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2017 Spectrum Sensing for Wavelet Packet Modulation Schemes in Wireless Communication Systems
Le, Ngon
Le, N. (2017). Spectrum Sensing for Wavelet Packet Modulation Schemes in Wireless Communication Systems (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/27853 http://hdl.handle.net/11023/3959 doctoral thesis
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Spectrum Sensing for Wavelet Packet Modulation Schemes
in Wireless Communication Systems
by
Ngon Thanh Le
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
GRADUATE PROGRAM IN ELECTRICAL AND COMPUTER ENGINEERING
CALGARY, ALBERTA
JULY, 2017
© Ngon Thanh Le 2017
Abstract
Orthogonal Frequency Division Multiplexing (OFDM) is the popular multicarrier modulation technique used in wireless communications, especially on fading channel. Having some advantages compared to OFDM, Wavelet Packet
Modulation (WPM) is a promising multicarrier candidate in Cognitive Radio
(CR) systems. An essential component of the CR systems is efficient and fast spectrum sensing (SS), which ensures that the secondary users (SUs) use the spectral resources at the right time and place. In this dissertation, based on an approximated Covariance Matrix of the noise-free received signal, several novel spectrum sensing methods are proposed for WPM systems. Dealing with the shortage of knowledge about the primary signal at the SUs’ receiver, another SS method based on Rao score’s test is also proposed. To reduce the sensing time, two sequential SS approaches are proposed and investigated. The excellent performances of the proposed methods are evaluated under various constraints, shown by some analytic and simulation results.
ii
Acknowledgements
First of all, I would like to express my deepest gratitude to Professor Abu
Sesay, my supervisor, for his valuable guidance, comments and criticism throughout the development of this dissertation under his supervision.
My sincere thanks go to Ammar Al-Masri for his useful discussions and suggestions, to Tung Lai for his long, kind friendship, and timely encouragement.
I would like to thank my colleagues for their help, the great lab environment and the splendid moments they shared with me.
I wish to make my feelings clear to my parents and my brother for their unconditional support and love. I never forget their sacrifice which gives me a chance to be born, grown-up and mature.
Finally, I would like to thank my wife, my parent in law, and my children for their support, patience and love. This period of time will be over soon, but love is growing up.
iii
Dedication
To my parents, my brother, my wife, my children and
to all others who have helped me. I love you all.
iv
Table of Contents
Abstract ...... II
Acknowledgements ...... III
Dedication ...... IV
Table of Contents ...... V
List of Figures...... IX
List of Abbreviations ...... XII
List of Symbols ...... XVI
CHAPTER 1 INTRODUCTION...... 1
1.1 BACKGROUND ...... 3
1.1.1 Radio channel characteristics ...... 3
1.1.2 Discrete Fourier transform (DFT) and fast Fourier transform (FFT) ...... 6
1.1.3 Short term Fourier transform (STFT) ...... 7
1.1.4 Wavelet Transform (WT)...... 9
1.1.5 Wavelets in communications ...... 18
1.1.6 Detector and Receiver operating characteristic (ROC)...... 22
1.2 DISSERTATION OBJECTIVES AND CONTRIBUTIONS ...... 25
1.3 DISSERTATION OVERVIEW ...... 28
CHAPTER 2 MULTICARRIER SYSTEMS, COGNITIVE RADIO SYSTEMS AND CONVENTIONAL SPECTRUM
SENSING METHODS ...... 31
2.1 CONVENTIONAL OFDM AND WPM WIRELESS SYSTEMS ...... 31
2.2 CONVENTIONAL RADIO AND COGNITIVE RADIO SYSTEMS ...... 37 v
2.2.1 Underlay ...... 38
2.2.2 Overlay ...... 39
2.2.3 Interweave ...... 40
2.3 CONVENTIONAL SPECTRUM SENSING METHODS ...... 41
2.3.1 Energy detector (ED) ...... 43
2.3.2 Matched filter ...... 45
2.3.3 Feature detection ...... 46
2.3.4 Cooperative spectrum sensing ...... 52
2.4 SOME SPECIAL SS METHODS ...... 54
2.4.1 Compressed sensing ...... 55
2.4.2 Information theoretic criteria (ITC) ...... 56
2.4.3 Markov model-based SS ...... 56
2.4.4 Entropy-based SS ...... 57
2.5 SOME FUNDAMENTAL LIMITS OF SPECTRUM SENSING AND EXPERIMENT RESULTS ON COGNITIVE RADIO ...... 58
2.6 SPECTRUM SENSING STANDARDIZATION ...... 63
2.7 CHAPTER CONCLUSIONS ...... 65
CHAPTER 3 SPECTRUM SENSING FOR WPM BASED COGNITIVE RADIO MULTI-CARRIER SYSTEMS USING
PILOT SIGNALS ...... 67
3.1 WPM SYSTEMS, COGNITIVE RADIO AND SPECTRUM SENSING ...... 68
3.2 WAVELET PACKET MODULATION BASED MULTI-CARRIER SYSTEMS ...... 70
3.3 PROPOSED SPECTRUM SENSING METHODS ...... 73
3.3.1 The optimal detection method ...... 75
3.3.2 ATDSC ...... 76
3.3.3 AWDSC ...... 77
vi
3.3.4 Approximated Covariance Matrix (ACM)-based method ...... 78
3.3.5 GLRT approaches for unknown parameters ...... 81
3.3.6 The Receiver Operation Characteristic (ROC) ...... 85
3.4 SIMULATION RESULTS ...... 89
3.5 CONCLUSION ...... 96
CHAPTER 4 SPECTRUM SENSING FOR WPM SYSTEMS BASED ON RAO’S SCORE TEST ...... 98
4.1 CONVENTIONAL RAO’S SCORE TEST-BASED SPECTRUM SENSING METHODS ...... 100
4.2 A NOVEL TEST STATISTIC BASED ON RAO TEST FOR WPM SYSTEMS ...... 102
4.3 PERFORMANCE ASSESSMENT ...... 110
4.4 SIMULATION RESULTS ...... 113
4.5 CONCLUSION ...... 117
CHAPTER 5 SEQUENTIAL SPECTRUM SENSING FOR WPM SYSTEMS ...... 118
5.1 SEQUENTIAL PROBABILITY RATIO TEST (SPRT) AND ITS APPLICATIONS ...... 120
5.2 SOME TRADITIONAL DETECTION METHODS FOR DETERMINISTIC SIGNALS AND RANDOM SIGNALS ...... 124
5.2.1 Matched filter for deterministic signals ...... 124
5.2.2 Estimator-Correlator for random signals ...... 125
5.3 THE NOVEL SEQUENTIAL SPECTRUM SENSING METHOD FOR WPM SYSTEMS ...... 126
5.4 THE TEMPORAL UPDATE RECURSION FOR THE TEST STATISTIC ...... 128
5.4.1 Deterministic signals ...... 128
5.4.2 Random signals ...... 128
5.5 BOUNDS FOR THE OVERALL PROBABILITIES Pfa AND Pmd ...... 130
5.6 PROCEDURES TO SETUP THE PARAMETERS OF THE SEQUENTIAL SENSING METHOD WHEN THE OVERALL PROBABILITIES Pfa
AND Pmd ARE GIVEN ...... 133
vii
5.6.1 Procedure 1 ...... 133
5.6.2 Procedure 2 ...... 135
5.7 SIMULATION RESULTS ...... 142
5.8 CHAPTER CONCLUSION ...... 152
CHAPTER 6 CONCLUSION AND FUTURE WORKS ...... 154
6.1 DISSERTATION SUMMARY AND CONCLUSIONS ...... 154
6.2 FUTURE WORK ...... 156
REFERENCES ...... 158
APPENDIX A ...... 190
viii
List of Figures
Figure 1.1 STFT resolutions for small (left) and large (right) fixed window sizes 9
Figure 1.2 An illustration of the spectrum of a signal covered by an infinite number of dilated wavelets ...... 13
Figure 1.3 Arbitrary tiling of the time-frequency plane using wavelet transform 13
Figure 1.4 One stage of the discrete wavelet transform (DWT) ...... 15
Figure 1.5 One stage of the inverse discrete wavelet transform (IDWT) ...... 15
Figure 1.6 Range of Wavelet Applications for Wireless Communications ...... 18
Figure 2.1 Block diagram of a conventional OFDM multicarrier system ...... 30
Figure 2.2 Block diagram of a conventional WPM multicarrier system ...... 31
M Figure 2.3 Transmitter structure for a WPM based multi-carrier system with 2 sub-channel full tree...... 32
M Figure 2.4 Receiver structure for a WPM based multi-carrier system with 2 sub-channel full tree...... 32
M Figure 2.5 The equivalent block diagrams of IDWT and DWT with 2 sub- channel full tree ...... 34
Figure 2.6. The diagram of the conventional energy detector ...... 42
M Figure 3.1 An equivalent representation of a WPM transceiver structure with 2 sub-channels...... 68
M Figure 3.2 The pilot placements of a WPM block with 2 sub-channels ...... 72
ix
Figure 3.3 ROC curves of the ACM-based method for different SNRs ...... 88
Figure 3.4 ROC curves of ACM-based methods in frequency-selective channel at
SNR=-20dB ...... 89
Figure 3.5 ROC curves for several methods in frequency-selective channel at
SNR=-10dB ...... 90
Figure 3.6 ROC curves for several methods in frequency-selective channel at
SNR=-20dB ...... 91
Figure 3.7. Pd performance comparison of various methods for Pfa 0.01 93 ......
Figure 4.1 ROC curves for several methods in frequency-selective channel at SNR
= -15 dB ...... 111
Figure 4.2 The detection performances of different methods for Pfa 0.01 at SNR
= -15 dB in frequency-selective channel ...... 113
Figure 5.1 Detection performances of sequential and non-sequential methods for deterministic signals ...... 141
Figure 5.2 ROC performance of sequential and non-sequential methods for deterministic signals with different number of blocks ...... 142
Figure 5.3 The coefficients i for different stages in sequential method for random signals ...... 143
C Figure 5.4 Nine eigenvectors of i for nine different stages in sequential method for random signals ...... 144
x
Figure 5.5 Detection performances of sequential and non-sequential methods for random signals ...... 145
Figure 5.6 ROC performances of non-sequential method and the first approach for random signals ...... 147
Figure 5.7 ROC performances of non-sequential method and the second approach for random signals ...... 148
xi
List of Abbreviations
Abbreviation Meaning
ACM approximated Covariance Matrix
ADC Analog-to-Digital Converter
ADSL Asymmetric Digital Subscriber Lines
AIC Akaike information criterion
ARMA autoregressive moving average
ATDSC Accumulated Time-Domain Symbol Cross-Correlation
AWDSC Accumulated Wavelet-Domain Symbol Cross-Correlation
AWGN Additive White Gaussian Noise
BER Bit Error Rate
BS Base Station
CAF Cyclic Autocorrelation Function
CDF Cumulative Distribution Function
CDMA Code Division Multiple Access
CFAR Constant False Alarm Rate
CIR Channel Impulse Response
CLT Central Limit Theorem
CP Cyclic Prefix
xii
CPE Customer Premises Equipment
CR Cognitive Radio
CSCG Circularly Symmetric Complex Gaussian
CSD Cyclic Spectral Density
CWT Continuous Wavelet Transform
Db xx Daubechies xx
DFT Discrete Fourier Transform
DSP Digital Signal Processor
DWT Discrete Wavelet Transform
FBMC Filter Bank Multicarrier
FCC Federal Communications Commission
FDM Frequency Division Multiplexing
FFT Fast Fourier Transform
FT Fourier Transform
FWT Fast Wavelet Transform
GLRT Generalised Likelihood Ratio Test
GPS Global Positioning System
IBI Inter-Block Interference
ICI Inter-Channel Interference
IDFT Inverse Discrete Fourier Transform
xiii
IDWT Inverse Discrete Wavelet Transform
IFFT Inverse Fast Fourier Transform
ISI Inter-Symbol Interference
ITC Information theoretic criteria
LLR Log-Likelihood Ratio
LMP Locally Most Powerful
MDL Minimum Description Length
OFDM Orthogonal Frequency Division Multiplexing
OLA Overlap-Add
PDF Probability Density Function
PSD Power Spectral Density
P/S Parallel to Serial
PSK Phase Shift Keying
PU Primary User
QAM Quadrature Amplitude Modulation
QMF Quadrature Mirror Filters
QoS Quality of Service
RF Radio Frequency
ROC Receiver Operating Characteristic
S/P Serial to Parallel
xiv
SPRT Sequential Probability Ratio Test
SS Spectrum Sensing
STFT Short Time Fourier Transforms
SU Secondary User
WPM Wavelet Packet Modulation
WSS Wide-Sense Stationary
WT Wavelet Transform
xv
List of Symbols
Symbol Definition a a vector
A a matrix
A(i,j) the (i,j)th element of A s scaling factor
translation factor
t mother wavelet function
s,, t, jk t scaling and translation wavelet
t scaling function
j scaling function coefficient at the scale j
j wavelet function coefficient at the scale j cov covariance matrix
T transpose matrix
H Hermitian transpose matrix
* complex conjugate h n low-pass filter of the QMF pair g n high-pass filter of the QMF pair
H, G components of QMF pairs at the receiver
xvi
1 1 H, G components of QMF pairs at the transmitter
2 L R the Lebesgue space
Wi n the overall response of the i th sub-channel at the receiver
Wi n the overall response of the i th sub-channel at the transmitter
E x average value of random variable x
Rex real part of complex variable x
Imx imaginary part of complex variable x
Pr probability of event
xvii
Chapter 1: Introduction 1
CHAPTER 1
INTRODUCTION
Generations of wireless communication systems are being continually developed to satisfy the need for increased capacity and improved quality. To overcome the challenges and the restrictions of wireless systems, many multicarrier modulations schemes such as orthogonal frequency division multiplexing (OFDM), wavelet packet modulation (WPM) and Filter Bank
Multicarrier (FBMC) are being investigated. By transforming a frequency- selective channel into multiple narrowband frequency-flat sub-channels, these techniques provide more bandwidth efficiency and better immunity to the effects of the fading channel. The fast digital transforms in these techniques make them becomes promising candidates for current and future applications.
Unlike Fourier-based OFDM, used in many existing wireless systems and services, in wavelet transform the data blocks overlap in time, which helps WPM offer higher sub-channel spectral containment [1]. As a result, the spectra of
WPM sub-channels still overlap but the received signal has an insignificant amount of interference from neighbouring sub-channels compared to OFDM signal. Therefore, WPM can eliminate narrowband interferences better than
OFDM.
Chapter 1: Introduction 2
To achieve higher speeds, higher energy and spectral efficiency, the next generation systems must be flexible and reconfigurable so they can adapt to their radio environments. In other words, these systems can optimize the radio parameters such as power and spectrum while guaranteeing a desirable quality of service (QoS). Compared to Fourier transform, wavelet-based systems offer many advantages for the design of wireless communications. Firstly, we can easily design the orthogonal basis wavelet functions and the WPM tree for various scenarios. These degrees of freedom help wavelet-based system have adaptive time-frequency resolution. These advantages of WPM systems over traditional
OFDM systems can be widely customized to satisfy different requirements of the systems. In addition, unlike sine and cosine basis wave which are infinite support, compact support wavelet basis functions (that is, they are zero outside of a compact set) provide promising potential advantages. Wavelet-based systems, such as cognitive radio, can adaptively reconfigure to maximize resource utilization and interference mitigation. Furthermore, unlike OFDM, WPM does not employ a cyclic prefix, and therefore it is more bandwidth efficient [2].
WPM systems have recently received research attention (see [3] and references therein) and found some potential applications, which include wireless communications, edge and transients detection, code division multiple access
(CDMA), asymmetric digital subscriber lines (ADSL), high speed copper wire communications [1], [4], and power line communication [5], [6].
Chapter 1: Introduction 3
The rest of the chapter is organized as follows. First some background information is introduced in Section 1.1. The main objectives and contributions of the dissertation are then presented in Section 1.2. Finally, Section 1.3 provides a chapter-wise overview of this dissertation.
1.1 Background
In this chapter, we first describe the characteristics of the radio channel.
This is followed by an overview of discrete Fourier transform (DFT), fast Fourier transform (FFT), and short time Fourier transform (STFT). Then, this section provides the concept of wavelet transform (WT), and some applications of wavelet in communications. Finally, we discuss the detector and its receiver operating characteristic (ROC).
1.1.1 Radio channel characteristics
Due to reflections, diffractions and scatterings from the wireless channel, the received signal suffers from various impairments. To classify the wireless channel and quantify these effects, the following parameters are defined:
Doppler shift: this is due to the relative movements of the transmitter, the receiver, and/or the obstacles between them. The maximum Doppler shift is expressed as follows [7]:
v f d ,max (1.1)
Chapter 1: Introduction 4 where v denotes the relative speed of the receiver with respect to the transmitter and is the carrier signal’s wavelength.
Coherence time Tc : the wireless channel is considered to be invariant over this time period. It is defined as follows [7]:
9 Tc (1.2) 16 fd ,max
The root mean square (RMS) delay spread Trms : this is the standard deviation of the multipath propagation time delays normalized with respect to their powers [7]:
L L 2 2 Pll P ll l1 l 1 Trms L L (1.3) Pl P l l1 l 1 where Pl and l denote the power and the propagation delay of the l th component, l1,..., L ( L is the number of paths), respectively.
Coherence bandwidth Bc : the wireless channel is considered flat over this bandwidth. Coherence bandwidth is inversely proportional to the RMS delay spread Trms .
Based on these parameters, the wireless channel can be classified as [7]:
Slow fading and fast fading. When the symbol duration Ts is greater than the coherence time Tc of the channel, the channel is said to be fast fading, otherwise, it is slow fading.
Chapter 1: Introduction 5
Frequency selective fading and frequency flat fading. The channel is considered to be frequency selective fading when Bs B c and frequency flat fading when Bs B c where Bs is signal bandwidth.
Propagation models that capture these parameters are generally classified into path loss, large-scale fading and small-scale fading. Path loss refers to the power reduction between the transmitted and received signals. Path loss is often modeled as an exponential function of the separation distance and also influenced by terrain contours, environment, and propagation medium as well as the height and location of antennas. Large-scale fading refers to the received signal power fluctuation due to diffraction around large objects in the propagation path. Due to the presence of reflecting and scattering, multiple versions of the transmitted signal arrive at the receiver with different amplitudes, phases and angles of arrival. This multipath phenomenon results in rapid fluctuations of the amplitudes, phases or propagation delays of the received signal and is referred to as small-scale fading. Path loss, large-scale fading and small-scale fading combine at the receiver to cause signal fading or distortion. In this dissertation, we only focus on small-scale fading, which characterizes the rapid fluctuations of the received signal over short distances or short time durations.
Chapter 1: Introduction 6
1.1.2 Discrete Fourier transform (DFT) and fast Fourier transform (FFT)
By passing through a discrete Fourier transform, a finite complex-valued sequence of signal in time domain is converted into a complex-valued sequence of equivalent length signal in the frequency domain as follows:
N1 2jkn / N Xk xne , kZ ,0 kN 1 (1.4) n0
This transform is widely used in many practical applications such as spectral analysis, radio transmission, data compression, image processing, and discrete cosine transform. The frequency domain signal can be converted back to the time domain by using an inverse discrete Fourier transform (IDFT) as follows:
N1 1 2jkn / N xn Xke , nZ ,0 nN 1 (1.5) N k0
To reduce computational complexity, fast Fourier transform (FFT), introduced in 1965 [8], can be used to compute the DFT and its inverse IDFT.
2 Specifically, FFT helps reduce the complexity of DFT and IDFT from O n to
O nlog n , where n is the length of the signal and O g n denotes the order of approximation. That is, for all positive values of N , there exists a positive real number M such that the computational complexity does not exceed M* g n for all n N .
FFT can be found in many applications in mathematics, engineering and science and can be implemented in computer software or hardware.
Chapter 1: Introduction 7
1.1.3 Short term Fourier transform (STFT)
The DFT cannot tell at what time a specific frequency component occurs, i.e., the time and frequency information cannot be seen at the same time. For example, a signal and its reverse version in time domain cannot be distinguished in the frequency domain. Another example is a signal which contains different frequency components at different intervals. More specifically, the locations and timings of individual frequency components are hidden in the complex phase. To obtain the time-frequency representation of the signal, the short term Fourier transform (STFT) was introduced as a solution. The short term Fourier transform is obtained when the Fourier transform (FT) is applied to each windowed signal. The continuous-time and discrete-time STFT can be expressed as follows:
j t X, x t w t e dt (1.6) j n Xm, xnwnme n where x t and x n are the input signals in time domain, w t and wn m are the windows for continuous-time and discrete-time, respectively.
From the representation on the windowed segments of the signal in the frequency domain, the STFT provides resolutions in both time and frequency domains. In other words, STFT helps to determine the frequency and phase content of local sections of an input signal as it changes over time. Typically, the
Chapter 1: Introduction 8 window, such as Gaussian and rectangular, is real and symmetric in practice.
As it slides along the time axis, the window can overlap. We can reconstruct the signal perfectly when these windows are designed properly. One of the most common technique to implement the inverse STFT is the overlap-add (OLA) method. In this method, linear convolution is converted to circular convolution by adding some zeros samples to both of the windowed signal and the filter. As a result, the linear convolution can be implemented by using inverse fast Fourier transform (IFFT) followed by FFTs. The additional cost of this technique is small and can be neglected.
The window in STFT is compactly supported, that is, the window size is limited. The dilemma of resolution, created by the Heisenberg Uncertainty
Principle [9], states that the product of signal duration and bandwidth is bounded from below by a fixed limit. However, in STFT, this window size is fixed, therefore arbitrarily good resolution in both time and frequency cannot simultaneously be obtained. As shown in Fig. 1.1, STFT with a narrow window provides good time resolution but poor frequency resolution (left), while STFT with a wide window provides better frequency resolution but poor time resolution
(right). In the time-frequency domain, a signal can be geometrically located in rectangles, as shown in Fig 1.1. According to the Uncertainty Principle, the area of these rectangles is not less than 1/ 4. As a result, it is difficult to use STFT to represent chirp signals, in which the frequency increases or decreases with
Chapter 1: Introduction 9
2 2 time, such as ft sin t . In addition, the basis functions for STFT are not orthogonal when successive windows overlap, that is, the hop size is less than the window length.
1.1.4 Wavelet Transform (WT)
To overcome the pitfalls of STFT, wavelet transform and multiresolution analysis was introduced, which provides good time and frequency resolution [10]-
[21]. The two dimensional (2D) function WT helps to analyze the signal waveform in both frequency and duration, while avoiding the drawbacks of STFT
(that is, the fixed window size). Specifically, the gross and small features can be captured by utilizing wavelet transform, with a large and small window, respectively. These multiple window sizes help WT better represent many real- world signals, thereby making WT well-suited for approximating data with sharp discontinuities. As a result, real signals can be stored more efficiently by using
Figure 1.1 STFT resolutions for small (left) and large (right) fixed window sizes
Chapter 1: Introduction 10
WT than by FT. Moreover, WT is faster than the Fast Fourier Transform
(FFT) [22].
A. Continuous wavelet transform (CWT)
To decompose a signal x t, we first choose a mother wavelet function
t, which can be designed based on the application and the nature of the input x t. Unlike the sine and cosine functions in Fourier transform, this mother wavelet t must be time-finite and of zero average. These basis wavelets are continuously scaled and translated versions of the mother wavelet t . The wavelet transform basis in the time and frequency domains are defined as follows:
1 t b jb a,, bt a b aae (1.7) a a
To form a complete orthonormal set, these basis wavelets functions must satisfy three conditions, namely, orthogonality, normalization and completeness.
The continuous wavelet transform X a, b of a function x t is the convolution of the input x t and the set of basis wavelets and can be defined by the following formula:
1 * X a, b x ta, b t dt (1.8) a
* where a and b are the scale and translate factors, respectively, with a 0 ; denotes the complex conjugate operation. The two dimensions in the wavelet domain are the scale factor a and the translation factor b , which represent frequency and time resolution, respectively.
Chapter 1: Introduction 11
To be invertible, the mother wavelet t must be admissible, that is,
t satisfies the following condition
ˆ 2 u C du u (1.9)
ˆ where u is the Fourier transform of t . This condition implies zero-mean ˆ mother wavelet 0 0 , that is, t is a high pass filter.
To reconstruct the continuous-time domain signal x t from X a, b, the inverse continuous wavelet transform is defined as follows
x t X a, b t da db a, b (1.10)
Similar to FT, wavelet transform defines an isometry over its image of the
2 space L R. In other words, both well-known FT and WT are distance- preserving transformations between metric spaces.
The CWT can provide an excellent picture of the signal such as showing the time and the frequency and identifying any transient events [17]. With the control of the scale and translation factors a and b , we have the ability to capture any features of the signal in both time and frequency domains. In addition, the mother wavelet function t can be designed for various purposes.
When the form of the input signal x t is known, e.g. the Global Positioning
System (GPS) signal, the CWT is excellent in capturing and storing most of its
Chapter 1: Introduction 12 properties. CWT can also capture Doppler shifts, delays, slew, chirping and kinematic behaviors.
Continuous wavelet transform (CWT) is extremely redundant and therefore, not very practical. All the functions X a, b in the wavelet domain with continuous values of a and b are not necessary to be computed, stored, and processed. Instead, the functions x t are converted into the wavelet domain with discretized values of a and b , as defined in the next Sub-section. The wavelet sampling theorem states that the signal x t can be perfectly expressed in the wavelet domain by using only some scales that are powers of 2, up to a certain level.
B. Discrete wavelet transform (DWT)
Unlike the CWT where the scale factor a and the translation factor b can be any real value with a 0 ; in the DWT, they are specified as in the definition of the basis wavelet functions:
j 1 t k 0 s 0 j, k t j (1.11) j s s0 0 where s0 is the dilation step and 0 is the translation step; both s0 and 0 are fixed numbers with s0 >1; j and k are integers. When the dilation step s0 is chosen to be 2, as is often the case, the wavelet function t is stretched by a factor of 2 and the corresponding spectrum is compressed by a factor of 2 as shown in Figure 1.2. Therefore, the dilated wavelets can be used to cover the
Chapter 1: Introduction 13
.
3 2 1 0
Figure 1.2 An illustration of the spectrum of a signal covered by
an infinite number of dilated wavelets. finite spectrum of the signal. To cover the spectrum of the signal at low frequencies, a scaling function t , a low-pass signal introduced by Mallat, can be used to replace infinite dilated wavelets [11]. In other words, the spectrum of the signal x t can be covered by using a finite number of wavelet functions and a scaling function.
Wavelet Packets, also known as Subband Tree, is a wavelet transform
.
Figure 1.3 Arbitrary tiling of the time-frequency plane using wavelet transform.
Chapter 1: Introduction 14 that offers a richer signal analysis. In this generalization of wavelet decomposition, the discrete-time signal is passed through many levels, usually through more filters than DWT. From the wavelet functions t and scaling function t , a set of orthonormal bases forms a library of wavelet packet bases, and each of the function is called a wavelet packet. When the dilation step s0 is chosen to be 2, the time-frequency plane can be divided into several rectangles with the same area; each corresponding to an input sequence. As shown in Fig.
1.3, the bandwidth and the time interval of these rectangles can be designed to be different, as long as the areas of these rectangles are the same. As a result, wavelets can help communication systems boost immunity to fading channels while minimizing the affects from tone and time impulse noise by designing the time-frequency tiling [23]. In addition, this flexibility allows the wireless systems have different rates in different bands, which is an advantage of wavelet-based system for the frequency-selective channel [3]. However, the major drawback of wavelet transform is the lack of shift invariance. A circular shift of the input signal corresponds to multiplying the output signal by a linear phase in DFT, while in DWT there is no such property. As a result, a shift of the input signal can make wavelet coefficients vary substantially.
Chapter 1: Introduction 15
For discrete-time input signal x n , the DWT is obtained by passing the input through multiple stages or levels. The main components of these stages in DWT are shown in Fig. 1.4, where h n and g n are low-pass and high- pass filters, respectively, followed by two down-samplers. In inverse discrete wavelet transform (IDWT), two up-samplers are followed by the low-pass filter
H
h(-n) ↓2 λ j-1
λ j
g(-n) ↓2 γ j-1
G
Figure 1.4 One stage of the discrete wavelet transform (DWT) H-1
λ j-1 ↑ 2 h(n)
λ j
γ j-1 ↑ 2 g(n)
G-1
Figure 1.5 One stage of the inverse discrete wavelet transform (IDWT).
Chapter 1: Introduction 16 h n and the high-pass filter g n, as shown in Fig. 1.5. In these figures, h n and g n are the time-reversed versions of h n and g n, respectively;
H, G and H-1, G-1 are the quadrature mirror filter (QMF) pairs at the decomposition and the synthesis stages, respectively. As depicted in this figure, the wavelet function coefficient and scaling function coefficient can be computed from the previous level in DWT. In IDWT, the coefficients and can be combined to compute the coefficients of the next level.
The mother wavelet t should be chosen appropriately such that the discrete wavelets can be orthonormal, that is,
t* t dt ( j m ) k n j,, k m n (1.12) where n denotes the Kronecker delta function.
To satisfy perfect reconstruction conditions, these two filters and two down-samplers form a quadrature mirror filters (QMF) and these operators must meet the following criteria:
1 1 H GG H 0 1 1 (1.13) H HG GI ,
In other words, the filters h n and g n must satisfy the following conditions
o the filter length of h( n ) must be even (i.e., 2N ),
o h n 2, n
Chapter 1: Introduction 17
o hnhn 2 k k , n
n o g n1 h 2 N 1 n , where 2N is the length of the filters and (k ) denotes the Kronecker delta.
In Fourier transform, a signal f t with limited bandwidth where
k M f , kZ can be represented by discrete samples . Similarly, in M wavelet transform, under some mild restrictions on the scaling functions, wavelet sampling theorem states that the function f t with limited bandwidth where
J 2 can be sampled and perfectly reconstructed by using a set of coefficients up to level J [24]. If the signal f t is not band-limited, we can obtain the reconstructed version f t such that the error between f t and f t is minimized.
According to [3], Daubechies wavelet family provides the best ROC performance compared to other families such as Coiffet, Discrete Meyer, and
Haar. Throughout this dissertation we only use the conventional Daubechies wavelet as the basis functions for WPM systems.
C. Multiresolution Analysis
The Fourier transform (FT) analysis of a signal only provides information about the frequency domain, while STFT only uses a fixed window size. By using a variable scale, WT provides multiresolution analysis, introduced by Mallat [11].
For signals containing different frequencies at different intervals, the wavelet transform analyzes the various time periods with different resolutions [25]. In the
Chapter 1: Introduction 18
2 Lebesgue space L R of square integrable functions (finite energy), a multiresolution analysis consists of a sequence of nested subspaces. This sequence of nested subspaces satisfies completeness, regularity relations, density, separation, and self-similarity relations in time/translate and frequency/scale.
1.1.5 Wavelets in communications
The areas of wireless systems where the wavelet technique can be applied are summarized as follows [3]:
• Modeling of time-variant wireless channels with wavelets as bases
Based on statistical impulse response, current wireless models perform pretty well for time-invariant channels. However, they fail to characterize the time-variant channels. From the ability to capture any features of the signal in both time and frequency domains, wavelet-based models can accurately represent time-varying and fast-fading channels with various advantages such as small error, fewer number of required coefficients, and fast convergence of estimate.
• Antenna design
• Mitigation of Interference, ISI and ICI
• Multiple access communications
• Green radio communications
• Application of Wavelet Technology in Cognitive and Cooperative
Wireless Communications
• UWB communication
Chapter 1: Introduction 19
• Wavelets for Single Carrier Modulation [2]
Instead of using raised-cosine filter, the wavelet and scaling functions are used for pulse-shaping. The spectral efficiency, power efficiency, and coding gain are shown to improve in these schemes.
• Wavelet Packet Modulation (WPM) for Multi Carrier systems
Similar to OFDM, multiple parallel narrowband orthogonal sub-channels help WPM systems achieve higher bandwidth efficiency and the frequency selective channel is converted into multiple flat fading channels. Besides some advantages as OFDM, WPM system has wider flexibility, which means that we can design the tree structure and basis function. In addition, they have small ICI and high bandwidth efficiency. The wavelet-based multicarrier is proven to outperform its OFDM-based counterpart with respect to bit error rate (BER).
Moreover, the fact that WPM does not utilize cyclic prefix (CP) help to improve spectral efficiency further compared to OFDM. Furthermore, WPM is less sensitive to synchronization error and effects of fading channel and high power amplifier (HPA).
• Wavelet based MIMO
In OFDM, signals are well represented as a sum of sinusoids. However, a non-continuous signal with an abrupt discontinuity requires an infinite number of
Fourier coefficients (the Gibbs phenomenon). Wavelets are much better in representing signals with discontinuities due to their time-localized
Chapter 1: Introduction 20
Figure 1.6 Range of Wavelet Applications for Wireless Communications [3]. characteristics. To improve spectral efficiency, wavelets can be used in wavelet shift keying [26] and for pulse shaping [26]-[29]. Via waveform and receiver design, the wavelet packet transform can improve multiple access communication throughputs [30].
Wavelet OFDM was adopted to be the modulation scheme in high definition power line communication (HD-PLC) [31], a communications technology developed by Panasonic, and in the IEEE 1901 standard. Major advantages of Wavelet OFDM over traditional FFT OFDM are the fact that
Wavelet OFDM achieves deeper notches and it does not require a guard interval, which improves spectral efficiency. According to the IEEE broadband wireless standard 802.16.3, wavelet-based OFDM is extremely flexible and modular design
Chapter 1: Introduction 21 channelization, data-asymmetry supported [32]. In addition, wavelet-based
OFDM has no known restrictions and the large stop-band attenuation provides great multi-service potential. Moreover, wavelet-based OFDM is much more robust to inter-channel, spectral spillage, and channel impairment than OFDM.
With no time resolution, FFT is only used to process stationary signals, whose joint probability distribution does not change over time. DWT can also accomplish tasks like DFT and do better for non-stationary signals. As shown in
[17], wavelets exhibit excellent ability over conventional Fourier-based methods in many applications, such as filtering white noise in a chirp signal, extracting binary signal from chirp noise or from white noise. In all of these examples, since both noise and signal appear at all frequencies, the conventional FT-based filter fails to remove noise from signals. For the former application, by only keeping the signal within a certain time and within a certain frequency range, wavelet- based approach can easily denoise very well. To extract binary signals from chirp noise, we can choose wavelet basis functions that match chirp noise such as the
Db40 (Daubechies 40). The binary signal can be extracted by keeping the signal within a certain time and within a certain frequency range. Even in the very low
SNR of -80 dB, it is shown that although the process is not perfect, the binary signal can still be reconstructed. In the case of binary signal with white noise,
STFT can be also used to separate noise and the signal. However, by using Haar basis functions, a good match to binary signal, the signal can be extracted much
Chapter 1: Introduction 22 better due to the dynamic resolution. In short, the performance of these processes would be very good if the form of either the signal or the noise is known in advance. Then, for best discrimination, the wavelet basis functions can be carefully chosen or designed to match either the input signal or the noise.
Simulation results have demonstrated the superior of wavelet-based techniques over FT-based. When the shapes of both the signal and the noise are unknown, a general wavelet such as Daubechies 4 or Daubechies 6 can be used. Wavelets can be found in many applications in other disciplines, such as UWB [33], data denoising/smoothing and lossless or lossy compression such as JPEG 2000, DjVu, and electrocardiograph (ECG) signals. Wavelets are used extensively in Signal and Image Processing, Medicine, Finance, Radar, Sonar, Geology and many other varied fields.
1.1.6 Detector and Receiver operating characteristic (ROC)
A binary classifier, also known as a detector, maps the input into one of two classes or regions. The input can be real-valued or complex-valued, discrete- valued or continuous-valued; and the binary outputs can be labeled as positive
(alternative hypothesis H1 ) and negative (null hypothesis H0 ). The detector decides whether the received signal consists of noise only ( H0 ) or a signal in noise
( H1 ). This binary hypothesis detection problem is a complicated task due to the effect of the wireless channel and noise from the ambient radiation. There are four possible cases as summarized in the table below:
Chapter 1: Introduction 23
In general, we wish to design a detector which has the probabilities of miss detection Pr H0 H 1 and the probabilities of false alarm Pr H1 H 0 as small as possible. However, we cannot reduce both of them simultaneously. Depending on the application and scenarios, we focus more on miss detection or false alarm or both.
Firstly, we have to form a test statistic T x , a function of the vector data x of length N . Then the test statistic T x is compared to a threshold to make a decision about the status of the object. In most cases, this threshold is pre-defined based on the probabilities of false alarm Pr H1 H 0 , which is denoted as Pfa . According to the Neyman-Pearson theorem [34], to maximize the probabilities of detection PD Pr HH1 1 for a given PFA , the alternative hypothesis H1 is decided if the log-likelihood ratio (LLR) exceeds a threshold, that is,
px H1 LLR x (1.14) px H0
Chapter 1: Introduction 24
where px H1 and px H0 are the conditional probability density function
(PDF) of the received vector signal x under H1 and H0 , respectively; and the threshold is found from the constraint Pfa .
According to the Neyman-Fisher factorization theorem [35], if the PDF px;θ can be factored as px;,θ g T x θ h x, then T x is a sufficient statistic of θ . In other words, the best test statistic T x is one that can extract all the information from the data x about θ .
However, in practice, the parameter vector θ in the PDFs under H1 and
H0 are unknown at the receiver, which make the LLR incomputable. To overcome this obstacle, the unknown parameters are replaced by their maximum likelihood estimates (MLE) under the corresponding hypothesis. This approach, known as generalised likelihood ratio test (GLRT), decides the alternative hypothesis H1 if px|θ , H 1 1 LG x (1.15) px|θ , H 0 0 where θ1 and θ0 are the MLEs of θ under H1 and H0 , respectively. Although
GLRT is not the optimum, it works pretty well in practice.
To evaluate the detection performances of a method, we can use the receiver operating characteristic (ROC), which is a plot of PD versus PFA by adjusting the threshold . From the ROCs of different methods, it is easy to decide the best detector.
Chapter 1: Introduction 25
1.2 Dissertation Objectives and Contributions
The overall objective of this dissertation is to design optimal detectors for
Wavelet Packet Modulation-based Cognitive Radio Multi-Carrier Systems at low
SNR regimes under different constraints. Firstly, we develop two methods based on the Accumulated Time-Domain and Wavelet-Domain Symbol Cross-
Correlation (ATDSC and AWDSC). To improve their performance, we propose a novel method based on partial knowledge of the pilot tone pattern. The optimal detection method involves with a large matrix when the number of WPM blocks and the number of sub-carriers are high. To reduce the computational complexity of the optimal detection method, we introduce an approximated Covariance
Matrix (ACM)-based method, in which all the elements of the covariance matrix of the noise-free signal are non-zero only at the pilot positions. This proposed method is combined with the GLRT approach for different scenarios, where the covariance matrix and the noise variance may be known or unknown at the receiver. Since the test statistics are scale-invariant under these conditions, the threshold can be designed to obtain a constant false alarm rate (CFAR) detector.
This fact about the CFAR proposed method makes it a suitable candidate for spectrum sensing in WPM-based cognitive radio systems. Secondly, a Rao-based test statistic for WPM system with pilot signal is derived for the situation when no prior information about the primary user is known. Furthermore, a sequential- based technique is proposed, which reduced sensing time while maintaining
Chapter 1: Introduction 26 acceptable detection performance. We introduce an update process for the test statistic. To obtain a pre-defined target performance, the thresholds for different levels are derived based on the upper and lower bounds of the probability of false detection. The detection performance of the proposed scheme is evaluated and compared with those of the conventional methods via simulations and analytical means. The ROC performance and PD performance as well as the sensing time of the proposed schemes with full information and partial information over the fading channel are also presented. The main contributions of this dissertation are summarized as follows:
Two new methods, Accumulated Time-Domain and Wavelet-Domain
Symbol Cross-Correlation (ATDSC and AWDSC), are proposed for CR
systems in frequency-selective channels. To further exploit partial
knowledge of the pilot tone pattern, an approximation of the optimal test
statistic is derived in Chapter 3. Analytical and simulation results are
presented to demonstrate the advantage of the proposed methods with
respect to detection performance and computational complexity.
To deal with this situation where no prior information about the primary
signal is known, a new method is proposed in Chapter 4, which is based on
the Rao’s score test. Under low SNR regimes, the actual value of the
parameter of interest is close to the particular value, which is zero, under
the null hypothesis. This fact makes Rao’s score test become a strong
Chapter 1: Introduction 27
candidate since it is the most powerful test. Firstly, a Rao-based test
statistic is derived for WPM system with pilot signal. The detection
performance assessment and the ROC performance of the novel test
statistic are provided to validate the advantage of the novel approach. It
is shown that the proposed Rao-based test maintains good performance
while significantly reducing the computational complexity. The fact that
the proposed Rao-based method does not require maximum likelihood
estimation (MLE) of some unknown parameters makes it more practical.
To solve the trade-off between the detection performance and sensing
time, a sequential-based spectrum sensing method is proposed in Chapter
5. This approach makes a decision about the status of the primary users
when there is enough confidence from the samples or when the time for
sensing is over. Firstly, we introduce the proposed method and the update
process for the test statistic. Then upper and lower bounds for the
probability of false alarm and the probability of miss detection are derived
analytically. Based on these bounds, we propose two procedures to setup
the thresholds for the test statistic at different stages. The simulation
results show that the proposed technique can achieve the overall detection
performance while minimizing the sensing time. It should be noted that
this method works well for dependent received signal. The fact that the
Chapter 1: Introduction 28
thresholds at different stages do not depend on the level helps to reduce
significantly the computational complexity.
1.3 Dissertation Overview
Chapter 1 provides backgrounds on some of the fundamental concepts.
Firstly, Sub-section 1.1.1 presents the characteristics of the wireless channel.
Next some orthogonal transforms are described. These include discrete Fourier transform (DFT) and fast Fourier transform (FFT), short term Fourier transform (STFT), and wavelet transform (WT) in Sub-sections 1.1.2, 1.1.3 and
1.1.4, respectively. The overview of wavelet transform in communications is provided in Sub-section 1.1.5. Finally, Sub-section 1.1.6 provides the background of detectors and ROC performance.
Chapter 2 introduces the conventional OFDM and WPM wireless systems.
This chapter first provides mathematical description of the conventional OFDM and the WPM systems. This is followed by a description of conventional radio and cognitive radio systems, which include underlay, overlay, and interweave classes. Next, some conventional and special spectrum sensing methods are presented. The advantages and disadvantages as well as the requirements of these methods are also given in this chapter. The fundamental limits of detection and experiment results are provided, followed by some current standards for cognitive radio systems.
Chapter 1: Introduction 29
Chapter 3 provides a derivation of an approximated version of the covariance matrix. By neglecting the small components of this matrix, we derive a sub-optimal test statistic. Test statistics are also developed when the receiver has no information about the covariance matrix and the noise power. The ROC performance of the proposed technique is studied via analytical and simulation methods. The computational complexities of our proposed methods and the optimal method are also compared. The comparisons are made between the ROC performance of the proposed methods and those of conventional methods.
Chapter 4 deals with detection problems when the receiver at the SUs has limited knowledge about the primary signal. Firstly, we present the motivation for proposing alternative approaches to the GLRT-based test. Next, a test statistic based on a Rao test for WPM system using pilot signal with unknown covariance matrix is derived. From the derived test statistic, we assess the detection performance of the new method. Comparisons are made between the derived Rao’s test-based technique and some conventional spectrum sensing methods under various constraints.
In Chapter 5, sequential spectrum sensing approaches are derived, which reduce sensing time. Firstly, the novel sequential detection techniques for WPM system are described. Then, we discuss the temporal update recursion for the test statistic. It is followed by two procedures to setup the sets of thresholds, which are based on the upper and lower bounds for the overall target probabilities.
Chapter 1: Introduction 30
Finally, simulation results of these approaches and traditional methods are presented to verify their abilities.
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 31
CHAPTER 2
MULTICARRIER SYSTEMS, COGNITIVE RADIO
SYSTEMS AND CONVENTIONAL SPECTRUM
SENSING METHODS
The models of OFDM and WPM multicarrier systems are presented in this chapter. Section 2.1 describes the conventional OFDM and WPM systems.
In this Section, the WPM system model for fading channels, which will be used in this dissertation, is also established. Next, Section 2.2 describes the conventional radio and cognitive radio systems. Conventional spectrum sensing methods for multicarrier systems are described in Section 2.3, followed by some special spectrum sensing methods in Section 2.4. Fundamental limits of detection and experimental results are presented in Section 2.5, followed by cognitive radio standards in Section 2.6. Finally, the chapter is concluded in Section 2.7.
2.1 Conventional OFDM and WPM wireless systems
Figure 2.1 shows the block diagram of a conventional OFDM system. In this figure, the constellation modulation can be M-Phase Shift Keying (M-PSK) or M- Quadrature Amplitude Modulation (M-QAM); S/P and P/S denote serial- to-parallel and parallel-to serial conversion, respectively; IFFT and FFT denote
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 32 inverse fast Fourier transform and fast Fourier transform, respectively; while
CP denotes cyclic prefix.
In OFDM, binary data sequence is first grouped and mapped onto complex signals by a constellation modulation as shown Fig. 2.1. After passing the S/P block, the input signal is converted from frequency to time domain via the IFFT block. The IFFT divides the bandwidth of the input stream into multiple parallel narrowband orthogonal sub-channels. These overlapped sub- channels help OFDM system achieve higher bandwidth efficiency than Frequency
Division Multiplexing (FDM). When the space between consecutive sub-channels is designed properly, the signals on each stream undergo frequency-flat fading channel. As a result, the frequency selective channel is converted into multiple flat fading channels. This helps OFDM systems eliminate inter-channel
Constellation Add S/P IFFT P/S Modulation CP
OFDM transmitter
Remove Constellation S/P FFT P/S CP Demod
OFDM receiver
Figure 2.1 Block diagram of a conventional OFDM multicarrier system
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 33 interference (ICI). To avoid inter-block interference (IBI) due to multipath effects, the CP is introduced [36]. To ensure that inter block interference is eliminated, the length of the CP must be longer than that of the multipath channel. At the receiver, the CP part of the received signal, which is the overlap of two consecutive OFDM blocks, will be discarded. This helps OFDM systems eliminate IBI effects. By using CP, the linear convolution is converted into circular convolution, which can help avoid ICI and inter-symbol interference
(ISI). All of these make OFDM achieve good performance and become the standard in many current wireless systems.
Figure 2.2 shows a block diagram of the baseband equivalent conventional wavelet packet modulation (WPM) system. In this figure IDWT and DWT denote inverse discrete wavelet transform and discrete wavelet transform, respectively.
Constellation S/P IDWT Modulation
WPM transmitter
Post Constellation DWT P/S processing Demodulation
WPM receiver
Figure 2.2 Block diagram of a conventional WPM multicarrier system
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 34
x0( n ) 2 h( n ) 2 h( n ) x1( n ) 2 g( n ) x2( n ) 2 h( n ) 2 g( n ) x( n ) g( n ) 3 2 2 h( n ) y( n )
2 g( n ) xM ( n ) 2 h( n ) 2 4 2 h( n ) xM ( n ) 2 g( n ) 2 3 x2M 2( n ) 2 h( n ) 2 g( n ) x2M 1( n ) 2 g( n )
Level M LevelM -1 Level1
M Figure 2.3 Transmitter structure for a WPM based multi-carrier system with 2
sub-channel full tree
2 h n v0 n h n 2 2 g n v1 n
2 h n v2 n g n 2 2 g n v3 n
h n 2 r n
g n 2
h n 2 v n 2M 4 h n 2 g n 2 v n 2M 3
2 v n h n 2M 2 g n 2 2 v n g n 2M 1 M Figure 2.4 Receiver structure for a WPM based multi-carrier system with 2
sub-channel full tree
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 35
Instead of using IFFT and FFT as in OFDM system, the WPM system uses IDWT and DWT blocks to convert the input signal from wavelet domain to time domain and vice versa at the transmitter and the receiver, respectively. Like
OFDM systems, these IDWT and DWT blocks also divide the signal bandwidth into multiple, parallel narrowband orthogonal sub-channels. The orthogonal sub- channels can also overlap, letting WPM system offer high bandwidth efficiency.
However, the symbol duration Ts in WPM systems is longer compared to OFDM, which can make the fading channel become fast fading. DWT and IDWT do not have the circular convolution property like FFT and IFFT. Since there is no CP block in WPM system, WPM offers higher bandwidth efficiency than OFDM. To overcome the effects of ISI and ICI, the WPM system utilizes a post-processing block, which is generally more complex than that of the OFDM system.
54The IDWT and DWT structure for a WPM based multi-carrier system
M with 2 sub-channels is shown in Figure 2.3 and Figure 2.4, where M is the number of levels. The low-pass filter h n and high-pass filter g n form a
M quadrature mirror filter (QMF) pair. At the transmitter, 2 data streams x( n ), x ( n ), , x ( n ), x ( n ) M 0 1 2M 2 2 M 1 are multiplexed using levels of QMF pairs to produce the transmitted signal y( n ) as shown in Fig. 2.3. At the receiver, the received signal r n is demultiplexed using M levels of QMF pairs to produce
2M v(), n v (), n , v (), n v () n data streams 0 1 2M 2 2 M 1 , as shown in Fig. 2.4. These
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 36 data streams are then processed by a post-processing block, which combines the outputs of the DWT block to provide maximum signal-to-noise ratio (SNR).
The IDWT and DWT blocks can equivalently be represented as shown in
Fig. 2.5. In this figure, Wi n and Wi n are the overall responses of the i th sub-channel at the transmitter and the receiver, respectively.
Throughout this dissertation, an L path channel with an exponentially decaying power delay profile is assumed. The channel is assumed to be slow fading where it remains static over the transmit duration of several WPM blocks.
T The channel impulse response (CIR) is expressed by h 0 1 L 1 where l is the l th tap gain of h . These tap gains are assumed to be independent, zero mean circularly symmetric complex Gaussian random variables. For normalization purposes, the sum of their variances is set to be 1.
x n t n u n v0 n 0 M 0 0 M ↑ 2 w0 n w0 n ↓ 2
v n x1 n t1 n u1 n 1 ↑ 2M w n w n ↓ 2M 1 y n r n 1 Channel + n
n t v n xM n tM n uM n 2M 1 2 1 M 2 1 w n w n 2 1 M ↑ 2 2M 1 2M 1 ↓ 2
IDWT DWT
M Figure 2.5 The equivalent block diagrams of IDWT and DWT with 2 sub-
channel full tree
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 37
2.2 Conventional radio and cognitive radio systems
Conventional radio systems were built for the analog model, where the spectrum bands are pre-assigned for each user. Studies by Federal
Communications Commission (FCC) and many independent measurements show that most radio frequency spectra were inefficiently utilized [37]. Cellular bands are overloaded, but other frequency bands such as televisions, radio stations, mobile carriers and air traffic control are insufficiently utilized. Spectrum utilization depends on time and geographical location, and the fixed spectrum allocation system prevents other users from using the free bands.
In response to the increasing demands for transmission bands, cognitive radio (CR) systems was first proposed by Joseph Mitola III in 1998 [38]-[39]. This is an intelligent radio whose radio-system parameters such as waveform, protocol, operating frequency, and networking can be programmed and configured dynamically. In this dissertation, we focus on the operating frequency of the users. In 2008, the unused radio frequency (RF) bandwidth was made available for public use. However, to exploit temporarily the white space, the devices must have the ability to sense the spectrum to prevent interference with the primary users (PUs). These primary users, who are assigned the band, have the highest priority in using the spectrum. At the same time, the secondary users (SUs), who have lower priority, can use the spectrum such that their transmissions do not cause interference to the PUs. The awareness by the SUs about the spectrum
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 38 usage can be obtained by beacons, by geolocation and database or by spectrum sensing at SUs’ receiver. In this dissertation we focus on spectrum sensing (SS) at the SUs’ receiver since this approach does not require any modification on the
PUs’ system, especially the legacy ones. First, the SUs sense the primary signals, which may be code division multiple access (CDMA) or orthogonal frequency- division multiplexing (OFDM). Then these transceivers detect the best available channels in the wireless spectrum in its vicinity, and optimize its transmission or reception parameters. The SUs must continuously monitor the chosen band to detect any change in the status of the PUs. When the primary signals are detected, the transmission between SUs must be stopped immediately. This procedure improves spectral usage by allowing more users to operate simultaneously. Enabling more effective wireless communications systems, CR technology will be the next major step in the near future [40]-[47].
To automatically detect available channels in a wireless spectrum, the SUs must know and utilize some characteristic information or prior knowledge about the primary signals. To be practical, the amount of the information should be minimal. Based on the knowledge that the SUs have, CR systems are divided into three categories (classes), namely, Underlay, Overlay and Interweave.
2.2.1 Underlay
In this approach, simultaneous transmissions from PUs and SUs are allowed to exist, as long as the interference level at the PUs remains acceptable
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 39
[48]-[53]. In other words, the SUs can utilize the occupied spectrum if they can make sure that the interference at the PUs does not exceed a predefined tolerable threshold, otherwise the SUs transmission may significantly degrade the performance of the PUs. Recently, many techniques have been proposed for underlay CR such as beamforming and spread spectrum. Beamforming, a signal processing technique used for directional signal transmission or reception by using multiple antennas, helps spatially reduce the interference at the PUs. On the other hand, the spread spectrum technique uses a spreading code to obtain a signal with wider band and lower power density. To satisfy the restriction on the interference level at the primary side, the SUs can also limit their transmission power, which is common for short range communications.
2.2.2 Overlay
Overlay model also allows the SUs to share the channel simultaneously with the PUs [54]-[57]. The PUs share some information about their signals with the SUs, such as transmitted data sequence (message) and the codebook (how to encode the sequence). The SUs may enhance and assist the PUs’ transmission, while also obtaining some additional communication for their own. Specifically, the SUs decode the messages from the PUs and use these messages either to eliminate the interference from the PUs at the SUs’ receiver or to aid the transmission of the PUs through relaying the messages to the primary receiver.
There are many optimization problems, which involve the trade-off between the
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 40 interference induced on the PUs and the performance improvements on the
SUs. In short, underlay CR can only be used when the primary signal is strong, while overlay CR is more suitable for weak primary signals.
2.2.3 Interweave
In this approach, the SUs are allowed to use the spectrum only when it is left vacant by the PUs [58]-[60]. The SUs and PUs can simultaneously transmit only if the spectrum sensing procedure of the SUs fails to detect of the PUs’ signal. In order to be aware of the existence of the PUs’ signal, the SUs must have the ability to periodically monitor and identify the surrounding environment. Different dimensions such as frequency, time and space domains can be explored to find the spectral gaps. Besides these conventional dimensions, other dimensions such as code and angle of arrival may be exploited.
Underlay and overlay approaches require the PUs to broadcast or register information such as the spectrum, the duration, the transmit power. These techniques help to simplify the SUs’ receivers but need to modify the PUs’ transmitters as well. The interweave method has received more attention due to its low cost and compatibility with legacy PUs’ transmitters. Spectrum sensing techniques for interweave cognitive radio systems have been well studied in the literature (see the next Section 2.3 and 2.4). In this dissertation, we focus on spectrum sensing on the SUs of the interweave cognitive radio system.
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 41
2.3 Conventional spectrum sensing methods
As shown in Section 2.2, in interweave cognitive radio system, the SUs must determine the presence or absence of the PUs’ signal via spectrum sensing.
Spectrum sensing can be implemented in the time, frequency, spatial, and wavelet domains. The spectrum sensing at the SUs can be modeled as a binary hypothesis testing problem, that is
w n H0 y n (2.1) sn w nH1 where yn, w n and s n are received signal, noise and noise-free received signal, respectively, H0 and H1 are the hypotheses corresponding to the absence and the presence of the primary signals, respectively. The noise w n are assumed to be zero-mean circularly symmetric complex Gaussian (CSCG)
random variable, i.e. wn CN 0, Cw . Based on the Neyman-Pearson lemma, the null hypothesis H0 is rejected when the log-likelihood ratio (LLR)
fy H 1 y log , where fy Hi is the probability density function fy H 0
(PDF) of y given the hypothesis Hi , i 1,2 , and is a pre-designed threshold.
To evaluate the performance of the test statistic, the false alarm probability Pfa
and the detection probability Pd are used as the metrics, which are defined as follows:
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 42
PHfa Pr 0 (2.2) PHd Pr 1
When the threshold increases (decreases), both Pfa and Pd decreases
(increases). This threshold must be designed properly, since large Pfa causes the
SUs to lose the opportunity to access undetected spectral holes, while small Pd introduces more interference to the PUs’ transmission. The threshold can be
selected for an optimal trade-off between Pfa and Pd . However, in most cases, it is difficult to determine what the noise power and the PDFs of the test statistics are under both hypotheses. Therefore, in practice, the threshold is determined
based on a fixed Pfa . A detector is called constant false alarm rate (CFAR) if Pfa does not depend on noise power and signal power. In this case, to achieve a pre-
defined Pfa , the threshold can be designed with no information about the noise power. This property can be determined based on the expression of the test statistic. In particular, the value of the test statistic T x does not change when the received signal is multiplied by a scale factor (that is, Tx T x where is a constant). In many practical applications, CFAR is a desirable property, when the noise power information is not available at the receiver or the system is under noise uncertainty.
Depending on the amount of information the SUs have about the primary signals, these spectrum sensing methods can be divided into three main categories: energy detector, matched filter, and feature-based. Different
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 43 techniques have different requirements and advantages as well as disadvantages. As a result, each of them is applicable to different scenarios. The more information available at the SUs receiver, the better the performance of the detector. In practice, some detection techniques can be combined to improve the detection performance. For example, to reduce the sensing time and computational complexity, wavelet-based or energy detector can be performed first to obtain a coarse scan for wideband signal followed by an accurate method for a fine scan, which takes more time to perform compared to the coarse stage.
2.3.1 Energy detector (ED)
When the SUs have no information about the primary signals, received signal power is the only information available to the SUs. Therefore, ED is the only detector that can be used for any zero-mean primary signal [61]-[65]. The received signal is assumed to be zero-mean circularly symmetric complex
2 2 2 Gaussian, that is, y~CN 0 ,s I , under H1 and y~CN 0I , under H0 ,
2 2 where s and are the power of noise-free signal and of noise, respectively.
Then, the log-likelihood ratio (LLR) is defined as
N 2 2 2 Py | H1 y y LLR(y ) log log exp Py | H 2 2 2 2 2 (2.3) 0 s s
From Eq. (2.3), the energy detector, also known as the radiometer, is the optimal non-coherent detector. As shown in Fig. 2.6, the continuous or digital received signal is pre-filtered, squared and followed by an integrator. Then, this
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 44
2 Pre-filter ( ) Ʃ or ʃ Test statistic
Figure 2.6. The diagram of the conventional energy detector measured energy of the received signal over a specified finite time duration and bandwidth is compared with a pre-designed threshold to determine the state of the PUs. Applying the Parseval’s theorem, the energy test statistic can also be computed in the frequency domain. Specifically, the detector compares the sum of
2 the power spectrum X f within the bandwidth of interest to a pre-designed threshold.
The probability of false alarm PFA and the probability of detection Pd are given by
2 PFA 1 F 2 2/ 2 N 2 2 (2.4) Pd1 F 2 2/ s 2 N
2 where and F 2 2 / are the threshold and Chi-squared distribution with 2 N degree of freedom 2N , respectively.
The main advantage of the simplest spectrum sensing technique is the low computational complexity with no prior information needed about the PUs’ transmission. It can detect any type of signal and is suitable for wideband spectrum sensing in practice. However, this method does not exploit any potential information about the primary signal. Hence, given a target
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 45
performance (i.e., Pfa and Pd ), the required number of samples N is in the
order of SNR 2 . That is, at low signal to noise ratio (SNR), the ED method requires a large number of samples. Because of the limit of sensing time, this main drawback limits the usage of this method. In addition, the noise power should be known or estimated accurately, which is difficult in practice. As a result, noise and interference power uncertainty significantly degrade the performance of the ED method. More specifically, with noise uncertainty, the ED spectrum sensing method fails to achieve the target performance when the SNR is below the SNR wall, regardless of the length of the sample sequence [66]. In other words, the SUs cannot distinguish between H0 and H1 when the systems are under low SNR regime and noise uncertainty. Moreover, the technique works inefficiently for spread spectrum primary signals because of the low power of the
PU signals. Furthermore, the ED method cannot distinguish between the primary signals, noise and interference.
While the ED method is a good candidate when the SUs have no information about the primary signals, other more complicated methods exploit knowledge about the primary signals and can be used to improve detection performance.
2.3.2 Matched filter
Contrary to the ED method, in matched filter method, the SUs know some patterns in the primary signals. These patterns, which are used for
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 46 synchronization or other purposes, include spread sequences, preambles, midambles and pilot patterns. This sensing method computes the test statistic T by correlating the received signal and the known pattern as follows:
H T Reys (2.5) where y and s are the received signal and known transmitted signal,
H respectively; denotes Hermitian transpose operator. The SUs’ receiver then compares the test statistic to a threshold to accept or reject the hypotheses [67]-
[69].
Since the signal to noise ratio (SNR) of the received signal is maximized by matched filtering, it has been shown that this technique is the best detector in terms of detection performance and convergence time [67]. Given a target
performance (i.e., Pfa and Pd ), the required number of samples is in the order of
1 O SNR . However, the method is prone to synchronization errors. In addition, the requirements of perfect knowledge about the primary signals and the fading channel response make this approach impractical. When these conditions are not met, its performance will degrade dramatically. Therefore, the use of this method is severely limited.
2.3.3 Feature detection
Feature detection methods detect spectral holes by detecting certain features embedded in the signal. In this Sub-section, some feature detection-based techniques are described. By exploiting the feature information embedded in the
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 47 received signals, these methods outperform the energy detectors, while overcoming the uncertainty of the model of the matched filter. In addition, these techniques can also distinguish among different types of transmissions and primary systems.
A. Cyclostationarity-based Spectrum Sensing
By exploiting the cyclostationarity features of the received signals, these methods can help SUs to detect spectral holes. The cyclostationarity features come from the first or second order statistics or the periodicity of the primary signals [70]-[72]. These periodic patterns are related to the cyclic prefix (CP) in
OFDM, symbol rate, chip rate, and channel code that can be appropriately modeled as a cyclostationary random process. These periodic patterns can also be caused by intentionally introduced periodicity used to aid channel estimation or
* spectrum sensing. If the time covariance function, R n, E y n y n , is periodic with respect to the time index n , the signal y n is called a second- order cyclostationary signal. This time covariance function R n, can be expressed as a Fourier series. First, the cyclic autocorrelation function (CAF)
R , which represent the coefficients of the Fourier series of R n,, is calculated as follows [69]:
R E y n y* n e j 2 n (2.6) where is the cyclic frequency, which is assumed known or can be estimated,
E is the expectation over the time index n . Because of the periodic behavior
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 48 of the primary signals, the CAF function has some periodic components. The cyclic spectral density (CSD) function can be computed by taking the Fourier transform of CAF as follows:
S f, R e j2 f (2.7)
From the information about the cyclic frequency of the primary signal, the presence or absence of the primary signals can be determined by comparing the peak values of the CSD function to a threshold. Since the wide-sense stationary (WSS) noise has no cyclostationarity features, this technique outperforms the ED methods in differentiating the primary signal from the noise.
However, the cyclic frequencies of the primary signal must be estimated or known, which is not realistic in many scenarios. Moreover, its computational complexity is high and it requires excessive Analog-to-Digital Converter (ADC).
Moreover, the imprecision of the clocks at the transceivers may make the received signal not really cyclostationary. Channel effects such as fading and
Doppler effects signals having second order cyclostationarity can reduce the detection performance. Furthermore, the use of pulse-shaping filters will diminish the cyclostationary nature of the signals [73].
B. Autocorrelation
The patterns in the primary signals; related to the cyclic prefix in OFDM, symbol rate, chip rate, and channel code; make the average autocorrelation R non-zero at certain values of . The average autocorrelation R is defined as
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 49
* R E y n y n (2.8)
Based on the function R , many test statistics have been proposed for spectrum sensing. Mort et al. proposed the ratio of sum of weighted R , 0 and R 0 as a test statistic [74]; while the test statistic in [75] is based on R and the power of received signal. To sense a transmitted primary TV signal, whose power spectral density (PSD) is supposed to be known at the receiver, the
Fourier transform of R is correlated with the known PSD [76]. To detect
OFDM signals, Zhongding et al. proposed using the average value of R over
[77]. The value R TD is used as a test statistic [78], where TD is the useful symbol data. To deal with synchronization problems and unknown noise variances, an approximation of the GLRT is used as a test statistic to sense the spectrum, based on the autocorrelation function as follows [79]:
N N c d 2 Rˆ n T max n1 ,I ,0 NN 2 (2.9) c d 1 2 Rˆ n Re R ˆ i R ˆ n N n Sc i S n S
1 K1 ˆ ˆ where Rn rnkNN c d . Being a CFAR, this method obtains good K k0 performance by taking the non-stationarity of x n into account.
These techniques can work under low SNR regime and outperform the ED method, however, they have been only evaluated on additive white Gaussian noise (AWGN) channel or slow fading channel. It is shown in [78] that the autocorrelation-based approach outperforms the cyclostationarity-based methods
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 50 using one cyclic frequency, but cyclostationarity-based methods using two cyclic frequencies are superior to the autocorrelation-based approach. This approach is robust to noise uncertainty and channel dispersion, however, it requires very high sampling rates, has high computational complexity, and is prone to sampling time error and frequency offset.
C. Covariance Matrix
Because of the dispersive channels, oversampling and the utilization of multiple antennas at the transmitter and the receiver, the received signal is correlated. Spectrum sensing can also be implemented by using the covariance matrix of the received signal, which is defined as follows
H C E yy (2.10) where y is the received signal. Since almost all received signals are zero-mean, this second-order information is adequate for spectrum sensing [80].
Under H0 , C is the covariance matrix of the noise, which is a diagonal matrix for white Gaussian noise. Under H1 , due to the oversampled signal, and the correlation between the transmitted symbol as well as the effects of the wireless channel, C is not a diagonal matrix. From this fact, many methods have
p norm x been proposed by using the ratio of p of the weighted off-diagonal
p norm x elements and p of the diagonal elements of the covariance matrix
1/ p n p p p norm x x [81]-[84], where is equal to 1 or 2 and is defined as p i . i1
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 51
It is shown that the methods are better than ED when the received signal is highly correlated and the noise power is unknown [82].
D. Eigenvalues of Covariance Matrix
The covariance matrix can be used for spectrum sensing by extracting its
2 eigenvalues. As seen in the previous sub-section, under H0 , C CN 0I, and all of its eigenvalues are the same. Under H1 , the values of its eigenvalues have a wide range. Therefore, many test statistics have been proposed based on the eigenvalue ratios. These eigen-based methods are summarized below [85].
T 1 1 Maximum-eigenvalue trace 1 N trace(C ) [86]-[89] i N i1
1 Maximum-minimum eigenvalue T where 1 2 ... N are N the descending arrangement of eigenvalues of C [88], [90]-[94].
E y 2 n Energy-minimum eigenvalue T [92] N
1 Maximum eigenvalue-geometric mean T 1/ N N i i1
1 Maximum eigenvalue-harmonic mean T 1 1N 1 N i1 i
N N 2 i/ i i1 i 1 Contra-harmonic mean-minimum eigenvalue T N
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 52
N i det C i1 Spherical Test based detector T N N 1 1 N trace C i N N i1
[95]
Generally, these test statistics can be expressed as the ratio between two
x/ x norms of the eigenvalues r p , where r and q are integer numbers and
1/ p n p p norm x x is defined as p i [96]-[97]. i1
Several analytical and simulation performances have been evaluated by using Tracy-Widom distribution [90], using the distribution of the ratio of the extreme eigenvalues of complex Wishart matrix [94], and using asymptotic random matrix theory [98]-[99]. These eigenvalue-based methods exhibit a significant improvement on performance compared to ED, robust to the carrier frequency offset and work well in low SNR regimes. However, to obtain a good approximation of the covariance matrix C , eigenvalue-based approaches require a large number of received samples, which leads to longer sensing time and low efficiency of cognitive radio system. In addition, the computational complexity of eigenvalue decomposition is high.
2.3.4 Cooperative spectrum sensing
Cognitive radio systems require reliable spectrum sensing to reduce the interference to the primary users and to increase the chance of data transmission of the secondary users. The performance of spectrum sensing is highly dependent
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 53 on channel parameters such as path loss, shadowing, multipath, interference and noise uncertainty as well as receiver uncertainty. These obstacles may
increase Pfa or reduce Pd of the spectrum sensing, especially in the low SNR regime.
To overcome this problem, many methods have been proposed based on the cooperation of many secondary users. From the different spatial locations, the received signals from many SUs with different strengths can be combined to make more accurate decision about the status of the primary users. Taking the advantages of spatial diversity, the performance of cooperative spectrum sensing can be improved significantly compared to that of spectrum sensing from a single
SU [100]-[107].
After sensing the spectrum, each SU reports hard decision, soft decision, or quantized information to the network [108]-[109]. There are three main topologies, centralized, distributed and relay-based networks. In the centralized network a central node makes the cooperative decision based on some of the received reports from the SUs, which have the best detection performance [110]-
[113]. This topology is easy for coordination, fast execution, reduced conflicts and it achieves good performance. However, it is unsuitable for large number of SUs and highly dependent on the central node. In the decentralized model, there is no central node and all the SUs communicate among themselves to obtain better decision [114]. This model helps eliminate the burden on the central node,
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 54 however the SUs may not have a unified decision since some SUs may not be reachable. In the relay-based topology, the SUs send reports to the central node through a relay node.
Exploiting spatial diversity helps achieve a better overall detection performance. However, when the channels are highly correlated (i.e., SUs are under the same obstacles), the performance will be degraded [115]. In practice, the reporting channels also experience fading and shadowing, which reduce the reliability of the local decisions. In addition, cooperation between SUs requires cooperation overhead, which increases computational complexity, delay, sensing time and energy consumption [116]-[118]. Moreover, the exposed node problem also degrades the performance of cooperative spectrum sensing [119].
Furthermore, the cooperative networks must have the ability to detect attacks of selfish SUs, who occupy all or part of the resources, prohibiting other SUs from accessing the cognitive radio resources [120]-[121]. Finally, further investigations need to be done for advanced SS techniques, since almost all methods in literature are only based on the simple ED and flat fading channel.
2.4 Some special SS methods
Besides the three main spectrum sensing categories discussed in Section
2.3, other spectrum sensing approaches are presented in this Section.
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 55
2.4.1 Compressed sensing
For all the previous SS methods discussed in the previous Section, the SU receivers must obtain enough samples in order to improve the sensing performance. However, the more the received signal samples, the more computational complexity and the longer the sensing time. Compressed sensing is based on two main principles, which are sparsity and incoherence. The former implies that the degrees of freedom (i.e., the number of basis ) of the signal is much smaller than its length; while the latter implies that the signal must be spread out in the domain they are acquired [122]. Put differently, the signals can be reconstructed with a number of measurements smaller than the signal length under some assumptions. One of the main difficulties of the CS-based approach is how to extract the non-zero entries of the signal.
Recently CS has received much attention in the literature [123]-[126].
Based on the sparsity of the signal, by using relatively few measurements, CS has become a strong candidate for wideband SS [124], [126]-[131]. Compressed sensing-based SS methods can help avoid using high speed analog-to-digital converter (ADC) and digital signal processor (DSP) by working at sub-Nyquist rates. This frequency is much lower than the Nyquist frequency, which is twice
the highest wideband frequency fmax , present in the signal. As a result, the computational complexity would be significantly decreased.
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 56
2.4.2 Information theoretic criteria (ITC)
The key idea of this method is the detection of the finite number of signals embedded in a noise, introduced by Akaike (Akaike information criterion or AIC)
[132] and by Schwartz and Rissanen (minimum description length or MDL) [133].
Unlike other SS methods, no subjective threshold levels are required in the hypothesis tests. By minimizing either the AIC or the MDL criteria, the number of significant eigenvalues can be determined [134]. Based on a Bayesian model, the MDL approach shows more consistent results than those of AIC [135]. In
[136], the first and second values of the cost functions of either AIC and MDL model are compared to determine the presence of the primary signal.
2.4.3 Markov model-based SS
In [137], the channel status statistics is captured by using a Markov chain model. Jason et al. apply a partially observable Markov decision process to minimize the sensing time and the false alarm [138]. A Markovian optimal stopping time problem is solved to minimize the scanning cost [139]. The presence of the primary signal can be modelled as a two-state Markov chain
[140]. The Markov model-based approach can also be used to estimate some primary signal parameters [141] and to detect malicious users, who intentionally send wrong reports to the fusion centre [142]. Thao et al. proposed a spectrum sensing method using a general model for the geometric dwell time distribution of a standard hidden Markov model [143].
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 57
2.4.4 Entropy-based SS
Shannon entropy is the expected value of the information contained in a flow of information. It is used as a measurement of the uncertainty of a discrete message, which is defined as follows:
N N HX PxIx ii Px ilog Px i (2.11) i1 i 1 where xi is the possible value of a discrete random variable X and the base of logarithm can be any positive number except 1. For a continuous valued signal, the differential entropy is defined as follows:
h X f xlog f x dx (2.12)
Assuming the waveform of the primary signal is known at the SUs’ receiver, an entropy-based SS method, using matched filters, is proposed in [144].
In this paper, the outputs of the matched filters are divided into L bins of equal
th length and xi in Eq. (2.11). is the probability that the output sample is in the i bin. Based on the fact that the entropy of a signal is maximized if the signal is
Gaussian, the empirical entropy is then used as a test statistic to determine the presence of the primary signal.
Inspired by the work in [144], some entropy-based methods have been proposed to deal with the noise uncertainty. In [145], the spectrum magnitude is used instead of the outputs of the matched filter. These magnitude values are also divided into L bins of equal length and xi in Eq. (2.11) is the probability
th that the magnitude value is in the i bin. It is shown that the discrete entropy of
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 58 the spectrum magnitude of white Gaussian noise is a constant and this frequency-domain detector outperforms ED and cyclostationary-based methods in low SNR regime. When the white Gaussian noise is passed through a wavelet packet transform, the energies of all sub-channels are the same. Based on this fact, Zi et al. in [146] proposed a SS method based on the probability density distribution of normalized wavelet energy of the sub-channels. Thanks to the fast wavelet transform (FWT), this method has simple structure and low computational complexity. Entropy can also be combined with cyclic features to sense the wideband spectrum [147]. Under generalized Gaussian noise, differential entropy for continuous random variable is used to sense the spectrum [148]. Since no prior knowledge of the noise power and primary signal is required at the SUs’ receiver, these entropy-based approaches are robust to noise uncertainty.
2.5 Some fundamental limits of spectrum sensing and experiment results on cognitive radio
There are many issues in designing and implementing the spectrum sensing function of SUs [149]-[152]. In this Section some requirements, challenges, and implementation issues as well as fundamental limits of spectrum sensing are discussed.
A wireless transceiver cannot transmit and receive the signal simultaneously. Therefore, the SUs can only transmit the secondary signal after a certain sensing and decision time. This fact may make the information about the
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 59
PUs’ status outdated, which may cause interference with the transmission of the PUs. In practice, the sensing time constraint depends on the type of the PUs
(e.g., small for public safety spectrum and large for TV channels).
Noise uncertainty is a huge problem that degrades the detection performance of the SUs [66]. In most of the sensing methods, the noise power needs to be known or estimated. Without noise uncertainty, the ED and matched
2 filter SS methods can acquire the desired performance by using O SNR and
1 O SNR samples, respectively. In other words, the noise at the receiver is assumed to be Gaussian and its power is known or estimated exactly. However,
2 2 in practice, the noise power is believed to be in a range / , where
x/10 10 and x denotes the uncertainty level. Since the noise power can be any value in this range, the ED method could not distinguish the two hypotheses when SNR is lower than a certain value, even with longer sensing time. To be
2 energy 1 specific, there exists a SNR threshold SNRwall , below which the ED method cannot sense the signal; and this limitation cannot be countered by infinite number of samples. Other approaches, such as feature detection-based and matched filter-based method, can still detect the primary signal even with noise uncertainty. It comes from the ability of feature-based techniques to differentiate between signal and noise.
The effects of SNR, time-bandwidth product and peak-to-peak noise uncertainty on the detection performance are shown in [153]. In [154], Josep et al.
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 60 showed that there exist non-uniform sampling walls in wideband signal detection in the presence of noise uncertainty. When the signal is sampled with the sampling rate below the sampling wall, at a given value of SNR, the performance target ( Pfa and Pd ) cannot be guaranteed even with infinite number of samples. When a PUs sends a signal between two consecutive sensing epochs
(that is, within a sensing period), the primary signal may not be detected [155].
The longer the sensing time, the more likely the occurrence of miss detection.
The uncertainty of the wireless channel is also a challenge for spectrum sensing. Due to path-loss and shadowing, Pfa and Pd depend on the instantaneous value of SNR. The low instantaneous signal strength may make a null hypothesis while the PU is actually transmitting and the SUs are nearby. As a result, a higher detection requirement must be fulfilled at the SUs to overcome the channel uncertainty. Using a signal uncertainty model, Miguel et al. proposed the average Pd , as a more useful performance parameter [156]. It can also be solved by using multiple SUs to form a cooperative group. The diversity gain from cooperative SUs can improve the overall detection performance significantly.
The effect of limited channel knowledge on the capacity of CR system is evaluated in [157].
There are other issues that affect the sensing performance such as carrier frequency offset, timing synchronization error, and quantization error. Carrier frequency offset may degrade the detection performance of ED and matched filter
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 61 methods, especially for matched filter methods. However, since the covariance matrix of the received signal is preserved under carrier frequency offset, this issue would not affect any covariance matrix-based approach, including eigenvalue- based. Timing synchronization error is introduced when the receiver samples the signal asynchronously. It affects all the spectrum sensing methods and can be dealt with by oversampling the received signal. Quantization error, also known as quantization noise, comes from the analog-digital converter (ADC) at the receiver. Traditionally this error is modeled as a uniformly distributed random variable with the power depending on the quantization step-size. Simulation
2 results show that the number of required samples is also O SNR for ED detector. For both non-coherent and coherent detectors, quantization noise simply results in SNR loss.
In the near future, with approved standards for cognitive radio, more and more cognitive radio systems will be deployed. As a consequence, spectrum sensing on the SUs must have the ability to deal with the primary signal and multiple interferences from other CR systems.
Some experimental results are conducted to evaluate the detection performance of spectrum sensing methods under various imperfect conditions.
According to [152], it is difficult to detect zero-mean primary signals at low SNR.
In addition, the information about the modulation scheme does not improve the detection performance. Using a sub-optimal detector to sense pilot-embedded
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 62 signals shows nearly optimal detection result at low SNR. It is also shown that the performance degradation caused by imperfect synchronization is small compared to that between the ED-based and coherent detectors. The effects of noise uncertainty, DC offset, and hardware temperature as well as close high- power signal are analyzed in [158]. In [159], real-time experimental results for
FFT-based spectrum sensing are demonstrated. The alternative hypothesis is decided when k out of N samples of the power spectral density (PSD) exceed the threshold. The detection performances of ED-based, covariance matrix-based and maximum-minimum eigenvalues-based methods are examined in [160]. As expected, the experimental results show that the last two approaches are robust to noise uncertainty and superior to ED-based in large margins. Under the presence of noise uncertainty, analog impairments and interference, some SS methods based on ED, matched filter and collaboration are evaluated in [161]. It is shown that there exists a SNR wall for matched filter-based approach in the presence of frequency offset. For ED method, increasing the size of FFT can improve the SNR wall. In collaborative detection, measurement results show an improving performance when the radios experience the independent multipaths.
Experimental results of ED-based, cyclostationary-based and collaborative SS methods are presented in [162]. This paper shows the relationship between the target performance and the sensing time of ED-based method in frequency domain. The cyclostationary-based methods do not have sufficient robustness to
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 63 sampling clock offset, hence this method requires tight synchronization. It is also shown that collaborative method can significantly improve the sensing performance. In [163], soft decision-based methods are shown to be superior hard- decision-based techniques for cooperative SS in the presence of reporting channel errors. However, soft decision schemes require lots of resources for feedback to the fusion centre, which reduces the efficiency of the CR systems.
2.6 Spectrum Sensing Standardization
The 802.22 standard is the first standard to consider cognitive radio spectrum sensing for improving spectral efficiency by using white spaces in the television (TV) frequency spectrum while ensuring that no interference is introduced to the PUs [164]-[167]. After six years of development, this standard was finally completed in July 2011. It helps extend broadcast access to rural and remote areas in developed countries as well as developing countries. Exploiting the propagation characteristics of the TV channels in the VHF and UHF bands between 54 and 862 MHz, each channel has a bandwidth of 6 MHz, 7MHz or
8MHz; mobile services can be provided in an area 20 to 40 km from the base station, and up to 100 km under good propagation conditions. The maximum bit rate is approximate 19 Mbit/s at a 30 km distance by using just one TV channel of 6 MHz.
To attain this goal, spectrum sensing is implemented inside the Customer
Premises Equipments (CPEs). The CPEs perform in-band and out-of-band
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 64 measurements, both are implemented by using fast sensing and fine sensing.
Fast sensing is performed within 1 millisecond per channel; then from the result of fast sensing, fine sensing will be performed which takes 25 milliseconds per channel. CPEs send information about the signal strengths on the channels of interest via a wireless link to the base station (BS), which controls the medium access for these CPEs. From these reports and some information such as geo- location, and auxiliary information from the network manager, BS evaluates, makes a decision about the channel status, and determines to stay transmitting and receiving or to change to a new channel.
IEEE 802.22 WRAN does not specify any spectrum sensing techniques as mandatory in either CPEs or BS. However, the detection performance and reporting structure are standardized. Specifically, both false alarm probability Pfa and missed-detection probability Pmd must be under 10%, even under a very low
SNR regime such as -20 dB SNR with a signal power of 116 dBm and a noise floor of 96 dBm. The maximum detection latency, which includes sensing time and subsequent processing time, is limited to 2 s.
Besides the 802.22 standard, efforts to establish some standards for devices operating in white bands may be found in [168]-[169]. Launched in March 2013, the IEEE P802.22.1 standard enables spectrum sharing in the band from 2 GHz to 4 GHz, provides harmful interference protection for low power licensed devices operating in TV Broadcast Bands, and helps support a wide variety of wireless
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 65 applications [170]. IEEE P802.22.2 is a recommended practice for the installation and deployment of IEEE 802.22 Systems. Since 2014, the standard
IEEE P1900.6 focuses on interfaces for spectrum sensing and data structures for dynamic spectrum access and other advanced radio communication systems [171]-
[172]. Specifically, this standard defines the information exchange between spectrum sensors and their clients in radio systems. The ECMA-392 specifies the
MAC and PHY layers for personal/portable wireless networks using in TV bands
[173]. This standard provides some protection mechanisms which may be used to meet regulatory requirements. IEEE SCC41 is a standard for cognitive radio and dynamic spectrum access management [174]. IEEE 802.11af, a wireless computer networking standard approved in Feb 2014, enables WLAN using TV white band in the VHF and UHF between 54 and 790 MHz [175].
2.7 Chapter conclusions
In this chapter, we have discussed multicarrier systems and the need for utilizing cognitive radio systems. This chapter also reviews three categories of cognitive radio systems and some conventional spectrum sensing methods. The discussion considers not only the detection performances but also many aspects of these techniques such as practical usability, sensing time, computational complexity, required information at the SUs’ receiver, effects of fading and
Doppler, as well as synchronization errors and frequency offset etc. None of them is optimal for all scenarios of multicarrier transmission systems. Finally, we
Chapter 2: Multicarrier systems, CR systems and conventional SS methods 66 provide an overview of fundamental limits of detection and experiment results, as well as some standards for cognitive radio systems.
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 67
CHAPTER 3
SPECTRUM SENSING FOR WPM BASED
COGNITIVE RADIO MULTI-CARRIER SYSTEMS
USING PILOT SIGNALS
In this chapter, we propose novel pilot-based spectrum sensing methods for wavelet packet modulation (WPM) based multi-carrier systems in frequency- selective fading channels. We develop two methods based on the Accumulated
Time-Domain and Wavelet-Domain Symbol Cross-Correlation (ATDSC and
AWDSC). To improve their performance, a novel method based on partial knowledge of the pilot tone pattern is proposed. In this novel method, a sub- optimal test statistic is introduced to reduce the computational complexity. To evaluate the merits of the proposed methods, we compare their performance to that of the conventional energy detector, the autocorrelation coefficient-based, and ATDSC-based methods. These comparisons demonstrate the effectiveness of the proposed methods over their comparatives.
The rest of the chapter is organized as follows. First, an introduction of
WPM systems, cognitive radio and spectrum sensing is provided in Section 3.1. It is followed by a mathematic model of the wireless system from transmitter to
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 68 receiver of WPM based multi-carrier systems for fading channels in Section
3.2. The proposed spectrum sensing methods for WPM based multi-carrier systems are then presented in Section 3.3. This is followed by simulation results in Section 3.4. Finally, the chapter is concluded in Section.3.5 Throughout the
* T H chapter, we use the notations Re , , , , , , and to, respectively, denote the real part, complex conjugation, transposition, Hermitian transposition, the convolution, the Kronecker product, and the Dirac delta function.
3.1 WPM systems, cognitive radio and spectrum sensing
Wavelet packet modulation (WPM), a promising multi-carrier technology alternative to orthogonal frequency division multiplexing (OFDM), has received recent research attention [3], [176]-[177]. Like OFDM, WPM mitigates the detrimental effects of the multi-path fading channel by dividing the wideband channel into multiple narrowband sub-channels. The side-lobe of WPM signal is much smaller than that of OFDM signal, which leads to lower inter-channel interference between adjacent channels. This advantage makes WPM become a strong candidate for the cognitive radio system.
Spectrum sensing, which has recently been receiving research interest, is one of the key elements of cognitive radio. There are several challenges for spectrum sensing: the required signal-to-noise ratio (SNR) may be very low, multipath fading and time dispersion of wireless channel as well as the
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 69 uncertainty of noise and interference power. There is a vast literature on the spectrum sensing problem for OFDM systems, however, only a few papers are related to wavelet. Wavelet based edge detection is another interesting method which exploits the advantages of excellent time-frequency localization properties of wavelet transform [178]. Wavelets and wavelet transform can also be applied in spectrum sensing by using coarse and fine sensing [179]. The spectrum sensing technique based on energy detector wavelet transform was investigated in [180]-
[181]. Chen et al. proposed a modified spectrum sensing method based on compressive sensing and wavelet for wideband cognitive radio [182]. However, these techniques are applicable to Rayleigh fading or AWGN channel only. In addition, these methods have poor performance in the low SNR regime.
The overall objective of this chapter is to design and analyze efficient spectrum sensing methods at the receiver for WPM-based multicarrier communication systems on cognitive radio networks. We apply wavelet transform in spectrum sensing technique, which is the key element of cognitive radio awareness. These methods, which can be applied in the low SNR regime and multipath frequency-selective fading channel, are based on the pilot signal of received signal. It is assumed that pilot patterns used to estimate the channel are known by secondary users. To evaluate the merits of the proposed spectrum sensing method, we compare the performance of the proposed spectrum sensing methods to the performance of the conventional spectrum sensing method such as
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 70 energy detection and the autocorrelation coefficient-based methods. These comparisons show that the proposed spectrum sensing method based on
Approximate Covariance Matrix (ACM) achieves superior performance improvements over its comparatives, in terms of receiver operating characteristic
(ROC), and also works well in the low SNR regime.
3.2 Wavelet packet modulation based multi-carrier systems
x n t n u n v0 n 0 M 0 0 M ↑ 2 w0 n w0 n ↓ 2
v n x1 n t1 n u1 n 1 ↑ 2M w n w n ↓ 2M 1 y n r n 1 Channel + n
n t v n xM n tM n uM n 2M 1 2 1 M 2 1 w n w n 2 1 M ↑ 2 2M 1 2M 1 ↓ 2
IDWT DWT
M Figure 3.1 An equivalent representation of a WPM transceiver structure with 2
sub-channels.
The structures of IDWT at the transmitter and DWT at the receiver in
WPM multi-carrier systems in Fig. 2.3 and Fig. 2.4 can be represented as in Fig.
3.1 [183].
At the transmitter, let Ti, m ( z ) denote the Z-transform of the filter impulse
th th response corresponding to the i sub-channel and the m level (refer to Fig. 2.3)
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 71
and Ti, m () z HzGz (), () , where H( z ) and G( z ) represent the Z-transforms of the low-pass and high-pass filters h( n ) and g( n ) , respectively. Then, in Fig. 3.1,
th wi ( n ) denotes the equivalent impulse response corresponding to the i
M (i 0, 1, , 2 1) sub-channel. The Z-transform of the equivalent impulse response wi ( n ) depends on the low-pass and high-pass filters as well as the branch that sub-channel belongs to and is given by
M 2m1 Wzi() Tz im, () (3.1) m1
At the receiver, the QMF pair formed by h( n ) and g( n ) is applied successively to demultiplex the received signal into parallel data symbol streams.
M Similar to IDWT, the 2 sub-channel DWT structure can also be represented as in Fig. 3.1. The filter wi ( n ) at the receiver is the time-reversed version of the
wi ( n ) v n filter at the transmitter. The sub-channel sequences i are obtained after applying DWT to the baseband equivalent WPM signal r n .
As shown in the block diagram of Fig. 3.1, we denote the up-sampled version of the input signal xi( n ) by ti( n ) , that is,
n M xi M if mod( n ,2 ) 0 ti n 2 (3.2) 0 otherwise
th Likewise, ui( n ) is used to denote the output of the i filter wi n and vi( n ) denote the down-sampled version of ui( n ) . From Fig. 3.1, uj n and vj n can be expressed as
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 72
2M 1 unj tnwn ii nwn j nwn j i0 2M 1 (3.3) ti n U ij n n w j n i0
M vj n u j 2 n (3.4) where Unwniji nwnwnwn j i j n is the overall
th th response from the i sub-channel at the transmitter to the j sub-channel at the receiver and n is the additive white Gaussian noise (AWGN).
Substituting Eq. (3.2) and Eq. (3.3) into Eq. (3.4) yields
2M 1 M vj n t iij k U2 n k j n i0 k 2M 1 M M xi k U ij2 n 2 k j n (3.5) i0 k 2M 1 M xi k U ij2 n k j n i0 k where j n is the noise filtered by wj n and down-sampled. Because of the orthogonality of the WPT and IWPT, assumed that the noise n is AWGN, the filtered noise j n is also AWGN and has the same power as that of the
M noise n . Let us denote Vij n U ij 2 n as the down-sampled version of
Uij n
2M 1 vj n x iij k V n k j n i0 k 2M 1 (3.6) xi n V ij n j n i0
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 73
For an ideal channel, we have n n and Vnij n ij , which results in perfect symbols vi n x i n at the receiver.
x n [ x n , x n ... x n ]T v n [ v n , v n ... v n ]T Let 0 1 2M 1 and 0 1 2M 1
th denote the data symbol and pre-processed signal vectors of the n WPM block
M from all of 2 sub-channels, respectively. We can rewrite Eq. (3.6) in matrix form as follows
v Hx η a η (3.7)
TT T T TT T T with x[ x 0 x 1 ... x N 1 ] and v[ v 0 v 1 ... v N 1 ] being vectors consisting of data symbols and pre-processed signals, respectively; N is the number of WPM blocks, a is the processed signal vector and H is a Toeplitz
th M M matrix where its i, j component is the 2 2 sub-matrix given by
Vij Vij V ij 0,0 1,0 2M 1,0 Vij Vij V ij 0,1 1,1 2M 1,1 Hi, j (3.8) V ijV ij V ij 0,21M 1,21 M 21,21 MM
3.3 Proposed spectrum sensing methods
In this Section, we propose three novel methods to detect the presence of the primary signal for cognitive radio systems. For the sake of simplicity, let us assume that the pilot signals are inserted in every sub-channels of a WPM block and in every b WPM blocks as illustrated in Fig. 3.2. The block type and
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 74 the comb type pilot arrangements correspond to 1 and b 1, respectively. The receiver is assumed to know the pilot pattern.
2M sub-channels
sub-channels WPM blocks WPM b WPM blocksWPM N
M Figure 3.2 The pilot placements of a WPM block with 2 sub-channels.
Let Sb r b s: r 0,1,..., N / b 1;0 s b and
M Sefei : 0,1,..., 2 / 1;0 f denote the set of indices of WPM blocks and sub-channels containing pilots, respectively, where N is the number of WPM blocks. Assume that xi n p i n d i n where the pilot symbol pi n p i is non-zero if i Si and n S b , and zero otherwise, while the data
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 75
symbol di n is zero if i Si and n S b , and non-zero otherwise. It can be rewritten in vector form as x p d .
3.3.1 The optimal detection method
Based on the assumption of the channel, the input data symbol and the fact that the number of sub-carriers is large enough, the received signal has
Gaussian distribution with zero mean. Assuming that noise is also Gaussian distributed with zero mean, the likelihood ratio test (LRT) becomes the
T r estimator-correlator (EC). Therefore, the optimal test statistic EC based on the received signal r to determine the presence or absence of a primary signal is given by:
r H 2 1 TEC rCCr r Ir (3.9)
2 where Cr is the covariance matrix of the noise-free received signal r , and is the noise power.
In general, the computational complexity of the receiver can be reduced by neglecting elements of the covariance matrix Cr with small values. However, as opposed to OFDM, after applying the IDWT to WPM sequence, one WPM block is spread into several consecutive WPM blocks. Therefore, the covariance matrix
Cr , in time domain, has only a few small elements. To overcome this disadvantage, we propose some methods in the next Sub-section.
Assuming that these WPM symbols have the same wavelet-domain pilot symbols, then spectrum sensing methods based on the Accumulated Time-
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 76
Domain Symbol Cross-Correlation (ATDSC) and Accumulated Wavelet-
Domain Symbol Cross-Correlation (AWDSC) can be applied. These proposed methods work based on the fact that the ATDSC and AWDSC have non-zero mean when several successive WPM symbols containing pilot signals use the same pilot symbols, which are unknown at the SUs’ receiver. The Accumulated
Time-Domain Symbol Cross-Correlation is inspired by the work in [184]. In this modified version, the number of terms is increased to improve the performance of the spectrum sensing method.
3.3.2 ATDSC
This spectrum sensing method is based on the received signal r . We assume that the same pilot symbols are inserted in every b WPM symbols.
Because all the filters applied to the data stream are linear time-invariant (LTI),
M the received signal also has a periodic term with period 2 b as the input data as illustrated in Fig. 3.2, given that the wireless multipath frequency-selective channel is slow fading. Therefore, the receiver can detect the presence of the signal by using the ATDSC function, TATDSC , defined as follows:
M 1 N/ b 1 (N / b )2 b 1 TRe r i r* i 2M b ATDSC (3.10) K 1i 0 where r i is the received signal at time i and the number of terms in Eq. (3.10)
M 1 is K2 N N / b 1 . The test statistic ATDSC can be rewritten as:
2M b 1 N / b 12 N / b 1 1 2 TATDSC ri ri (3.11) 2K i0 0 0
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 77
M th th where r i r i 2 b is the sample in the i group. As shown in Eq.
(3.11), to reduce the computational complexity, the received signal r n is
M divided into 2 b groups, each group containing N/ b samples. The computation of the ATDSC function is based on these groups, and no term involves two or more groups. As a result, the computational complexity of this
M 1 method is reduced significantly from K2 N N / b 1 multiplications of the
T 2M N b T test statistic ATDSC in Eq. (3.10) to multiplications of ATDSC in Eq.
(3.11). In addition, an important advantage of the ATDSC function is that it is independent of the initial sample time instance.
3.3.3 AWDSC
This AWDSC method is based on the pre-processed signal ν . Similar to the received signal r , the pre-processed signal ν also has a periodic term with
M period 2 b . As a result, similar to Eq. (3.11), the receiver can use the AWDSC function defined as
2M b 1 N / b 12 N / b 1 1 2 TAWDSC vi vi (3.12) 2K i0 0 0
M th th where v i v i 2 b is the sample in the i group. The pre-processed
M signal ν is also divided into 2 b groups, each group containing N/ b samples. Like the ATDSC function, the AWDSC function is also independent of the initial sample time instance and has low computational complexity.
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 78
Although the performances of these methods have been improved by modifying the TDSC, they still have some disadvantages as shown in the simulation results. To overcome their drawbacks, we develop a new method which is based on Approximated Covariance Matrix (ACM) to further exploit the correlation between the pilot signals in the next Sub-section.
3.3.4 Approximated Covariance Matrix (ACM)-based method
In this Sub-section, we propose a new test statistic to detect the presence or absence of the primary signal based on the pre-processed signal ν . In order to simplify the problem, we make the following assumptions:
E di n 0, * 2 Edndnij1 2 s nn 1 2 ij , * E pi d j n 0 (3.13)
E pi n 0, * Eppi j C p ij,
2 where the signal power s and the values C p i, j are assumed known at the receiver. The first two equations in (3.13) imply that the data symbols are zero
2 mean, temporally independent and have power s . As is usual in practical scenarios, the data symbols and pilot symbols are also independent. The pilot symbols pi are also zero mean and have the covariance matrix C p , as shown in the last two equations. It is noted that the pilot symbols pi themselves are unknown at the receiver. The problem then is to distinguish between two hypotheses:
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 79
Η0: v=η H 0 (3.14) Η1: v=a+η H 1
The pre-processed noise η is also white Gaussian as the noise n of the channel because the DWT at the receiver maintains the statistical properties of the noise. Similar to the received signal r , the pre-processed signal ν is also
Gaussian distributed with zero mean. Hence,
2 0, I under H0 v (3.15) 0C, 2 I under H a 1
2 where Ca is the covariance matrix of a , and is the noise power.
Consequently, the optimal test statistic Topt based on the pre-processed
ν signal for distinguishing between these hypotheses is given by [34]
H 2 1 : decide H1 Topt v Ca C a I v (3.16) : decide H0 where is a pre-defined threshold.
2 1 When the SNR is very low, the matrix Ca I can be approximated
2 by I . This approximation is valid when all the eigenvalues of the covariance
2 matrix Ca are small compared to the noise power , i.e., at the low SNR regime. It is noted that the computational complexity of Topt is high when the number of WPM blocks and the number of sub-carriers are high. To reduce the complexity, based on the approximate expression for the matrix Ca , a test statistic TACM is derived in appendix A:
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 80
M 2N 1 2 1 p, Sb 2 2 C , 2 s* a s d, Sb TvvACM v C a , v 2 2 2 0 0 , SSSSSi b b i b (3.17)
th th where v is the pre-processed signal in the sub-channel in the WPM
p, Sb d, Sb p block (see Fig. 3.1), Ca and C a are sub-matrices of the matrices Ca and
d p d C a , respectively. Ca and Ca are the covariance matrices of the pre-processed
d signal produced by only the pilot signal and the data signal, respectively, and C a is defined in Eq. (A.12) of appendix A.
p,, Sb p S b 2 Let Ca, C a ,/ s , the expression for TACM in Eq. (3.17) becomes
2 N 1 2M 1 2 2 s * p,, Sb d S b TvvACM Ca ,, v C a v 2 0 0 , SSSSSi b b i b
(3.18)
Under the assumption that the covariance matrix of the pilot signals Cp , the wavelet basis function h n and g n , and the channel power delay profile
2 C p, Sb C d, Sb k are known, the receiver has information about a and a by using Eq.
(A.10) and Eq. (A.17), then TACM can be easily obtained. Equation (3.18) shows that the test statistic TACM is the sum of the energy of the pre-processed signal ν
and the correlation between the pre-processed signals at pilot positions. Equation
(3.18) can be further simplified by using the approximation in Eq. (A.18) of appendix A as follows:
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 81
2 2 s * p, Sb TACM v v Ca , v 2 (3.19) SSSSSi or b , i b b
It can be seen in Eq. (3.19) that the test statistic is the sum of the energy of the pre-processed signal ν at the data positions and the correlation between the pre-processed signal ν at the pilot positions. It should be emphasized that approximating the covariance matrix Ca helps to significantly reduce the
M 2 r computational complexity from 2 N 2 multiplications of the test statistic TEC
M M M 2 in Eq. (3.9) to N 2 2/ 22/ multiplications of the test statistic
TACM in Eq. (3.19) while maintaining its good performance as shown in the simulation. By using the approximated version of the covariance matrix Ca , the test statistic can be obtained without having to deal with the large matrix manipulation.
3.3.5 GLRT approaches for unknown parameters
2 In general, the covariance matrix Ca and the noise variance are unknown. In this case the generalized likelihood ratio test (GLRT) can be used
[34]:
2 pv;,, Ca 1 H 1 : decide H1 LG v 2 (3.20) pv;, H : decide H0 0 1
2 where Ca is the maximum-likelihood estimate (MLE) of Ca under H1 , 1 and
2 2 0 are MLEs of under H1 and H0 , respectively. We shall examine the following three special cases:
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 82
2 A. The covariance matrix Ca unknown, the noise variance known
2 When is known, Eq. (3.20) becomes
H 2 : decide H1 TACM GLRT v Ca v / (3.21) : decide H0
The estimation of Ca is derived as follows. From (3.14), the PDF of v under H1 is given by
1 1 1 H 2 fv H1 N /2 exp v Ca I v (3.22) 2 2 2 det Ca I where det denotes the determinant of the matrix. Taking the logarithm in Eq.
(3.22), the log-likelihood function of Ca under H1 can be expressed as
N ln 2 N 21 H 2 1 lnfv H1 ln det Ca I v C a I v 2 2 2 (3.23)
2 Let R denote Ca I , Eq. (3.23) can be rewritten as
N ln 2 N 1 H 1 lnfHv1 ln det RR v v 2 2 2 (3.24)
1 Since R is positive semi-definite (i.e., R 0), R is also a positive semi-
1 definite. Equation (3.24) can be rewritten as a function of R :
N ln 2 N 1 1 fR1 ln det RR 1 vH v 2 2 2 (3.25)
Since det R and ln x are concave functions, it is easy to show that
1 1 ln det R is a concave function of R . Combining with the fact that
N ln 2 1 H 1 1 1 xR x is a linear function of R , we can conclude that f R 2 2
1 is a concave function of R . Then we can find the maximum likelihood
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 83
1 1 estimation (MLE) of R by letting the first derivation of f over R equal to df R1 0 zero (i.e., dR1 ) as follows:
N di j d 1 1 RM1 Rvv 1 1 1 ij, 1 ij (3.26) 2det RdR i, j ij,, dR 2 ij ij,, ij
1 th 1 th 1 where R is the i, j element of R and M i, j is the i, j cofactor of R . i, j
From Eq. (3.26), it is easy to show that
i j 1 M i, j 1 vvi j (3.27) det R1 N
th ˆ The left hand side of Eq.(3.27) turns out to be the i, j element of R , that is,
1 Rˆ vv ij, N ij (3.28)
Equation (3.28) can be rewritten as
N 1 ˆ 1 H R vv (3.29) N n0
It should be noted that the elements of Ca are non-zero only at the pilot positions, as shown in the appendix A, so the elements of R are non-zero only at the pilot positions and the diagonal positions. From Eq. (3.29), the estimated
H matrix Ca can be computed by averaging the elements of the matrix vv at the
2 pilot positions, in which s can be obtained based on the power of the pre- processed signal ν at the data positions. Regarding the computational
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 84
2 2 M M complexity, it should be noted that only about 2 / out of N 2 elements of Ca are needed to be estimated in this case.
2 B. The covariance matrix Ca known, the noise variance unknown
2 When the SNR is very low, the MLE of under H1 and H0 can be expressed as follows [34]:
M 1 N 2 1 2 2 max 0,v n tr C 1 M a N 2 n0 (3.30) N 2M 1 2 1 2 0 M v n N 2 n0
2 2 Substituting the estimation 1 and 0 in Eq. (3.21) into Eq. (3.19) with known Ca , the GLRT can be shown as
M2 2 1 1 HH 1 LGv N2 log0 log d i 1 v v v Ca v 2 2 2 (3.31) i 2 0 1 1 where di are the eigenvalues of the covariance matrix Ca as shown in Eq. (3.36)
To reduce the computational complexity, the last term in Eq. (3.31) can be obtained using the approximated version of Ca , similar to the test statistic TACM in Sub-section 3.3.4.
2 C. The covariance matrix Ca and the noise variance unknown
The test statistic TACM GLRT can be obtained by using the estimation of Ca ,
2 2 1 , and 0 similar to the approaches in parts A and B. It is emphasized that the test statistic TACM GLRT is scale invariant, which in turn makes its probability
2 density function (PDF) under H0 not to depend on the noise power . As a
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 85 result, the threshold can be chosen to obtain a constant false alarm rate
(CFAR) detector.
3.3.6 The Receiver Operation Characteristic (ROC)
To assess the performance of the proposed methods we consider the ROC performance. Under H0 , the pre-processed signal ν has a normal distribution
2 with the same variance as the noise power. It comes from the fact that the
DWT at the receiver preserves the power and the uncorrelated property of the
H input. Using eigendecomposition, the matrix Ca can be expressed as Ca RDR where R is a unitary matrix and D is a diagonal matrix. Using that expression, the test statistic TACM can be expressed as
H2 HH 2 H 2 TACMvCva / vRDRv / ρDρ d ii (3.32) i
H where the components i of the vector ρ R v / are uncorrelated normal variables with zero mean and unit variance, di is the i th diagonal element of the diagonal matrix D .
Under H1 , let Cv denote the covariance matrix of the pre-processed vector
2 v , where CCv a I . First, we make the linear transformation v=LQρ where
H L is a matrix defined as Cv LL , Q is a square matrix whose columns are the
H 2 eigenvectors of the matrix L CLa / . Then the test statistic TACM can be expressed as
H2 HHH 2 H 2 TACMvCva/ ρQLCLQρ a / ρAρ a ii (3.33) i
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 86
where the diagonal elements ai of the diagonal matrix A are the eigenvalues
H 2 of the matrix L CLa / , vector ρ is composed of uncorrelated normal random variables with zero mean and unit variance. From the relationship between the
H H 2 matrices Ca and A (i.e., A Q LCLQa / ), it is easy to show that their eigenvalues ai and di are related as follows
2 2 ai d i d i / (3.34)
Next the expression for the eigenvalues di of the covariance matrix Ca in
Eq. (3.32) is derived. For the sake of simplicity, we assume that the number of
WPM blocks N is divisible by the space b between two consecutive WPM blocks containing pilot symbols.
p From Eq. (A.10) in Appendix A, the matrix Ca can be expressed as
p T p, Sb Ca 11 Bt C a B f (3.35) where 1 is a N/ b 1 column vector, Bt and B f are b b and
th th matrices with all zero elements except the s, s and the f, f elements equal to 1, respectively, and denotes Kronecker product. It is easy to show that the
T non-zero eigenvalues of 11 F are N/ b times the non-zero eigenvalues of F
( N/ b is the length of the vector 1 ), and the non-zero eigenvalues of F Bt and F B f are equal to the non-zero eigenvalues of F , where F is an arbitrary
p, Sb matrix. Let us denote the set of eigenvalues of the sub-matrix Ca by
p M p p M di , i 1..2 / , then the set of eigenvalues of Ca is N/ bd i , i 1..2 / .
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 87
The expressions in Eq. (A.12) and Eq. (A.18) of appendix A show that
d th Ca is a diagonal matrix where its i diagonal element is zero in the pilot
M 2 positions (i2 j , Sb , j S i ) and s in the data positions. Since the pilot positions and data positions are exclusive, the set of eigenvalues di of the covariance matrix Ca is the union of the set of the eigenvalues of the covariance
p d matrix Ca and that of the covariance matrix Ca , that is,
Nbd/ p i 1..2 M / i 2 M M M di s i 2 / 1..2 / N 2 1 1/ b (3.36) M M M 0i 2 / N 2 1 1/ bN 1.. 2
From the set di , it is easy to obtain ai by using Eq. (3.34). Under H0 and H1 , Eq. (3.32) and Eq. (3.33) show that the value of TACM has a generalized chi-squared distribution with two different sets of parameters di and ai .
From the two PDFs of TACM under H0 and H1 , the ROC of the receiver can be easily determined. The threshold can also be determined based on the given probability of false alarm Pfa and the PDF of TACM under H0 .
p It is noted that the size of the matrix Ca is much bigger than that of the
p, Sb p sub-matrix Ca . Therefore, the derivation of the eigenvalues di of Ca from the
p p, Sb eigenvalues di of the sub-matrix Ca helps to reduce the computational complexity significantly with respect to evaluating the PDFs of TACM under H0 and H1 ; and especially the threshold when the number of samples is very large.
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 88
When the number of samples is very large, or the number of WPM blocks N is very large, the central limit theorem (CLT) can be applied to find the PDFs of TACM under H0 and H1 as follows
2 Νddi,2 i under H0 i i T (3.37) Ν aa, 22 under H i i 1 i i
Thus the detection probability Pd and the false alarm probability Pfa can be determined as
a i PPr T | HQ i d 1 2 2 ai i (3.38) d i PPr TH | 1 Q i fa 0 2 2 di i where is the decision threshold and Q is the standard Gaussian complementary cumulative distribution function (CCDF). When the false alarm probability Pfa is given, the threshold and the detection probability Pd can be determined as
2 1 di2 dQ i 1 P fa i i d a 2 dQP2 1 i i i fa (3.39) P Q i i i d 2 2 ai i
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 89
When the false alarm probability Pfa and the detection probability Pd are known, the number of required WPM blocks can also be obtained. From Eq.
(3.38), by canceling the threshold , the following equation is derived
2 1 2 1 di2 dQ i 1 P fa a i 2 aQP i d (3.40) ii ii
By substituting the values of di and ai in Eq. (3.36) and Eq. (3.34), the number of required WPM blocks N can be determined by solving Eq. (3.40) using any numerical methods.
3.4 Simulation results
In this Section, we present simulation results to assess the performance of the proposed spectrum sensing methods in frequency-selective channels.
Throughout this Section, we compare the performance of the proposed ACM- based spectrum sensing methods to the performance of the proposed ATDSC- based, AWDSC-based methods and those of the conventional methods such as energy detection-based and TDSC-based. Moreover, the proposed ACM-based methods are compared with the conventional spectrum sensing methods where the test statistics are based on the eigenvalues of the covariance matrix of the received signal as follows [185]-[186]
Teig_1 max / i i
Teig _ 2 max/ min (3.41)
Teig_ 2 i / min i
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 90
In addition, we also make performance comparisons with a method based on the autocorrelation coefficient [187]:
** Extxt / Extxt (3.42)
Throughout the simulations, the number of sub-channels is set to 64 (i.e.,
M 6 ). To obtain good bandwidth efficiency with low complexity, the length of the low-pass filter and high-pass filter of the conventional Daubechies basis functions are set to be 12. For the pilot pattern, the distances between pilots in the time-domain and wavelet-domain are chosen to be 4 (i.e., b 4 ). The pilot ratio is 1/16, which is better than that in [187]. The simulations are conducted by choosing the input signals of the IDWT block to be composed of independent and identically distributed 16-QAM symbols. The number of WPM blocks is set to 32. Since one WPM block on wavelet domain is spread into multiple blocks in time domain, the fading channel is assumed to be slow, frequency-selective fading where the channel remains static over the transmit duration of several WPM blocks The channel is assumed to follow an exponentially decaying power delay profile and the length of the fading channel is set to 10. Moreover, the performance of the described methods in this Section is
14 analyzed by using Monte Carlo methods of 2 iterations.
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 91
The first performance metric we consider is the Receiver Operation
Characteristic (ROC). Figure 3.3 shows that the ROC curves obtained from the analysis in Sub-section 3.3.6 are in strong agreement with the simulation results for different values of SNR. The ability of the ACM-based method in the low
SNR regime is also demonstrated in Fig. 3.3.
1
0.9
0.8
0.7 SNR=-20dB SNR=-25dB 0.6
d 0.5 P
0.4 SNR=-30dB 0.3
simulation 0.2 analytic
0.1
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P fa
Figure 3.3 ROC curves of the ACM-based method for different SNRs.
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 92
Figure 3.4 ROC curves for several methods in frequency-selective channel at
SNR=-20dB.
In Fig. 3.4, the performances of ATDSC-based and AWDSC-based methods are almost identical and significantly better than that of TDSC-based method. This is mainly because the number of terms used to sense the spectrum of TDSC-based method is much smaller than those of ATDSC-based and
AWDSC-based methods. By further exploiting more information about the pilot signals, the proposed ACM-based method outperforms ATDSC-based and
AWDSC-based methods, as shown in Fig. 3.4. Compared to the optimal method,
ACM-based approach maintains negligible performance degradation while significantly reducing the computational complexity. At a target Pfa of 0.01, the
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 93
ACM-based methods achieve Pd of 0.556, while the Pd of the optimal method,
TDSC-based, ATDSC-based, and AWDSC-based methods are 0.565, 0.018, 0.071,
0.085, and 0.205, respectively.
Figure 3.5 presents ROC curves for several methods in frequency-selective channel with SNR = -10dB. The eigenvalue-based and autocorrelation coefficient- based methods (see Eqs. (3.43) and (3.44)) do not utilize the structure of the covariance matrix of the received signal, which results in system performance degradation. That is why Fig. 3.5 is plotted with SNR= -10dB. It is shown that the proposed ACM-based method outperforms the conventional methods under
Figure 3.5 ROC curves for several methods in
frequency-selective channel at SNR=-10dB.
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 94 low SNR regime.
Next we compare the ROC performance of the proposed ACM-based method with full information and partial information (in Sub-section 3.3.5) in
Fig. 3.6 with SNR = - 20dB. From the family of ROC curves in the figure, it is noted that at a target Pfa of 0.01, the ACM-based method with full information achieve Pd of 0.556, while the Pd of the energy detector method, the ACM-based
2 2 method with (a) unknown n , (b) unknown Ca , (c) unknown Ca and n are
0.032, 0.548, 0.315, and 0.309, respectively. It is shown that the absence of knowledge of Ca leads to significant performance degradation as expected.
Figure 3.6 ROC curves of ACM-based methods with full and partial information
in frequency-selective channel at SNR=-20dB.
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 95
2 However, the absence of knowledge of the noise power n leads to negligible degradation in detection performance.
Given Pfa 0.01, the probability of detection Pd performances of the proposed method and different methods as a function of SNR are shown in Fig.
3.7. For the sake of clarity, the curves of various methods are plotted left to right in the order from best to worst performance. As shown in the last paragraph of
M Sub-section 3.3.4, with N 32 and 2 64, the ACM-based method using the test statistic TACM in (3.17) reduces the number of multiplications down to 0.0003
r of that of the optimal method using the test statistic TEC in (3.9), with negligible
Figure 3.7. Pd performance comparison of various methods for Pfa 0.01 .
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 96
degradation in detection performance as shown in Fig. 3.7. At a target Pd of
0.8, the proposed ACM-based method, with full information, achieve SNR gains of 0.23 dB, 1.56 dB, 1.75 dB, 5.32 dB, 5.57 dB, 8.16 dB, 11.36 dB, 11.9 dB, 11.15
2 dB, 12.9 dB, and 21.67 dB over the ACM-based with unknown n , unknown Ca ,
2 unknown Ca and n , AWDSC-based, ATDSC-based, energy detector, TDSC- based, autocorrelation coefficient-based, and eigenvalue-based methods, respectively.
3.5 Conclusion
In this chapter, we have proposed novel spectrum sensing methods for
WPM systems using pilot signal. First, we proposed ATDSC-based and AWDSC- based spectrum sensing methods which detect the spectrum vacancy by exploiting the periodicity of the received signal. These methods work well when the number of pilot signals is large, even at low SNR. However, under low SNR regime and sparse pilot signals in the time-wavelet plane, their performance deteriorates. The spread made by the IDWT and DWT in the WPM transceiver and the effect of the multipath frequency-selective channel also contribute to performance degradation. Next, we proposed the ACM-based spectrum sensing method to overcome the disadvantage of the proposed ATDSC-based and
AWDSC-based methods. The performance of the ACM-based method is compared to those of various methods. Analysis and simulation results show that
Chapter 3: SS for WPM based CR multi-carrier systems using pilot signals 97 this new method outperforms all of its comparatives and achieve significant improvement.
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 98
CHAPTER 4
SPECTRUM SENSING FOR WPM SYSTEMS BASED ON
RAO’S SCORE TEST
In the previous chapter, we proposed a spectrum sensing method based on an approximated Covariance Matrix (ACM) assuming some prior information about the primary signal at the SUs. However, in practice there is usually no prior information about the primary signal at the SUs. Use of an approximated covariance matrix with lack of information can degrade the detection performance significantly. To overcome this problem, we consider new approaches which are based on the following: GLRT proposed in the previous chapter, Rao’s score test, Wald test, and locally most powerful (LMP) test. These methods do not need prior knowledge of signals from the PUs (i.e., the unknown covariance matrix Ca of the primary signal). However, like GLRT, Wald test requires MLE of the parameter values under H1 , which makes it difficult to implement and also increases the computational complexity. The LMP test is used for a one-sided hypothesis test whose PDFs under H0 and H1 differ only by a scalar value and there are no nuisance parameters; while in our case, there are lots of parameter values in the unknown covariance matrix.
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 99
When the PDF of received signal under H0 and H1 are the same except for the set of parameter θ , the binary hypothesis testing problem in Eq. (2.1) can be modeled as follows:
θ0H 0 θ (4.1) θ0H 1
From Eq. (3.15), it is shown that the parameter vector θ0 is zero vector under H0 . Under H1 , the parameter vector θ consists of the elements of Ca .
In this chapter, we propose to apply the Rao’s score test [34]. This is the most powerful test when the true value of the parameter of interest θ is close to the particular value θ0 under the null hypothesis H0 . In our case, when Ca is unknown, the parameter of interest θ consists of the elements of Ca , and θ is indeed close to θ0 under low SNR regimes. Unlike the GLRT-based approaches,
Rao’s score test does not require a maximum likelihood estimation (MLE) of θ ˆ under hypothesis H1 . When the MLE value θ is difficult to obtain, this main advantage of Rao’s score test is obvious. Moreover, the performance of Rao’s score test is asymptotically equivalent to that of GLRT [34]. The Rao-based test constrains the probability of false alarm PFA and maximizes the probability of detection PD for all θ close to θ0 . However, this test is not the optimal for a large deviation of θ from θ0 , for which the GLRT-based test, in the previous chapter, is more suitable.
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 100
This chapter is organized as follows. Firstly, conventional Rao’s score test- based spectrum sensing methods are reviewed in Section 4.1. Then, the test statistic based on Rao’s score test is derived for WPM system with pilot signals in Section 4.2. Section 4.3 presents the detection performance assessment and the
ROC performance of the novel test statistic. Finally, the simulation result of this approach is presented in Section 4.4. The chapter is summarized in Section 4.5.
4.1 Conventional Rao’s score test-based spectrum sensing methods
In [190], composite hypothesis testing methods, based on Rao test, are applied to cooperative sensing. An optimal likelihood ratio test statistic, based on the Neyman-Pearson (NP) criterion, is derived at the fusion center. The authors also propose a modified Rao test statistic for decision making at the fusion center, which receives hard or quantized decisions from local detectors. They show that their proposed methods have lower computational complexity compared to the conventional NP-based method; they also do not require MLE estimation and have performance very close to the optimum. A novel method based on the Rao test is proposed for spectrum sensing task in the generalized
Gaussian distribution noise environments [191]. Without prior knowledge, the test statistic based on Rao test enhances the spectrum sensing performance over conventional energy detection in non-Gaussian noise. Thanks to the improved decision fusion rule, the Rao-based method is extended to a multi-user cooperative framework and shows significant performance enhancement over the
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 101 traditional energy detection method. A sub-optimal detector based on Rao test is proposed for situations when antenna correlation coefficients, PU signal power, and noise variance are unknown to the SU [192]. It is shown, analytically and by simulation, that the performance is superior to those of detectors for multiple antenna systems. Rao test is often used to overcome the fact that the covariance matrix of primary signal is unknown at the SUs [193]-[196]. In [193], a detector based on Rao test is proposed for MIMO radar and the simulation results show that the test is equivalent to the GLRT detector in Gaussian noise with known covariance matrix. In [194], a Rao test is combined with a subspace detector to form a new two-stage detector. This method possesses a constant false alarm rate
(CFAR) property and is more robust to mismatched signal (i.e., calibration and pointing errors, imperfect antenna shape, wavefront distortions) than their conventional counterparts. In [195], a Bayesian approach maximum a-posteriori
(MAP) estimation of the unknown covariance matrix is derived based on a suitable model for the probability density function (PDF) of the unknown covariance matrix. Bayesian versions of Rao and Wald tests are then used to detect the signal. These detectors are shown to outperform Kelly's GLRT and non-Bayesian Rao and Wald tests in several scenarios. In [196], a Rao-based method is used to detect a signal with unknown covariance matrix in the presence of Gaussian noise.
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 102
4.2 A novel test statistic based on Rao test for WPM systems
Given a PDF f v;θ which depends on θ (i.e., the unknown covariance
2 matrix CCv a I ) under H1 , the score is defined as follows: logf θ ; v S θ θ (4.2)
The Rao test statistic is defined as
H logfvθ ; log f vθ ; TSSH θIθθ1 Iθ 1 Rao 0 0 0θ 0 θ (4.3) θθ0 θθ 0
where Iθ0 is the Fisher information matrix at θ θ0 which is defined as [35]:
2 logfvθ ; log ff vθ ; log vθ ; Iθ E θ E θ i. j (4.4) ij i j
In Eq. (4.4), E denotes the expectation over the data v. It should be noted that the PDF of v under H0 is f v;θ 0 . From the simulation results in
2 the previous chapter, the lack of noise power information insignificantly degrades the detection performance. Therefore, in this Section, we propose to only focus on the elements of the covariance matrix Ca of the noise-free pre- processed signal a . From Eq. (A.10) in Appendix A, the covariance matrix Cv of the pre-processed signal can be expressed as
2p d 2 CCv a CC a a p2 2 d 2 Cas I s C a (4.5)
Tp, Sb 2 2 d 2 11 Bt Ca B fss I C a
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 103
where 1 is a N/ b 1 column vector, Bt and B f are b b and
th th matrices with all zero elements except the s, s and the f, f elements equal to 1, respectively, and denotes the Kronecker product.
The parameter vector θ only consists of the elements of the
2M 2 M C p, Sb complex matrix a , as defined by
p,,, Sb p S b p S b θC,a ,,,Re, Re C a , , ,,Im Im C a , (4.6) where Re and Im denote the real part and imaginary part, respectively.
p, Sb Since Ca is a Hermitian matrix, in Eq. (4.6), we are only concerned with the
th p, Sb , ( ) element of the matrix Ca . As a result, there are totally
2 2MMM 1 2 2 1 2 MM 2 2 M 1 1 elements in the parameter vector θ . 2 2
From now on, the symbols θi, θ j and θ, will be used interchangeably.
Next, we compute the Fisher information matrix Iθ and the inverse
1 matrix I θ0 in Eq. (4.4) and Eq. (4.3). First the PDF of the pre-processed signal v under the hypothesis H1 is given by
1 fv; H , C exp vH C1 v 1 v N 2M v (4.7) det Cv
M where N 2 is the length of received signal. Taking the logarithm of both sides, we obtain the following equation
M H 1 lnfv ;θ N 2ln lndet Cv v C v v (4.8)
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 104
th Taking the partial derivatives of both sides with respect to i , the i element of the score S θ can be expressed as
H 1 Si ln f v ;θ ln det Cv v C v v (4.9) i i i
Applying the following identities [197]
AB ABA B
1 C ln detC tr C (4.10) C C1 C 1 C 1 into Eq. (4.9), we obtain
1Cv H 1 C v 1 lnf v |θ tr Cv v C v C v v (4.11) i ii
Cv From Eq. (4.5) and (4.6), the partial derivative can be expressed as i
p, Sb CvT C a 11 Bt B f i i (4.12) T 11 Bt Θ i B f
p, Sb Ca where the matrix Θi can be derived from Eq. (4.6) as i
H M Θ, ee 1 2/ H H M Θ, ,Re eeee 1 2/ (4.13) H H M Θ, ,Im j ee j ee 1 2/
M In Eq. (4.13), e denotes a 2 / 1 vector consisting of all zero elements
th except the element, which is equal to 1; and j denotes the imaginary unit. It
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 105 can be seen that Θ is independent to the parameter of interest θ . As a result,
Cv is also independent to θ . i
From Eq. (4.11), taking partial derivatives yields
2 1CvH 1 C v 1 lnf v ;θ tr Cv v C v C v v (4.14) ij j i ji
Cv Using the identity trA tr A and the fact that is j j i independent of j , Eq. (4.14) can be rewritten as
2 CC CC lnf v ;θ tr C1v C 1 v vH C 1 v C 1 v C 1 v v v v v v ij ji ji (4.15) H 1Cv 1 C v 1 v Cv C v C v v i j
Using the identity trAB tr BA , Eq. (4.15) can be rewritten as
2 CC CC ln;f v θ tr C1v C 1 v tr C 1 v C 1 v C 1 vvH v v v v v ij ji ji (4.16) C C tr C1v C 1 v C 1 vvH v v v i j
th Evaluating expectation over v in Eq. (4.16), the i, j element of the
Fisher information matrix can be expressed as follow
2 C C Iθ Elnf vθ ; tr C1v C 1 v i, j v v (4.17) ij ij
2 Substituting θ0 for θ (equivalent to setting Cv I ) into Eq. (4.17), we obtain
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 106
1 C C I θ tr v v 0 i, j 4 (4.18) i j
From Eq.(4.18), it is easy to see that Iθ0 is a symmetric matrix. Next we evaluate the right hand side in this equation. Applying the identity
A BCD AC BD into Eq. (4.12), the following equation is obtained
Cv C v T T 11 11 Btt B ΘΘ ij BB ff (4.19) i j
Using the identity trAB tr A tr B , Eq. (4.19) can be rewritten as
2 C C N trv v tr11T 11 T tr B B trΘΘ tr BB tr ΘΘ tt ij ff ij (4.20) i j b where N/ b is the number of WPM blocks containing pilot symbols. Next we evaluate tr ΘΘi j . Substituting Eq. (4.13) into Eq. (4.20), the different values of tr ΘΘi j can be found as follows,
trΘΘ tr eeeeH H 1, 1 2 , 2 1 1 2 2 1 2 (4.21)
trΘΘ tr eeeeeeH H H 2 0 1, 1 2 , 2 ,Re 1 1 2 2 2 2 1 2 1 2 (4.22)
M M for 11 2/;1 2 2 2/ ,
trΘΘ tr eeeeeeHj H H 0 1, 1 2 ,,Im 2 1 1 2 2 2 2 (4.23)
M M for 11 2/;1 2 2 2/ ,
trΘ Θ trj eeeeH H j eeee H H 2 11,,Re,,Re 22 1111 2222 1212 (4.24)
trΘ Θ trj eeeeH H j eeee H H 2 11,,Im,,Im 22 1111 2222 1212 (4.25)
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 107
Combining Eqs. (4.18), (4.20) and (4.21)-(4.25), it is easy to see that
Iθ0 is a diagonal matrix. More specifically, the diagonal elements of the Fisher matrix Iθ0 can be expressed as follows:
2 1 N p, Sb 4 if i is a diagonal element of Ca b 2 2 N p, S Iθ if is the real part of a off-diagonal element of C b 0 i, i 4 i a (4.26) b 2 2 N p, S b 4 if i is the imaginary part of a off-diagonal element of Ca b
Substituting the values of Iθ0 in Eq. (4.26) into Eq. (4.3), the Rao- based test statistic becomes
H 1 TSSRao θIθ0 0 θ 0 b2 21 2 1 2 (4.27) SSS, ,,Re ,,Im 2 4 θ θ θ N 02 0 2 0
log f v Next we compute the term S i in Eq. (4.3) by i θ θ0 substituting θ0 for θ into Eq. (4.11) as follows:
1T 1 HT Si ln f v |θ 2 tr 11 B tif Θ B 4 v 11 B tif Θ B v i θ θ 0 X Yi Zi (4.28)
H i Evaluating the term v Z v in Eq. (4.28) gives
N N HiHi H i v Z v vm Zm,, n v n v m X m n Y v n m, n 1 m , n 1
N 2M (4.29) * i Xm,, n v p m Y p q v q n m, n 1 p , q 1
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 108
i M M i where Zm, n is a 2 2 submatrix of Z , implicitly defined in Eq. (4.28), vn
i th th and vq n are defined in Chapter 3, X m, n and Yp, q are the m, n and p, q
i elements of matrices X and Y , respectively. From the implicit definition of X in Eq. (4.28), it is easy to see that
1 if m , n Sb X m, n (1.30) 0 otherwise
Substituting Eq. (1.30) into Eq. (4.29) yields
* 2M N N H i i v Z v vmYp p, q vm q (4.31) p, q 1m1 m 1 m Sb m S b
i Similarly, from the implicit definition of Y in Eq. (4.28) and Eq. (4.13), we obtain
, 1 if pq f Yp, q 0 otherwise 1 if p fq and f , ,Re Yp, q 1 if pfqf and (4.32) 0 otherwise jpif fq and f , ,Im Yp, q jpif fq and f 0 otherwise
Substituting Eq. (4.31) into Eq. (4.28) yields
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 109
2 1 1 N S vm , 2 4 b f θ θ0 m1
m Sb * 2 N N S Re vmvm , ,Re 4 bf bf (4.33) θ θ0 m1 m 1 m Sb m S b * 2 N N S Im vmvm , ,Im 4 bf bf θ θ0 m1 m 1 m Sb m S b
Substituting Eq. (4.33) into Eq. (4.27) yields the final expression for the
Rao test statistic:
22 2 2 b2 N NN T vm2 2 vmvm Rao2 12 bf bf bf (4.34) N m1 mm 1 1 m S m S m S b b b
From the expression of TRao , it can be seen that the test statistic depends on the pre-processed signal at the pilot positions only. It should be noted that in the first term, the noise power is subtracted from the square of the sum before squaring, while the second term is not. This comes from the fact that in the first term, the sum is multiplied by itself (the same point in the wavelet domain), while in the second term, the sums at two different points in the wavelet domain are multiplied together. Moreover, in the second term, since the first index is less than the second index , that term must have a factor of 2.
Equation (4.34) can be further manipulated as follows:
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 110
22 2 2 b2 N N N T vm vmvm Rao2 12 bf bf bf N m1 m 1 m 1 m S m S m S b b b 2 b2 N 2M 22 Re v m 4 2 12 b f (4.35) N m1 m Sb 2 2 2 b2 N N 2M vm2 2 vm 4 2 12 bf bf N m1 m 1 m S m S b b
Then the test statistic TRao can be expressed as
2 M b 2 22 4 TRao2 12 TT R 2 R (4.36) N where
2
N TR vm bf (4.37) m1
m Sb
2 As seen in Eq. (4.36), TRao is a function of TR and . Therefore, when the
2 noise power is known or estimated, we can equivalently use TR in Eq. (4.37) as our new test statistic, to detect the presence of the PU’s signal.
4.3 Performance assessment
When the number of available samples for sensing is large, the PDFs of the test statistic TRao can be expressed as follows [34]:
a 2 under H 2lnT v r 0 Rao '2 (4.38) r under H1
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 111
2 '2 where r and r denotes central and non-central chi-squared distribution,
M 2 2 H respectively, with degrees of freedom r , θθ1 0 I θθθ 0 1 0 where θ1 is the true value under H1 and θ0 0 under H0 . The detection performance of TR is the same as that of TRao , as shown in Eq. (4.36).
Substituting θ by θ0 and I θ0 from Eq. (4.26), the parameter can be expressed as
2 p, Sb Ca , (4.39) ,
The test statistic TR in Eq. (4.37) can also be expressed as
H TR v p Dv p (4.40) where v p is the pre-processed signal v extracted at pilot positions only (i.e.,
M M v pv ,, S b S, D is a N/ b 2/ N / b 2/ matrix
D I 11H I 2M / defined as 2M / where 2M / is an identity matrix of size and
1 is a N/ b 1 column vector.
Under hypothesis H0 , the pre-processed signal ν p has a normal
2 distribution with the same variance as the noise power. Using
1 eigendecomposition, the matrix D can be expressed as D QDΛ D Q D where QD is a unitary matrix and ΛD is a diagonal matrix. From the definition of D , it is
M easy to show that the matrix D has non-zero 2 / eigenvalues of value N/ b while the rest are zero. Using that expression, the test statistic TR in Eq. (4.40) can be expressed as
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 112
2M / HH1 H N 2 TRpppvDv vQΛQDDD v p uΛu D u i (4.41) b i1
1 where the components ui of the vector u QD v p are uncorrelated normal variables with zero mean and unit variance.
Under hypothesis H1 , the vector v p has normal distribution with zero
C mean and covariance matrix v p , which is a matrix extracted from the covariance matrix Cv at pilot positions. Using eigendecomposition, the matrix
C C Q Λ Q 1 Q Λ v p can be expressed as v p p p p where p is a unitary matrix and p
1/2 is a diagonal matrix. First we make the linear transformation vp Q pΛ p Qu , where Q is a square matrix whose columns are the eigenvectors of the matrix
1/2H 1/2 Λp Q p DQΛ p p . Then vector u is composed of uncorrelated normal random variables with zero mean and unit variance. The test statistic TR in Eq. (4.40) can then be rewritten as follows:
H HHH1/2 1/2 H 2 TRppvDv uQΛ pppp Q DQΛ Qu uΛu ii u (4.42) i where the diagonal elements i of the diagonal matrix Λ are the eigenvalues of
1/2H 1/2 the matrix Λp Q p DQΛ p p .
Equation (4.41) and (4.42) show that the values of the test statistics TR under H0 and H1 have generalized chi-squared distributions with two different sets of weighted parameters N/ b and i . The ROC of the receiver can be easily determined from the two PDFs of TR under H0 and H1 . Finally, the threshold can also be determined based on the given probability of false alarm
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 113
Pfa and the PDF of TR under H0 . Like the asymptotical PDF of TR , the exact
PDFs of TR under H0 can be obtained without information of the covariance
C C H matrix v p (extracted from the covariance matrix v ). However, under 1 , the
T C asymptotical and the exact PDFs of R can only be obtained with known v p . It should be emphasized that using the Rao-based test statistic only needs
2M 2N 2M 1 1 2 additions and multiplications, while the ACM-based GLRT b i
M M2 M 2 M 2 2 2 approach needs N2 2 3 N multiplications. Without having to deal with the large matrix manipulation, our proposed Rao-based test significantly reduces the computational complexity while maintaining its good performance as shown in the simulation.
4.4 Simulation results
In this Section, the simulation results are presented to demonstrate the detection ability of the Rao-based test for WPM systems with unknown covariance matrix over ACM-based approach with known covariance matrix and the GLRT-based with unknown covariance matrix.
As in Chapter 3, Pfa and Pd would be used to evaluate the detection performance of the methods. Throughout the simulations, the input signals of the
IDWT at the transmitter are composed of independent and identically distributed (i.i.d.) 16-QAM symbols. The basis function used in the IDWT and
DWT at the transceiver is the conventional Daubechies with N=6 (i.e., db6) [10].
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 114
The distances between pilots in the time-domain and wavelet-domain are chosen to be 4 (i.e., b 4 ), so that 6.25% of the input data is the pilot signal.
Moreover, the number of sub-channels is set to 64 (i.e., M=6). Furthermore, for each burst of data, there are 32 blocks of data, each block containing 64 16-QAM
M symbols (i.e., 2 64 ). With a bandwidth of 20 MHz, the sensing time is 0.1024 milliseconds.
1
0.9
0.8
0.7
0.6
d 0.5 P
0.4 ACM-based method Optimal method 0.3 unknown C GLRT a unknown C Rao 0.2 a Energy detector 0.1
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P fa
Figure 4.1 ROC curves for several methods in
frequency-selective channel at SNR = -15 dB.
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 115
With longer low-pass and high-pass filters in the QMF pairs at the transmitter and receiver, the advantage of WPM signal over OFDM signal is higher sub-channel spectral containment. However, the complexity of the transceiver also increases and the channel is assumed to be static for a longer interval. In our simulations, the lengths of those filters are set to 12 in order to compromise these constraints. A wireless channel with an exponentially decaying power delay profile and a root mean square delay spread of 50 ns is assumed. All the simulation results in this chapter are obtained by using Monte Carlo method with 214 iterations.
The ROC results for WPM systems with an SNR of -15 dB for several methods are shown in Fig. 4.1. Here, we compare the ROC curve corresponding to the GLRT-based method in Chapter 3 to that corresponding to the proposed
Rao-based method where the covariance matrix of the primary signal is unknown. We also compare them with those corresponding to the optimal and the ACM-based methods with known covariance matrix Ca as “upper thresholds”. The simulation result shows negligible performance difference between the Rao-based and GLRT-based methods, as expected in [34]. For instance, when Pfa 0.01 and Pfa 0.02 , the proposed Rao-based method achieves Pd 0.249 and Pd 0.31 where the GLRT-based test achieves Pd 0.251 and Pd 0.311. It should be noted that the Rao-based approach only needs 0.4% of the number of multiplications of the GLRT-based test, while maintaining
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 116 negligible performance degradation compared to that of the GLRT-based. From
Fig. 4.1 and simulation results in Chapter 3, it is also shown that the proposed
Rao-based approach outperforms some traditional methods
Next the probability of detection Pd performances of the Rao-based method and different methods, as a function of SNR, are shown in Fig. 4.2, for
Pfa 0.01. Figure 4.2 also shows that the proposed Rao-based test statistic maintains good detection performance compared to GLRT-based test statistic and outperforms traditional ED method. At a target of Pd 0.8 , the proposed
0 10
-1 Optimal method d 10 P ACM-based method unknown C a unknown C Rao a Energy detector
-2 10 -35 -30 -25 -20 -15 -10 -5 0 5 SNR
Figure 4.2 The detection performances of different methods for Pfa 0.01
at SNR = -15 dB in frequency-selective channel.
Chapter 4: Spectrum sensing for WPM systems based on Rao’s score test 117
Rao-based test, with unknown covariance matrix Ca , exhibits SNR losses of -
2.066 dB, -2 dB, -0.086 dB over the optimal with known Ca , ACM-based with known Ca , GLRT-based with unknown Ca , respectively and SNR gain of 3.708 dB over the traditional energy detector.
4.5 Conclusion
In this chapter, a novel spectrum sensing method for WPM system using pilot signal with unknown covariance matrix is proposed based on the Rao test.
First, the motivation for proposing alternative approaches to the GLRT-based test is presented. Next, we derive a Rao-based spectrum sensing test to overcome the disadvantages of GLRT-based methods discussed in the previous chapter.
The performance of the proposed Rao-test method is compared to those of the conventional methods. Analysis and simulation results show that this new method achieves negligible degradation performance while significantly reducing computational complexity compared to the GLRT-based method with unknown covariance matrices.
Chapter 5: Sequential spectrum sensing for WPM systems 118
CHAPTER 5
SEQUENTIAL SPECTRUM SENSING
FOR WPM SYSTEMS
In cognitive radio (CR) system, only primary users (PUs) are licensed to use the spectral bands. To avoid interfering with the PUs, the secondary users
(SUs) must sense the channel before accessing the spectral bands. If the sensing time is too short, the detection performance at the SUs may be too poor and can result in high probability of false alarm or high probability of miss detection. The high probability of false alarm causes interference with the PUs while the high probability of miss detection reduces the chance of the SUs to transmit data. If the sensing time is too long, the time slot of the SUs for transmitting data becomes short, that is, the spectral efficiency of the CR system is seriously degraded. As a consequence, the sensing time must be carefully designed to improve the effectiveness of the cognitive radio systems while maintaining the quality of the SUs’ network.
In many traditional detection methods, the sensing time is pre-determined, regardless of the status of fading channel or the confidence of the test statistic. In other words, the test statistic is computed based on a fix number of samples, and
Chapter 5: Sequential spectrum sensing for WPM systems 119 then a hypothesis is determined based on the test statistic. However, sometimes, the decision can be made with fewer samples. This happens when the test statistic obtained from some initial samples provides enough confidence to decide the hypothesis.
In the previous chapter, we proposed some spectrum sensing methods based on an approximated covariance matrix (ACM) and Rao’s score test. In both of them, the sensing times are pre-determined and cannot adapt to various scenarios. To overcome this problem, we propose new sequential approaches which are based on the ACM method in Chapter 3. The stopping rule in these sequential techniques is a simple scheme based on thresholding. The hypothesis
H0 is accepted when the test statistic is too small while the hypothesis H1 is accepted when the test statistic is too large; otherwise the SUs’ receiver continues monitoring until one of the hypotheses is accepted. The novel method, proposed in this chapter, uses a variable number of samples in order to maintain the overall performance while maintaining a low average number of required samples.
In other words, the sensing time required for the same detection performance is significantly reduced.
This chapter is organized as follows. Firstly, an introduction and literature review about sequential probability ratio test (SPRT) are presented in Section
5.1. Next, some traditional SS methods for deterministic signals and random signals are reviewed in Section 5.2. Then, the new sequential SS techniques for
Chapter 5: Sequential spectrum sensing for WPM systems 120
WPM system are described in Section 5.3, followed by the temporal update recursion for the test statistic in Section 5.4. From the upper and lower bounds for the overall probabilities Pfa and Pmd in Section 5.5, two novel procedures to setup the parameters of the sensing method are described in Section 5.6. Finally, the simulation results of these approaches are presented in Section 5.7, followed by the conclusion in Section 5.8.
5.1 Sequential probability ratio test (SPRT) and its applications
Sequential probability ratio test, inspired by work of Neyman and
Pearson, is developed by A. Wald [198]-[200]. At any stage of the experiment, one of three decisions can be made, that is, accept the null hypothesis, reject the null hypothesis, and continue the experiment. This process is stopped when one of the hypotheses is chosen. Since the ratio test is a variable, the number of received signals required for the test is not pre-determined. In other words, the number of observation signals is a random variable since the ratio test depends on the observed samples. It has been shown that, in general, given the same detection performance (i.e., Pfa and Pd ), the average number of observation signals is considerably smaller than the fixed number of observation signals of the traditional most powerful test [200]. Specifically, SPRT only requires about 50% of the number of observation signals compared to the traditional most powerful test [198]. Assuming that the received signal is independent, it is interesting that
Chapter 5: Sequential spectrum sensing for WPM systems 121
the upper and lower thresholds used in Wald’s work depend only on Pfa and Pd as follows:
1 upper (5.1) lower 1 where and are the false alarm probability and the miss detection probability, respectively.
It can be seen that this test can be realized with no prior information about the probability distribution function of received signal under both hypotheses H0 and H1 . In addition, the computational complexity of this test is very low, which comes from the fact that the thresholds do not depend on the moment the test is performed. From the work of Wald, L. Aroian developed a direct way to obtain the receiver operating characteristic (ROC) and the average sample number (ASN) [201].
Several methods related to optimizing sensing time in CR networks have recently been studied [202]-[207]. The received signals at the SUs are modeled by an autoregressive moving average (ARMA) process, assuming that the coefficients are unknown [202]. Using this model, the sequential likelihood ratio test is derived to determine the hypotheses. In [203], Emad S. Hassan analyzes the trade-off between the sensing reliability and power efficiency in cooperative cognitive radio networks over fading channels. The drawback of this method is that it does not consider the false alarm probability, which may limit the chance
Chapter 5: Sequential spectrum sensing for WPM systems 122 of the SUs to use the licensed spectrum. In [204]-[206], the sensing time and other parameters such as transmission time and energy detector threshold are optimized to maximize the throughput while maintaining the interference below a certain level. The sensing time is also optimized to solve the trade-off between sensing reliability and power efficiency in cooperative cognitive radio [207].
Applications of sequential tests in cognitive radio systems have been well studied in the literature (see [208]-[209] and references therein). In [208], a blind wideband spectrum sensing method is proposed, which assumes that interference may appear and disappear on any channel at any time instant. In [210]-[211], sequential detection is applied in cooperative sensing to reduce the sensing time.
In [210], the unknown parameters, such as signal strength and noise variance, are sequentially estimated using MLE, and then the sequential detection is performed based on these estimated values. In [211], the local detectors also take the measurements, but they transmit the information to a fusion centre where the sequential test is used. Sequential and cooperative sensing is also used for multi- channel cognitive radios to maximize the effective average data rate of CR transmitters [212]. Energy and cyclostationarity-based tests are combined with sequential detection to track state dynamics of communication channels [213]. It is shown that this method achieves significant improvements in terms of small detection delays and higher spectral utilization. Sequential tests can be combined with cooperative compressed sensing, cyclic prefix-based autocorrelation
Chapter 5: Sequential spectrum sensing for WPM systems 123 detection, fuzzy logic, with weighted energy detector [214], higher-order statistics- based tests, or suprathreshold stochastic resonance (SSN)-based tests [215].
Sequential tests can also be applied to control the average number of reporting bits, to gain efficient use of the frequency band for the control channels, and to decide between multiple hypotheses. The reputation of SUs is used with sequential test at the fusion centre to reduce the number of sensing reports as well as improve sensing performance [216].
In this chapter, we develop inequalities for the detection performance of the test (i.e., Pfa and Pd ), propose a procedure to set up the upper and lower thresholds. We also proposed a sequential detection test based on the thresholds which are designed using another approach. It is interesting that similar to the result of Wald, under some assumptions, the thresholds at different stages do not depend on “the number/order of the stage”. This surprising result helps to significantly reduce the computational complexity. Simulation results show that the novel methods proposed in this chapter reduce the sample size required to meet a specified reliability target. These approaches can be applied for any test statistic whose probability density functions (PDFs) under both hypotheses H0 and H1 , with different number of samples, are known. In this chapter, we consider the detection problem for both deterministic signals and random signals under low SNR regimes, in which the PDFs of the test statistic for the deterministic signals under H0 and H1 are given [34]. The random signals, which
Chapter 5: Sequential spectrum sensing for WPM systems 124 are derived from the previous chapter, are also considered in this chapter. It should be noted that our work is different from that of A. Wald [198]-[200] as follows:
It is not necessary for the received signal to be independent.
The detection process must be stopped before a pre-defined stage.
5.2 Some traditional detection methods for deterministic signals and random signals
5.2.1 Matched filter for deterministic signals
In this case the problem is to distinguish between two hypotheses H0 and
H1 as follows
H0 :x = w no transmitted signal (5.2) H1 :x = s + w transmitted signal, where s is a known deterministic signal vector, w is white Gaussian noise
(WGN) vector, and x is the received signal vector. The test statistic for the matched filter for deterministic signals is as follows [34]:
N 1 T x n s n (5.3) n0
The detection performance can be found by noting that
N 1 2 2 N0, s n under H0 n0 T (5.4) N1 N 1 N s2 n, 2 s 2 n under H 1 n0 n 0
Chapter 5: Sequential spectrum sensing for WPM systems 125
For a fixed number of samples, the probability of false alarm and the probability of detection can be expressed as follows:
P Q fa N 1 2 2 s n n0 (5.5) N 1 s2 n P Qn0 D N 1 2 2 s n n0 where is the threshold used to decide the hypotheses, and
1 2 Q x exp t / 2 dt (5.6) x 2
5.2.2 Estimator-Correlator for random signals
The detection problem is to distinguish between the hypotheses H0 and
H1 as follows
H0 :x = w no transmitted signal (5.7) H1 :x = s + w transmitted signal, where s is zero-mean, Gaussian random process with known covariance matrix
Cs , w is a white Gaussian noise (WGN) vector, and x is the received signal vector. The conditional distributions are
N 0I,2 under H 0 x (5.8) N 0C, 2 I under H s 1
Chapter 5: Sequential spectrum sensing for WPM systems 126
The Neyman-Pearson detector for random signals is the Estimator-
Correlator, where the test statistic is given as follows:
1 T 2 T x Cs C s I x (5.9)
1 2 Denote CCCs s I . It is easy to show that C is a positive- definitive matrix. Equation (5.9) is rewritten as
T T x Cx (5.10)
2 When the SNR is very low as in the previous chapter, C Cs . When the matrix C is known, the PDFs and cumulative distribution functions (CDFs) of the test statistic T under H0 and H1 can be obtained as in the previous chapters.
5.3 The novel sequential spectrum sensing method for WPM systems
T N The procedure based on the optimal test statistic Ni using the first i
th blocks of samples at the i stage for distinguishing between these hypotheses is given by
Ni 1: decide H 1 T Ni : decide H Ni 0 0 (5.11) otherwise : continue
Ni Ni T where 1 and 0 are two thresholds for Ni . The procedure starts with a few blocks of signals. The number of blocks increases until the final stage (i.e.,
NN1 2 .... Nend ), or until we are confident enough about the status of the channels. Since our test statistic for random signals is based on the pilot signals,
Chapter 5: Sequential spectrum sensing for WPM systems 127
the number of blocks Ni should be divided by b , the distance between two consecutive WPM blocks containing pilot signals. For deterministic signals, there is no constraint on the number of blocks Ni .
Equivalently, from Eq.(5.11), the thresholds must satisfy the inequality
Ni N i 1 0 for all the values of Ni , where the equality holds only at the final stage
Ni Ni Ni Nend . These thresholds 1 and 0 determine the false alarm p fa and miss
Ni th detection pmd probabilities at the i stage, respectively, which in turn determine the overall Pfa and Pmd of the method. There is a trade-off between the sensing
Ni time and the overall performance of this method. The higher the thresholds 1
Ni or the lower the thresholds 0 , the lower the false detection and the miss
Ni Ni detection ( Pfa and Pmd ), the better the overall performance and the higher the average number of blocks needed to sense (the longer the sensing time) and vice versa.
Ni Ni The probabilities p fa and pmd at stage Ni can be defined in terms of following conditional probabilities
pNiPr T N i | H fa Ni 1 0 (5.12) pNiPr T N i | H md Ni 0 1
It should be noted that from the procedure of this method, the correct detection probabilities satisfy the following inequalities
Ni N i pd1 p md
Ni N i (5.13) p0 0 1 p fa
Chapter 5: Sequential spectrum sensing for WPM systems 128
Ni Ni where the equalities hold only at the final stage Nend ; pd and p0 0 denote the correct detection probability at stage Ni under H0 and H1 , respectively. The
Ni inequalities in (5.13) come from the fact that we choose the thresholds 1 and
Ni Ni N i 0 to satisfy 1 0 for all i .
5.4 The temporal update recursion for the test statistic
th When the detection procedure continues to the i stage, the test statistic
T T Ni can be computed based on the previous test statistic Ni1 as in the next Sub- section.
5.4.1 Deterministic signals
T For deterministic signals, from Eq.(5.3), it can be easily shown that Ni can be updated as follows
Ni 1 T T x n s n Ni N i1 (5.14) n Ni1
5.4.2 Random signals
For random signals, the test statistic can be updated using the following steps
T temp Step 1: Initialize the test statistic Ni and set the temporary variables and
temp T to 0.
temp Step 2: At stage i , update the temporary variables temp and T as follows
Ni 1 tempnew temp old v Ni1 Sb (5.15) temp H Tnew temp new** C temp new
Chapter 5: Sequential spectrum sensing for WPM systems 129
C p, Sb , C a where 2 .
T Step 3: Compute the test statistic Ni based on the new temporary variables as follows
2N12M 1 2 N 1 i2 i 2 TTs v s C d, Sb , vTT temp temp Ni N i1 2 2 a new old (5.16) Ni10 S i N i 1 Sb
T Ni Ni Step 4: Compare the test statistic Ni with the two thresholds 1 and 0 as in
(5.11) to continue to the next step or to decide the hypothesis H0 or H1 .
Step 5: Prepare for the next stage as follows
tempold temp new temp temp (5.17) Told T new
Step 6: Go back to step 2.
These temporal update recursions compute the test statistic for the next
Nend N end stage and help reduce the overall complexity. Since 1 0 , this procedure must terminate at stage Nend at the latest.
It should be emphasized that temporal update recursion for the test
T statistic Ni helps to significantly reduce the computational complexity. For deterministic signals, the recursions reduce from 3 Ni real multiplications of
T 3N the normal update procedure for Ni to end real multiplications of the recursive
T update procedure for Ni . For random signals, the recursions reduce from
2 2 M M M M M 1 2 2 M 1 2 2 2Ni 3 6 n l 2N 3 6 nl to real multiplications
Chapter 5: Sequential spectrum sensing for WPM systems 130
of the of the update procedure, where nl is the number of stages. This computational complexity reduction helps the procedure to be more suitable for real-time applications. Instead of dealing with large matrices, this approach updates the test statistic regularly and has low computational complexity.
P P 5.5 Bounds for the overall probabilities fa and md
This Section is based on the PDF f and the cumulative distribution
N N F T P i P i P Ni function (CDF) of the test statistic Ni . Let fa , con , and md denote the
th false alarm, continuity, and miss detection probabilities after the Ni stage of the procedure, respectively, and defined as follows:
N N PNiPrj T j jiT 1, 1 , N i H fa 0 Nj 1 N i 1 0 N N PNi Prj T j ji 1, , under HH or con0 N j 1 0 1 (5.18) N N PNiPrj T j jiT 1, 1 , N i H md0 Nj 1 N i 0 1
The correct detection probabilities are defined as follows
N N PNiPrj T j jiTH 1, 1 , N i 0 0 0Nj 1 N i 0 0 N N (5.19) PNi Prj T j jiTH 1, 1 , N i d0 Nj 1 N i 1 1
From the definitions in (5.18) and (5.19), it is easy to derive the following relationship between these probabilities:
PNi1 P N i 1 P N i 1 H Ni 0 0con fa 0 Pcon Ni1 N i 1 N i 1 (5.20) Pmd P con P d H1
The overall false alarm and miss detection probability are the sum of the false alarm and miss detection probabilities at all the stages, i.e.
Chapter 5: Sequential spectrum sensing for WPM systems 131
Ni Pfa P fa i
Ni (5.21) Pmd P md i
It is noted that the following equation, Eq. (5.22), holds only for the first stage, since at the higher stages, there are some cases which are decided in the previous stages.
PN11 F N 1 fa N1 1 PFNi N1 F N 1 con N11 N 1 0 (5.22) PNi F N1 md N1 0
F T where N1 is the CDF of the test statistic N1 .
Ni Ni It should be noted that the probabilities p fa and pmd in Eq. (5.12) and the correct detection probabilities in Eq. (5.13) can be determined based on the
T CDFs of the optimal test statistic Ni . However, the probabilities in Eqs. (5.18) and (5.19) are intractable. Therefore, we shall develop upper bounds and lower bounds for them. The procedure to setup the parameters of the sensing method will be introduced to meet the requirements (i.e. the overall probabilities Pfa and
Pmd ).
F t Ti It can be shown that the conditional CDF Ti T j of given that
T i j 0 F t j where is always greater than the unconditional CDF Ti , that is,
F tFt T Ti T j i (5.23)
Similarly, the following inequality is introduced
Chapter 5: Sequential spectrum sensing for WPM systems 132
F tFt T Ti T j i (5.24)
Ni1 Ni Replacing t and by 1 and 0 into Eq. (5.23) under H0 , respectively, the inequality in Eq. (5.23) becomes
Ni N i Pcon P fa PpNNii1 1 P N i fa faNi1 fa (5.25) Pcon
Ni1 Ni Similarly, replacing t and by 1 and 1 into Eq. (5.24) under H0 , respectively, the inequality in Eq. (5.24) becomes
Ni N i Pcon P0 0 PNi1 p N i 1 fa fa Ni1 (5.26) Pcon
From the definitions in (5.18), it is obvious that
Ni1 N i Pfa P con (5.27)
Combining the inequalities in (5.25)-(5.27), the upper and lower bounds
Ni1 for Pfa can be obtained as follows
NNii NN ii PPcon fa PP con 0 0 Ni1 NN ii 1 NN ii 1 pfaN P fa P fa min P con , p fa N (5.28) Pi1 P i 1 con con
Ni1 Similarly, the following upper and lower bounds for P0 0 can be obtained
NNii NN ii PPcon0 0 PP con fa Ni1 NN ii 1 NN ii 1 p0 0N P 0 0 P 0 0 min Pcon , p 0 0 N (5.29) Pi1 P i 1 con con
Ni1 Ni1 Under H1 , the following upper and lower bounds for Pmd , and Pd can be obtained
Chapter 5: Sequential spectrum sensing for WPM systems 133
NNii NN ii PPcon d PP con md Ni1 NN ii 1 NN ii 1 pd P d P d min P con , p d PNi1 P N i 1 con con (5.30) NNii NN ii PPcon md PP con d Ni1 NN ii 1 NN ii 1 pmd P0 0 P md min P con , p 0 0 PNi1 P N i 1 con con
Ni Ni Assuming that all of the thresholds 1 and 0 are known, the upper
Ni Ni bounds and lower bounds of p fa and pmd can be computed based on the
Nend inequalities in Eq. (5.28)-(5.30), and finally those of the probabilities p fa and
Nend pmd can be obtained.
5.6 Procedures to setup the parameters of the sequential sensing
P P method when the overall probabilities fa and md are given
In this Section, we propose two procedures to setup the parameters of the sensing method.
5.6.1 Procedure 1
In the first approach, from the bounds for the overall probabilities Pfa and
Pmd derived in the previous Section, the following procedure is introduced to choose the thresholds for stages.
Step 1: First the minimum number of blocks N fix for the fixed-length
T method is obtained based on the PDFs/CDFs of the test statistic N fix which are described in the previous Section.
Step 2: The numbers of blocks N i can be chosen. Obviously, the number of blocks at the last stage Nend can be chosen to be equal to N fix , that is
Chapter 5: Sequential spectrum sensing for WPM systems 134
Nend N fix . The more the number of stages, the sooner the sensing procedure
Nend Nend stops for a given particular target p fa and pmd . However, the computational complexity also increases.
Ni Ni Step 3: Next, the thresholds 1 and 0 are designed from the first stage to the last stage such that the upper bounds of the overall Pfa and Pmd do not exceed the given target. If one of the upper bounds exceeds the target, we can choose the thresholds at the previous stages or repeat step 2.
Besides the overall Pfa and Pmd of the method, another metric we need to evaluate is the sensing time, which is proportional to the number of blocks needed for sensing. The average number of blocks can be expressed as follows:
Ni N i NPi fa P0 0 H 0 i Naverage (5.31) Ni N i NPi md P d H1 i
Ni1 Ni1 Ni1 Ni1 From the inequalities for Pfa , P0 0 , Pmd , and Pd in Eq. (5.28)-(5.30), the upper and lower bounds for Naverage can be obtained, assuming that all the
Ni Ni thresholds 1 and 0 are selected. From the way we choose N i (i.e.
NN1 2 .... Nend N fix ), it is easy to show that the average number of blocks needed for the same performance is always less than the number of blocks for the fixed-length method. The longer we observe the received signals, the more information we have to decide about the hypothesis. Consequently, the thresholds
Ni Ni NiNj N i N j 1 and 0 are chosen such that 1 1, 0 0 i j . Using this
Chapter 5: Sequential spectrum sensing for WPM systems 135 approach, the sensing process can achieve a given target with the smaller average number of blocks.
It should be noted that, given the covariance matrix C , all of the previous procedures are independent of the received signal stream. Hence, all of the thresholds can be designed offline, which makes this method more practical.
Next, we consider some special cases of the covariance matrix C of random signals. When the matrix C is diagonal, it is easy to see that the temporal update recursion and the performance analysis are the same as those of the deterministic signal detection. When the covariance matrix C is a Toeplitz matrix, we can treat it as a special case of the generalized Toeplitz matrix.
5.6.2 Procedure 2
In this Sub-section, we propose another approach to the design of the
T thresholds for the test statistics Ni of the random signals only, for all the stages
Nend but the last one ( 1 is determined by the pre-defined target performances already). Consider the last stage with Nend samples, to decide the hypothesis, we
T Nend N end recall that the test statistic Nend is compared to the thresholds 1 0 . With
T N N the test statistic Ni from a few samples i end , sometimes we ensure whether
T Nend N end T T Ni Nend 1 0 if Ni is large enough ( Ni 1 ). This is different from the previous approach in Sub-section 5.6.1 where the hypothesis is determined from
T T Nend N end Ni when we are almost certain whether Nend 1 0 or not. In other
T Ni T Nend N end words, in this new approach, event Ni 1 implies event Nend 1 0 ,
Chapter 5: Sequential spectrum sensing for WPM systems 136
T Ni whereas in the previous approach in Sub-section 5.6.1, event Ni 1 only tells
T Nend N end us that the probability of event Nend 1 0 is large.
Ni T N N To derive the threshold 1 of the test statistic Ni , the end end
C covariance matrix Nend N end in Eq.(5.14), is partitioned as follows:
CNN C12 iNN C C 12 1 C 0 C ii ii i NNi i Nend N end H H (5.32) CC12 22 C 12 C 22 0 0 C i where i is a coefficient belonging to [0,1].
T The test statistic Nend can then be expressed as follows
C C T TT Ni N i 12 x1 TNx C NN x x1 x 2 end end end H x (5.33) C12 C 22 2 where x1 is the first Ni samples and x 2 is the remaining ( Nend N i samples).
T Substituting Eq. (5.32) into Eq. (5.33), the test statistic Nend can be rewritten as follows
C C T1 xT C x x T i NNi i 12 x Nend iNN 1 i i 1 H C12 C 22 (5.34) 1 TxT C x i Ni i
C If the coefficient i is a value such that i is a positive semi-definitive
xT C x x matrix, the second term i in Eq. (5.34) is non negative for all values of .
T T Nend As a consequence, to ensure that Nend is greater than the threshold Nend 1 ,
Ni we can set up the threshold 1 as follows:
Ni1 N end 1 1 (5.35) 1i
Chapter 5: Sequential spectrum sensing for WPM systems 137
T It is recalled that from the CDF of Nend , given the target performances,
Nend 1 can be easily determined. When the coefficient i is obtained, the threshold
Ni T C 1 for Ni is obtained from Eq.(5.35). Within infinite values of i such that i is a positive semi-definitive matrix, the minimum value of i is chosen which
Ni results in the minimum value of 1 . Hence, given the targeted performance, the number of blocks needed for sensing, which is proportional to the sensing time, is minimized.
min It should be noted that all of the values of i for each stage are also independent of the received signal stream. Hence, similar to procedure 1, all of the thresholds can be designed offline and utilized even in real-time applications.
This fact makes this approach more practical.
C Next we study the positive semi-definiteness of the matrix i where i is treated as the sole variable, i 0,1, then we propose a procedure for
min C C determining the value of i . Consider two matrices i and i where i i
C and i is a positive semi-definitive matrix. From the Weyl's inequality in matrix theory, it is well-known that [217]
det AB det A (5.36) where A and B are two arbitrary positive semi-definitive matrices. Applying Eq.
(5.36) to
C 0 i i NNi i C C (5.37) i i 0 0
Chapter 5: Sequential spectrum sensing for WPM systems 138
detC det C we obtain i i . By considering the upper-left square sub-matrices
C C of i and i , the following inequalities are derived
detC 1 n det C 1 nnN 1, i i end (5.38)
C 1 n C1 n n n C where i and i are the upper-left sub-matrices of i and
C n i , respectively, which are obtained by only taking the first rows and the first n C C C columns of i and i . Given that i is a positive semi-definitive matrix, the right hand side of Eq. (5.38)is non-negative for all n1, Nend . Therefore, the left hand side of Eq. (5.38) is also non-negative for all n1, Nend , which results
C 1 0 C in the positive semi-definitive matrix i . In short, given i i , if i is
C a positive semi-definitive matrix, then so is i .
Consider a 2 2 submatrix of a positive semi-definitive matrix C by
th th taking the i and j rows and columns of C . Since every principal sub matrix is positive semi-definite, it is easy to derive that Cij CC ii jj ij, . Then we can
C conclude that i is not a positive semi-definitive matrix. In addition, i 0
C C iN end N end is a positive definitive matrix. i 1
From these two facts it can be concluded that for each value of
min Ni,1 NN i end , there exists a value i 0,1 such that
positive semi definitive if min C i i i min (5.39) not positive semi definitive if i i
Hence, for each value of Ni , we can use any root-finding method such as
min the bisection method or the interpolation method to find the value of i . An
Chapter 5: Sequential spectrum sensing for WPM systems 139
min interesting fact is that the differences between the values of i are small when
Ni is not too small or too big ( Ni/ N end is not too close to 0 and 1). Hence, to
min reduce the complexity of finding the value of i1 , we can choose the two initial
min min Nend values for root-finding method around i . From i and 1 , using Eq.
Ni (5.35), the threshold 1 is obtained.
min Next we consider the variation of i when Ni increases via evaluating
C i the eigenvalues and eigenvectors of i for different values of . For the sake of simplicity, we assume that the Nend N end matrix C is a Toeplitz matrix where the element Ci, j ci j is zero for all i, j such that i j m. We will show
min that if we choose Ni such that m Ni for all the values of i , then i are almost the same for all i . To prove it, let us consider two consecutive matrices
C C i and i1 and set both i and i1 to be . We are only concerned about the smallest eigenvalues of these matrices, which determines whether or not they are positive semi-definite, and the corresponding eigenvectors of these matrices.
Let us denote i1 and vi1 as the smallest eigenvalue and its
C v corresponding eigenvector of i1 , respectively. Then i1 and i1 satisfy the following equation
C v v i1 i1 ii 1 1 (5.40)
C The matrix i1 can be partitioned into four sub-matrices and the eigenvector vi1 can be partitioned into 2 sub-vectors as follows:
Chapter 5: Sequential spectrum sensing for WPM systems 140
top top A B vi1 v i 1 H bottom i1 bottom (5.41) B C vi1 v i 1 where ABD,, are Ni N i , NNi end N i , and Nend NN i end N i
top bottom matrices, respectively; vi1, v i 1 are Ni 1 and Nend N i 1 vectors, respectively; and Ni N i1 N i . Equation (5.41) can be rewritten as follow
top bottom top Avi1 Bv i 1 ii 1 v 1 H top bottom bottom (5.42) B vi1 Dv i 1 ii 1 v 1
top From the assumptions about the Nend N end matrix C , vi1 is very small
bottom compared to vi1 and can be negligible. Equation (5.42) can be rewritten as follow
bottom Bvi1 0 bottom bottom (5.43) Dvi1 ii 1 v 1
v C Similarly, i and i are the eigenvector and eigenvalue of i , respectively. Then vi and i satisfy the following equation
C v v i i ii (5.44)
C The matrix i can be partitioned into four sub-matrices as follow
1 D B C i (5.45) 1 1 BH A
ABD,, C where are sub-matrices of i1 , denotes the transpose matrix over the secondary diagonal. The eigenvector vi1 can be partitioned into 2 sub- vectors as follows:
Chapter 5: Sequential spectrum sensing for WPM systems 141
1 D B top top vi v i bottom i bottom (5.46) 1 1 v v BH A i i
top bottom where vi, v i are Nend N i 1 and Ni 1 vectors, respectively. Equation
(5.46) can be rewritten as follows:
1 Dvtop B v bottom v top i i ii (5.47) 1H top 1 bottom bottom B vi Av i ii v
bottom From the assumptions about the Nend N end matrix C , vi is also very
top small compared to vi and can be negligible. Equation (5.47) can be rewritten as follow
top top Dvi ii v
H top (5.48) B vi 0
Comparing Eq. (5.48) and Eq. (5.43), it is easy to see that for
C i i1 , the eigenvectors of i1 are the right-shifted versions of those of
C C C i ; and the eigenvalue i of i and the eigenvalue i1 of i1 are almost the
C same. It comes from the fact that from the definition of i in Eq. (5.32), it can
C C be seen that i1 is almost the shifted-version of i (shift right and down
N N C i1 i samples). As a result, if i is a semi-positive definite matrix, so is
C min i i1 . In other words, i are almost the same for all . From Eq. (5.35), the
Ni threshold 1 is also the same for all stages. This fact helps to avoid computing the thresholds for different stages, which in turn reduces the computational
Chapter 5: Sequential spectrum sensing for WPM systems 142 complexity. It is interesting that this result is similar to that of Wald [198]-[200], where the threshold does not depend on the number of sensing samples. However, it should be noted that the work of Wald is only proposed for independent received signals.
As the first approach, this second approach also helps reduce the sensing time while maintaining a given target. Compared to the first approach, this approach sets up the thresholds more strictly. Consequently, the sensing time of this method is longer than that of the general approach, but it is still shorter than that of the conventional methods. However, in contrast to the first approach, where we only derived the lower and upper bounds of the detection performance; in this method the pre-defined detection performance can be obtained exactly, as shown in the simulation results.
5.7 Simulation results
In this Section, we present simulation results to evaluate the performance of the proposed spectrum sensing method. We compare the performance of the proposed sequential methods to the performance of the conventional spectrum sensing methods (i.e., fixed length method), based on the same test statistic.
Throughout the simulations, the number of sub-channels is set to 64 (i.e.,
M 6 ). The lengths of the low-pass and high-pass filters of the conventional
Daubechies basis functions are set to 12. For the case of deterministic signals, s n are chosen as Acos 2 fn where all of the parameters A,, f are
Chapter 5: Sequential spectrum sensing for WPM systems 143 known. For the case of random signals, we use the same pilot pattern which is introduced in Chapter 3. The simulations are conducted by choosing the input signals of the IDWT block to be composed of independent and identically distributed 16-QAM symbols. The number of stages was set to 10, where the comparisons are made after every 100 blocks of signals. The test statistics used to detect deterministic and random signals are defined in Eq. (5.3) and Eq.(5.9), respectively. The 10-tap frequency-selective channel with an exponentially power delay profile is assumed. Moreover, the performance of the described methods in this Section is analyzed by using Monte Carlo methods of 220 iterations for both deterministic and random signals.
Figure 5.1 shows the performance results of the traditional and proposed methods on detecting the deterministic signals for different number of samples. In this figure, the probabilities of the hypotheses H0 and H1 are assumed to be the same (i.e., PH0 P H 1 1/ 2 ). Therefore, the error is defined as the arithmetic mean of the probability of false alarm and that of miss detection (that is, error Pfa P md / 2 ). When only 100 samples are used for sensing, there is no difference between the traditional and sequential approaches. Therefore, the performances are the same. It can be seen that the proposed method can achieve the same value of error while only requiring significantly smaller number of samples. From the performance curves in Fig. 5.1, it is noted that at a target error of 0.125, the traditional method needs 850 samples, while the average
Chapter 5: Sequential spectrum sensing for WPM systems 144 number of samples required for the proposed method is only 550. Due to the limitation of the sensing time, the detection procedure must be stopped before or when the SUs’ receiver gets 1000 samples. As a result, the total error of the proposed sequential technique asymptotically approaches the lowest total error of the traditional method (at 1000 samples). It should be noted that the proposed method only needs 888 samples to asymptotically achieve the best performance
(the lowest error) of the traditional methods needs 1000 samples. This performance can be obtained if the probabilities of false alarm and miss detection of all the previous stages are set to be small enough.
Performances of different methods on detection of deterministic signals 0.35 Proposed method Traditional method
0.3 md
P 0.25 2 + a f P = 0.2 rror er
0.15
0.1 100 200 300 400 500 600 700 800 900 1000 Average number of samples
Figure 5.1 Detection performances of sequential and non-sequential methods
for deterministic signals.
Chapter 5: Sequential spectrum sensing for WPM systems 145
6 0.9 5.6 5.2 4.7 4.2 0.85 3.6
3
0.8 2.4 d P 1.9 0.75 1.5
1.2 0.7 1
0.65 0.15 0.2 0.25 0.3 0.35 P fa
0.906 8.28.1 87.97.87.7 7.6 7.4 7.2 7.1 0.904 6.8
6.6 0.902
d 0.9 6.3 P
0.898
0.896
0.894
0.114 0.116 0.118 0.12 0.122 0.124 0.126 0.128 P fa
Figure 5.2 ROC performances of sequential and non-sequential methods
for deterministic signals with different number of blocks.
Chapter 5: Sequential spectrum sensing for WPM systems 146
Figure 5.2 shows the ROC performance of the traditional and proposed methods with different number of blocks for deterministic signals (each block contains 100 samples). Ten blue curves represent the ROC performances of the traditional method for ten values of number of blocks, that is,
Ni 100* ii , 1 10 . For the sake of clarify, the data is presented in two figures, where the top figure shows nine blue ROC curves corresponding to nine values of i , 1i 9 and the bottom figure shows two blue ROC curves corresponding to two values of i , 9i 10 . Each red point represents a ROC performance with its corresponding average number of blocks. Each set of upper and lower thresholds
Ni N i 1, 0 creates a red point in the figure. It is shown that the proposed method can achieve better ROC performance while utilizing smaller average number of
1
0.9
0.8
0.7
0.6
i 0.5
0.4
0.3
0.2
0.1
0 1 2 3 4 5 6 7 8 9 i
Figure 5.3 The coefficients i for different stages in sequential method
for random signals
Chapter 5: Sequential spectrum sensing for WPM systems 147 blocks. Specifically, the proposed method with an average required number of blocks of 3.1 blocks and 6.3 blocks can have ROC performance better than that of the traditional method with 4 blocks and 9 blocks, respectively. Due to the limitation of the sensing time, the sensing procedure must be stopped before or at stage 10. Figure 5.2 shows that the proposed sequential method asymptotically approaches the best ROC performance of the traditional method ( i 10 ) with smaller average number of samples.
Next, the coefficients i are shown in Fig. 5.3 for different stages for random signals. This figure shows that the values of i are almost the same (i.e.,
C Figure 5.4 Nine eigenvectors of i for nine different stages
in sequential method for random signals
Chapter 5: Sequential spectrum sensing for WPM systems 148
Performances of different methods on detection of random signals 0.25 Proposed method Traditional method
0.2 md
P 0.15 2 + a f P = 0.1 rror r er
0.05
0 100 200 300 400 500 600 700 800 900 1000 Average number of samples
Figure 5.5 Detection performances of sequential and non-sequential methods
for random signals
i ) for nine different stages corresponding to nine values of i , 1i 9 . The
C i nine eigenvectors of i for nine different values of are shown in Fig. 5.4. These nine eigenvectors correspond to the smallest eigenvalues of nine covariance
C matrices , which determines whether or not i is positive semi-definite matrix.
C These nine eigenvectors of nine matrices i are plotted left to right in the ascending order of i , 1i 9 . It is shown in this figure that these eigenvectors of
C C i1 are right-shifted versions of those of i . Figures 5.3 and 5.4 confirm the
C conclusion about the eigenvalues and the eigenvectors of i , which helps reduce the computational complexity of computing the thresholds for different stages.
Chapter 5: Sequential spectrum sensing for WPM systems 149
Figure 5.5 shows the performance results of the traditional and proposed methods in detecting the random signals for different number of samples. As deterministic signals, the probabilities of the hypotheses H0 and H1 are assumed to be the same (i.e., PH0 P H 1 1/ 2 ), and error Pfa P md / 2 . It can be seen that the proposed method can achieve the same error value while only requiring significantly smaller number of samples. From the performance curves in the figure, it is noted that at a target error of 0.05, the traditional method needs 520 samples, while the average number of samples required for the proposed method is only 320. Due to the limitation of the sensing time, the detection procedure must also be stopped before or when the SUs’ receiver gets
1000 samples for random signals. Similar to Fig. 5.1, Fig. 5.5 shows that the total error of the proposed sequential technique asymptotically approaches the lowest total error of the traditional method (at 1000 samples). It should be noted that the proposed method only needs 820 samples to asymptotically achieve the best performance (the lowest error) of the traditional method of 1000 samples. Similar to Fig. 5.1, this performance in Fig. 5.5 can be obtained if the probabilities of false alarm and miss detection of all the previous stages are set to be small enough.
Chapter 5: Sequential spectrum sensing for WPM systems 150
1 8.68.78.88.9 9.19.29.38.28.38.48.57.98.17.77.5 7.397.16.86.5 6.15.7 5.2 4.8 4.2
3.7
0.95 3.2
2.7
0.9 2.2
d 1.8 P
0.85
1.5
1.3 0.8
1.1
1 0.75
0 0.05 0.1 0.15 0.2 0.25 P fa
Figure 5.6 ROC performances of non-sequential method and the first approach
for random signals
Figure 5.6 shows the ROC performance of the traditional method and the proposed sequential method using procedure 1 with different number of blocks for random signals. Similar to Fig 5.2, ten blue curves represent the ROC performances of the traditional method for ten values of the number of blocks, that is, Ni 100* ii , 1 10 . Each red point represents a ROC performance with its corresponding average number of blocks. It is shown that the proposed method can achieve better ROC performance while utilizing smaller average number of blocks. Specifically, the proposed method with 2.7 and 5.2 blocks can
Chapter 5: Sequential spectrum sensing for WPM systems 151 have ROC performance better than those of the traditional method with 4 and 8 blocks, respectively.
Figure 5.7 shows the ROC performance of the traditional and proposed method using procedure 2 with different number of blocks for random signals.
For each given target performance ( Pfa and Pd ), the thresholds are computed and the average number of blocks are shown. It is shown that the proposed method can achieve the target ROC performance while utilizing smaller average number of blocks. Specifically, the proposed method with an average of 7.5, 8.2,
. ROC performance of the proposed SPRT method with different number of blocks 1 8.8 8.2 7.5 6.7 5.9
0.95 4.9
4
0.9 3 d P
0.85 2
0.8
1 0.75
0 0.05 0.1 0.15 0.2 P fa
Figure 5.7 ROC performances of non-sequential method and the second approach
for random signals
Chapter 5: Sequential spectrum sensing for WPM systems 152 and 8.8 blocks can achieve the same performance as those of the traditional method with 8, 9, and 10 blocks, respectively. It can be seen that for the first four stages, this method gains no reduction on the average number of required blocks. It comes from the fact that we do not have enough information to conclude about the presence or absence of the primary signals within only the first four blocks. When the receiver gets more blocks of samples, the hypothesis can be determined before the receiver gets all of 10 blocks. Therefore, the average number of blocks can be reduced compared to the traditional method. Since the way procedure 2 sets the thresholds is stricter than that of procedure 1, procedure 2 reduces the average number of samples less than that of procedure 1.
However, procedure 2 ensures the exact target performance of the detector, not an interval as in procedure 1. It is shown in Fig. 5.7 that all the points are on the
ROC curves, where the exact ROC performances are obtained, while all the ROC points in Fig. 5.6 are not. Moreover, the computational complexity of procedure 2 for finding the thresholds is much simpler than that of procedure 1. It should be noted that all of the thresholds can be designed offline for both procedures.
5.8 Chapter conclusion
In this chapter, we have proposed new spectrum sensing methods for
WPM systems based on a sequential probability ratio test. We first describe the proposed method and the update process for the test statistic for deterministic and random signals. Then the upper and lower bounds for the probability of false
Chapter 5: Sequential spectrum sensing for WPM systems 153 alarm and the probability of miss detection are derived. From those bounds, a procedure is proposed to setup the thresholds for the test statistic at different stages. The test is derived to achieve an overall probability of false alarm and an overall probability of miss detection, such that the average number of samples is minimized. Simulation results show that the proposed techniques can achieve the same target detection performance by using a significantly smaller average number of samples compared to non-sequential traditional SS methods. In other words, with the same average number of required samples, the proposed approaches offer better ROC performance when compared to the conventional SS methods. Furthermore, it is also shown that the thresholds are almost the same for different stages, which reduces significantly the computational complexity of computing the thresholds.
Chapter 6: Conclusion and future works 154
CHAPTER 6
CONCLUSION AND FUTURE WORKS
This chapter concludes the whole dissertation. Section 6.1 provides a dissertation summary and conclusions, followed by some suggestions for future work in Section 6.2.
6.1 Dissertation Summary and conclusions
The main objective of this dissertation is to design sub-optimal spectrum sensing at the SUs’ receivers for WPM-based cognitive radio multi-carrier systems in the low SNR regime under various constraints and analyze the detection performance of those proposed techniques. There are three main contributions in this work. The first contribution, presented in Chapter 3, establishes two SS methods based on the Accumulated Time-Domain and
Wavelet-Domain Symbol Cross-Correlation (ATDSC and AWDSC) for CR systems in frequency-selective channels. By exploiting the periodicity of the received signal, these two methods can sense the spectrum well when the number of pilot signals is large enough. Next, assuming partial knowledge of the pilot tone pattern, a novel method based on approximated covariance matrix was proposed. GLRT approaches for unknown parameters such as the covariance matrix and the noise variance are also proposed and evaluated. Simulation results show that the proposed spectrum sensing methods achieve superior performance
Chapter 6: Conclusion and future works 155 improvements over various comparatives, in terms of receiver operating characteristic (ROC) and Pd performance, and also work well in the low SNR regime. It is also shown that the missing information of the noise power leads to negligible degradation in detection performance. The second contribution in
Chapter 4 is the derivation of the test statistic based on Rao test for pilot- embedded WPM system with unknown covariance matrices. Under low SNR regime, Rao score’s test is the most powerful test since the PDF of the received signal under H1 is close to that under H0 . This proposed method is suitable since the covariance matrix does not need to be known or estimated. The detection performance assessment and the ROC performance of the novel test statistic are also presented in Chapter 4. Simulation results demonstrate its ability to detect
PUs’ signal with low computational complexity compared to the GLRT-based methods. In Chapter 5, to reduce the sensing time, the third contribution is the investigation of sequential probability ratio test (SPRT)-based method for WPM system. The update process is presented to reduce the computational complexity.
Based on the upper and lower bounds for Pfa and Pmd , the first procedure for setting the set of thresholds for different stages is proposed. By partitioning the covariance matrix, the second procedure is proposed for setting the set of thresholds. The two approaches help to reduce the average number of samples, which in turn improve spectral efficiency while maintaining the quality of the network. Simulation results show that these proposed methods significantly
Chapter 6: Conclusion and future works 156 reduces the number of required samples to satisfy the specified reliability target for both deterministic and random signals. The fact that the set of thresholds can be designed offline makes these approaches more practical. It is also shown that with pre-designed set of thresholds, these approaches work well with dependent primary signals. The advantage and disadvantage of the two procedures are discussed. It should be noted that the proposed methods in Chapter 5 can be used for both OFDM and WPM systems, while the novel techniques in Chapter 3 and Chapter 4 can only be used for WPM systems.
6.2 Future work
Due to some system model assumptions made in the analysis and simulation, several aspects of system designs are not considered in this dissertation. Consequently, the dissertation can be extended, in some ways, with the following suggestions for future work:
Throughout the dissertation, the channel is assumed to be slow, frequency-
selective fading where the channel remains static over the transmit
duration of several WPM blocks. The work presented in this dissertation
can be extended, to fast fading channels, where the channel remains static
over the transmit duration of one WPM block or varies within the
transmit duration of one WPM block.
In this dissertation we also assume that the synchronization at the receiver
is perfect. However, in a realistic scenario, those assumptions are
Chapter 6: Conclusion and future works 157
impossible. Thus, the theoretical and simulation analyses for WPM
systems can be realized with time offset estimation at the receiver. The
performance degradations arising from the non-ideality of channel
estimation can also be investigated through theoretical and simulation
analyses.
To increase the capacity and improve the quality of the WPM system,
WPM technique can be combined with multiple-input multiple-output
(MIMO) technique. Combining with these techniques, time, frequency, or
space diversities can be exploited to increase SNR at the receiver, to
mitigate fading, increase data rate within the requirement of bandwidth,
reduce interference by exploiting spatial diversity, etc.
In Chapter 3 and in Chapter 4, the GLRT-based and Rao-based
approaches still assume some known knowledge of primary signals such as
the signal frame and the wavelet basis functions. One can develop some SS
methods for various data structures and wavelet basis functions.
References 158
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Appendix 190
Appendix A
In this appendix, we derive the expression for the test statistic TACM . We start by first deriving the expression for the covariance matrix.
H HH Ca E aa E Hxx H (A.1)
Since H , p and d are independent, the covariance matrix Ca can be written as
HH H CaEE HxxH E HCH x
H H H EHCpd CH EE HCH p HCH d (A.2) p d Ca C a where CCCCa,,, x p d are the covariance matrices of the vectors axpd,,, , respectively.
As shown in Eq.(3.13), Cd is a diagonal matrix with all the diagonal
2 elements being equal to zero at the pilot positions, and s otherwise. Equation
th th i, j 2M 2 M n, n (3.13) also shows that the element of 1 2 sub-matrix
C C p n1, n 2 of the matrix p can be expressed as
C i, j if n , n S and i , j S C i, j p1 2 b i p n1, n 2 (A.3) 0 otherwise
Csub C n, n S C Let us denote by p the sub-matrix p n1, n 2 , 1 2 b , of the matrix p
th th p since it does not depend on its position n1, n 2 . The , sub-matrix Ca ,
p H of the matrix Ca E HCH p can be expressed as
Appendix 191
Cp E HCH subH a,,, n1 p n 2 (A.4) n1, n 2 Sb
th p where , 0..N 1. The , element of the matrix Ca , can be expressed as
Cp,,,, E HCH iij sub * j a,,, n1 p n 2 n,, n S i j S 1 2 b i (A.5) sub * Cp i, j E Vi, n 1 V j , n 2 n1,, n 2 Sb i j S i
M where , 0 : 2 1.
M It is noted that Vij n U ij 2 n is the down-sampled version of Uij n .
Substituting into Eq.(A.5), we obtain
**M M EVVij, n 1 , n 2 E U i , 2 n 1 U j , 2 n 2 (A.6)
It is noted that Unwniji nwnwnwn j i j n . Let
Wi, n w i n w j n be the total response of the i th sub-channel filter at the transmitter and the j th sub-channel filter at the receiver. Equation (A.6) can be rewritten as
L1 L 1 **M M EVij, n 1 V , nEW 2 i , 2 nkkW 1 * j , 2 nkk 2 k0 k 0 L1 2 M M ki*W, 2 nkW 1 * j , 2 nk 2 k0 (A.7)
L1 2 2 where k is the power of the k th tap of the wireless channel k 1 . k0
Substituting Eq. (A.7) into Eq. (A.5), we obtain
Appendix 192
L1 p2 M M Ca ,, C pki ij , * W , 2 nkW 1 * j , 2 nk 2 nnSijS1, 2b , i k 0
(A.8)
Because of the orthogonality of the WPM system and the fact that the
M number of channel paths L is small compared to the number of sub-channels 2 , the components of the matrix H are concentrated along the main diagonal. As a
p consequence, the value of the element Ca , , is only significant if , Sb
p and , Si . We approximate the matrix Ca by considering only the elements of
p the matrix Ca in these pilot positions.
Using that approximation, Eq. (A.8) can be rewritten as
L1 2 M M , Sb and C pij, ki * W, 2 nkW 1 * j , 2 nk 2 if p , S C , 0nnNijS1 , 2 ,i k 0 i a , b|n , n 1 2 0 otherwise
(A.9) where b |n1 , n 2 denotes that n1 and n2 are divisible by b .
p It should be noted that the value of the element Ca , , is
p, Sb independent of the value of and as long as , Sb . Let us denote by Ca
p the sub-matrices Ca , with , Sb , that is
p, Sb p Ca , if ,SSb and , i Ca , , (A.10) 0 otherwise
d H Next we derive the expression for the covariance matrix Ca E HCH d
CdHEE HCH H2 ICH HH 2 EE HH 2 HCH H ad s d s s d (A.11)
Appendix 193
where C d is the diagonal matrix in which its diagonal elements are 1s at pilot positions and 0s otherwise.
When the symbols of the input of the WPM transmitter with no pilot are
2 independent and have identical power s , the outputs ν of the DWT at the
2 receiver are also independent and have identical power s . This property of the
E HHH I WPM system results in N 2M . Equation (A.11) can be rewritten as
Cd2 I 2 E HCH H asN 2M s d (A.12) d C a
th d d The , sub-matrix C a , of the matrix C a is given by
d H Ca,,,, E HCH n d n n n (A.13) n Sb
th d The , element of the matrix C a , can be expressed as
d H Ca,,,,,,,, E HCH n i d n n ii n i n Sb i S i * EH,,n,, i H n i (A.14) n Sb i S i * EVi,, n V i n n Sb i S i
M Similarly, substituting Vij n U ij 2 n into Eq. (A.14), we obtain
L1 * 2 M M EVii,,,, n V n ki* W 2 nkW * i 2 nk (A.15) k0
p Similar to the matrix Ca , , for the sake of simplification, we neglect all
d th d the values of the matrix C a , except the , element of the matrix C a , where Sb and Si as follows
Appendix 194
L1 L 1 d2 2 M 2 2 M C a ,,,, ki *W 2 nk ki * Wnk 2 (A.16) nSiSkbi0 bniSk | i 0
d It should be noted that the value of C a , , does not depend on the
d, Sb WPM block index as long as Sb and Si . Let us denote by C a the
d sub-matrices C a , with Sb , that is
d, Sb d C a , if SSb and i C a , , (A.17) 0 otherwise
We can further simplify Eq. (A.17) by approximating the elements of the
d, Sb matrix C a . When Si , the filter response Wi, ni,, Si i can be
M considered to be zero. Usually the chosen number of sub-carriers 2 is much greater than the number of channel paths L . Furthermore, the channel response
L1 2 is normalized such that k 1. Under these assumptions, when i , the k0
d element C a , , can be approximated as 1. In short, the , th element of
d the sub-matrix C a , can be rewritten as
d 1 if SSb and i C a , , (A.18) 0 otherwise
p d d From the derivation of the expression of the matrices Ca , Ca and C a , the covariance matrix Ca of the vector a can be expressed as
p d p2 2 d CCCCa a a as I s C a (A.19)
Finally, using the approximated version of Ca , the test statistic Topt in Eq.
(3.16) becomes TACM as follows
Appendix 195
H2 H p 2 2 d 2 H p 2 H 2 2 H d 2 2 TACMvCva/ vC a ss I Cv a //// vCv a vv s vCv a s (A.20)
Using Eq. (A.10) and Eq. (A.17), the three items in Eq. (A.20) can be expanded as
H C p, Sb vCvHp/2 vCv Hp / 2 vCv H p, Sb / 2 v a v a a, a 2 ,,SSSSb b b b (A.21) p, Sb * Ca , v v 2 , SSSi b b
2 N 1 2M 1 2 vH 2 Iv / 2 s v s 2 (A.22) 0 0
2 2 2 2 d,,, S 2 d S d S 2 vCvHdss vCv H b sv C b ,, s C b v a2 2 a 2 a 2 a SSSSSb b i i b (A.23)
Substituting Eq. (A.21)-(A.23) into Eq. (A.20), we obtain
M 2N 1 2 1 p, Sb 2 2 C , 2 s* a s d, Sb TvvACM v C a , v 2 2 2 0 0 , SSSSSi b b i b (A.24)
When Eq. (A.18) is valid, Eq. (A.24) can be further simplified as
2 p, Sb 2 s * Ca , TACM v v v 2 2 (A.25) SSSSSi or b , i b b