Rxx(k,k +
?) := E{x(k)xH(k +?)} which has a Fourier series representation [59, 67]
Rxx(k,k +?) =
summationdisplay
??A
Rxx(?;?)ej2??k (4.5)
where the nonconjugate cyclic autocorrelation function (CAF) Rxx(?;?) at cycle frequency
? ? A is given by
Rxx(?;?) = lim
M??
1
M
Msummationdisplay
k=1
Rxx(k,k +?)e?j2??k. (4.6)
Similar comments apply to conjugate CAFs.
4.3 Large Sample Statistics of Sample CAF
In this section we investigate large sample statistics of estimated CAF when we have
colored Gaussian noise as observations. These results are exploited later in Sec. 4.4 to
propose detectors for detection of cyclostationary PU signals. Our focus will be on the
58
binary hypothesis testing problem involving p-dimensional processes (p ? 1)
x(k) =
?
??
??
n(k) : H0
s(k)+n(k) : H1
(4.7)
where n(k) is zero-mean, spatially uncorrelated, temporally colored Gaussian and s(k) =
summationtextLc
l=0 h(l)spu(k ? l) where spu(k) is a scalar cyclostationary signal (emitted by a primary
user) with at least one known cycle frequency ? and at least one known lag ? such that
Rspuspu(?;?) negationslash= 0, and p-column h(l) is the complex channel impulse response. Our approach
(similar to [59, 67, 64, 73]) is to test the estimated CAF of the observations at pairs (?i,?i),
i ? 1, to check whether it is nonzero. However, unlike [59, 67, 64, 73], we explicitly exploit
the known statistics of observations under H0 (namely, we have colored Gaussian noise with
possibly unknown correlation function). We are interested only in nonzero cycle frequencies.
4.3.1 Nonconjugate CAF
These results are appropriate for OFDM signals.
Spatially White and Temporally Correlated Gaussian Noise
Consider a p?1 WSS proper (circularly symmetric) complex-valued zero-mean colored
Gaussian sequence x(k) = n(k) with correlation function Rxx(k,k + ?) = Rnn(k,k + ?) =
Rnn(?) = diag{Rii,n(?), 1 ? i ? p}. Also, since n(k) is proper, E{n(k)nT(k + ?)} ? 0.
Given an observation length of M samples, we estimate the (nonconjugate) CAF Rxx(?;?)
as
?Rxx(?;?) = 1
M
Msummationdisplay
k=1
x(k)xH(k +?)e?j2??k. (4.8)
59
It is easy to see that limM?? E{?Rxx(?;?)} = Rxx(?;?). Let (xi is the i-th component of
x)
?Rik,x(?;?) := 1
M
Msummationdisplay
m=1
xi(m)x?k(m+?)e?j2??m. (4.9)
We prove Lemma 1 in the Appendix Sec. A.
Lemma 1 . For x(k) = n(k), a WSS proper zero-mean Gaussian sequence, we have
lim
M??
Mcov
parenleftBig?
Rij,n(?;?1), ?R?kl,n(?;?2)
parenrightBig
= ??ij,n(?;?1 ??2)?i,k?j,l??,? (4.10)
and
lim
M??
Mcov
parenleftBig?
Rij,n(?;?1), ?Rkl,n(?;?2)
parenrightBig
= lim
M??
Mcov
parenleftBig?
Rij,n(?;?1), ?R?lk,n(??;??2)
parenrightBig
= ??ij,n(?;?1 +?2)?i,l?j,k??,??. (4.11)
where Rik,x(?) := E{xi(t)x?k(t+?)} and
??ij,n(?;?1 ??2) :=
?summationdisplay
m=??
Rii,n(?m)Rjj,n(m+?1 ??2)e?j2??m ? (4.12)
Finite Memory Colored Gaussian Noise
We now make a further assumption that correlation function of noise is ?effectively?
zero for lags > Ln where (the upperbound) Ln is known, i.e.
Rnn(?) = 0 for |?| > Ln. (4.13)
By (4.10), (4.11) and (4.13), for ?,? ? A, we have
lim
M??
Mcov
parenleftBig?
Rij,n(?;?1), ?R?kl,n(?;?2)
parenrightBig
=
?
???
???
??ij,n(?;0)?i,k?j,l??,?, for ?1 = ?2
0 for |?1 ??2| > 2Ln,
(4.14)
60
lim
M??
Mcov
parenleftBig?
Rij,n(?;?1), ?Rkl,n(?;?2)
parenrightBig
= 0. (4.15)
Also, by [59, 67, 64], asymptotically (as M ? ?), ?Rik,x(?;?) is a Gaussian random variable
(complex-valued but not necessarily circularly symmetric) and ?vectorized? matrix ?Rxx(?;?)
is a Gaussian random vector for any ? and ?, for x(t) under either H0 or H1 and whether
x(t) is finite memory or not. Invoking asymptotic Gaussianity, it then follows that
lim
M??
?M ?R
ij,n(?;?) ? Nc (0,??ij,n(?;0)) ?i,j. (4.16)
Furthermore, ?Rij,x(?;?) is asymptotically independent of ?Rkl,x(?;?) if i negationslash= k or j negationslash= l.
Moreover, ?Rij,x(?;?1) is asymptotically independent of ?Rkl,x(?;?2) if ? negationslash= ? (?,? ? A) or
?1 negationslash= ?2, for any i,j,k,l. As discussed in Sec. 2.2.1, for OFDM signals the nonconjugate CAF
peaks at ? = ?Nc (Nc = number of subcarriers). If we take ?1 = Nc and ?2 = ?Nc, we need
to have Ln < 2Nc for (4.14) to hold true.
4.3.2 Conjugate CAF
These results are appropriate for GMSK signals. Given an observation length of M
samples, we estimate the conjugate CAF Rxx(?)(?;?) := limM?? 1M summationtextMk=1 E{x(k)xT(k +
?)}e?j2??k as
?Rxx(?)(?;?) = 1
M
Msummationdisplay
k=1
x(k)xT(k +?)e?j2??k. (4.17)
Spatially White and Temporally Correlated Gaussian Noise
Mimicking Sec. 4.3.1 we obtain Lemma 2 whose proof is given in the Appendix Sec. B.
Lemma 2 . For x(k) = n(k), a WSS proper zero-mean Gaussian sequence, we have
lim
M??
Mcov(?R(?)ij,n(?;?1), ?R?(?)kl,n(?;?2)) =
?summationdisplay
m=??
[Rik,n(m)Rjl,n(m+?2 ??1)
+Ril,n(m+?2)Rjk,n(m??1)]ej2??m??,? ? (4.18)
61
Finite Memory Colored Gaussian Noise
For GMSK signals, the nonconjugate CAF peaks at ? = 0. Confining our attention to
?1 = ?2 = 0, for finite-memory, spatially-uncorrelated, colored Gaussian noise, we obtain
lim
M??
Mcov
parenleftBig?
R(?)ij,n(?;0), ?R?(?)kl,n(?;0)
parenrightBig
=
?
????
????
????
????
2summationtextLnm=?Ln R2ii,n(m)ej2??m??,? if i = k = l = j
summationtextLn
m=?Ln Rii,n(m)Rjj,n(m)e
j2??m??,?
if (i,j) = (k,l) but i negationslash= j, or (i,j) = (l,k)
0 otherwise .
(4.19)
Invoking asymptotic Gaussianity, it then follows that for finite-memory colored Gaussian
noise
lim
M??
?M ?R
(?)ij,n(?;0) ? Nc
parenleftBig
0,
Lnsummationdisplay
m=?Ln
(1 +?i,j)Rii,n(m)Rjj,n(m)ej2??m
parenrightBig
?i,j. (4.20)
NotingthatsincelimM?? ?R(?)ij,x(?;?) = limM?? ?R(?)ji,x(?;?)and ?R(?)ij,x(?;0) = ?R(?)ji,x(?;0),
unlike nonconjugate CAFs, in case of conjugate CAFs, ?R(?)ij,x(?;?) is asymptotically inde-
pendent of ?R(?)kl,x(?;?) for i negationslash= k or j negationslash= l if i ? j and k ? l (or if i ? j and k ? l).
4.4 Test Statistics
Based on the large sample statistics of CAFs of colored Gaussian noise discussed in
Sec. 4.3, we now propose ad hoc CFAR (constant false alarm rate) detectors for detection
of nonzero nonconjugate as well as conjugate CAFs. They are ad hoc as they do not follow
any optimality criterion. Since we do not assume any knowledge about the structure of the
colored noise (its variance or correlation function) or the underlying channel (flat or frequency
selective), an optimal detector based on the likelihood ratio or generalized likelihood ratio
test, does not appear to be possible. We will assume that the finite-memory assumption
62
(4.13) holds true in this section. The desired PU signals are assumed to have nonzero CAF
at certain known non-zero cycle frequencies and lags whereas CAF of colored noise is zero at
all lags and nonzero cycle frequencies. The proposed tests exploit the large sample statistics
of the estimated CAF to yield CFAR detectors.
4.4.1 Nonconjugate CAF: OFDM Signals
Consider the binary hypothesis testing problem (4.7). The PU channel impulse response
and the noise correlation function are unknown. We do know that noise is zero-mean complex
(proper) Gaussian, spatially uncorrelated and temporally possibly colored (and of finite
memory). Also, spu(t) in (4.7) is a scalar cyclostationary signal (emitted by PUs) with K ? 1
known cycle frequency and lag pairs (?k,?k), k = 1,2,??? ,K, such that Rspuspu(?k;?k) negationslash= 0
and |?1 ??2| > Ln so that (4.14) holds true. The binary hypothesis testing problem in this
case can be formulated as
H0 : Rij,x(?k;?k) = 0 ?i,j,k
H1 : Rij,x(?k;?k) negationslash? 0 for some i,j,k.
(4.21)
Under both hypotheses, we have limM?? E{?R(?;?)} = R(?;?). Under H0, by the results
of Sec. 4.3.1, ?Rij(?k;?k) has zero-mean ?i,j,k, and under H1, it has non-zero mean for some
i,j,k; therefore, |Rij(?k;?k)| > 0 for some i,j,k. Define (see also (4.12))
??ij,n(?k) :=
Lnsummationdisplay
m=?Ln
Rii,n(?m)Rjj,n(m)e?j2??km (4.22)
and let ?r denote the Kp2 ? 1 vector composed of elements ?Rij,x(?k;?k)/radicalbig??ij,x(?k) for all
distinct triplets (i,j,k). Then by (4.16), we have limM???M?r H0? Nc (0,I) leading to
2M?rH?r H0? ?22Kp2 where ?2n denotes the central chi-square distribution with n degrees of
freedom (dof). Since |Rij(?k;?k)| > 0 for some i,j,k, we have E{?rH?r|H1} > E{?rH?r|H0}.
Furthermore, since the true values ??ij,n(?k), (k = 1,2,??? ,K), are not available, we replace
63
them by estimates ??ij,x(?k) and consider the following ad hoc test motivated by 2M?rH?r and
E{?rH?r|H1} > E{?rH?r|H0}:
T := 2M
Ksummationdisplay
k=1
psummationdisplay
i=1
psummationdisplay
j=1
|?Rij,x(?k;?k)|2
??ij,x(?k)
H1
greaterlessequal
H0
? (4.23)
where
??ij,x(?k) :=
Lnsummationdisplay
m=?Ln
?Rii,x(?m)?Rjj,x(m)e?j2??km, (4.24)
?Rii,x(m) := 1
M ?m
M?msummationdisplay
k=1
xi(k)x?i(k +m), m ? 0, (4.25)
?Rii,x(?m) = ?R?ii,x(m), and the threshold ? is picked to achieve a pre-specified probability of
false alarm Pfa = P{T ? ?|H0}. By the results of Sec. 4.3.1, we can find (shown next)
asymptotic distribution of T under H0 and design a CFAR test. Such CFAR tests are not
optimal in any sense but do yield a constant false-alarm rate, a desirable property, under
model parameter uncertainties. CFAR tests have been used in radar and other problems
under parameter uncertainty [61, Sec. 8.1].
We now turn to calculation of ?. It follows from (4.16) that asymptotically (as M ? ?),
2M|
?Rij,x(?k;?k)|2
??ij,x(?k)
H0? ?2
2. (4.26)
Eqn. (4.14) applied to finite-memory colored Gaussian noise also implies that as M ? ?,
underH0, ?Rii,x(m) converges in the mean-square sense, hence in probability (i.p.), to Rii,x(m)
[70, Appendix B]. Together with (4.26) this results implies that asymptotically [70, Lemma
B.4]
2M|
?Rij,x(?;?)|2
??ij,x(?k)
H0? ?2
2. (4.27)
As noted in Sec. 4.3.1, any element of ?Rxx(?k;?k) is asymptotically independent of all other
elements and the same holds true for distinct cycle frequencies, therefore, in (4.23) we have
(asymptotically) mutually independent random variables for distinct (i,j,k) triples. Since
64
there are Kp2 such terms in T , asymptotically
T H0? ?22Kp2. (4.28)
4.4.2 Conjugate CAF: GMSK Signals
In this case too the binary hypothesis testing problem is formulated as (4.21) except
that we use R(?)ij,x(?k;?k) instead of Rij,x(?k;?k) and we will take ?k = 0 ?k since that is
where the GMSK CAF is the strongest [68]. Following Secs. 4.3.2 and 4.4.1, we propose the
test statistic
Tconj := 2M
Ksummationdisplay
k=1
psummationdisplay
i=1
psummationdisplay
j=i
|?R(?)ij,x(?k;0)|2summationtext
Ln
m=?Ln(1+?i,j)?Rii,x(m)?Rjj,x(m)ej2??km
H1
greaterlessequal
H0
?conj. (4.29)
Using the results of Sec. 4.3.2 and noting that there are K(p2 + p)/2 terms in the double
summation in (4.29), it follows that
Tconj H0? ?2K(p2+p) (4.30)
which allows us to calculate the test threshold ?conj corresponding to a specified Pfa.
4.4.3 Single Antenna Case: Nonconjugate CAF
Proposed Test
For p = 1 (single antenna), the proposed nonconjugate CAF test reduces to
2M
Ksummationdisplay
k=1
|?R11,x(?k;?k)|2summationtext
Ln
m=?Ln |?R11,x(m)|2e?j2??km
H1
greaterlessequal
H0
?. (4.31)
65
4.5 Detection Probability
In this section we derive the asymptotic distribution of the proposed test statistics of Sec.
4.4 under H1 for a specified PU signal. We will assume a non-random channel; for random
channels, our results offer conditional detection probability conditioned on the channel.
4.5.1 Nonconjugate CAF
Under H1, we have Rij,x(?k;?k) negationslash= 0 for at least one pair (i,j) and ?k. Let ?r denote the
Kp2 ?1 vector composed of elements ?Rij,x(?k;?k) := ?Rij,x(?k;?k)/radicalbig??ij,x(?k) for all distinct
triples (i,j,k), with m1 := E{?r} composed of elements Rij,x(?k;?k)/
radicalBig
?(1)ij (?k) (for large
M, i.e. M ? ?) where ?(?)ij (?k) := summationtextLnm=?Ln R(?)ii,x(?m)R(?)jj,x(m)e?j2??km and R(?)ii,x(m) :=
E{xi(t)x?i(t + m)|H?}, ? = 0 or 1. Let ?rre = Re(?r), ?rim = Im(?r), ?c = [?rTre ?rTim]T and
mc = [Re(mT1 ) Im(mT1 )]T. Then invoking asymptotic Gaussianity [59, 67, 64], we have
limM???M (?c?mc) H1? Nr (0,?c1) where ?c? := limM??(1/M)cov(?c,?c|H?)= conditional
covariance, ? = 0 or 1. Define
SNR = lim
M??
(1/M)
Msummationdisplay
k=1
E{bardbls(k)bardbl2}[E{bardbln(k)bardbl2}]?1. (4.32)
If SNR=0, then the hypothesis H1 reduces to H0, with m1 = 0 = mc, ?(1)ij (?k) = ?(0)ij (?k),
and by (4.14)-(4.15), ?c0 = I. Furthermore limSNR?0 ?c1 = ?c0 = I. UnderH0, complex ?r is a
proper random vector, hence, ?0 = I where ?? := limM??(1/M)cov(?r,?r?|H?)= conditional
covariance, ? = 0 or 1. Therefore, under H1,
lim
SNR?0
lim
M??
(1/M)cov(?r,?r?) = lim
SNR?0
?1 = ?0 = I. (4.33)
Again invoking asymptotic Gaussianity [59, 67, 64], under H1 and for ?low? SNRs (i.e.
SNR ? 0),
lim
M??
?M (?r?m
1)
H1? N
c (0,?1). (4.34)
66
(Note that limM?? Mcov
parenleftBig?
Rij(?;?), ?Rij(?;?)
parenrightBig
negationslash= 0 when the PU signal is present, hence
?Rij(?;?) is not proper, but is approximately proper for ?low? SNRs.) Then for large M, i.e.
M ? ?,
2M?rH??11 ?r H1? ?22Kp2(?1), (4.35)
where ?2n(?) denotes noncentral chi-square distribution with n degrees of freedom and non-
centrality parameter ?, and ?1 = 2MmH1 ??11 m1. Under low SNR, ?1 ? I ? 2M?rH??11 ?r ?
T and ?1 ? 2MmH1 m1. Thus, asymptotically,
T H1? ?22Kp2(?1), (4.36)
where (assuming s(t) =summationtextLcl=0 h(l)spu(t?l))
?1 = 2M
psummationdisplay
i=1
Ksummationdisplay
k=1
psummationdisplay
j=1
|Rij,x(?k;?k)|2summationtext
Ln
m=?Ln R
(1)
ii,x(?m)R
(1)
jj,x(m)e?j2??km
, (4.37)
R(1)ii,x(m) = Rii,n(m)+E{si(t)si(t+m)|hi(l), 0 ? l ? Lc}. (4.38)
Therefore, given a specified false alarm probability Pfa, the threshold ? satisfies Pfa =
integraltext?
? fT (x)dx where T ? ?
2
2Kp2 and fT is the probability density function of T . The resultant
probability of PU detection Pd is then Pd = P{T ? ?|H1, h(l), 0 ? l ? Lc} =integraltext?? fT (y)dy
where T ? ?22Kp2(?1). Note that (4.34) is valid only as SNR ? 0 since ?Rij(?;?) is not proper
when PU signal is present; hence, our results may not be accurate for ?high? SNRs. The
simulation results in Sec. 4.7.2 show that the low SNR results are quite good for all SNRs
for the presented example.
4.5.2 Conjugate CAF
WewillfollowthesamelineofargumentsasinSec.4.5.1. UnderH1, wehaveR(?)ij,x(?k;0) negationslash=
0 for at least one pair (i,j) and ?k. Let ?r(?) denote the K(p2+p)2 ? 1 vector composed
67
of elements ?R(?)ij,x(?k;0) := ?R(?)ij,x(?k;0)/
radicalBigsummationtext
Ln
m=?Ln(1+?i,j)?Rii,x(m)?Rjj,x(m)ej2??km for
1 ? i ? j ? p, withm2 := E{?r(?)}composed of elements R(?)ij(?k;0)/
radicalBigsummationtext
Ln
m=?Ln(1+?i,j)?m
where ?m = R(1)ii (m)R(1)jj (m)ej2??km(for large M). Let limM??(1/M)cov(?r(?),?r?(?)) =: ?2.
Then invoking the asymptotic Gaussianity [59, 67, 64], under H1 and for low SNRs,
lim
M??
?Mparenleftbig?r
(?) ?m2
parenrightbig? N
c (0,?2). (4.39)
(Note that, as in Sec. 4.5.1, limM?? Mcov
parenleftBig?
R(?)ij,x(?;0), ?R(?)ij,x(?;0)
parenrightBig
negationslash= 0 when the PU
signal is present, hence ?R(?)ij,x(?;0) is not proper, but is approximately proper for ?low?
SNRs.) Then for large M,
2M?rH(?)??11 ?r(?) H1? ?2K(p2+p)(?2), (4.40)
where ?2 = 2MmH2 ??12 m2. Under low SNR, ?2 ? I ? 2M?rH(?)??12 ?r(?) ? Tconj and ?2 ?
2MmH2 m2. Thus,
Tconj H1? ?2K(p2+p)(?2) (4.41)
where
?2 = 2M
Ksummationdisplay
k=1
psummationdisplay
i=1
psummationdisplay
j=i
vextendsinglevextendsingleR
(?)ij,x(?k;0)
vextendsinglevextendsingle2
summationtextLn
m=?Ln(1+?i,j)R
(1)
ii,x(m)R
(1)
jj,x(m)ej2??km
. (4.42)
4.6 Computational Complexity
Here we will compare computational complexity of our proposed tests with that of [59]
(used in the single antenna case (p = 1) for testing a single cycle frequency at a single
lag in [67] and extended to multiple cycle frequencies in [64]), [73] (extension of [59] to
the multiple antennas case (p > 1)), and [71]. We will express computational complexity in
terms of number of complex multiplications needed to implement the various approaches; the
respective counts for various approaches are listed in Table 4.1. Given an observation sample
size of M samples, for computing the CAF at given ? and ?, we need 2M multiplications. To
68
Table 4.1: Number of complex multiplications for M samples with colored Gaussian noise
Proposed [59, 64, 73] Tani-Fantacci [71]
p=1, K=1 2M + (Ln + 1)M 3M + M2 log2M + 4Lw 3M
nonconjugate CAF
p? 1, K ? 1 2KMp2 +p(Ln + 1)M Kp2parenleftbig3M + M2 log2M + 4Lwparenrightbig
conjugate CAF
p? 1, K ? 1 KMp(p+ 1) +p(Ln + 1)M Kp2+p2 parenleftbig3M + M2 log2M + 4Lwparenrightbig
compute the (stationary) autocorrelation at any lag we need M multiplications, and there
are Ln + 1 lags to be considered. This yields a count of 2M + (Ln + 1)M multiplications
for the proposed test (4.31) for single antenna and single pair (?,?), and a count of 2MK +
(Ln + 1)M for single antenna and K pairs (?k,?k) (ignoring computations for (4.24) given
estimates of correlations). For the proposed multi-antenna detectors (4.23) or (4.29) with p
antennas, we have to calculate p2 CAFs (cyclic auto- and cross-correlations) and p stationary
autocorrelation functions at Ln +1 lags, yielding a count of KMp(2p)+p(Ln +1)p complex
multiplications for (4.23) and a count of KMp(p+1)+p(Ln+1)p multiplications for (4.29).
For the test of [59] described in Sec. 2.4, one needs 2M multiplications to calculate the
CAF at given ? and ?, (M/2)log2(2M) multiplications for the FFT involved (to compute the
cyclic periodogram), and M+4Lw multiplications for smoothing the two cyclic periodograms.
This yields the number of complex multiplications needed as 3M + M2 log2 M + 4Lw for a
single pair (?,?). The extension to p > 1 given in [73] requires p2parenleftbig3M + M2 log2 M +4Lwparenrightbig
complex multiplications for nonconjugate CAF-based test and p2+p2 parenleftbig3M + M2 log2 M +4Lwparenrightbig
multiplications for conjugate CAF-based test for a single pair (?,?); extension to K pairs
(?k,?k) increases the multiplication count by a factor of K. Using the results of [71, Sec. IV-
C], the test (2.30) for a single pair (?,?) requires 3M complex multiplications; its extension
to K pairs (?k,?k) is not available in [71, Sec. IV-C].
4.7 Simulation Examples
We now present some computer simulation examples using OFDM and GMSK signals
to illustrate our detectors and to compare their performance and computational complexity
69
with some existing detectors. In all presented examples noise variance/correlation function
is assumed to be unknown.
4.7.1 Example 1: Analytical Threshold Selection, p = 1, K = 1, Colored Gaus-
sian Noise
The proposed test as well as tests of [59, 64] rely on asymptotic distribution of the
test statistic under H0 in order to calculate the test threshold for a specified false-alarm
probability Pfa. How accurate is this distribution for finite data lengths? Parameters (L,?)
of the smoothing Kaiser window (see Sec. 2.4) have a strong influence on this distribution
(whereas there is no such issue with our approach). Ref. [64] has extensively used (L,?) =
(2049,10) and it is claimed therein that for M ? 1000, it is quite accurate. We generated
scalar (p = 1) colored Gaussian noise by passing white zero-mean circularly symmetric
complex-Gaussian noise through a 3-tap linear filter with coefficients {0.3,1,0.3}. Fig. 4.1(a)
with (L,?) = (2049,1) and Fig. 4.1(b) with (L,?) = (65,1) show the actual and design
Pfa?s based on M = 2000 and 2000 Monte Carlo runs where the thresholds were based on
?22 distribution of the respective test statistics. It is seen that for [59, 64], (L,?) = (65,1)
works much better than the other choice; the choice (L,?) = (2049,10) was poor also (not
shown). On the other hand the proposed approach needs no such parameters but needs Ln
which was picked to be 5, and works quite well.
4.7.2 Example 2: Single Antenna Receiver, Colored Gaussian Noise, K =1 or
4, OFDM PU signal (IEEE 802.11a/g)
The primary signal is an OFDM signal satisfying IEEE 802.11a/g WLAN standard with
number of subcarriers Nc = 64 and cyclic prefix of length Ncp = 16. The number of OFDM
symbols were taken to be Nofdm = 50 or 250 with the corresponding observation size of
M = Nofdm(Nc +Ncp) = 4000 or 20000. The subcarrier modulation was 64-QAM. The false
alarm probability is 0.01 and all results are averaged over 2000 Monte Carlo runs. For PU
signal detection we picked either K = 1 with ? = 1/(Nc + Ncp) and ? = Nc, or K = 4 with
70
? = 1/N,2/N, and ? = ?Nc. We used colored Gaussian noise as in Example 1. The OFDM
signal was passed through a constant gain channel. In applying the proposed test we set
Ln = 5. In Fig. 4.2 we compare the theoretical performance based on (4.28) and (4.36) with
the simulations-based performance for the proposed detector when p = 1 and the channel is
unit gain. It is seen that the agreement between theory and simulations is excellent.
4.7.3 Example 3: Multi-antenna Receiver (p = 1,2,4,8), OFDM PU signal
(IEEE 802.11a/g), Colored Gaussian Noise, Rayleigh flat-fading channel,
K =1,2,4
We consider the example of Sec. 4.7.2 (64 subcarrier OFDM signal from IEEE 80211a/g
WLAN standard) except with variable number of antennas (p), Rayleigh flat-fading sub-
channels and independent colored Gaussian noise at each antenna. For PU signal detection
we picked either K = 1 with ? = 1/(Nc + Ncp) and ? = Nc, or K = 4 with ? = 1/N,2/N
and ? = ?Nc, or K = 2 with two different combinations of cycle frequency and lags:
(?,?) = (1/N,Nc),(2/N,Nc) or (?,?) = (1/N,Nc),(1/N,?Nc). In Figs. 4.3 and 4.4 we
compare our proposed approach (Ln = 5) with that of [59], [64] or [73] (modified to handle
nonconjugate CAFs, an extension of [59] to multiple antenna receiver) via simulations for
M = 20000 (250 OFDM symbols) based on 2000 runs; for [59, 64, 73], we used a Kaiser
smoothing window with parameters (Lw,?) = (65,1). It is seen in Figs. 4.3 and 4.4 that for
K = 4 and any value of p, the approaches of [59, 64, 73] yield a much larger value of Pd than
the design value 0.01 of Pfa at very low SNRs (-30 dB) whereas there is no such problem
with our proposed approach; as SNR ? 0, H1 becomes H0. This reflects the discussion of Sec.
4.7.1 regarding analytical threshold selection: our proposed detector does a much better job
of yielding the design Pfa compared to the approaches of [59, 64, 73]. It is seen from Figs.
4.3 and 4.4 that the proposed approach is slightly superior (higher Pd for a given SNR) to
[59, 64, 73] for every value of p and K = 1 and 2, discounting K = 4 for which Pfa is not
maintained by [59, 64, 73]. Note also that there is an SNR advantage of about 2 dB (same
Pd with a 2dB smaller SNR) in using K = 4 over K = 1, and about 1 dB in using K = 2
71
over K = 1, when p = 1, and a somewhat smaller advantage when p is larger. On the other
hand, for a fixed value of K, each increase in the value of p by a factor of 2 (i.e. p from 1 to
2, or from 2 to 4, etc.), yields an SNR advantage of about 2dB.
Using the results of Table 4.1, we find that the approaches of [59, 64, 73] need 2.54, 3.05,
3.63 and 4.16 times more complex multiplications than the proposed approach for p =1, 2,
4 and 8, respectively, when K = 1, and 2.54, 3.05, 3.63 and 4.16 times more complex
multiplications for p =1, 2, 4 and 8, respectively, when K = 4.
In Fig. 4.5 we compare our proposed detector with that of [76] under the same conditions
as for Fig. 4.4(a). We implemented the detector of [76, Secs. III, V] and for our approach, we
used a single lag and cycle frequency pair at (?,?) = (1/N,Nc). The test threshold for the
detector of [76] was computed via simulations (based on 50000 runs) to achieve Pfa = 0.01.
It is seen from Fig. 4.5 that the proposed approach significantly outperforms [76].
4.7.4 Example 4: Single Antenna Receiver, Colored Gaussian Noise, K =1,
OFDM PU signal (WiMAX)
We now turn to comparison with the Tani-Fantacci test (2.30) [71]. Here we used an
OFDM PU signal (similar to examples in [71]) satisfying WiMAX standard with QPSK
subcarrier modulation, Nc = 256, Ncp = 64, M = (Nc + Ncp)256 = 81920, L = Ncp + 2,
? = 1/N, ? = Nc, Sbin = 4 (see Sec. 2.4); for shorter observation lengths (such as M = 2000),
performance of [71] is erratic and poor, and is not shown here. In Fig. 4.6 we compare Pd
performance (as a function of SNR for Pfa = 0.05) of the proposed test (4.31) with the
Tani-Fantacci test (2.30) [71] under colored Gaussian noise (chosen as in Sec. 4.7.1). It is
seen from Fig. 4.6 that the Tani-Fantacci test (2.30) yields a much lower actual Pfa than the
design value of 0.05 when the theoretical expression provided in [71] is used; this results in
a poorer Pd performance. We also selected the test threshold based on simulations (10000
Monte Carlo runs). This leads to an improved performance but it is still inferior to our
72
proposed test (about 3-4 dB SNR penalty to achieve the same performance). Simulation-
based threshold selection for colored Gaussian noise is not practical since the noise correlation
will not be known in practice and since the distribution of the test statistic is dependent
upon the noise correlation (even asymptotically). Note that the analysis in [71] assumes
white Gaussian noise.
4.7.5 Example 5: Multi-Antenna Receiver (p = 4), GSM GMSK PU signal,
colored Gaussian Noise, K = 1
Now we consider a GMSK PU signal (GSM standard) with N=2 samples per symbol.
We seek to detect the PU signal using conjugate CAF at the normalized cycle frequency
? = 14 and lag ? = 0. The channel was flat Rayleigh fading, with circularly symmetric
complex Gaussian additive noise. Fig. 4.7 shows the Pd performance (as a function of SNR
for Pfa = 0.01) of our proposed approach and that of [73] for Rayleigh fading channel based
on 2000 runs when number of antennas p = 4 and M=540 (1 ms duration). For the approach
of [73], we used Kaiser window with parameters (?,Lw) = (1,65). As seen from Fig. 4.7,
the performance of our proposed detector is slightly better than that of [73]. Using the
results of Table 4.1 for conjugate CAFs, we find that the [73] needs 1.82 times more complex
multiplications than the proposed approach for M =540; thus the proposed approach is to
be preferred when both performance and computational complexity are considered.
4.7.6 Example 6: Computational Complexity
Here we compare computational complexity of various schemes using the analysis of Sec.
4.6. The complex multiplications counts are shown in Fig. 4.8 for varying sizes of observation
length M when nonconjugate CAFs are used. It is seen that the computational complexity
of the proposed approach is lower than that of [59, 64] in the single antenna case, and that
of [73] for multiple antennas (p=4), by a factor of 2.0 to 6. (The length L of the frequency-
domain Kaiser smoothing window needed in [59, 67, 64] was picked to be L = 65 whereas in
73
[64] it has been picked to be L = 2049 regardless of the sample size M which would result
in still larger computational load for the approaches of [59, 64, 73].)
4.8 Conclusions
Detection of cyclostationary PU signals in colored Gaussian noise for cognitive radio
systems was considered based on looking for single or multiple cycle frequencies at single or
multiple time lags in the CAF of the noisy PU signal. We explicitly exploit the knowledge
that under the null hypothesis of PU signal absent, the measurements originate from colored
Gaussian noise with possibly unknown correlation. Our formulation allows us to simplify
the spectrum sensing detector, and obviates the need for estimating an unwieldy covariance
matrix needed in the Dandawate-Giannakis [59, 64] and related approaches. We considered
both single and multiple antenna receivers and both nonconjugate and conjugate CAFs.
A performance analysis of the proposed detectors was carried out. Supporting simulation
examples were presented to compare our detectors with those of [59], [73], [64] and [71].
Our proposed approaches are computationally cheaper than the Dandawate-Giannakis and
related approaches [59], [64], [73] while having quite similar detection performance for a fixed
false alarm rate. Our proposed approach significantly outperforms [71].
74
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.05
0.1
0.15
0.2
0.25
Design Pfa
Actual P
fa
M=2000; 2000 runs; Kaiser window (2049,1); p=1; CGN
Dandawate?Giannakis
proposed
theory
(a)
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0.02
0.04
0.06
0.08
0.1
0.12
Design Pfa
Actual P
fa
M=2000; Kaiser window (65,1); CGN; p=1; 2000 runs
Dandawate?Giannakis
proposed
theory
(b)
Figure 4.1: Example 1. Analytical threshold selection performance: colored Gaussian noise,
M = 2000, 2000 Monte Carlo runs. The approach labeled ?Dandawate-Giannakis? is used
in [59, 67, 64].
75
?25 ?20 ?15 ?10 ?5 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Pfa = 0.01; M=20000 or 4000; ACGN channel; p=1; K=1 or 4; 2000 runs
Simulation: 1 cycle freq, 1 lag, 250 symbols
Theory: 1 cycle freq, 1 lag, 250 symbols
Simulation: 2 cycle freqs, 2 lags, 250 symbols
Theory: 2 cycle freqs, 2 lags, 250 symbols
Simulation: 1 cycle freq, 1 lag, 50 symbols
Theory: 1 cycle freq, 1 lag, 50 symbols
Simulation: 2 cycle freqs, 2 lags, 50 symbols
Theory: 2 cycle freqs, 2 lags, 50 symbols
Figure 4.2: Example 2: Performance analysis: theoretical and simulation-based performance
comparison of probability of detection Pd versus SNR for the proposed single-antenna spec-
trum sensing: OFDM signal, non-random channel with unit gain, 2000 runs, Pfa = 0.01,
p = 1, K = 1 or 4, colored Gaussian noise, M = 4000 or 20000.
76
?30 ?25 ?20 ?15 ?10 ?5 0 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Pfa = 0.01; M=2000; p=1; Rayleigh fading;2000 runs
DG/L: 1 ?, 1 ?
Prop: 1 ?, 1 ?
DG/L: 1 ?, 2 ?
Prop: 1 ?, 2 ?
DG/L: 2 ?, 1 ?
Prop: 2 ?, 1 ?
DG/L: 2 ?, 2 ?
Prop: 2 ?, 2 ?
(a)
?30 ?25 ?20 ?15 ?10 ?5 0 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Pfa = 0.01; M=2000; p=2; Rayleigh fading;2000 runs
DG/L: 1 ?, 1 ?
Prop: 1 ?, 1 ?
DG/L: 1 ?, 2 ?
Prop: 1 ?, 2 ?
DG/L: 2 ?, 1 ?
Prop: 2 ?, 1 ?
DG/L: 2 ?, 2 ?
Prop: 2 ?, 2 ?
(b)
Figure 4.3: Example 3: Simulation-based comparison of Pd versus SNR with variable number
of antennas: OFDM signal, M = 20000 (250 OFDM symbols), Rayleigh flat-fading channel,
2000 runs, Pfa = 0.01, p = 1 in (a) and p = 2 in (b), K = 1, 2 or 4, colored Gaussian noise.
The approach labeled ?DG/L? refers to the ones based on [59, 64, 73] and that labeled
?prop? refers to our proposed detector. the labels ??? and ??? refer to cycle frequency and
lag, respectively.
77
?30 ?25 ?20 ?15 ?10 ?5 0 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Pfa = 0.01; M=2000; p=4; Rayleigh fading;2000 runs
DG/L: 1 ?, 1 ?
Prop: 1 ?, 1 ?
DG/L: 1 ?, 2 ?
Prop: 1 ?, 2 ?
DG/L: 2 ?, 1 ?
Prop: 2 ?, 1 ?
DG/L: 2 ?, 2 ?
Prop: 2 ?, 2 ?
(a)
?30 ?25 ?20 ?15 ?10 ?5 0 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Pfa = 0.01; M=2000; p=8; Rayleigh fading;2000 runs
DG/L: 1 ?, 1 ?
Prop: 1 ?, 1 ?
DG/L: 1 ?, 2 ?
Prop: 1 ?, 2 ?
DG/L: 2 ?, 1 ?
Prop: 2 ?, 1 ?
DG/L: 2 ?, 2 ?
Prop: 2 ?, 2 ?
(b)
Figure 4.4: Example 3: Simulation-based comparison of Pd versus SNR with variable number
of antennas: OFDM signal, M = 20000 (250 OFDM symbols), Rayleigh flat-fading channel,
2000 runs, Pfa = 0.01, p = 4 in (a) and p = 8 in (b), K = 1, 2 or 4, colored Gaussian noise.
The approach labeled ?DG/L? refers to the ones based on [59, 64, 73] and that labeled
?Proposed? refers to our proposed detector.
78
?30 ?25 ?20 ?15 ?10 ?5 0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Pfa = 0.01; M=2000; p=4; Rayleigh fading;2000 runs
Proposed
Axell?Larsson
Figure 4.5: Example 3: Simulation-based comparison of Pd versus SNR with four antennas:
OFDM signal, M = 20000 (250 OFDM symbols), Rayleigh flat-fading channel, 2000 runs,
Pfa = 0.01. The approach labeled ?Axell-Larsson? is from [76] and that labeled ?Proposed?
refers to our proposed detector.
?30 ?25 ?20 ?15 ?10 ?5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Pfa = 0.05; (256+64)*256 samples; Colored Gaussian noise;1000 runs
Tani?Fantacci: Simulation?based threshold
Tani?Fantacci: Theoretical threshold
Proposed
Figure 4.6: Example 4: Simulation-based comparison of Pd versus SNR with single antenna
under non-random channel with constant gain: Pfa = 0.05, p = 1, K = 1, M = 81920,
2000 Monte Carlo runs, OFDM signal (number of subcarriers Nc=256, cyclic prefix length
Ncp = 64), colored Gaussian noise. The approach labeled ?Tani-Fantacci? is that of [71].
79
?20 ?15 ?10 ?5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
P d
Pfa=0.01; M=540; p=4; CGN; GMSK signal;2000 runs
Proposed
Zhong et al
Figure 4.7: Example 4: Simulation-based comparison of probability of detection versus SNR
for GMSK signal with 2 samples per symbol, with p = 4 antennas under Rayleigh flat-fading,
colored Gaussian noise and M = 540: Pfa = 0.01, 2000 runs. The approach labeled ?Zhong
et al? is from [73].
0 1000 2000 3000 4000 5000 6000 7000 8000 900010
2
103
104
105
106
107
number of samples
# of complex multiplications
Computational complexity
Proposed: p=1, K=1
Proposed: p=4, K=1
Dandawate?Giannakis: Lw =65, K=1
Zhong et al: p=4, Lw =65, K=1
Proposed: p=1, K=4
Proposed: p=4, K=4
Lunden et al: Lw =65, K=4
Zhong et al: p=4, Lw =65, K=4
Figure 4.8: Example 5: Computational complexity of different spectrum sensing algorithms
for varying sizes of observation length M, (L,?) = (65,1) when nonconjugate CAFs are
used.
80
Chapter 5
Design of Sensing and Power Allocation Strategies for Energy Aware Multi-Channel
Cognitive Radio Networks
Frequency spectrum and energy are two key resources of green cognitive radio network-
s with battery-powered wireless terminals. The issues of how to utilize sparse frequency
spectrum and limited energy resource pose challenges to the design of sensing and power
allocation strategies that affect both throughput and energy consumption. In this chapter,
we first construct an utility function that incorporates throughput as reward and energy
consumption as cost for a time slotted multi-channel cognitive radio network. An optimiza-
tion problem to maximize the utility function is formulated involving optimization of both
sensing parameters (sensing duration, local test threshold) and power allocation strategy.
The problem is non-convex, however, we decouple it into two separate convex problems and
propose an iterative algorithm to obtain a suboptimal solution. The simulation results show
that our iterative algorithm converges fast and performs better than a ?only power allocation
optimization? approach and an existing approach that ignores energy efficiency.
5.1 Introduction
Thegoalofcognitiveradioistoalleviatetheshortageoffrequencyspectrumforemerging
wireless communication applications. To maximize spectrum utilization while maintaining
an upper bound on the interference to licensed users, spectrum sensing is a key step of
cognitive radio network (CRN) that identifies spectrum opportunity. Existing spectrum
sensing techniques [56] include matched filter, energy detector [22] and feature detector
[74]. On the other side, energy efficiency is an objective of green cognitive radio networks
with battery for power supply [77] and has recently attracted more attention than before.
81
However, most current research on CRN focuses on sensing accuracy and maximization of
throughput [56, 22, 78, 79]. This motivates the need for designing advanced power-aware
spectrum sensing, MAC protocols and cross layer approaches such as PHY-MAC design.
In recent years, energy-aware spectrum sensing and access related research has been
gaining importance when power limitation becomes a key feature of green CRN. In [80, 81],
energy efficient sequential sensing is studied for multi-channel PU (primary user) networks.
However, since SU (secondary user) can only sense and transmit on a single PU channel at
a time, power allocation issues are not involved. An energy efficient power allocation and
transmission duration design is proposed in [82] for multi-channel simultaneous transmission.
It is assumed that the activity states (active or idle, ON or OFF) of slotted PU channels are
independent and identically distributed. However, perfect sensing is assumed in [82], thus
spectrum sensing optimization is not involved.
In [22], using periodic sensing, a sensing-throughput trade off optimization problem is
formulated and the optimal sensing duration is proven to exist based on energy detection.
Joint optimization of soft-decision cooperative spectrum sensing, channel access strategy and
power allocation is proposed in [79] for an overlay multichannel cognitive radio networks.
The optimization aims to maximize the secondary users? sum instantaneous throughput with
constraints on interference to primary users. Ref. [78] investigates a single SU and multiple
channel CRN where the SU can sense and access multiple PU channels simultaneously. To
maximize the overall throughput, an optimization problem involving joint optimization of
sensing duration, sensing detector threshold and transmit power allocation is formulated
under constraints on interference to the PU network and total transmit power over multiple
channels. However, energy consumption is considered in none of [22, 79, 78].
Energy consumption is considered as a constraint in constrained throughput optimiza-
tion in [80, 83]. In [80], a limited energy budget is allocated to sensing, probing for channel
quality and data transmission, and the variables associated with these factors are then joint-
ly optimized to maximize the achievable average throughput of the secondary users in a
82
CRN composed of multiple PU channels. In [83] distributed sensing and access strategies
for opportunistic spectrum access under constraints on energy consumption by the SUs and
interference probability to the PU, is formulated as a sequential decision making problem
in the framework of partially observable Markov decision processes for a slotted PU net-
work. However, the sensing duration optimization and power allocation are not discussed in
[80, 83].
Energy consumption is incorporated into the optimization problems in [84, 82, 85, 81].
Ref. [84] considers designing optimal sensing policy to maximize the expected reward that
rewards potential transmission throughput and penalizes energy consumption in sensing and
transmission. The optimization problem is formulated under a partially observable Markov
decision process formulation, but power allocation for SU transmission is not considered in
[84]. An energy efficient power allocation and transmission duration optimization is proposed
in [82] for multi-channel simultaneous transmission in a slotted CRN. However, perfect sens-
ing is assumed in [82], thus spectrum sensing optimization is not involved. Ref. [85] studies
how to choose sensing duration to balance energy consumption and system throughput. An
utility function incorporating throughput as reward and energy consumption as penalty is
formulated to find optimal sensing duration under a collision probability constraint for PU
protection. However, only single PU channel is considered and power allocation issues are
omitted. Maximizing energy efficiency is considered in [81] for a CRN in which a secondary
user sequentially senses multiple licensed PU channels before it transmits on one idle PU
channel. Energy consumption includes consumption during sensing and transmission. Sens-
ing strategy deciding when to stop sensing and transmitting, and power level allocation for
transmission are jointly considered. The optimization problem is formulated as a sequential
decision-making problem. An alternative parametric formulation of the original optimization
problem is proposed to solve the posed problem, which rewards throughput and penalizes
83
energy consumption. However, the sensing duration and sensing test threshold are not op-
timized. Moreover, power allocation for multiple channel simultaneous transmission is not
considered as the SU is allowed to transmit only over one channel.
For balancing throughput maximization and energy consumption minimization in mul-
tiple channels CRN, joint design of spectrum sensing and power allocation has rarely been
explored in the literature; this is the topic of this chapter. Based (partially) on the ap-
proaches of [85, 81] and [82], we consider both sensing and power allocation optimization for
simultaneous multiple channel sensing and transmission. We aim to design spectrum sensing
parameters (sensing duration, local test threshold) and power allocation strategy to balance
throughput and energy consumption.
In this chapter, we consider a CRN composed of a single SU and multiple PU chan-
nels. We assume that the primary users access the PU channels in a synchronous time slot
structure [86, 87]. The activity states of the PU channels are independent from each other
and are constant during one slot. One SU can sense all PU channels simultaneously with
wide-band spectrum sensor at the beginning of each slot. Then the SU can access multiple
idle PU channels using advanced spectrum access technology such as OFDM.
Sensing duration, local test threshold and power allocation will affect both throughput
and energy consumption. Long sensing duration means high detection accuracy, thus prob-
ability of collision with PU and related energy consumption for collided transmission are
reduced. However, longer sensing duration also means shorter potential transmission time
for SU. How to balance power allocation among different idle PU channels determines how
effectively the temporal transmission opportunity and limited power budget is exploited. By
treating effective throughput as reward and energy consumption as cost, we first propose an
utility function that incorporates throughput and energy consumption with a price weight.
The energy consumption includes sensing and transmission energy consumption where we
only consider effective transmission throughput without collision with PU. The sensing du-
ration, local test threshold and power allocation strategy are the optimization parameters to
84
maximize the utility function. The utility function is non-convex, however, we decouple it
into two separate convex problems and propose an iterative algorithm to obtain suboptimal
solutions. Both sensing parameters (sensing duration, local test threshold) and power allo-
cation are optimized under flat fading channel condition, total transmission power constraint
and interference constraint to protect PU activity.
Notation: Complex scalar hpm denotes the flat fading coefficient of subchannel m
from the PU on subchannel m (if active) to the SU sensing subchannel m while hsm is the
flat fading coefficient of subchannel m from the SU transmitter to SU receiver on subchannel
m. In the following hpm and hsm are assumed to be known before the SU senses subchannel
m. ?2xm is the power of received PU signal on subchannel m. ?2 is the noise power on each
subchannel. ?2 and ?2xm are assumed to be perfectly known a priori. Nonnegative real ?
denotes the energy price and [x]+ := max(x,0). x? is the optimal value for x optimizing
some function of x. Ptot is the total transmit power constraint. Pdmin denotes the minimum
detection probability requirement and Pfmax denotes the maximum allowable false alarm
probability for spectrum sensing. Tsmin is the lower bound on sensing duration Ts. P{?}
denotes probability of an event {?}.
5.2 System Model
In this chapter, we consider a CRN composed of single SU and Mp PU channels. We
assume the primary users access the PU channels in a synchronous time slotted structure
of duration T [86]. The activities of PU on each channels stay either idle or busy during
one slot, and are independent from each other. The SU senses and accesses PU channels
in a slotted manner synchronized to the PU network. We assume the maximum allowable
sensing duration is T/2, which is reasonable considering sparsity of transmission duration
for SU. At the beginning of each slot, SU senses all Mp PU channels with duration Ts. If the
sensing result shows the channel is idle, SU will access the channel in the rest of the slot and
the transmission time is T ?Ts. If the sensing result shows the channel is busy, SU will not
85
access the channel in this slot and will wait for the next slot to sense again. All the channel
gains either between PU and SU or between SU transmitter and receiver pair are assumed
to be known a priori. Knowledge of the average noise power and signal power of PU on each
PU channel is assumed to be available.
5.2.1 Primary network
Consider a cognitive radio network consisting of Mp PU channels. For simplicity we
assume that all channels have the same bandwidth (this assumption is not essential) and
each of them occupies a bandwidth of B Hz. The primary users access these channels in
a synchronized slotted manner, each slot of duration T. Assume that the activity states of
PU on each channel stays constant, either idle or busy, and are independent on different PU
channels and time slots. Let S = [s1,s2,....,sMp] denote the channel activity state vector, of
which sm is the state of mth PU channel on a single slot; sm = 0 and sm = 1 denote mth
channel is idle or busy, respectively. Let ?0m and ?1m be the stationary probabilities that
the mth channel is idle and busy, respectively:
?0m = P{sm = 0},
?1m = P{sm = 1}. (5.1)
5.2.2 Spectrum Sensing
Similar to [78], with wide-band spectrum sensing, there are N (time-)samples for each
PU channel. Let ym[n] be the received signal at mth (1 ? m ? Mp) PU channel at the nth
(1 ? n ? N) sampling instant. A binary hypothesis test is performed by the SU to decide
whether any PU activity is present
H1m : ym[n] = hpmxm[n] +wm[n],
H0m : ym[n] = wm[n]. (5.2)
86
H1m and H0m denote hypotheses that the activity state of PU at the mth channel is sm = 1
and sm = 0, respectively. xm[n] is the PU signal at mth channel and nth sampling instant,
which is assume to be i.i.d. white with mean zero and variance ?2xm. The noise wm[n]
is assumed to be circularly symmetric complex Gaussian distributed with zero mean and
variance ?2, for all Mp PU channels. Complex-valued flat channel fading coefficient hpm stays
constant during the entire slot. The average SNR of PU signal on mth channel received by
SU during one slot is deduced as ?m = |hpm|2?2xm?2 .
With energy detector [22], the sensing test statistic Lm is defined as
Lm = 1N
Nsummationdisplay
n=1
|ym[n]|2. (5.3)
Suppose the sensing frequency (number of samples per second) is fs. Based on [22], the false
alarm probability Pfm and detection probability Pdm are given by
Pfm = Q
parenleftbiggparenleftBig?
m
?2 ?1
parenrightBigradicalbig
Tsfs
parenrightbigg
, (5.4)
Pdm = Q
parenleftBiggparenleftBig
?m
?2 ??m ?1
parenrightBigradicalBigg Tsfs
2?m +1
parenrightBigg
(5.5)
where ?m is local test threshold at mth channel detector and Q(x) = 1?2? integraltext?x e?t22 dt. Based
on (5.4) and (5.5), we can express Pfm as a function of Ts and Pdm as
Pfm = Q
parenleftBigradicalbig
2?m +1Q?1(Pdm)+?m
radicalbig
Tsfs
parenrightBig
. (5.6)
Based on the energy detector?s result, we acquire the estimated channel states as ?S =
[?s1,?s2,...,?sMp], of which ?sm = 0 and ?sm = 1 represent the event mth PU channel is detected
87
as idle or busy, respectively. The relation between sm and ?sm is given by
P{?sm = 0|sm = 0} = 1?Pfm, P{?sm = 1|sm = 0} = Pfm, (5.7)
P{?sm = 0|sm = 1} = 1?Pdm, P{?sm = 1|sm = 1} = Pdm. (5.8)
For protection of PU at each PU channel, the detection probability at the SU is required
to have a lower bound Pdmin. Similarly, we impose an upper bound Pfmax on the false alarm
probability. In a realistic CRN scenario, it is required that Pfmax ? 0.5 and (hopefully)
Pdmin ? 0.9. In this chapter, we assume that the spectrum sensing detector is designed is
to keep the detection probability to be equal to the minimum requirement Pdmin in order to
protect PU activities, resulting in an upper bound of 1?Pdmin on the collision probability
(both SU and PU transmit simultaneously). On the other hand, we aim to minimize the false
alarm probability Pfm with the upper bound Pfmax to obtain more transmission opportunity
for SU when PU is inactive on mth channel. Therefore, the relevant constraints are as follows
Pdm = Pdmin and Pfm ? Pfmax (5.9)
where m = 1,2,...,Mp.
We first clarify the range of sensing duration Ts. First we need to demonstrate that
Q(x) is a monotonically decreasing function with Q(0) = 0.5. With a fixed Pdmin, it follows
from (5.6) that the constraint Pfm ? Pfmax < 0.5 implies
?m
radicalbig
Tsfs +Q?1(Pdmin)radicalbig2?m +1 > 0, m = 1,...,Mp. (5.10)
The minimum sensing duration Tsm,min for the mth PU channel that satisfies (5.9) is given
by
Tsm,min = (Q?1(Pdmin))22?m +1(?
m)2fs
, m = 1,2,...,Mp. (5.11)
88
As noted earlier in the chapter, the upper bound on Ts is T/2. For those PU channels
that need Tsm,min > T/2, we would not transmit and allocate any power to them since their
sensing requirement can not be met within the T/2 sensing duration. Let the set O denote
the set of PU channels for which Tsm,min ? T/2, and let the size (number of elements) of
O be M. For convenience, we (re-)index the PU channels in O as ranging over 1,2,...,M.
Consequently, the lower bound for Ts is deduced as
Tsmin = max
1?m?M
(Tsm,min). (5.12)
with Ts picked to be Ts ? (Tsmin, T2].
5.2.3 Spectrum Access
The access strategy is that the SU will only transmit on the mth PU channel that
is detected as idle (1 ? m ? M), and the SU?s transmission on this channel is allocated
transmit power Pm ? 0. Under the channel gain |hsm|2 and channel noise with power ?2, the
potential achievable rate Rm bps/Hz (instantaneous capacity) on the mth idle PU channel
is given by
Rm = Blog2(1+ |hsm|
2Pm
?2 ). (5.13)
The sum of SU?s transmission power over various channels has an upper bound of Ptot. Let
Pm denote the power allocated to the mth PU channel for SU?s transmission. Then we must
have
Msummationdisplay
m=1
Pm ? Ptot. (5.14)
89
5.3 Problem Formulation
5.3.1 Throughput and Energy Consumption
In this chapter, we consider energy consumed during sensing and transmission, and
the effective throughput when a PU?s absence on the mth channel is correctly detected.
Since PU?s activity on M channels are independent, we analyze the energy consumption
and throughput on the mth PU channel under the following four cases. Assume the power
consumed during sensing is Ps.
? Case 1 (sm,?sm) = (0,0): Under hypothesis H0, PU is detected as absent, and the
SU data is successfully transmitted after sensing. The throughput is (T ?Ts)Rm. The
energy consumed due to sensing is PsTs and energy consumed during transmission is
Pm(T ?Ts). The probability for this case is ?0m(1?Pfm).
? Case 2 (sm,?sm) = (0,1): Under hypothesis H0, PU is detected as present due to false
alarm. Based on the spectrum access policy, there would be no SU transmission. As
a result the effective throughput and energy consumed during transmission are both
0. The energy consumed during sensing is PsTs, and the probability for this case is
?0mPfm.
? Case 3 (sm,?sm) = (1,0): Under hypothesis H1, PU is detected as absent (inactive).
The SU would transmit data but it would collide with the PU signal. We assume
the SU receiver can not correctly decode the collided data, so there is no effective
throughput. The energy consumed during sensing and collided transmission is PsTs
and Pm(T ?Ts), respectively. The probability for this case is ?1m(1?Pdm).
? Case 4 (sm,?sm) = (1,1): Under hypothesis H1, PU is detected as present (active).
Thus no SU data is transmitted after sensing. The throughput and energy consumed
are the same as in case 2. The probability for this case is ?1mPdm.
90
Based on the above four cases, the achievable throughput during one slot duration is
given by
C =
Msummationdisplay
m=1
Rm(T ?Ts)?0m(1?Pfm) (5.15)
and the potential energy consumption during one slot is deduced as
E = PsTs +
Msummationdisplay
m=1
Pm(T ?Ts)[?0m(1?Pfm)+?1m(1?Pdm)]. (5.16)
We are interested in choosing Ts,{?m},{Pm} to balance throughput C maximization
and energy consumption E minimization. Inspired by [84, 85, 81], we aim to formulate an
utility function that incorporates throughput as reward and energy consumption as cost.
Assuming an unit energy price ?, the energy consumption cost is given by ?E while the
throughput reward is C. With high energy price ?, energy consumption minimization gains
more importance than throughput, wheres with low ?, we pay more attention to throughput
maximization whilereducingtheimportanceofenergyconsumption. Tobalancethethrough-
put and energy consumption, we define an utility function UG(Ts,{?m},{Pm}) = C ? ?E,
which should be maximized to balance throughput maximization and energy consumption
minimization:
UG(Ts,{?m},{Pm})
=
Msummationdisplay
m=1
bracketleftBig
(Blog2
parenleftBig
1+ |hsm|
2Pm
?2
parenrightBig
??Pm)?0m(1?Pfm)
??Pm?1m(1?Pdm)
bracketrightBig
(T ?Ts)??PsTs. (5.17)
We wish to find optimal energy-aware sensing parameters and power allocation set (Ts,
{?m}, {Pm}) to maximize the utility function subject to (s.t.) all relevant constraints. This
91
optimization problem is
Problem P1: max
Ts,{?m},{Pm}
UG(Ts,{?m},{Pm})
s.t. Pdm = Pdmin ?m
Pfm ? Pfmax ?m
Pm ? 0 ?m
?Mm=1Pm ? Ptot. (5.18)
[We note that this is the parametric formulation in [81] where theultimate goal is to maximize
energy efficiency defined as bits per sec per Joule. In this chapter we use UG(Ts,{?m},{Pm})
as defined above to retain more flexibility compared to [81] where an unique price ? leads
to maximization of their energy efficiency metric.] This utility function is non-convex in
Ts,{?m},{Pm}, thus an optimal set (Ts,{?m},{Pm}) may exist but there is no obvious effec-
tive way to find it. However, we can decouple this problem into two separate optimization
problems each one of them is convex. Based on the separate convex problems, we propose an
iterative optimization approach to find locally optimal solution (Ts,{?m},{Pm}). The two
optimization problems are detailed in the following section.
5.4 Iterative Algorithm for optimizing Sensing Parameters and Power Alloca-
tion
5.4.1 Power Allocation Optimization
With a fixed (Ts,{?m}) satisfying Pdm = Pdmin and Pfm ? Pfmax, it is easy to prove
that UG(Ts,{?m},{Pm}) is strictly convex in {Pm} (m = 1,...,M. With fixed Ps, Ts, ? and
T, the term ??PsTs and the coefficient T ? Ts in UG(Ts,{?m},{Pm}) are both constants.
Ignoring the fixed terms, we only consider the remaining part UP({Pm}) of the original utility
92
function, defined as
UP({Pm}) =
Msummationdisplay
m=1
bracketleftBig
(Blog2(1 + |hsm|
2Pm
?2 )??Pm)?0m(1
?Pfm)??Pm?1m(1?Pdm)
bracketrightBig
. (5.19)
The corresponding optimization problem is reformulated as
Problem P2: max
Pm
UP({Pm}) (5.20)
s.t. Pm ? 0,
Msummationdisplay
m=1
Pm ? Ptot
With ? denoting the (non-negative) Lagrange multiplier, the Lagrangian function L(?) to
be minimized is given by
L(?) =
Msummationdisplay
m=1
bracketleftBigparenleftBig
?Pm ?Blog2(1 + |hsm|
2Pm
?2 )
parenrightBig
?0m(1?Pfm)
+?Pm?1m(1?Pdm)
bracketrightBig
+?
parenleftbigg Msummationdisplay
m=1
Pm ?Ptot
parenrightbigg
. (5.21)
The second-order partial derivatives of the objective function are given by
?2UP({Pm})
?P2m =
?B?0m(1?Pfm)parenleftbig
?2
|hsm|2 +Pm
parenrightbig2 ln2, (5.22)
?2UP({Pm})
?Pm1?Pm2 = 0 ?m1 negationslash= m2 (5.23)
where 1 ? m,m1,m2 ? M. Since Pfm ? Pfmax < 1, it follows that the Hessian ?2UP({Pm})
is negative-definite, implying that UP({Pm}) is concave in {Pm} [88].
93
With a fixed ?, the optimal power allocation is obtained by solving ?L?Pm = 0 to obtain
Pm =
bracketleftbigg B?
0m(1?Pfm)
{?[?0m(1?Pfm)+?1m(1?Pdm)]+?}ln2
? ?
2
|hsm|2
bracketrightbigg+
, ?m = 1,...,M. (5.24)
Let{P?m}and ?? denote the optimal power allocation and optimal value of the Lagrange mul-
tiplier ?. then the optimal solution must satisfy the Karush-Kuhn-Tucker (KKT) conditions
[88]
Msummationdisplay
m=1
P?m ?Ptot ? 0, (5.25)
?? ? 0, (5.26)
??(
Msummationdisplay
m=1
P?m ?Ptot) = 0, (5.27)
bracketleftbigg
?? B(?2/|h
sm|2 +P?m)ln2
bracketrightbigg
?0m(1?Pfm)
+??1m(1?Pdm)+?? = 0 (5.28)
where m = 1,...,M.
The ?maximum? power allocation is {P?m}|??=0 (P?m is Pm given by (5.24) with ? re-
placed with ??), which is the optimal solution for problem P2 without any transmit power
constraints. As a result, when the total power constraint Ptot is not less than ?Mm=1P?m|??=0,
based on (5.27) and (5.26), ?? = 0 and the resulting optimal power allocation is {P?m}|??=0.
When Ptot < ?Mm=1P?m|??=0, based on (5.25), (5.26) and (5.27), the optimal ?? > 0 im-
plying that ?Mm=1P?m ? Ptot = 0. Note that ?Mm=1P?m ? Ptot is a subgradient of L w.r.t. ?,
??L = ?Mm=1Pm ?Ptot, evaluated at Pm = P?m. Since the optimal ?? minimizes the Lagrange
dual function, we must have ??L|?=??,Pm=P?m = 0. Since there is only single Lagrange multi-
plier ?, we use line search for the optimal ?? that satisfies ??L|?=??,Pm=P?m = 0. The transmit
94
power Pm is related to ? via (5.24). The lower limit for the line search is ? = 0 such that
??L > 0 and the upper-limit is any positive value for which ??L < 0.
5.4.2 Optimization of Sensing Parameters Ts and ?m
With fixed power allocation Pm (m = 1,...,M) that satisfiessummationtextPm ? Ptot and Pm ? 0,
we aim to find optimal sensing parameters Ts and {?m}. Since our sensing policy is to keep
detection probability Pdm = Pdmin ? 0.9 and minimize false alarm probability Pfm ? Pfmax <
0.5, the local test threshold {?m} is a function of Ts and Pdmin, based on (5.5). As a result,
we just have to optimize w.r.t. sensing duration Ts.
Set Pdm = Pdmin into UG(Ts,{?m},{Pm}) in problem P1 and reformulate it as below
UG(Ts,{?m},{Pm})
=
Msummationdisplay
m=1
bracketleftBigparenleftBig
Blog2(1+ |hsm|
2Pm
?2 )??Pm
parenrightBig
??0m(1?Pfm)(T ?Ts)
??Pm?1m(1?Pdmin)(T ?Ts)??PsMTs
bracketrightBig
(5.29)
where Pfm is defined in (5.6). With a fixed power allocation {Pm}, energy price ?, stationary
channel state distribution {?0m},{?1m} and channel noise power ?2, we define the following
constants for convenience to formulate the optimization problem for Ts:
c1m :=
bracketleftBig
Blog2
parenleftBig
1 + |hsm|
2Pm
?2
parenrightBig
??Pm
bracketrightBig
?0m, (5.30)
c2m := ?Pm?1m(1?Pdmin), (5.31)
c3 := ?PsM. (5.32)
Obviously, {c2m} and c3 are all positive. We will assume that the value of ? is such that
{c1m} is positive for all m. We need this condition for Lemma 1 to hold true. Intuitively,
95
if this condition is not satisfied, then we lose any rationale for SU transmission because c1m
represents the reward we obtain from balancing throughput and energy consumption during
transmission on the mth PU channel.
While {?m} can be expressed as a function of Ts and Pdmin based on (5.5), we transform
optimization in problem P1 into optimization of Us(Ts) that only relies on Ts, as follows:
Problem P3: max
Ts
Us(Ts) =
Msummationdisplay
m=1
[c1m(T ?Ts)(1?Pfm)
?c2m(T ?Ts)?c3Ts] (5.33)
s.t. Pfm ? Pfmax
Pdm = Pdmin
where m = 1,2,...,M and Pfm is a function of Ts with fixed Pdmin as in (5.6).
Now we prove that Us(Ts) is a concave function in Ts in its range (Tsmin,T/2]. First, we
define um as the utility function for the mth PU channel
um = c1m(T ?Ts)(1?Pfm)?c2m(T ?Ts)?c3Ts. (5.34)
Lemma 1 . The utility function um is a concave function in Ts ? (Tsmin, T2] .
Proof: Based on [22], Pfm in (5.6) is decreasing and convex in Ts ? (Tsmin, T2] when
Pfm ? Pfmax < 0.5. We calculate the second partial derivative of um w.r.t. Ts as
?2um
?T2s = 2c1m
?Pfm
?Ts ?c1m(T ?Ts)
?2Pfm
?T2s . (5.35)
96
Since c1m > 0 and Pfm is decreasing and convex in Ts, we have
?2um
?T2s < 0. (5.36)
Thus um is concave in Ts ? [Tsmin, T2] square
Since sum of concave functions is concave [88] and Us(Ts) =summationtextMm=1 um, it follows that Us(Ts)
is concave in Ts. Then we can use line search on [Tsmin, T2] to obtain the optimal Ts and
corresponding {?m} that satisfies Pdm|Ts,?m = Pdmin.
5.4.3 Iterative algorithm for power allocation and sensing parameters optimiza-
tion
First we use an initial sensing duration and test threshold set that satisfies the inter-
ference constraints Pdm = Pdmin and Pfm < Pfmax. Then we use the Lagrange optimization
method to obtain the optimal power allocation. Second, based on power allocation results,
we further optimize Ts and ?m using convex optimization. Repeat these two steps until max-
imum iteration number or the improvement is below a predefined threshold. First choose
a random initial Ts ? (Tsmin, T2], then choose initial local test threshold ?m that satisfies
Pdm = Pdmin. The initial power allocation strategy is an equal allocation such that Pm = PtotM
with m = 1,2,...,M. Let i denote step sequence number; (Ts,i,{?m,i},{Pm,i}) denote the
optimal spectrum sensing parameters set in ith step. The iterative optimization algorithm
is detailed below.
ITERATIVE OPTIMIZATION ALGORITHM
1) Set i = 0 and initialize (Ts,i,{?m,i},{Pm,i}).
2) Use the Lagrange optimization method to solve the convex optimization problem P2
for optimal power allocation {P?m} based on (Ts,i,{?m,i}).
97
3) For the fixed power allocation {P?m}, apply the convex optimization method of Sec.
5.4.2 to find a stationary point T?s and ??m to maximize the utility function Us(Ts) in
problem P3.
4) Set (Ts,i+1,{?m,i+1},{Pm,i+1})=(T?s ,{??m},{P?m}). If the improvement of utility func-
tion UG(Ts,i+1,{?m,i+1},{Pm,i+1}) compared with UG(Ts,i,{?m,i},{Pm,i}) in problem
P1 is less than a predefined threshold or if the algorithm reaches a maximum number
of iterations, then stop, else go back to step 2.
5.5 Simulation Examples
Consider a wideband CRN composed of Mp = 8 ?narrowband? PU channels. The slot in
the slotted system has a fixed length T = 100ms. Assume the primary user idle probabilities
on each channel are ?0 = [0.7,0.5,0.8,0.7,0.6,0.9,0.5,0.7]. The sampling rate of energy
detection is fs = 6M samples/s. The noise power is ?2 = ?40dBW. The minimum required
detection probability is Pdmin = 0.9 and maximum false alarm probability is Pfmax < 0.5.
The channel power gains |hsm|2 and |hpm|2 are all ergodic, stationary and exponentially
distributed with mean of 1, but they are independent from each other. The average SNR
vector of the primary users? signal on each channel is {?m} = [?21.5 ? 21 ? 20.5 ? 20 ?
19.5?19?18.5?18]dB with mean |hpm|2 = 1. We use 10000 slots simulation runs with the
channel power gains picked independently over different slots.
5.5.1 Utility function versus total power constraint
The average utility function UG versus total power constraint Ptot is shown in Figs. 5.1
and 5.2 on a per slot basis for two different values of energy price ? (=10 and 150, respec-
tively). In these figures the approach labeled ?Only power allocation? is the utility function
value with a random Ts located in (Tsmin,T/2] and only power allocation optimization, and
98
the approach of [5] corresponds to our approach with ? = 0. It is seen that the iterative
algorithm converges quickly and outperforms the case where only power allocation is opti-
mized. Fig. 5.2 shows that in the low Ptot region, utility increases as Ptot increases. However,
when Ptot becomes larger than approximately ?12dBW, the utility function plateaus out;
this phenomenon is not present in Fig. 5.1. The reason for this leveling off is the increase
in energy price which discourages higher transmit power even when the transmit power con-
straint Ptot is increased. Compared with the throughput maximization scheme of [78] that
does not consider energy consumption, our proposed iterative algorithm can achieve larger
utility value, especially with high Ptot.
?25 ?20 ?15 ?10 ?5600
800
1000
1200
1400
1600
1800
2000
2200
Total Transmit Power Constraint (dBW)
Average Utility Function
Utility Function versus total power constraint: ?=10
Only Power Allocation Optimization
One iteration
Two iteration
Final Iteration
Throughput maximization algorithm [5]
Figure 5.1: Utility function versus the total power constraint, when the minimum required
detection probability is fixed at 0.9, energy price ? = 10. The final iteration denotes the result
obtained after 10 iterations. The ?Only power allocation? is the utility function value with
a random Ts located in (Tsmin,T/2] and only power allocation optimization. The approach
of [5] corresponds to our approach with ? = 0.
5.5.2 Energy Efficiency versus energy price ?
We define energy efficiency as the ratio of throughput and energy consumption, and
further define normalized energy efficiency as energy efficiency normalized to have a peak
99
?24 ?22 ?20 ?18 ?16 ?14 ?12 ?10 ?8 ?6300
400
500
600
700
800
900
1000
1100
1200
1300
Total Transmit Power Constraint (dBW)
Average Utility Function
Utility Function versus total power constraint: ? =150
"Only Power Allocation" Optimization
1st iteration
2nd iteration
Final Iteration
Throughput maximization algorithm [5]
Figure 5.2: Utility function versus the total power constraint, when the minimum required
detection probability is fixed at 0.9, energy price ? = 150. The final iteration denotes
the result obtained after 10 iterations. The ?Only power allocation? is the utility function
value with a random Ts located in (Tsmin,T/2] and only power allocation optimization. The
approach of [5] corresponds to our approach with ? = 0.
value of one. In a similar fashion we also define normalized energy consumption per slot and
normalized throughput per slot. Fig. 5.3 shows the normalized energy efficiency, normalized
energy consumption and normalized throughput versus price of energy ?. The variation of
? will affect throughput and energy consumption in two ways. First, for small values of ?,
throughput is more important than the energy consumption. As energy price ? increases,
energy consumption begins to become a significant factor, finally outweighing throughput.
As a result, as ? increases, the throughput decreases to save energy. There exists a ?balance?
point at which normalized energy efficiency is maximized.
5.6 Conclusions
In this chapter we studied how to balance throughput and energy consumption for a
slotted multi-channel CRN. An optimization problem involving sensing parameters (sensing
100
0 1000 2000 3000 4000 5000 6000 70000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Price of Energy ?
Normalized Criterion
Normailize energy efficiency versus energy price ?
Normalized Throughput
Normalized Energy Consumption
Normalized Energy Efficiency
Figure 5.3: Normalized criteria versus the price of energy ?, when the total power constraint
is fixed at ?20dBW.
duration and local test threshold) and power allocation was formulated to maximize a utili-
ty function that rewards throughput but penalizes energy consumption. To obtain (locally)
optimal sensing parameters and power allocation, an iterative algorithm was proposed. Nu-
merical simulations demonstrate that the iterative algorithm converges fast. It performs
better than a ?only power allocation optimization? approach and an existing approach [78]
that ignores energy efficiency.
101
Chapter 6
On Energy Efficient MIMO-Assisted Spectrum Sharing for Cognitive Radio Networks
We formulate the design of energy efficiency maximization for a single secondary link
in an underlay spectrum sharing cognitive radio network under an SU (secondary user)
transmit-power constraint and an upperbound on the interference power at the PU (primary
user). We propose energy efficient precoding (beamforming) for the SU transmitter to maxi-
mize energy efficiency defined as the transmission rate to power ratio, when the terminals in
the system are equipped with multiple antennas. The underlying channels are assumed to be
known at the SU transmitter. The nonlinear optimization (fractional programming) prob-
lem is transformed into a parametrized convex optimization problem and the corresponding
solution is discussed. Computer simulation examples are provided to illustrate the proposed
approach and to explore the trade-off between energy efficiency and spectrum efficiency.
6.1 Introduction
Cognitive radio techniques aim to alleviate the shortage of frequency spectrum as wire-
less communication applications proliferate dramatically. Spectrum sharing is central to
cognitive radio which permits (unlicensed) secondary users (SUs) to access the licensed
spectrum as long as the interference to (licensed) primary users (PUs) 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 [98]. This chapter is concerned with underlay systems.
102
To enhance spectrum utilization, MIMO (multiple-input multiple-output) techniques have
been employed in [97] for a single secondary link transmission in underlay CRNs (cognitive
radio networks).
Besides the capacity/transmission rate issues, energy consumption is a critical factor
that deserves attention. For mobile terminals with battery for power supply, the energy con-
sumption issues are more serious because development of battery technology lags behind the
development of wireless communication technologies. Energy efficiency and related aspects
for wireless communication systems and networks have recently been reviewed in [94, 93].
An energy efficient design aims to enhance energy efficiency of the transmitter defined as the
transmission rate to power ratio.
In this chapter, we aim to optimize the transmit signal covariance matrix (precoding ma-
trix) to maximize the energy efficiency for the SU transmission in an MIMO-based underlay
CRN. Our work is inspired by [89] where energy efficient MIMO precoding is investigated for
point-to-point MIMO wireless communications. Optimal energy efficient precoding strate-
gies are proposed for both static and fast-fading MIMO channels to maximize the energy
efficiency defined as the transmission rate to power ratio. However, cognitive radio net-
work aspects requiring limiting interference to PUs and circuit power consumption are not
considered in [89, 92].
Recently, several investigations on energy efficient MIMO CRNs have been reported in
[92, 90, 95]. Minimization of total transmission energy consumption for point-to-point MI-
MO transmission is studied in [92] under sum rate constraint; however, [92] does not consider
the interference power constraint to PU. For a TDMA-based cognitive radio MIMO multius-
er multiple access network, [90] studies how to optimize time allocation and beamforming
among SU transmitters to minimize total transmission energy consumption while guarantee-
ing the SUs? rate requirement and the interference power constraints. However, the focus of
[92, 90] is minimization of energy consumption rather than maximization of energy efficiency.
A Stackelberg game theory based power control involving both SU and PU is considered in
103
[95] to maximize the long-term energy efficiency. However, the energy efficiency is defined
as an utility function rather than the more commonly accepted and utilized definition of the
transmission rate to power ratio (see [89] and references therein). Moreover, MIMO related
transmission is not involved in [95]. In addition, circuit power issues are also not considered
in [92, 90, 95].
In this chapter, we optimize the transmit signal covariance matrix (precoding matrix)
to maximize the energy efficiency for the SU transmission in an MIMO-based underlay
CRN. We construct an energy efficiency function that incorporates both transmission and
circuit power consumption. Then an energy efficiency maximization problem is formulated
for a single secondary link in a CRN under SU transmit-power and PU interference power
constraints. It turns out to be a fractional programming problem that is iteratively solved
via a parametrized convex optimization problem formulation.
In Sec. II, the system model is introduced and the problem formulation is presented. A
fractional programming based iterative semidefinite programming (SDP) algorithm is pro-
posed and analyzed in Sec. III. In Sec. IV we present a multichannel formulation. Computer
simulations are presented in Sec. V to illustrate the proposed approaches.
Notation: The determinant of the square matrixSis denoted by|S|andS followsequal 0 denotes
that S is a positive semi-definite matrix. The notation M? denotes the conjugate transpose
of a matrix M and E{?} denotes the statistical expectation. The space of n ? m matrices
with complex entries is denoted by Cm?n. For circularly symmetric complex Gaussian vector
x with mean vector ? and the covariance matrix ?, the probability distribution is denoted
as x ? CN(?,?). The notation tr(S) denotes the trace of square matrix S, IM denotes the
M ?M identity matrix, and x? denotes the optimal solution with respect to (w.r.t.) x of
some objective function g(x).
104
6.2 System Model and Problem Formulation
6.2.1 System Model
In this chapter, we consider a CRN composed of a single SU transmitter and receiver
pair that shares the same bandwidth for transmission with a PU receiver. Assume multiple
antennas are employed by SU and PU such that the SU transmitter has Nt,s antennas, the
SU receiver has Nr,s antennas, and the PU receiver has Nr,p antennas. We consider a block
channel fading model such that the channel state would be constant in the frame duration.
The frame duration is assumed to be long enough to allow us to adapt the transmit signal
covariance matrix so as to maximize the energy efficiency of secondary link while keeping
the interference power measured at PU below a predefined threshold. It is assumed that the
channel state information from the SU transmitter to the SU receiver and to the PU receiver
are precisely known at the SU transmitter.
Let n denote the time index. The complex baseband received signal y[n] at the SU
receiver is given by
y[n] = Hsx[n] +z[n] (6.1)
where y[n] ?CNr,s?1, x[n] ?CNt,s?1 is the transmitted signal from the SU, Hs ?CNr,s?Nt,s is
the channel matrix from the SU transmitter to the SU receiver, and z[n] ?CNr,s?1 is additive
white noise assumed to be complex Gaussian with zero mean and covariance matrix ?2zINr,s.
Assume that the elements of Hs are mutually independent and identically distributed as
CN(0,?2Hs).
Let Q = E{x[n]x?[n]} denote the transmit covariance matrix (precoding matrix) of the
SU transmitter. Assume that an ideal Gaussian code-book with infinitely large number of
codeword symbols is used at the SU transmitter such that x[n] ? CN(0,Q). Assume that
the rank of Q is d. Then the eigenvalue decomposition of Q is
Q = V?V? (6.2)
105
where V ?CNt,s?d contains the orthonormal eigenvectors of Q as its columns so that V?V =
Id and ? = diag{?21, ?22, ??? ,?2d} is the d?d diagonal matrix with the eigenvalues ?2i (1 ?
i ? d) of Q along its diagonal. Each column of V is the precoding vector for one transmit
data stream of the SU transmitter where d ? Nt,s denotes the degree of spatial multiplexing.
The transmit power is given by tr(Q) = summationtextdi=1 ?2i . Clearly Q followsequal 0 since it is a covariance
matrix. We assume that the transmit power for SU has an upper bound Pt such that
tr(Q) ? Pt. The corresponding achievable rate for the SU transmission link (when Q and
Hs are known to the transmitter) is given by [96, Chap. 8]
C(Q) = log2|INr,s +?HsQH?s| bps/Hz where ? = 1?2
z
. (6.3)
Let Hp ? CNr,p?Nt,s denote the channel fading coefficients from the SU transmitter to
the PU receiver. For the PU receiver, with Nr,p receive antennas, the interference power
constraint is given by
tr(HpQH?p) ? ? (6.4)
where ? is a predefined interference power upperbound.
6.2.2 Energy Model
It is shown in [93] that besides transmit power consumption, the circuit power con-
sumption is another factor that affects the energy efficiency of the system. Consequently,
we consider both transmit power and circuit power consumption as parts of our energy con-
sumption model. In this chapter the circuit power is taken to be a constant Pc regardless of
transmit power. The overall power consumption at the SU transmitter is then given by
P(Q) = ? tr(Q)+Pc (6.5)
where ? is the reciprocal of the drain efficiency of the power amplifier [92].
106
6.2.3 Problem Formulation
As in [89] and others, we consider bits per Joule as the metric for energy efficiency. The
energy efficiency is defined as the ratio of transmission rate to energy (power) consumption,
denoted by ?(Q). The goal of this chapter is to optimize transit signal covariance matrix
(the precoding matrix) Q to maximize ?(Q) under SU transmit power constraint and PU
interference power constraints. The optimization problem is then given by
P1 : max
Qfollowsequal0
?(Q) = log2|INr,s +?HsQH
?
s|
? tr(Q)+Pc (6.6)
s.t. tr(Q) ? Pt (6.7)
tr(HpQH?p) ? ? (6.8)
Q followsequal 0. (6.9)
6.3 Solutions based on Concave Fractional Program and Related Parametric
Problem
The problem P1 is nonlinear in Q and it is not clear how to obtain an explicit solution.
However, it will be shown to be a concave fractional program [105], which can then be solved
with an equivalent parametric formulation resulting in a concave optimization problem.
Lemma 1. The optimization problem P1 is a concave fractional program. ?
Proof: Concave fractional programming problems are very briefly reviewed in the Ap-
pendix C following [105]. Since the numerator and the denominator of ?(Q) in problem P1
are concave and affine, respectively, in Q, problem P1 is a concave fractional program [105].
square
Consider a parameter ? ?R. Following [105], problem P1 transformed to the following
parametrized concave objective function
F(Q,?) = log2|INr,s +?HsQH?s|??(? tr(Q)+Pc) (6.10)
107
with the corresponding optimization problem P2, given by
P2 : max
Qfollowsequal0
F(Q,?)
s.t. tr(Q) ? Pt (6.11)
tr(HpQH?p) ? ? (6.12)
Q followsequal 0. (6.13)
Since problem P1 is a concave fractional program, problem P2 is a parametric concave
program. The problemP2is more tractable than problemP1because of its simpler structure
in which the numerator and denominator of ?(Q) are separated in (Q,?) with the help of
?. Since Q is semi-definite, the concave optimization problem P2 can be solved by utilizing
mature SDP (Semidefinite Program) software. In this chapter, we employ SeDuMi that is
embedded in the CVX Matlab software package [91].
We would like to show that the optimal solution to problem P1 can be obtained from
the optimal solution(s) to the parametrized problem P2. Define
?(?) = F(Q?(?),?) (6.14)
where
Q?(?) = arg
braceleftbigg
max
Qfollowsequal0
F(Q,?)
bracerightbigg
(6.15)
satisfying the constraints (6.11)-(6.13), that is, ?(?) denotes the optimal objective function
value for problem P2 for a given ?. Then we have the following theorem to demonstrate the
relation between optimal solutions of P1 and P2.
Theorem 1:LetQ? betheoptimalfeasibletransmitcovariancematrixthatachievesthe
optimal energy efficiency, that is, the maximum value of the concave fractional program inP1
subject to all constraints. Then we have ?(Q?) = ?? if and only if ?(??) = F(Q?(??),??) =
maxQfollowsequal0 F(Q,??) = 0. ?
108
Proof: Based on the inequality constraints of P1, dom{?} is compact. Let Q? be an
optimal solution of P1 and let ?? = ?(Q?) = C(Q?)/P(Q?). When ?(Q?) = ??, we have
?(Q) = log2|INr,s +?HsQH
?
s|
? tr(Q)+Pc ? ?
?
and
?(Q?) = log2|INr,s +?HsQ
?H?
s|
? tr(Q?)+Pc = ?
?.
Since ? tr(Q) + Pc > 0, it follows that F(Q,??) ? 0 and F(Q?(??),??) = 0. Conversely,
when ?(??) = F(Q?(??),??) = 0, we have ?(Q?) = ?? and also F(Q,??) ? 0 implying
?(Q) ? ??. Thus Q? ?solves? problem P1. square
It follows from [105] that ?(?) has the following properties:
?(?) > 0 iff ? < ?? (6.16)
?(?) = 0 iff ? = ?? (6.17)
?(?) < 0 iff ? > ?? (6.18)
Furthermore, the optimal solution of problem P1 is equivalent to the optimal solution of
problemP2with parameter ? = ??. Consequently, seeking the root of the nonlinear equation
?(?) = 0 is the key to solving P1. It turns out [105] that ?(?) is convex, continuous and
strictly decreasing in ? with lim??? ?(?) = ? and lim???? ?(?) = ??. Finally, there
exists an unique solution ? = ?? for ?(?) = 0.
The subgradient of ?(?) can also be obtained through solving problem P2. With
optimal solution Q? of problem P2 with ? = ??, the following value
log2|INr,s +?HsQ?H?s|??(? tr(Q?)+Pc) (6.19)
109
is a tangent to ?(?)|?=??, which means that ?(? tr(Q?) + Pc) is a subgradient of ?(?) at
?? (that is also the derivative if ?(?) is differentiable at ??).
With these characteristics of P1, a bisection method can be employed for solving the
nonlinear equation ?(?) = 0. One can search for the optimal ?? over an interval [?min ?max]
which is known to contain the optimal value ??. Since ?(Q) is always positive, the possible
?? = ?(Q?) is positive too. Thus we initially set ?min = 0. The upperbound?max is an
arbitrary value such that ?(?max) in P2 is negative. During the iterative bisection algo-
rithm, we first solve the concave optimization problem P2 at the midpoint ? = ?min+?max2 .
Depending on the feasibility of P2, we determine whether the optimal value is located in
the lower or upper half of the interval, then update the interval accordingly. This would
generate a new search interval that is half of the former one. The iteration would continue
until the width of the interval is small enough: ?max ? ?min ? ? where ? is the tolerance
to control the accuracy of the algorithm. The iterative bisection algorithm is summarized
below:
BISECTION ALGORITHM FOR PARAMETRIC OPTIMIZATION
1) Given ?min = 0 and ?max (? ??), tolerance ? > 0.
2) Set ? = ?min+?max2 . Solve the concave optimization problem P2.
3) If ?(?) > 0 then ?min = ?, else if ?(?) < 0, then ?max := ?.
4) If ?(?) = 0 or ?max ??min ? ?, stop; otherwise go to step 2.
6.4 Multichannel Energy Efficient Precoding
In this subsection, we consider a general multichannel cognitive radio network where
both secondary and primary users transmit on multiple parallel subchannels with multi-
ple antennas. This scenario can be viewed as that of MIMO-OFDMA in which multiple
110
subcarriers are utilized. Assume that there are K parallel OFDM subcarrier tones. Let
Hsk ? CNr,s?Nt,s and Hpk ? CNr,p?Nt,s denote the channel matrices for the kth subchan-
nel (1 ? k ? K) from the SU transmitter to the SU and PU receivers, respectively. Let
Qk ?CNr,s?Nt,s denote the signal covariance matrix (precoding matrix) on the kth subchan-
nel. Suppose that each subchannel has the same bandwidth BHz. The transmit power and
interference power constraints on each subchannel are identical and are the same as in the
single-channel case discussed in Sec. III. The achievable transmission rate for the SU on the
kth channel is
Rsk = Blog2|INr,s +?HskQH?sk| bps. (6.20)
Let wk denote a predefined nonnegative priority weight for transmission over the kth
channel (1 ? k ? K) and ?Kk=1wk = K. Then the weighted sum rate is given by
C({Qk}Kk=1) =
Ksummationdisplay
k=1
wkB log2|INr,s +?HskQH?sk|. (6.21)
The total power consumption at the SU is given by
P({Qk}Kk=1) = ?
Ksummationdisplay
k=1
tr(Qk)+KPc. (6.22)
The corresponding energy efficiency, defined by ?({Qk}Kk=1) = C({Qk}Kk=1)/P({Qk}Kk=1), is
given by
?({Qk}Kk=1) =
summationtextK
k=1 wkB log2|INr,s +?HskQH
?
sk|
?summationtextKk=1 tr(Qk)+KPc . (6.23)
The transmit and interference power constraints are given by
tr(Qk) ? Pt ?k (6.24)
tr(HpkQkH?pk) ? ? ?k (6.25)
Qk followsequal 0 ?k (6.26)
111
The overall optimization problem can be formulated as
P3 : max
{Qk}Kk=1
?({Qk}Kk=1)
s.t. tr(Qk) ? Pt ?k (6.27)
tr(HpkQkH?pk) ? ? ?k (6.28)
Qk followsequal 0 ?k. (6.29)
The corresponding parametric objective function is given by
F({Qk}Kk=1,?) =
Ksummationdisplay
k=1
wkBlog2|INr,s +?HskQH?sk|
??
parenleftBigg
?
Ksummationdisplay
k=1
tr(Qk)+KPc
parenrightBigg
. (6.30)
Consequently, we consider the following parametric problem associated with P3:
P4 : max
{Qk}Kk=1followsequal0
F({Qk}Kk=1,?)
s.t. tr(Qk) ? Pt ?k (6.31)
tr(HpkQkH?pk) ? ??k (6.32)
Qk followsequal 0 ?k. (6.33)
Define
F({Qk},?) = wkBlog2|INr,s +?HskQH?sk|??(?tr(Qk)+Pc). (6.34)
Then it follows that
F({Qk}Kk=1,?) =
Ksummationdisplay
k=1
F({Qk},?). (6.35)
112
Notice that the constraints in the problem P4 is separable. Therefore, the solution to
problem P4 is equivalent to solutions to the following K subproblems:
P5 : max
Qkfollowsequal0
F({Qk},?)
s.t. tr(Qk) ? Pt (6.36)
tr(HpkQkH?pk) ? ? (6.37)
Qk followsequal 0. (6.38)
The solution algorithm for problem P5 is similar to that for the problem P2.
6.5 Simulation Examples
It is assumed that the SU transmitter is equipped with Nt,s = 2 or 3 antennas and
both SU and PU receivers have 1, 2 or 3 antennas. The elements of the channel fading
matrices Hp and Hs from the SU transmitter to the primary and secondary receivers are
both mutually independent complex Gaussian random variables distributed as CN(0, 0.1)
and CN(0,1), respectively. It is assumed that the power amplifier efficiency is 38% for the
secondary transmitter, leading to ? = 1/0.38 in (6.5). The static circuit power consumption
is Pc = 20mW. At the PU receiver, the maximum interference power constraint is ? = 1mW
. The noise power is assumed to be ?2z = 1mW (=0 dBm). The reported results are based
on 1000 Monte Carlo runs with 1000 randomly generated channel states Hp and Hs.
6.5.1 Energy Efficiency and Capacity Comparisons
Figs. 6.1 and 6.2 show the capacity and energy efficiency, respectively, of the secondary
link under two cases of energy efficiency maximization as discussed in Sec. III and spectral
efficiency (capacity) maximization without any consideration of power consumption, as dis-
cussed in [97]. Three different antenna settings were considered: 2?1 MISO with Nt,s = 2,
Nr,s = 1 and Nr,p = 1, 2 ? 2 MIMO with Nt,s = 2, Nr,s = 2 and Nr,p = 2 and 3 ? 3
113
MIMO with Nt,s = 3, Nr,s = 3 and Nr,p = 3. The transmit power constraint Pt for the SU
transmitter was varied from 0dBm to 22dBm. It is seen that compared with the convention-
al spectrum efficiency maximization design, the energy efficient design can enhance energy
efficiency dramatically, particularly in the higher transmit power range but at the cost of
smaller throughput. In addition, more receive antennas result in higher energy efficiency
and capacity because spatial diversity is utilized.
0 2 4 6 8 10 12 14 16 18 20 220
2
4
6
8
10
12
Transmit Power Constraint (dBm)
capacity (bps/Hz)
Capacity comparison
3x3 MIMO: spectral eff. criterion
3x3 MIMO: energy eff. criterion
2x2 MIMO: spectral eff. criterion
2x2 MIMO : energy eff. criterion
2x1 MISO: spectral eff. criterion
2x1 MISO : energy eff. criterion
Figure 6.1: Capacity comparison: In the energy efficiency criterion we maximize energy
efficiency ?(Q) as in problem P1 whereas in the spectral efficiency criterion we maximize
the channel capacity as in [97]. In 2 ? 1 MISO system we used Nt,s = 2, Nr,s = 1 and
Nr,p = 1, the 2?2 MIMO system had Nt,s = 2, Nr,s = 2 and Nr,p = 2, and the 3?3 MIMO
system had Nt,s = 3, Nr,s = 3 and Nr,p = 3.
6.5.2 Multichannel Energy Efficient MIMO Transmission
In this case, two subchannels are utilized and each subchannel has independent and
identically distributed channel gain matrix. Furthermore, each subchannel has independent
transmit power and interference power constraints for which the parameter settings are the
same as for the single channel scenario discussed in Sec. V-A. The priority weights of the
two channels are 0.6 and 1.4, respectively. The bandwidth B is assumed to be 1 Hz. Figs.
6.3 and 6.4 show the capacity and energy efficiency, respectively, of the secondary link under
two cases of energy efficiency maximization as discussed in Sec. IV and spectral efficiency
114
0 2 4 6 8 10 12 14 16 18 20 220
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Transmit Power Constraint (dBm)
energy efficiency (bits/Joule)
Energy Efficiency Comparison
3x3 MIMO: spectral eff. criterion
3x3 MIMO: energy eff. criterion
2x2 MIMO: spectral eff. criterion
2x2 MIMO: energy eff. criterion
2x1 MISO: spectral eff. criterion
2x1 MISO: energy eff. criterion
Figure 6.2: Energy efficiency comparison: In the energy efficiency criterion we maximize
energy efficiency ?(Q) as in problem P1 whereas in the spectral efficiency criterion we
maximize the channel capacity as in [97]. In 2?1 MISO system we used Nt,s = 2, Nr,s = 1
and Nr,p = 1, the 2?2 MIMO system had Nt,s = 2, Nr,s = 2 and Nr,p = 2, and the 3?3
MIMO system had Nt,s = 3, Nr,s = 3 and Nr,p = 3.
(capacity) maximization without any consideration of power consumption. We used two
MIMO antenna configurations: the 2?2 MIMO system with Nt,s = 2, Nr,s = 2 and Nr,p = 2,
and the 3?3 MIMO system with Nt,s = 3, Nr,s = 3 and Nr,p = 3. The conclusions drawn
in Sec. V-A apply here too.
6.6 Conclusions
In this chapter, we proposed an energy-efficient transmit covariance matrix (precoder)
design for spectrum sharing underlay cognitive radio networks when multiple antennas are
employed at the secondary user transmitter. We constructed an energy efficiency function
that incorporates both transmission and circuit power consumption. Then an energy effi-
ciency maximization problem was formulated for a single secondary link in a CRN under
SU transmit-power and PU interference power constraints. It turned out to be a fractional
programming problem that was iteratively solved via a parametrized convex optimization
problem formulation. A multichannel extension of the problem was also investigated. The
115
0 2 4 6 8 10 12 14 16 18 20 220
2
4
6
8
10
12
14
16
18
20
22
24
26
28
Transmit Power Constraint (dBm)
capacity (bps/Hz)
Capacity comparison
3x3 MIMO: spectral eff. criterion
3x3 MIMO: energy eff. criterion
2x2 MIMO: spectral eff. criterion
2x2 MIMO: energy eff. criterion
Figure 6.3: Capacity comparison for the multichannel case with K = 2: In the energy
efficiency criterion we maximize energy efficiency ?({Qk}Kk=1) as in problem P3 whereas in
the spectral efficiency criterion we maximize the channel capacity as in [97]. We used two
MIMO systems: the 2?2 MIMO system with Nt,s = 2, Nr,s = 2 and the 3?3 MIMO system
with Nr,p = 2, and Nt,s = 3, Nr,s = 3 and Nr,p = 3.
trade-off between energy efficiency and spectrum efficiency was explored via computer sim-
ulations.
116
0 2 4 6 8 10 12 14 16 18 20 220
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Transmit Power Constraint (dBm)
energy efficiency (bits/Joule)
Energy Efficiency Comparison
3x3 MIMO: spectral eff. criterion
3x3 MIMO: energy eff. criterion
2x2 MIMO: spectral eff. criterion
2x2 MIMO: energy eff. criterion
Figure 6.4: Energy efficiency comparison for the multichannel case with K = 2: In the energy
efficiency criterion we maximize energy efficiency ?({Qk}Kk=1) as in problem P3 whereas in
the spectral efficiency criterion we maximize the channel capacity as in [97]. We used two
MIMO systems: the 2?2 MIMO system with Nt,s = 2, Nr,s = 2 and the 3?3 MIMO system
with Nr,p = 2, and Nt,s = 3, Nr,s = 3 and Nr,p = 3.
117
Chapter 7
On Precoding for Maximum Weighted Energy Efficiency of MIMO Cognitive Multiple
Access Channels
We study weighted energy efficiency maximization for multiple-input multiple-output
cognitive multiple access channels under both secondary user transmit power constraint
and primary user interference power constraint. Energy efficiency is defined as the ratio
of weighted sum rate and energy consumption including both transmission and circuit en-
ergy consumption. The nonlinear (fractional programming) energy efficiency optimization
problem is transformed into a series of parametrized convex optimization problems. A com-
bination of bisection search method and cyclic coordinate ascent-based iterative water-filling
algorithm is proposed to solve the parametrized problem. Computer simulation examples
are provided to illustrate the proposed approach and to explore the trade-off between energy
efficiency and spectrum efficiency.
7.1 Introduction
Cognitive radio techniques aim to alleviate the shortage of frequency spectrum as wire-
less communication applications proliferate dramatically. Spectrum sharing is central to
cognitive radio which permits (unlicensed) secondary users (SUs) to access the licensed
spectrum as long as the interference to (licensed) primary users (PUs) 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 the interference power received at primary
users should be below the predefined interference threshold [24]. In the second scheme (spec-
trum overlay) secondary users need to detect spectrum holes and then access spectrum white
118
space in an nonintrusive way [24]. This chapter is concerned with energy efficient precoding
for cognitive multiple access in spectrum underlay cognitive radio networks.
Current research on wireless communication channel precoding mainly focuses on opti-
mizing transmit signal covariance matrix (precoding) to maximize spectral efficiency (capac-
ity) [106, 107, 100, 101] where MIMO (multiple-input multiple-output) techniques have been
employed to enhance spectrum utilization. The water filling algorithm for Gaussian multiple
access channel under individual transmit power constraint was first proposed in [106]. Ref.
[100] further proposed a sum power iterative water-filling for multiantenna Gaussian broad-
cast channel (BC) based on [106], which solves an equivalent dual MAC (multiple access
channel) capacity maximization with sum power constraint. A similar problem as [100] was
solved alternatively with a dual decomposition method by [107] with the same system model
as [100]. For weighted sum rate maximization of the Gaussian BC channel under sum power
constraint, the iterative water filling algorithm in [100] was extended to solve dual MAC
capacity maximization under sum power constraint based on BC-MAC duality [101]. For
cognitive MAC with multiple antennas, in [99] maximization of the weighted sum rate under
weighted sum power constraint (interference power to PU) was solved with a dual decompo-
sition algorithm [107]. However, none of the algorithms proposed in [106, 107, 100, 101, 99]
consider energy efficiency (i.e. they omit any consideration of power consumption).
Besides the capacity/transmission rate issues, energy consumption is a critical factor
that deserves attention. For mobile terminals with battery for power supply, the energy con-
sumption issues are more serious because development of battery technology lags behind the
development of wireless communication technologies. Energy efficiency and related aspects
for wireless communication systems and networks have recently been reviewed in [93, 111].
An energy efficient design aims to enhance energy efficiency of the transmitter defined as the
transmission rate to power ratio. Energy efficient precoding has been considered in [89, 108].
The energy efficient precoding for point to point MIMO wireless communication was first
studied in [89]. For multiuser BC, an energy efficient precoding scheme was proposed [108].
119
However, cognitive radio network aspects requiring limiting interference to PUs and circuit
power consumption are not considered in [89] and [108].
In this chapter, we optimize the transmit signal covariance matrix (precoding matrix)
to maximize the energy efficiency for a MIMO cognitive multiple access channel (C-MAC).
The secondary BS is informed of the channel states from SUs to both secondary BS and PU;
SUs are assumed to obtain the optimal power allocation information from BS via an addi-
tional control channel. To our best knowledge, there is currently no paper on energy efficient
precoding for MIMO C-MAC. We formulate a weighted energy efficiency (EE) optimiza-
tion problem under transmit power constraint and interference power constraint for MIMO
cognitive MAC channels. EE is defined as the ratio of weighted sum capacity and energy
consumption that includes both transmission power and static circuit power consumption.
With circuit power consumption, the energy efficiency versus transmission power is a bell
shape function [111], such that the maximal energy efficiency can be achieved in moderate
transmission rate. This is different from the conclusion that energy efficiency is maximized
at very low or even minimum transmit power when only transmission power consumption
is considered [89]. The EE objective function is nonlinear but quasi-convex, which can be
transformed into a series of parametrized concave energy-aware utility functions, following
[105]. We prove that such transformation guarantees that the optimal energy efficient power
allocation is obtained; a similar proof can also be found in [110]. A combination of bisection
search method and cyclic coordinate ascent-based iterative water-filling algorithm is pro-
posed to solve the parametrized problem. Computer simulation examples are provided to
illustrate the proposed approach and to explore the trade-off between energy efficiency and
spectrum efficiency.
In Sec. II, the system model is introduced and the problem formulation is presented. A
fractional programming based bisection search algorithm is proposed in Sec. III. In Sec. IV
we present a cyclic coordinate ascent based semidefinite programming (SDP) algorithm to
120
optimize transmit covariance matrix iteratively. Computer simulations are presented in Sec.
V to illustrate the proposed approaches.
Notation: We use bold letters x to denote vectors and bold capital letters H to
denote matrices. tr{H} denotes the trace of matrix H. x ? 0 denotes that x is a non-
negative vector (all elements are non-negative). |S| denotes the determinant of the general
matrix S. S followsequal 0 denotes S is positive semidefinite matrix. The notation M? denotes the
conjugate transpose of a matrix M. The space of m ? n matrices with complex entries is
denoted by Cm?n. Let R+ denote the field of positive real numbers. For circularly symmetric
complex Gaussian vector x with mean vector ? and the covariance matrix ?, the probability
distribution is denoted as x ? CN(?,?). IM denotes the M?M identity matrix. x? denotes
the optimal solution with respect to (w.r.t) x of some objective function g(x). ?Subject to?
is abbreviated as s.t.
7.2 System Model and Problem Formulation
7.2.1 System Model
We consider an MIMO cognitive multiple access channel where K secondary users with
Nt,s transmit antennas communicate with a secondary BS with Nr,s receiver antennas. The
K SUs and the secondary BS share spectrum with a primary user with Np,r receive antennas.
We consider a block channel fading model where the channel state is constant during the
frame duration which is assumed to be long enough to allow us to adapt the transmit signal
covariance matrix (precoding matrix) so as to maximize the energy efficiency of cognitive
MAC while keeping the interference power measured at PU below a predefined threshold.
It is assumed that the channel state information from the SU transmitter to the secondary
BS and to the PU receiver is precisely known at the secondary BS. The K SUs can commu-
nicate simultaneously to the secondary BS where successive interference cancellation (SIC)
technique [104] is used to decode the received K SUs? signal.
121
The secondary BS?s complex baseband received signal at time n, denoted y[n], is mod-
eled as
y[n] =
Ksummationdisplay
k=1
Hk,sxk[n] +z[n] (7.1)
where y[n] ? CNr,s?1 denotes received signal vector, xk[n] ? CNt,s?1 denotes transmitted
signal from the kth SU,Hk,s ?CNr,s?Nt,s denotes channel matrix from the kth SU transmitter
to secondary BS, and noise z[n] ? CNr,s?1 is distributed as CN(0,?2zINr,s). Let Hk,p ?
CNr,p?Nt,s denote the channel matrix from kth SU to PU. Let Qk = E{xk[n]x?k[n]} denote
the transmit covariance matrix (precoding matrix) of the kth SU transmitter. Assume that
an ideal Gaussian code-book with infinitely large number of codeword symbols is used at the
SU transmitter such that xk[n] ? CN(0,Qk). Assume that the rank of Qk is dk. Then the
eigenvalue decomposition of Qk is
Qk = Vk?kV?k (7.2)
where Vk ? CNt,s?dk contains the orthonormal eigenvectors of Qk as its columns so that
V?kVk = Idk and ?k = diagbraceleftbig?21, ?22, ??? ,?2dkbracerightbig is the dk ? dk diagonal matrix with the
eigenvalues ?2i (1 ? i ? dk) of Qk along its diagonal. Each column of Vk is the precoding
vector for one transmit data stream of the kth SU transmitter where dk ? Nt,s denotes the
degree of spatial multiplexing. The transmit power is given by tr(Qk) = summationtextdki=1 ?2i . Clearly
Qk followsequal 0 since it is a covariance matrix. We assume that the transmit power for kth SU has
an upper bound tr{Qk} ? Pst. Under the spectrum sharing scenario, the interference power
to PU has an upperbound ? such that ?Kk=1tr{Hk,pQkH?k,p} ? ?. For the channel matrices
Hk,s and Hk,p, we consider a block fading model where Hk,s and Hk,p keep constant for one
block and are independently distributed among the blocks. Moreover, Hk,s and Hk,p are
perfectly known to the secondary BS.
It is assumed each SU k has a priority weight wk [104, 101, 99]. With successive inter-
ference cancellation (SIC) [104], the received K SU signals are decoded successively. The
122
corresponding achievable rate is given by [99, Eqn. 5]
R?k = Blog2 |INr,s +?
summationtextk
j=1 H?j,sQjH
?
?j,s|
|INr,s +?summationtextk?1j=1 H?j,sQjH??j,s| bps , ?k (7.3)
where ? = 1?2
z
(?2z is the noise power), B is the bandwidth and ?1,...,?K, a permutation of
(1,...,K), represents a decoding order (?K is decoded first and ?1 is decoded last).
According to [104], the optimal decoding order for SIC is obtained by sorting priority
weights in a non-increasing order as w?1 ? w?2 ? ??? ? w?K. Consequently, without loss of
generality the optimal decoding order is assumed to be ?k = k [101, 99]. The weighted sum
rate for all K users is given by
C({Qk}Kk=1) =
Ksummationdisplay
k=1
?kBlog2|INr,s +??kj=1Hj,sQjH?j,s| (7.4)
where ?k = wk ?wk+1, and wK+1 = 0.
As demonstrated in [93], both transmission and circuit energy consumption contribute
to total energy efficiency. The circuit power is treated as a constant Pc. Power amplifier
efficiency is considered for SU?s transmission power. The overall power consumption of K
SUs is given by
E({Qk}Kk=1) = ?
Ksummationdisplay
k=1
tr{Qk}+KPc (7.5)
where ? is the reciprocal of the drain efficiency of the power amplifier.
Similar to [89], the energy efficiency denoted as ?({Qk}Kk=1), is defined as the ratio of
weighted sum rate and energy consumption, with units bits per Joule. We aim to optimize
transmit covariance matrices {Qk}Kk=1 to maximize ?({Qk}Kk=1) under both transmit power
123
constraint and interference power constraint to PU, which is given by
P1 : max
{Qk}Kk=1
?({Qk}Kk=1) = C({Qk}
K
k=1)
E({Qk}Kk=1) (7.6)
s.t. tr{Qk} ? Pst and Qk followsequal 0, ?k (7.7)
Ksummationdisplay
k=1
tr{Hk,pQkH?k,p} ? ?. (7.8)
7.3 Solutions: Concave Fractional Program and Related Parametrized Problem
The problem P1 is nonlinear and it is hard to obtain explicit solution directly. However,
it turns out to be a concave fractional program which can be solved with an equivalent
parametrized program.
Lemma 1. The optimization problem P1 is a concave fractional program. ?
Proof: Since the numerator and denominator of ?({Qk}Kk=1) in problem P1 are concave and
affine, problem P1 is a concave fractional program based on the definition and property of
concave fractional program [105]. square
Consider a parameter ? ? R. Following [105] problem P1 is related to the following
concave objective function
?({Qk}Kk=1,?) =
Ksummationdisplay
k=1
?kBlog2 |INr,s +??kj=1Hj,sQjH?j,s|?? (?
Ksummationdisplay
k=1
tr{Qk}+KPc) (7.9)
and the corresponding optimization problem P2
P2 : max
{Qk}Kk=1
?({Qk}Kk=1,?) (7.10)
subject to (7.7)-(7.8). Since P1 is a concave fractional program, P2 is a parametrized concave
program [105]. P2 is more tractable than P1 because of its simpler structure where the
numerator and denominator are separated with the help of ?.
124
We now show that the optimal solution to P1 can be obtained from the optimal solution
to P2. Define
?(?) = ?({Q?k(?)}Kk=1,?) (7.11)
where
{Q?k(?)}Kk=1 = arg
braceleftbigg
max
{Qkfollowsequal0}Kk=1
?({Qk}Kk=1,?)
bracerightbigg
(7.12)
satisfying (7.7)-(7.8). Then we have the following theorem linking the optimal solutions of
problems P1 and P2.
Theorem 1: Let {Q?k}Kk=1 be the optimal feasible transmit covariance matrix that achieves
the optimal energy efficiency, that is, the maximum value of the concave fractional program
in P1 subject to all constraints. Then we have ?({Q?k}Kk=1) = ?? if and only if ?(??) =
?({Q?k(??)}Kk=1,??) = max{Qkfollowsequal0}K
k=1
?({Qk}Kk=1,??) = 0. ?
Proof: Based on the inequality constraints of P1, dom{?} is compact. Let {Q?k}Kk=1 be
an optimal solution of P1 and let ?? = ?({Q?k}Kk=1) = C({Q?k}Kk=1)/E({Q?k}Kk=1). When
?({Q?k}Kk=1) = ??, we have
?({Qk}Kk=1) =
summationtextK
k=1 ?kBlog2|INr,s +??
k
j=1Hj,sQjH
?
j,s|
?summationtextKk=1 tr{Qk}+KPc ? ?
? (7.13)
and
?({Q?k}Kk=1) =
summationtextK
k=1 ?kBlog2|INr,s +??
k
j=1Hj,sQ
?
jH
?
j,s|
?summationtextKk=1 tr{Q?k}+KPc = ?
?. (7.14)
Since?summationtextKk=1 tr{Qk}+KPc > 0, itfollowsthat?({Qk}Kk=1,??) ? 0and?({Q?k(??)}Kk=1,??) =
0. Conversely, when ?(??) = ?({Q?k(??)}Kk=1,??) = 0, we have ?({Q?k}Kk=1) = ?? and also
?({Qk}Kk=1,??) ? 0 implying ?({Qk}Kk=1) ? ??. Thus Q? ?solves? problem P1. square
It follows from [105, Sec. 2.2.4] that ?(?) has the following properties: ?(?) > 0 iff ? <
??, ?(?) = 0 iff ? = ?? and ?(?) < 0 iff ? > ??. Moreover, the optimal solution of P1
is equivalent to the optimal solution of P2 with parameter ? = ??. Consequently, seeking
the root of the nonlinear equation ?(?) = 0 is the key to solving P1. It turns out [105]
125
that ?(?) is convex, continuous and strictly decreasing in ? with lim??? ?(?) = ? and
lim???? ?(?) = ??. Finally, there exists an unique solution ? = ?? for ?(?) = 0.
With these characteristics of P1, a bisection method can be employed for solving the
nonlinear equation ?(?) = 0. The optimal ?? can be searched over the interval [?min ?max]
which is known to contain ??. Since ?({Qk}Kk=1) is always positive, the possible ?? =
?({Q?k}Kk=1) is positive too. Thus we initially set ?min = 0. The upperbound ?max is an
arbitrary value satisfying ?(?max) < 0. The iteration would continue until the algorithm
accuracy is reached such that ?max ??min ? ? (? is accuracy tolerance) [105]. The iterative
bisection algorithm is summarized below:
Bisection Algorithm for Parametrized Optimization
1) Given ?min = 0, ?max ? ??, tolerance ? > 0.
2) Set ? = ?min+?max2 . Solve the concave optimization problem P2.
3) If ?(?) > 0 then ?min = ?;, else if ?(?) < 0 ?max := ?.
4) If ?(?) = 0 or ?max ??min ? ?, stop; otherwise go to step 2.
7.4 SDP-based Cyclic Coordinate Ascent for Problem P2
Due to convexity of ?({Qk}Kk=1,?), problem P2 can be solved by standard interior point
algorithms that unfortunately suffer from high computational complexity. Utilizing the in-
herent structural properties of ?({Qk}Kk=1,?), we now propose a simple iterative water-filling
algorithm. It is a generalization of the cyclic coordinate ascent algorithm proposed in [100]
for optimizing transmit covariance matrix with sum power constraint in broadcast channels.
Due to the coupled interference power constraint involving all K SUs? transmit covariance
matrices {Qk}Kk=1, the classical water-filling [106] for individual power constraint cannot be
126
directly used here. Due to coupled power constraint, all K SUs need to do water-filling
simultaneously with the same water level. On the other hand, the objective function in
problem P2 is also coupled on all K transmit covariance matrices {Qk}Kk=1, therefore, an
equivalent Lagrangian problem based on gradient methods cannot be directly applied. The
reason is that all {Qk}Kk=1 are correlated with each other in their Lagrangian primal-dual
solution. In our proposed generalized iterative water-filling algorithm, we first use cyclic
coordinate ascent algorithm to decompose the coupled objective function into a sum of K
non-coupled functions involving only individual Qk?s. The other transmit covariance ma-
trices are treated as interference and all Qk?s are simultaneously updated based on results
from the previous iteration. The Lagrangian primal-dual method is used to solve the op-
timization problem in each step (iteration) which decomposes weighted interference power
constraint into separable constraints. Finally, we decompose the optimization problem with
separable objective functions and constraints into K individual semi-definite programming
(SDP) problems. We first describe the cyclic coordinate ascent algorithm that decomposes
coupled objective function in problem P2 into separable functions.
Cyclic Coordinate Ascent Based Iterative Water-Filling
1) Initialize Q(0)k = INt,s ?k
2) At iteration m, compute for k = 1,...,K and i ? k the noise and interference matrix
coefficients
?(m)k,i = INr,s +?
isummationdisplay
j=1,jnegationslash=k
Hj,sQ(m?1)j H?j,s (7.15)
and the sum power of all other users except user k
?k =
Ksummationdisplay
i=1,inegationslash=k
tr{Q(m?1)i } (7.16)
127
3) Water-filling step: Treat the matrices ?(m)k,i and variables ?k as parallel, noninterfer-
ing ones and obtain the new transmit covariance matrix {Sk}Kk=1 by water-filling under
interference power constraint ?Kk=1tr{Hk,pSkH?k,p} ? ?, as described next. Define in-
termediate variable ?({Sk}Kk=1) as below
?({Sk}Kk=1) =
Ksummationdisplay
k=1
{
Ksummationdisplay
i=k
?iBlog2|?Hk,sSkH?k,s +?(m)k,i |
??(?(tr{Sk}+?k)+KPc)}. (7.17)
Let {S(m)k }Kk=1 be the solution of
{S(m)k }Kk=1 = arg max
Skfollowsequal0,tr{Sk}?Pstsummationtext
K
k=1 tr{Hk,pSkH
?
k,p}??
?({Sk}Kk=1). (7.18)
4) Update transmit covariance matrices as
Q(m)k = 1KS(m)k + K?1K Q(m?1)k , k = 1,...,K.
The proof of convergence of the proposed iterative water-filling algorithm is similar to
that in [100, 101] and is omitted. Owing to the convexity of ?({Sk}Kk=1), the ?water-filling
step? is a convex optimization problem that has a unique solution. We first consider the
Karush-Kuhn-Tucker (KKT) conditions for the problem in (7.18). The Lagrangian function
[88] is given by
L({Sk}Kk=1,?) = ?({Sk}Kk=1)??(
Ksummationdisplay
k=1
tr{Hk,pSkH?k,p}??) (7.19)
s.t. Sk followsequal 0, tr{Sk} ? Pst, ?k (7.20)
128
where ? is the dual variable. By the KKT conditions for (7.18)
Ksummationdisplay
k=1
tr{Hk,pSkH?k,p}?? ? 0 (7.21)
?(
Ksummationdisplay
k=1
tr{Hk,pSkH?k,p}??) = 0 (7.22)
? ? 0 (7.23)
??({Sk}Kk=1)/?Sk ??H?k,pHk,p = 0, 1 ? k ? K (7.24)
where
??({Sk}Kk=1)
?Sk =
Ksummationdisplay
i=k
?iBH?k,s(?Hk,sSkHk,s +?(m)k,i )?1Hk,s ???INt,s (7.25)
The dual function of (7.18) is problem P3
P3 : g(?) = max
{Sk}Kk=1:Skfollowsequal0
tr{Sk}?Pst
L({Sk}Kk=1,?) (7.26)
Because the problem in (7.18) is convex, it is equivalent to the following minimization prob-
lem
min
?
g(?), s.t. ? ? 0 (7.27)
We choose initial ?, calculate Sk(?) and resulting g(?), and then update ? according to the
descent direction of g(?). This process repeats until the algorithm converges.
Since g(?) may not be differentiable, we use subgradient direction to update ?. The
subgradient is ? ?summationtextKk=1 tr{Hk,pSkH?k,p}. When summationtextKk=1 tr{Hk,pSkH?k,p} > ?, increase ? and
vice versa. With a fixed ?, the optimal transmit covariance matrices{Sk}Kk=1 can be obtained
by solving problem P3. We reformulate the objective function in P3 as
L({Sk}Kk=1,?) =
Ksummationdisplay
k=1
{
Ksummationdisplay
i=k
?iBlog2 |?Hk,sSkH?k,s +?(m)k,i |??(?(tr{Sk}+?k)+KPc)
129
??(tr{Hk,pSkH?k,p}? 1K?)}. (7.28)
Let L?(Sk,?) denote kth ?element? in L({Sk}Kk=1,?):
L?(Sk,?) =
Ksummationdisplay
i=k
?iBlog2|?Hk,sSkH?k,s +?(m)k,i |??(?(tr{Sk}+?k)+KPc)
??(tr{Hk,pSkH?k,p}? 1K?). (7.29)
It is clear that objective function as well as the constraints in P3 are separable. Therefore,
the solution to problem P3 is equivalent to solutions to the following K subproblems:
P4 : max
Skfollowsequal0
tr{Sk}?Pst
L?(Sk,?). (7.30)
The objectiveL?(Sk,?) is a linear combination of concave functions on Sk and the constraints
on Sk are also convex. Based on [88], problem P4 is a concave optimization problem in
positive semidefinite matrix Sk. In a realistic MAC scenario, it is reasonable to assume that
the BS has more antennas than the SUs. Consequently, the possible MAC channel models
include SISO, SIMO and MIMO models. For SISO and SIMO models, an SU has just one
transmitter antenna and only the transmit power sk needs to be optimized; however, it is
hard to get an explicit solution for sk using a gradient method directly [99, 101]. Due to
convexity of L?(Sk,?), sk can be obtained using a simple line search in range (0, Pst). For
general MAC with multiple antennas at both transmitter and receiver, we need to optimize
transmit covariance matrix Sk. However, the solution for Sk cannot utilize a water-filling
algorithm as in [106, 100, 108] because Sk cannot be diagonalized. Fortunately, Sk is a
positive semi-definite matrix, therefore the concave optimization problem P4 can be solved
by utilizing the SDP software, in particular, SeDuMi embedded in the CVX Matlab software
package.
130
Let {S?k}Kk=1 and ?? denote the optimal energy efficient power allocation and optimal
value of the Lagrange multiplier ?. Then {S?k}Kk=1 and ?? satisfy the KKT conditions.
7.5 Simulation Examples
We consider a C-MAC composed of a single PU, 2 SUs with multiple transmit antennas
and a single secondary BS with multiple antennas. The priority weights for two SUs are
fixed at (w1,w2)=(1.4,0.6). The bandwidth B is normalized to be 1. The elements of the
channel fading matrices {Hk,p}2k=1 and {Hk,s}2k=1 from the SU transmitter to the primary
and secondary receivers are both mutually independent complex Gaussian random variables
distributed as CN(0, 0.1) and CN(0,1), respectively. It is assumed the power amplifier
efficiency is 38% for the secondary transmitter?s RF amplifier, leading to ? = 1/0.38 in
(7.5). The static circuit power is assumed to be 100 mW. The value of interference power
constraint on PU is ? = 1 mW. The background white Gaussian noise power is assumed
to be 1 mW (0dBm). The reported results are based on 100 Monte Carlo runs with 100
randomly generated {Hk,p}2k=1 and {Hk,s}2k=1.
Figs. 7.1 and 7.2 show the capacity and energy efficiency, respectively, of the cognitive
MAC channel under two cases of energy efficiency maximization as discussed in Sec. III and
spectral efficiency (capacity) maximization without any consideration of power consumption,
as discussed in [99]. Our proposed scheme and the spectral efficiency maximization of [99] are
denoted as ?EE Max? and ?SE Max? respectively in these figures. Three different antenna
settings were considered: 1 ? 1 SISO with Nt,s = 1, Nr,s = 1 and Nr,p = 1, 1 ? 2 SIMO
with Nt,s = 1, Nr,s = 2 and Nr,p = 2 and 2 ? 2 MIMO with Nt,s = 2, Nr,s = 2 and
Nr,p = 2. The transmit power constraint Pst for the SU transmitter was varied from 0 mW
to 150 mW. It is seen that compared with the conventional spectrum efficiency maximization
design for multiple access channel, the energy efficient design can enhance energy efficiency
dramatically, particularly in the higher transmit power range but at the cost of smaller
131
throughput. Inaddition, morereceiveantennasresultinhigherenergyefficiencyandcapacity
because spatial diversity is utilized.
7.6 Conclusions
In this chapter, we proposed an energy-efficient transmit covariance matrix (precoder)
design for MIMO based cognitive multiple access channel in spectrum sharing underlay
cognitive radio networks. We constructed an energy efficiency function that incorporates
both transmission and circuit power consumption. Then an energy efficiency maximization
problem was formulated for C-MAC under individual SU transmit-power and PU interference
power constraints. It turned out to be a fractional programming problem that was iteratively
solved via a parametrized convex optimization problem formulation. We further decompose
the coupled parametrized convex optimization into K parallel non-interfering sub convex
optimization problem solved by simple line search or mature SDP algorithm. Compared
with existing spectrum efficiency maximization [99], our proposed energy efficient scheme can
enhance energy efficiency significantly under different receiver antenna number. The trade-off
between energy efficiency and spectrum efficiency was explored via computer simulations.
132
0 50 100 1500
2
4
6
8
10
12
14
16
Transmit Power Constraint (mW)
capacity (bits per second)
Capacity comparison
1x1 SISO EE Max
1x1 SISO SE Max
1x2 SIMO?MAC EE Max
1x2 SIMO SE Max
2x2 MIMO EE Max
2x2 MIMO SE Max
Figure 7.1: Capacity comparison: In the energy efficiency criterion we maximize energy effi-
ciency ?({Qk}Kk=1) as in problem P1 whereas in the spectral efficiency criterion we maximize
the channel capacity as in [99]. In 1?1 SISO system we used Nt,s = 1, Nr,s = 1 and Nr,p = 1,
the 1?2 SIMO system had Nt,s = 1, Nr,s = 2 and Nr,p = 2, and the 2?2 MIMO system
had Nt,s = 2, Nr,s = 2 and Nr,p = 2.
0 50 100 1500
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Transmit Power Constraint (mW)
energy efficiency (bits/Joule)
Energy Efficiency Comparison
1x1 SISO SE Max
1x1 SISO EE Max
1x2 SIMO SE Max
1x2 SIMO EE Max
2x2 MIMO SE Max
2x2 MIMO EE Max
Figure 7.2: Energy efficiency comparison: In the energy efficiency criterion we maximize
energy efficiency ?({Qk}Kk=1) as in problem P1 whereas in the spectral efficiency criterion
we maximize the channel capacity as in [99]. In 1?1 SISO system we used Nt,s = 1, Nr,s = 1
and Nr,p = 1, the 1 ? 2 SIMO system had Nt,s = 1, Nr,s = 2 and Nr,p = 2, and the 2 ? 2
MIMO system had Nt,s = 2, Nr,s = 2 and Nr,p = 2.
133
Chapter 8
Conclusion and Future Work
8.1 Conclusion
This dissertation covers two main topics: 1) low complexity cyclostationary spectrum
sensing, and 2) energy efficient resource allocation for cognitive radio network.
In Chapter 2 through Chapter 5, secondary users need to first sense the activities of
primary users and then decide whether to transmit or not based on presence or absence of
primary users. This secondary access scheme is called overlay spectrum sharing for cogni-
tive radio network. Low-complexity cyclostationary spectrum sensing under white Gaussian
noise and colored Gaussian noise are studied in Chapter 3 and Chapter 4, respectively. In
addition to sensing complexity, energy efficiency has become a ?hot? research topic due to
the trend of sustainable energy and green wireless communication and networks. A joint
energy efficient sensing and transmission for multichannel based cognitive radio works is
proposed in Chapter 5. The energy efficient resource allocation is also studied under another
spectrum sharing cognitive radio network named underlay CRN in Chapter 6 and Chapter
7. With underlay access, secondary users can transmit simultaneously with primary users
under limited interference power constraint to primary users. To utilize spatial diversity and
multiplexing, MIMO-assisted transmission is incorporated into transmission scheme. We de-
sign optimal transmit power allocation for energy efficiency maximization in single secondary
link and multi secondary links based cognitive radio network.
In Chapter 3, detection of cyclostationary primary user (PU) signals in white Gaussian
noise for cognitive radio systems was considered based on looking for a cycle frequency at
a particular time lag in the cyclic autocorrelation function (CAF) of the noisy PU signal.
We explicitly exploit the knowledge that under the null hypothesis of PU signal absent,
134
the measurements originate from white Gaussian noise with possibly unknown variance.
Our formulation allows us to simplify the spectrum sensing detector, and obviates the need
for estimating an unwieldy covariance matrix needed in the Dandawate-Giannakis [59] and
related approaches. We considered both single and multiple antenna receivers and both
nonconjugate and conjugate CAFs. A performance analysis of the proposed detectors was
carried out. Supporting simulation examples were presented to compare our detectors with
those of [59], [73], [63] and [71], and to demonstrate the capability to discriminate among
signals exhibiting cyclostationarity at different cycle frequencies and/or time lags. Our pro-
posed approaches are computationally cheaper than the Dandawate-Giannakis and related
approaches [59], [73] while having quite similar detection performance. Our proposed ap-
proach significantly outperforms [71] while having same computational complexity. In white
noise channels, the single-antenna detector of [63] has a 3dB SNR advantage for achieving a
detection probability of 0.9 over our proposed detector but requires five times more complex
multiplications. Also, Jallon?s test [63] fails to reject a class of cyclostationary interfering
signals whereas our proposed detector is successful.
In Chapter 4, detection of cyclostationary PU signals in colored Gaussian noise for cog-
nitive radio systems was considered based on looking for single or multiple cycle frequencies
at single or multiple time lags in the CAF of the noisy PU signal. We explicitly exploit the
knowledge that under the null hypothesis of PU signal absent, the measurements originate
from colored Gaussian noise with possibly unknown correlation. Our formulation allows us
to simplify the spectrum sensing detector, and obviates the need for estimating an unwieldy
covariance matrix needed in the Dandawate-Giannakis [59, 64] and related approaches. We
considered both single and multiple antenna receivers and both nonconjugate and conjugate
CAFs. A performance analysis of the proposed detectors was carried out. Supporting sim-
ulation examples were presented to compare our detectors with those of [59], [73], [64] and
[71]. Our proposed approaches are computationally cheaper than the Dandawate-Giannakis
135
and related approaches [59], [64], [73] while having quite similar detection performance for
a fixed false alarm rate. Our proposed approach significantly outperforms [71].
In Chapter 5, we studied how to balance throughput and energy consumption for a
slotted multi-channel CRN. An optimization problem involving sensing parameters (sensing
duration and local test threshold) and power allocation was formulated to maximize a utili-
ty function that rewards throughput but penalizes energy consumption. To obtain (locally)
optimal sensing parameters and power allocation, an iterative algorithm was proposed. Nu-
merical simulations demonstrate that the iterative algorithm converges fast. It performs
better than a ?only power allocation optimization? approach and an existing approach [78]
that ignores energy efficiency.
In Chapter 6, we proposed an energy-efficient transmit covariance matrix (precoder)
design for spectrum sharing underlay cognitive radio networks when multiple antennas are
employed at the secondary user transmitter. We constructed an energy efficiency function
that incorporates both transmission and circuit power consumption. Then an energy effi-
ciency maximization problem was formulated for a single secondary link in a CRN under
SU transmit-power and PU interference power constraints. It turned out to be a fractional
programming problem that was iteratively solved via a parametrized convex optimization
problem formulation. A multichannel extension of the problem was also investigated. The
trade-off between energy efficiency and spectrum efficiency was explored via computer sim-
ulations.
In Chapter 7, we proposed an energy-efficient transmit covariance matrix (precoder)
design for MIMO based cognitive multiple access channel in spectrum sharing underlay
cognitive radio networks. We constructed an energy efficiency function that incorporates
both transmission and circuit power consumption. Then an energy efficiency maximization
problem was formulated for C-MAC under individual SU transmit power and PU interference
power constraints. It turned out to be a fractional programming problem that was iteratively
solved via a parametrized convex optimization problem formulation. We further decompose
136
the coupled parametrized convex optimization into K parallel non-interfering sub convex
optimization problem solved by simple line search or mature SDP algorithm. Compared
with existing spectrum efficiency maximization [99], our proposed energy efficient scheme can
enhance energy efficiency significantly under different receiver antenna number. The trade-off
between energy efficiency and spectrum efficiency was explored via computer simulations.
8.2 Future Work
The research work on energy efficient precoding in underlay MIMO single secondary link
and power allocation in SIMO cognitive MAC is an initial step of our research on energy
efficient resource allocation in CRN. There are still future research topics that merit study:
? On Robust Energy Efficient MIMO-Assisted Spectrum Sharing
In Chapter 6, we proposed energy efficient transmit covariance matrix (precoder) for
MIMO-assisted single secondary link. However, our algorithm requires exact channel
state information (CSI) from secondary transmitter to both secondary receiver and
primary receiver. Due to channel noise, fading and/or shortage of feedback between
transmitter and receiver, CSI may be partially known or even unknown. This brings
challenge to our proposed algorithm in Chapter 6 and inspires us to design robust
energy efficient precoding (beamforming) for the SU transmitter to maximize energy
efficiency when CSI is not perfectly known.
? On Energy Efficient MIMO Broadcast (BC) in Underlay Cognitive Radio
Networks
In underlay broadcast channel, how to optimize the transmit signal covariance matrix
at the base station (BS) for multiple secondary user receivers to maximize capacity
have attracted attention and efforts from academia. However, how to design energy
efficient broadcast (BC) has received less attention. We aim to enhance energy ef-
ficiency of BC transmission by optimizing transmit signal covariance matrix at BS.
137
The secondary users is equipped with either single or multiple antennas. Moreover,
different secondary receivers have different nonnegative priority weights, thus the total
BC capacity is a weighted sum. Consequently, the final energy efficiency is defined as
a weighted capacity versus energy consumption. The BC transmission is constrained
by the interference power upper bound to the primary users.
138
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Appendices
148
Appendix A
Proof of Lemma 1 of Chapter 4
We have
E{?Rik,x(?;?)} = Rik,x(?)[ 1M
Msummationdisplay
m=1
e?j2??m] (A.1)
and
E{xi(m1)x?j(m1 +?1)e?j2??m1x?k(m2)xl(m2 +?2)ej2??m2}
= cum4parenleftbigxi(m1),x?j(m1 +?1),x?k(m2),xl(m2 +?2)parenrightbig
?e?j2?(?m1??m2) +Rij,x(?1)R?kl,x(?2)e?j2?(?m1??m2)
+Rik,x(m2 ?m1)Rlj,x(m1 ?m2 ??2 +?1)e?j2?(?m1??m2). (A.2)
For Gaussian sequences, the 4th cumulants are identically zero. Hence,
cov(?Rij,x(?;?1), ?R?kl,x(?;?2))
= E{?Rij,x(?;?1)?R?kl,x(?;?2)}?E{?Rij,x(?;?1)}E{?R?kl,x(?;?2)}
= 1M2
Msummationdisplay
m1=1
Msummationdisplay
m2=1
[Rik,x(m2 ?m1)Rlj,x(m1 ?m2 ??2 +?1)
?e?j2?(?m1??m2)] =: A. (A.3)
Setting m1 ?m2 = m in (A.3) and with x(k) = n(k), we have
A = 1M2
M?1summationdisplay
m=?(M?1)
e?j2??mRik,n(?m)Rlj,n(m+?1 ??2)?[
Msummationdisplay
m1=|m|+1
e?j2?(???)m1]. (A.4)
149
We have
1
M
Msummationdisplay
k=1
e?j2?(???)k =
??
????
???
??
1 for ? = ? ? A
e?jpi(???)(M+1)
M
sin(?(???)M)
sin(?(???))
for ? negationslash= ?, ?,? ? A.
(A.5)
It then follows that for any fixed m,
lim
M??
1
M
Msummationdisplay
k=|m|+1
e?j2?(???)k = lim
M??
1
M[
Msummationdisplay
k=1
e?j2?(???)k ?
|m|summationdisplay
k=1
e?j2?(???)k] = ??,?. (A.6)
Using (A.3), (A.4), (A.6) and the fact that n(k) is spatially uncorrelated, we obtain (4.10).
Using (4.9) it follows that
?R?ik,x(?;?) = ?Rki,x(??;??) = ?Rki,x(1??;??). (A.7)
Therefore, from (4.10) and (A.7), we obtain (4.11). square
150
Appendix B
Proof of Lemma 2 of Chapter 4
When n(k) is WSS, circularly symmetric, complex-valued, zero-mean colored Gaussian
and x(k) = n(k), it follows easily that E{?Rnn(?)(?;?)} = 0 and
cov(?R(?)ij,n(?;?1), ?R(?)kl,n(?;?2))
= E{?R(?)ij,n(?;?1)?R(?)kl,n(?;?2)} = 0 (B.1)
where R(?)ik,x(?) := E{xi(m)xk(m+?)}. Mimicking Sec. 4.3.1 (and setting m2 ?m1 = m),
we have
cov(?R(?)ij,n(?;?1), ?R?(?)kl,n(?;?2))
= 1M2
Msummationdisplay
m1=1
Msummationdisplay
m2=1
[Rik,n(m2 ?m1)Rjl,n(m2 ?m1 +?2 ??1)
+Ril,n(m2 ?m1 +?2)Rjk,n(m2 ?m1 ??1)]e?j2?(?m1??m2)
= 1M2
M?1summationdisplay
m=?(M?1)
ej2??m[Rik,n(m)Rjl,n(m+?2 ??1)
+Ril,n(m+?2)Rjk,n(m??1)]?[
Msummationdisplay
m2=|m|+1
e?j2?(???)m2]. (B.2)
Eqn. (4.18) then follows. square
151
Appendix C
Fractional Programming
In [105], fractional programming is introduced. Consider the following fractional pro-
gram
CF1 : max
x?C1
F1(x) where F1(x) = f(x)g(x), (C.1)
C1 ?Rn is a compact convex set, f(x) is nonnegative and concave on C1, and g(x) is positive
and convex on C1.
Problem CF1 is said to be a concave fractional program if the numerator f(x) of F1(x)
is concave on C1, and g(x) and C1 are convex function and set, respectively. In addition it is
assumed that f(x) is nonnegative on C1 if g(x) is not affine. square
152