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UBC Theses and Dissertations

Performance of collaborative spectrum sensing in a cognitive radio system Wang, Geng 2009

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Performance of Collaborative Spectrum Sensing in a Cognitive Radio System by Geng Wang B.Sc., Queen’s University, Canada, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  April 2009  ©  Geng Wang, 2009  Abstract Cognitive radio (CR) is a novel approach to improving the spectral efficiency of licensed radio frequency bands by opportunistically accessing unused portions of the band without introducing undue interference to a licensed user. To reliably identify unused portions in a dynamic environment, a collaborative spectrum sensing (CSS) approach is known to be advantageous. In this thesis, we investigate two issues related to CSS. A weighted energy fusion scheme for secondary users (SUs) with different sensing channel conditions is shown to achieve good sensing performance. To analyze the per formance, a numerical approach utilizing a result in the probability density function of the weighted sum of noncentral chi-square random variables is used. Simulation results confirm the viability of the proposed numerical approach. The performance degradation resulting from imperfect reporting channels and energy measurement quantization in a CSS system is investigated. Simulation results show that the sensing performance can be significantly degraded. To reduce the performance degradation, unequal error protection of transmitted symbols through unequal power allocation (UPA) is employed. Simulation results are provided to quantify the gain provided by UPA.  11  Table of Contents  Abstract Table of Contents List of Tables  V  List of Figures List of Abbreviations List of Symbols Acknowledgements 1  2  Introduction 1.1  Motivation and Goal  1.2  Contributions  1.3  Thesis Organization  Background 2.1  Spectrum Sensing  2.2  Energy Detector  2.3  Collaborative Spectrum Sensing  111  3  Performance Analysis of Weighted Gain Combining 3.1  Introduction  3.2  System Model  3.3  PDFs of Decision Statistic  3.4  Analysis of Truncation Error 3.4.1  0 Hypothesis. H  3.4.2  1 Hypothesis. H  3.5  General Comment  3.6  Performance Evaluation  4 Performance Analysis of Imperfect Reporting Channels  5  4.1  Introduction  4.2  System Model  4.3  Simulation Results  Conclusion  11  27  38  5.1  Main Thesis Contributions  39  5.2  Recommendations for Future Work  39  Bibliography  40  iv  List of Tables 3.1  PDF values and truncation errors for decision statistic, Z, under H 0 hypothesis, for K values of 20 and 25 (N  =  5, 2TW  =  10,  0 i  =  F= [1, 2, 3, 4, 5] dB) 3.2  18  1 PDF values, and truncation errors for decision statistic, Z, under H hypothesis, for K values of 20 and 25 (N P  4.1  =  =  5, 2TW  =  10, ,u  =  20  [1, 2, 3, 4, 5] dB)  Weight values w at different average SNRs  V  32  List of Figures 2.1  Block diagram of an energy detector  5  2.2  Collaborative spectrum sensing  8  3.1  Centralized collaborative spectrum sensing system  3.2  Plot of numerically obtained PDF against simulation histogram for 0 hypothesis (N Z under H I’  3.3  =  [1, 2, 3, 4, 5] dB, K  =  =  =  10, o  =  v/3, K  =  25, 19  25)  Plot of numerically obtained PDF against simulation histogram for 1 hypothesis (N Z under H F  3.4  5, 2TW  14  =  [1, 2, 3, 4, 5] dB, K  =  =  5, 2TW  =  10,  j =  v/3, K  =  25, 21  25)  Complementary ROC performance comparison between different num ber of SUs at a fixed system average SNR (N  =  [1, 2, 3, 5], P  =  3 23  dB) 3.5  Complementary ROC performance comparison between different sys tem average SNRs at a fixed number of SUs with each SU having a different SNR (N  3.6  =  5, F  =  [3, 4, 5] dB)  24  Complementary ROC performance comparison between different sys tem average SNRs at a fixed number of SUs with all SUs having the same SNR (N  =  5, F  =  [3, 4, 5] dB)  vi  25  3.7  Complementary ROC performance comparison between several fusion techniques (N = 5, I’  4.1  [1, 2, 2, 6, 7] dB)  26  System model for collaborative spectrum sensing with imperfect sens 29  ing and reporting channels 4.2  Complementary ROC curves for quantized energy measurements trans 3 = 10 dB, mitted over perfect and Rayleigh faded reporting channels (F Fr  31  10 dB)  4.3  Weight values w at different average SNRS  33  4.4  Complementary ROC curves for quantized energy measurements trans mitted over Rayleigh faded reporting channels with and without UPA 34  (L10dB,ir10dB) 4.5  Complementary ROC curves for quantized energy measurements trans mitted over Rayleigh faded reporting channels at different reporting 8 = 5 dB, channel average SNRS (F  4.6  r  = [3, 5, 7, 10, 121 dB)  35  Complementary ROC curves for quantized energy measurements trans mitted over Rayleigh faded reporting channels at different reporting 3 = 5 dB, channel average SNRs with and without UPA (F  r  [5, 101 36  dB)  vii  List of Abbreviations AWGN  Additive White Gaussian Noise  CR  Cognitive Radio  CSS  Collaborative Spectrum Sensing  DF  Decision Fusion  EF  Energy Fusion  EGC  Equal Gain Combining  FC  Fusion Centre  FCC  Federal Communications Commission  LRT  Likelihood Ratio Test  LSB  Least Significant Bit  MSB  Most Significant Bit  MSE  Mean Square Error  OSS  Opportunistic Spectrum Sharing  PDF  Probability Density Function  PU  Primary User  RF  Radio Frequency  RV  Random Variable  SNR  Signal to Noise Ratio  SU  Secondary User  UEP  Unequal Error Protection  viii  UPA  Unequal Power Allocation  WGC  Weighted Gain Combining  ix  List of Symbols r (t)  Received signal process at the input of an energy detector  s(t)  Primary user signal process  n(t)  Noise process  h  Amplitude gain of a channel noise variance  0 N  One-sided noise power spectral density  Y  Decision statistic at the output of an energy detector  f  Carrier frequency  T  Observation time inverval (second)  W  One-sided bandwidth (Hz)  TW  Time bandwidth product Instantaneous SNR for ith secondary user Average SNR for ith secondary user System average SNR Average sensing channel SNR Average reporting channel SNR  x  A central chi-square variate with c degrees of freedom A noncentral chi-square variate with a degrees of freedom and non-cent rality parameter 3 Decision threshold  x  0 H  Hypothesis 0 corresponding to PU inactive  1 H  Hypothesis 1 corresponding to PU active  Pd  Probability of detection of one SU  Pm  Probability of miss detection of one SU  Pf  Probability of false alarm of one SU  Pd,! adin 9  Average probability of detection in a fading channel PDF of instantaneous SNR  m Q  Probability of miss detection of CSS  Qf  Probability of false alarm of CSS Threshold for k-out-of-N fusion rule  Z  Decision statistics of energy fusion Energy measurement at ith SU Weight for ith SU  / weight  for ith bit in a bit vector  N  Number of SUs  M  Number of quantization bits  xi  Transmitted M-bit bit vector at ith SU Received M-bit bit vector at ith SU  xi  Acknowledgements First and foremost, I would like to express my sincere appreciation to Dr. Cyril Leung, who has provided me with the opportunity to pursue graduate studies in wireless communication. He has taught me the fundamentals of problem solving and provided me with invaluable suggestions and support during the course of my study. I would like to thank my aunt and uncle, Liying Liu and Erbin Dai for their guidance and support since the day I began my education in Canada. Finally, I would like to thank my parents, Lina Liu and Tongsheng Wang, who have supported me throughout my years of education. Without their endless love and immense encouragement, this thesis would never have been written.  GENG WANG  The University of British Columbia Vancouver, Canada April 2009  xii  Chapter 1 Introduction The electromagnetic radio spectrum is a valuable and limited natural resource. In traditional spectrum management, the radio spectrum is divided into bands which are allocated to different types of services and users with exclusive usage rights. With the rapidly growing use of wireless applications and the resulting increase for bandwidth resources, this fixed spectrum management scheme is problematic. A 2002 report by FCC (Federal Communications Commission) Spectrum Policy Task Force indicates that many portions of the spectrum is inefficiently utilized for significant periods of time [1]. A more efficient use of these “white spaces”, both temporally and geographically, could create opportunities for new technologies. One way to efficiently utilize the empty spectrum is by employing opportunistic spectrum sharing (OSS), which allows secondary users (SUs) to use a primary user (PU) band on a non-interference basis.  1.1  Motivation and Goal  To realize opportunistic spectrum sharing, a promising paradigm [2,3j is cognitive ra dio (CR). A CR is an intelligent radio system which performs three tasks: spectrum sensing, dynamic frequency allocation and transmission power control. Spectrum 1  sensing is the process for discovering spectrum holes which create transmission op portunities for a SU. A spectrum hole is a band of frequencies, assigned to a PU which is not being utilized by that PU at a particular time and specific location. The primary objective of spectrum sensing is to maximize a SU’s transmission opportunity while minimizing interference that may introduced to the PU by the SU. The PU signal received at the sensing device may be degraded by various channel impairments, such as fading and noise, making the task of reliable spectrum sens ing more challenging. To improve the performance, collaborative spectrum sensing (CSS) has been investigated [4—7]. The idea behind CSS is to allow multiple SUs to perform spectrum sensing and then make a collaborative decision regarding the activity state of the PU transmitter. Since it is unlikely that all SUs will experience the same channel degradation at the same time, CSS can substantially improve the performance. We consider a centralized CR system consisting of two or more SUs and a fusion centre (FC). The FC is responsible for making a final decision regarding the activity state of a PU based on the sensing reports from SUs and according to a certain fusion rule. Many different fusion rules have been proposed and studied [4—9]. There are two main types, decision fusion (DF) and energy fusion (EF). A decision fusion rule is employed at the FC when only binary decisions are reported by the SUs, whereas an energy fusion rule is employed when energy measurements are reported. CSS has attracted a lot of attention in recent research literature. In this the sis, we study two problems in CSS. In Chapter 3, we analyze the performance of the weighted gain combining energy fusion scheme in [7]. The performance analysis in  [71  is based on Monte Carlo simulation. We will use a simpler, non-fading, system  model to examine a numerical approach for analyzing the performance. In Chapter 4, we analyze the performance degradation of energy fusion based CSS with imper  2  feet reporting channels and explore the improvement achievable with unequal power allocation for reporting channel transmissions.  1.2  Contributions  The main contributions of the thesis are: • The application of a recent result from the statistics literature on the distri bution of the sum of weighted noncentral chi-square random variables which allows numerical evaluation of the sensing performance of weighted gain energy fusion in CSS. • The performance of CSS with quantized energy measurements reported over imperfect reporting channels is studied. The use of unequal power allocation (UPA) on quantized energy bits is investigated.  1.3  Thesis Organization  The thesis is organized as follows. In Chapter 2, we provide an overview of spectrum sensing and energy detector. CSS and various fusion techniques are also described. In Chapter 3, we study a SNR based weighted energy fusion scheme for collaborative spectrum sensing. We propose a numerical method for sensing performance analysis. In Chapter 4, we look at CSS with imperfect reporting channels. To improve detection performance, we examine the use of UPA on the reporting channels. In Chapter 5, we conclude the thesis with a list of main contributions. A few topics for further study are also suggested.  3  Chapter 2 Background 2.1  Spectrum Sensing  Spectrum sensing can be viewed as a binary hypothesis testing problem in which 1 indicates that hypothesis H 0 indicates that the PU is inactive whereas hypothesis H a PU is active. If we denote the signal received at a SU by r(t), we can write  r(t)  =  I n(t),  0 H  (2.1)  1 hs(t)+n(t), H where ri(t) is a noise process, h is the amplitude gain of the channel and .s(t) is the PU’s transmitted signal. The performance of spectrum sensing can be characterized by the probability of false alarm (Pf) and the probability of detection (Pd). The term Pf is the probability that a SU decides the PU is active when the PU is actually inactive. It reflects the level of missed access opportunity for the SU. The term Pd is the probability that a SU decides that the PU is active when the PU is actually active. The probability of miss detection Pm  =  1  —  Pd indicates the level of interference introduced to the PU  by a SU. Typically, Pm is restricted to be below an acceptable level to protect the PU. 4  Based on different prior knowledge about the PU signal characteristics, the most commonly used techniques for spectrum sensing include the matched filter detector, the cyclostationary feature detector and the energy detector [10]. A matched filter detector is known to be the optimum approach for PU signal detection if the signal characteristic is known. A cyclostationary feature detector explores the periodicity in the received signals to identify a PU’s state. However, this approach also requires prior information about the PU signal. For detection of unknown signals, an energy detector is often used.  2.2  Energy Detector  An energy detector makes a decision about the PU activity state by comparing an energy measurement (decision statistic) against a predefined threshold. A PU active is declared if the decision statistic is greater than the threshold. A block diagram of the energy detector is shown in Figure 2.1. The received signal process, r(t), is first pre-filtered using a bandpass filter of bandwidth W to remove out-of-band noise. The filter output is then squared and integrated over a time interval, T, to produce the decision statistic Y.  r(t)  Bandpass  (  TJ ( It  )2  Figure 2.1: Block diagram of an energy detector According to the sampling theorem, additive white Gaussian noise (AWGN) pro cess n(t) can be represented as [llj °°  sin(2irW(t i/2W)) 2irW(t i/2W) —  (t)  —  —  —  5  (2 2)  where  n  n() is the outcome of a Gaussian random variable with zero mean  and variance u 2  =  W and N 0 N 0 is the one-sided noise power spectral density. Over  the interval (0, T), the energy of  n(t)  can be approximated by a finite sum of 2TW  terms [11], i.e. fn2(t)dt  If we define ñ  (2.3)  the decision statistic Y at the energy detector output can be  =  written as [11] 2TW 2  (2.4)  The decision statistic under hypothesis H 0 follows a central chi-square distribution with 2TW degrees of freedom. For hypothesis H , the received process is 1 hs(t) + m(t) with  s  =  r(t)  =  The channel amplitude gain h is assumed to be  (*)  deterministic at this point. The decision statistic follows a noncentral chi-square distribution with 2TW degrees of freedom and a non-centrality parameter 21’, where 1’ is the instantaneous sensing channel SNR. A standard notation is  =  0 H  XTW,  [41 (2.5)  1 ), H 1’ X2TW(2  and the probability density function (PDF) of Y is given by [4]  f()  , 2 2TWF(TW)Ye =  (  0 H  (2.6)  1 ITw_l(/P), H  where F(.) denotes the gamma function [12], Ia(.) is the first kind modified Bessel function of degree c. The performance of an energy detector under AWGN channel can be character ized by the following conditional probabilities Pf  =  P(Y  )IHo)  (2.7)  Pd  =  P(Y  ) 1 AIH  (2.8)  6  where A denotes a decision threshold. Closed form expressions for Pf and Pd are derived in  [41, P(TW,) P(TW)  —  -  Pd  =  QTW(/f,  29 (.)  v5)  (2.10)  where F(a, b) is the upper incomplete gamma function [12] and Qa(b, c) is the gener alized Marcum Q-function [13]. With fading, the channel amplitude gain h in (2.1) is time-varying. If we assume that the gain does not change significantly during a sensing period T or over 2TW samples, the probability of detection can be obtained by averaging the instantaneous SNR-conditioned Pd (2.10) as d,fading =  f  P fr(x)dx  (2.11)  where fr(x) denotes the PDF of SNR. Since Pf is independent of the SNR, it is still given by (2.9). For Rayleigh fading, the instantaneous SNR P follows the exponential PDF fr(’y) where P  =  =  exp  (2.12)  0  E(P) is the average SNR. The average probability of detection can be  TW—2 =  e / X  —  [41  calculated from (2.10)-(2.12) as  2.3  (-),  I e  -Q)  +(11) TW—2  2(1+F) —  TW—1  — — — /  Al’ I n! \2(1+1’) it  (2.13) \fl  Collaborative Spectrum Sensing  The fading and noisy nature of a wireless communication channel places a major challenge in spectrum sensing. Since sensing decisions based on a single SU measure 7  ments may be unreliable, the idea of collaborative spectrum sensing has attracted a lot of research interest. A typical scenario for CSS is shown in Figure 2.2. The communication paths between the PU and the SUs may be subject to independent fading and the motivation behind CSS is that it is unlikely for all SUs to simultaneously experience a severe fade.  User 1  ((c  )))  Secondary User 2  Primary User Transmitter  Secondary User 3  Figure 2.2: Collaborative spectrum sensing CSS can be realized by using either a decentralized or a centralized CR system. In the decentralized case, the SUs form an ad-hoc system to exchange sensing in formation. A final decision is made at each SU based on an evaluation of its own sensing result as well as results received from other SUs. In this thesis, we focus on the centralized case, where individual sensing results are reported to a common FC which makes the final decision regarding PU activity. A SU may be required to make a local binary decision regarding the PU activ ity state and forward the decision to the FC. In this case, the FC uses a decision fusion rule to make a final decision. Alternatively, a SU may forward a local energy measurement to the FC, and the FC uses an energy fusion rule. When it is important to reduce reporting channel overhead, DF is usually pre 8  ferred. A general DF fusion rule is the k-out-of-N rule, where a final decision 1 is made when k-out-of-N SUs reported 1. When k to the OR rule. When k  =  =  1, the k-out-of-N rule is equivalent  N, the decision rule becomes the AND rule. By selecting  different values of k, different detection performances are obtained. For cognitive radio, the OR fusion rule is often chosen [5] since it provides maximum protection to the PU. For SUs with identical local spectrum sensing performances, i.e. Pf and Pd are the same for all SUs, the system probability of false alarm  Q  and system  probably of detection Qd can be calculated by [14]  ()  =  Qd  where  =  >  pk(  ()P(i  p)Nk  -  Pd)  (2.14)  (2.15)  represents the value of choice for k.  i  In energy fusion, the FC makes a final decision by comparing a weighted sum of the received local energy measurements against a predefined threshold. This decision process can be expressed as 1 H  N  (2.16)  where Y, i w, i  =  =  1, 2,.  .  .  ,  N, are the individual SU energy measurement reports, and  1,2,... , N, is the weight for Y. In the case where w is the same for all Y,  we have equal gain combining (EGC). The theoretical performance of ECC under AWGN channel can be evaluated by [4] Qd,EGC  =  Qf,EGC =  where  t  =  /5)  (2.17)  TW, )/2) F(N TW)  (2.18)  QN.TW  F is the sum of the SNRs for all SUs. For Rayleigh fading, an  average probability of detection for EGC can be computed with (2.11) and (2.17) 9  as[4]  Qd,EGC =  f  QN.TW  (‘ )•  (N  _  P  l)!Nt  ()  (2.19)  dF,  where the second term is the gamma PDF. In the case where weights  w,  are dif  ferent for Y, we have weighted gain combining (WGC). In general, since energy measurements contain more information than 1-bit decisions, EF has a better sensing performance than DF [6,7]. Furthermore, for independent SUs, the best CSS performance is obtained using the likelihood ratio test (LRT) [15]. The decision statistic of LRT can be expressed as [15] =  A(Y)  where  =  (Yi, Y2,... YN)  N  1 H A,  =  (2.20)  is a vector of individual SU energy measurement reports.  We note that implementation of (2.20) requires knowledge of the conditional PDFs of f(yH ) and f(yIHo) which may not be available. 1  10  Chapter 3 Performance Analysis of Weighted Gain Combining In this chapter, we investigate an energy fusion scheme for a situation in which SUs have different reliabilities. In particular, we propose to apply two probability density functions of the sum of weighted central chi-square RVs [16] and the sum of weighted noncentral chi-square RVs [17] to characterize the decision statistic.  From these  PDFs, we numerically evaluate the detection performance of weighted gain combining in terms of the probability of false alarm and the probability of miss detection.  3.1  Introduction  An accurate and reliable spectrum sensing scheme can ensure a high level of protection to the PU while providing maximum transmission opportunities for the SUs. In energy fusion, a group of SUs participate in spectrum sensing by measuring the received energy on a PU channel of interest and then forwarding results to the FC. The FC is responsible for making a final decision about the activity state of the PU using some energy fusion rule. As mentioned in Chapter 2, this decision process can be viewed  11  as a hypothesis testing problem. The decision statistic is 1 H  Z=wY  (3.1)  where Y is the energy reported by SU i and w is the associated weight. Based on a decision threshold ) chosen to achieve a target miss detection probability. The decision rule is to declare H 1 if Z  . 0 ?.; otherwise, declare H  It is shown in [6, 7] that energy fusion has better performance than the OR decision fusion. However, in most papers, it is assumed that all SUs have the same average SNRs. This may not always be so in practice since SUs may be at different distances from the PU transmitter. Since a SU with a high average SNR is likely to provide more reliable information about the activity state of the PU, it is beneficial to consider the reliability of each SU in making the final decision. In [7], a weighted gain combining scheme is proposed, in which the weight w for SU i is computed as  w=  i  / 3.2  —  ri 1 I  —  where F denotes the average SNR of SU  i.  It follows from (3.2) that  N  1 Z  =  1.  Simulation results in [7] shown that the WGC scheme yields better detection perfor mance than traditional EGC and OR decision fusion rule and it provides performance similar to that of the optimal LRT. For EGC, it is shown in [4] that the decision statistic Z for N SUs is a central 0 while Z is a chi-square RV with N• 2TW degrees of freedom under hypothesis H noncentral chi-square RV with N 2TW degrees of freedom and non-centrality pa rameter 2F  =  . However, when energy measurements 1 2F under hypothesis H  are weighted and summed, the decision statistic may no longer be a chi-square RV. In [18], a Gaussian approximation on the local energy measurement Y at each 12  SU is made. Expressions for the system probability of false alarm probability of miss detection  Qm  Q  and the system  are then derived. However, this approximation may  not be very accurate for an energy detector with a small number of degrees of freedom, e.g. 2TW < 10, as assumed in [4,6,7,19]. To the best of my knowledge, no work in the context of CR has addressed the evaluation of the distribution of the decision statistic for WGC. In this chapter, we use two existing results on the PDFs of the sum of weighted central chi-square RVs and the sum of weighted noncentral chi-square RVs to characterize the decision statis tic. We show through numerical evaluation that the proposed PDFs can accurately approximate the detection performance of WGC in AWGN channel. This chapter is organized as follows: The system model is described in Section 3.2. In Section 3.3, we consider the PDFs of the sum of weighted central chi-square RVs and the sum of weighted noncentral chi-square RVs. In Section 3.4, we study truncation errors associated with these PDFs. In Section 3.5, we compare the detec tion performances of WGC obtained numerically and by simulation.  3.2  System Model  We consider a centralized CR system consisting of N SUs and one FC located within the transmission range of a PU transmitter as shown in Figure 3.1. All SUs are considered to be potential interferers to the PU; SUs refrain from transmitting when the PU is detected active. SUs are equipped with identical energy detectors as de scribed in Figure 2.1. Energy measurements collected by SUs are forwarded to the FC through idealized reporting channels which introduce no delays or errors. The PU ) with probability 0.5 and inactive (hypothesis 0 is assumed to be active (hypothesis H ) with probability 0.5. The activity state is independent from one sensing interval 1 H to the next. In practice, the activity pattern of the PU will depend on the nature 13  of the PU transmitter. For example, the activity state changes for a TV transmitter may not be as frequent as the state changes in a cellular base station transmitter. It is mentioned in Chapter 2 that the central and noncentral chi-square distributions directly correspond to the decision statistic at the output of an energy detector un der both hypothesis for AWGN channel. For ease of analysis, we adopt the average SNR based weighting scheme in [7] with the assumption that the sensing channels are AWON with known SNR I’, i  =  1,2,... , N. The weight w can then be computed as ,i 1 = Zl  Fi  (3.3)  Primary User (PU) Transmitter  Figure 3.1: Centralized collaborative spectrum sensing system.  3.3  PDFs of Decision Statistic  In Chapter 2, it was noted that when the PU is inactive, the energy detector output is distributed according to the central chi-square distribution with 2TW degrees of freedom. When PU is active, the energy detector output is distributed according 14  to the noncentral chi-square distribution with 2TW degrees of freedom and noncentrality parameter 2P. However, when the energy measurements Y, i  =  1,2,.  .  .  ,  N  are weighted and summed at the FC, the decision statistic Z no longer follows the , it becomes a weighted sum of N 0 chi-square distributions. Under hypothesis H central chi-square RVs, each with 2TW degrees of freedom. Under hypothesis H , 1 it becomes a weighted sum of N noncentral chi-square RVs, each with 2TW degrees of freedom and non-centrality parameter 2F. Since simple closed-form expressions for these corresponding PDFs are not available, we investigate the use of two infinite series expressions in [16, 17]. Let fz(zHo) and fz(zIHi) denote the conditional PDFs of Z under hypothesis 0 and H H 1 respectively. From [16], fz(zHo) can be expressed as a sum of Laguerre polynomials  fz(zlHo)  (D  )k!ckL(v/21) 2 P(u/ =  (2/2  P(/2)  0 >0, V  (3.4)  where coefficient ck satisfies a recurrent relation Ck  k>1,  =  =  (i-) N  3 d  =  v/2 N  2  11(1+ i  (3.5)  — (_  / (  )  —v/2 ,  j  ,  1.  (3.6)  (3.7)  The other terms in (3.4) through (3.7) are . k  L(x)  =  m—0  k (k  ) -;:  m  a  >  0,  is the kth generalized Laguerre polynomial, • ii  =  N 2TW is the total number of degrees of freedom for N SUs. .  15  (3.8)  • w is the weight for SU i, computed by (3.3), .  max(w) + min(w)  =  •  (39)  > 0 is a parameter which is suitably chosen to improve the computation  efficiency of (3.4) (jto  =  v/3 is used in the thesis),  • F(.) is the gamma function  [131.  From [17], the conditional PDF fz(zjHi) can be written as v/2—i  fz(zIHi)  F  2k’ 2 ) Lv/21) (vz)  (2/2 F(v/2)  where coefficient  Ck  (3.10)  k—i  1,  k  =  ‘/2 Co =  (3.11)  (  iL Sw(v/2—o) exP_+(/ ) 2  N  • II  —vj/2  (1 + i  (_  —  i))  (3.12)  ,  N  3 d  0 >0, V  satisfies the following recurrent relation 1  Ck  00  =  —  w)i_i (iio  ,  + w(v/2  —  )  j1.  (3.13)  l+jl)  j=i  The parameter 6, in (3.12) denotes the non-centrality parameter for SU i, i.e. 6,  =  2F,.  Parameters v, w.j, 3 and u 0 in (3.10) through (3.13) are as previously defined. (3.12) and (3.13) can be viewed as a generalization of (3.6) and (3.7). By substituting 6,  =  0 for all i into (3.12) and (3.13), (3.4) and (3.10) become equivalent. Moreover,  by setting v  2TW, w  =  1 and N  =  1, (3.4) and (3.10) reduce to the PDFs of  central chi-square distribution and noncentral chi-square distribution respectively. 16  Analysis of Truncation Error  3.4  Since both (3.4) and (3.10) involve an infinite sum, it is important to study the truncation error when the infinite sum is truncated in a computer implementation. In order to get numerical results, we use the following approximation for both (3.4) and (3.10), )k!ckL(v/21) 2 P(v/  fz(z)  (2):/2  0 >0, V  F(v/2)  (3.14)  where the infinite sum is upper limited by a chosen value of K. In the following, we study the truncation error base on a CSS system with N the number of degrees of freedom equals 2TW  =  =  5, P  =  [1, 2, 3, 4, 5] dB,  10 for all SUs and po  =  v/3. The  0 and H H 1 cases are considered in Subsection 3.4.1 and 3.4.2 respectively.  3.4.1  0 Hypothesis H  If only the K + 1 terms in the infinite sum in (3.14) are used, the truncation error can be computed by evaluating (3.4) from k  =  K + 1 to cc. In our study, instead  of evaluating an infinite sum, we estimate a section of the truncation error, e, , by 1 evaluating (3.4) from k bound,  2 , 0 E  =  K + 1 to a sufficiently large value Kmax and we show a  on the truncation error for the rest of the terms (from Kmax + 1 to cc)  is insignificant. The reason for this approach is that we would like to show a small value of K is sufficient to evaluate the PDF. Table 3.1 lists the PDF values with the associated truncation errors evaluated at different values of Z. The PDF values are computed by evaluating (3.14) and (3.5) to (3.7) for K  =  20 and 25. The first section of the truncation errors are computed by  evaluating (3.4) from k  =  21 and 26 to Kmax  =  150 (replace cc with Kmax in (3.4)),  and the error bounds are computed by [16] —z -  (z,Kmax,o,) 2 eo,  kp  v/2—1  )V/ ( Z 2 P(/ exp  (8) —  17  maz+  2 k k!P(v/2)  (3.15)  where 1—w/  _max  1+  -  (w//3)(-  316)  -1)  It can be observed from the table that the computed PDF values are the same for K  20 and K  =  25 for all values of Z. This is because the truncation errors are  small for these two values of K. It can be seen that the truncation errors in the th 5  and  rd 3  columns can be either positive or negative. This is due to the term (_)m  in (3.8) which is positive or negative depending on whether m is even or odd. The first section of truncation errors can be reduced by using a larger value of K, for instance, at Z  =  25, a significant reduction in the magnitude of the truncation error  can be achieved with K  =  25 compared to K  =  20. However, a larger value of K also  linearly increases the computation time. In general, an adequate value of K should be selected considering both accuracy and computation time. The last column of Table 3.1 lists the error bounds for terms from Kmax + 1 to cc. It can be seen that the error bounds are insignificantly small at all values of Z. This confirms that Kmax  =  150 is  a reasonable choice. Table 3.1: PDF values and truncation errors for decision statistic, Z, under H 0 hy pothesis, for K values of 20 and 25 (N = 5, 2TW = 10, p = v/3, 1’ = [1, 2, 3, 4, 5] dB). PDF (K 20) 3.36 x iU’ 4.28 x 10 1.89 x 10’ 1.50 x 10—2 1.13 x iO 2.74 x i0 3.47 x 10b0  Z 1 5 10 15 20 25 30  (K = 20) 3.51 x 10—22 —7.32 x i0’ 1.41 x 1012 —1.09 x 1012 2.63 x 10 1.83 x 10” 1 E,  1.29  x  1012  PDF (K = 25) 3.36 x io’ 4.28 x iO 1.89 x 10’ 1.50 x 10—2 1.13 x iO 2.74 x iO 3.47 x 10°  (K = 25) 1.46 x 1024 1.68 x iO’ —2.86 x iO’ 8.47 x iU’ —2.28 x iO’ 4 —3.18 x 10—14 —1.47 x iO”  Figure 3.2 plots the PDF values computed by (3.4) for N ho  =  v/3, I’  =  =  2.65 1.47 1.26 1.09 5.54 6.01 2.44  5, 2TW  x x x x x x x  i0 i0’ i0 10—32 10—31 iO° 10—29  =  10,  [1, 2, 3, 4, 5] dB and K = 25. It can be seen that the numerically 18  obtained PDF curve matches the simulation histogram quite well.  0.16-  0.12—  0.08—  0.04-  0.00  5  10  I 20  15  I 30  25  35  PDF  Figure 3.2: Plot of numerically obtained PDF against simulation histogram for Z under H 0 hypothesis (N = 5, 2TW = 10, o = u/3, K 25, I’ = [1, 2, 3, 4, 5] dB, K=25).  3.4.2  1 Hypothesis H  0 hypothesis, Table 3.2 lists the By following the same approach as described in H PDF values with their associated truncation errors,  ei,  The PDF values  and  are computed by evaluating (3.14) and (3.11) to (3.13) for K  =  20 and 25. The first  section of the truncation errors are computed by evaluating (3.10) from k 26 to Kmax  =  =  21 and  150 (replace oo with Kmax in (3.10)) and the error bounds for terms 19  from Kmax + 1 to oo are computed by [17] (Z,KmaxjLQ,/3) 2 Ei,  (23)/  / •(—)  v/2  ñ  +  \i’qJ  (3.17)  (  (w/)  i)  —  1+(w/i3)(-—1) 1 \\ 4= k/2k+/2k+VN 2k  VZ  —  =  —  N  / •exP J 8 where ö  )  (_ ( exp(—)expI——  F(/2)  _i/2  )  )  max+  Sj denotes the sum of N non-centrality parameters. The term q follows  the same definition as (3.16). It can be seen that the truncation errors in the 3’ and th 5  columns can be either positive or negative, this can be explained by the same  reasoning as given in hypothesis H . The order of magnitude of the truncation errors 0 also reduces by using a larger value of K. For instance, at Z of magnitude in error reduction at K  =  25 compare to K  =  =  30, there is one order  20. The bounds  61,2  for  the errors after Kmax + 1 are listed in the last column of Table 3.2. It can be seen that Kmax  =  150 is a sufficient choice for H 1 hypothesis as well.  1 Table 3.2: PDF values, and truncation errors for decision statistic, Z, under H v/3, 1’ = [1, 2, 3, 4, 5] hypothesis, for K values of 20 and 25 (N = 5, 2TW = 10, dB). Z 1 5 10 15 20 25 30  PDF (K = 20) 4.72 x 10—19 1.44 x 10 4.13 x 10_2 1.29 x 10_i 2.63 x 10_2 1.23 x i0 2.33 x i0  (K = 20) 4.59 x iO 6.29 x iO’ —5.14 x 10b0 —4.33 x 10_8 6.51 x 10—8 5.57>< i0— —1.74 x 10—8 1 Ej  PDF (K = 25) 2.52 x i0’ 1.44 x i0 4.13 x 10—2 1.29 x 10_i 2.63 x 10—2 1.23 x io— 2.33 x i0  6 (K = 25) —8.47 x 10_21 —3.27 x io’ 4.56 x 10_li —4.32 x 10” —4.03 x 10 —9.41 x iO —2.80 x 10  Figure 3.3 plots the PDF values computed by (3.10) for N ito  =  v/3, F  =  =  61,2  4.50 2.49 2.14 1.84 9.40 1.02 4.14  5, 2TW  x x x x x x x  i0 10 10—25 10_22 10_21  io’ 10_19  =  10,  [1, 2, 3, 4, 5] dB and K = 25. As expected, the numerically obtained  PDF curve matches the simulation histogram quite well.  20  0.16—  0.12-  0.08-  0.04-  0.0 0  5  10  I  I  I  20  25  30  35  PDF  Figure 3.3: Plot of numerically obtained PDF against simulation histogram for Z = v/3, K = 25, F = [1, 2, 3, 4, 5] dB, under H 1 hypothesis (N = 5, 2TW = 10, K=25).  21  3.5  General Comment  Numerical computation in MATLABTM uses the IEEE floating-point standard arith metic which provides an accuracy of approximately 16 decimal digits. With the particular computer used in this study, we encountered inaccuracies due to round-off errors for calculations involving very large numbers, e.g. computation of factorials in MATLABTM. As an alternative approach, the PDF plots and the truncation error tables presented in this chapter were obtained with MAPLETM. However, to per form the same calculations in MATLAB , a possible approach would be taking the TM logarithm of the large numbers.  3.6  Performance Evaluation  In this section, we evaluate the detection performance of WGC and compare our numerical results to simulation results. We use the complementary receiver operating characteristic (ROC) curve to characterize the sensing performance, which is a plot of the probability of miss detection against the probability of false alarm. The system probability of false alarm, detection,  Qm,  Q,  and the system probability of miss  of WOC in AWGN sensing channels can be numerically computed as pA =  P(Z  ) 0 AjH  1 —  J J  fz(zHo)dz  (3.18)  fz(zHi)dz  (3.19)  0  pA  =  1  —  P(Z  ) 1 )H =  0  where fz(zHo) and fz(zIHi) are given in (3.4) and (3.10) respectively. following analysis, we assume 2TW new variable, system average SNR,  =  10,  0 t  =  v/3 and K  =  For the  25. We introduce a  which is defined as an average of the SNR  values among N SUs. Figure 3.4 shows the complementary ROC curves for several different numbers of 22  SUs. In each case, P is fixed at 3 dB. It can be seen that the performance of WGC improves as the number of SUs increases. The figure shows that the complementary ROC curves obtained by using the numerical method closely match the simulation results.  100  Th_  -  .  ‘Vv VV:  v:::  &*c :.*\:V:  c10  :::::::::::::::::::::::::;::::::::::::::::::::::N *v  \  \- :v  :‘  :is:  ‘V  \ .D  10  :  —2  X\  :  N = 1 Numerical SNR = 3dB N = 1, Simulation, SNR = 3dB — N = 2, Numerical, SNR = [2, 4] dB * N = 2, Simulation, SNR = [2, 4]dB — — — N 3, Numerical, SNR = [2, 3, 4] dB U N=3,Simulation,SNR=[2,3,4]dB N=5,Numerical,SNR=[1,2,3,4,5]dB o N=5,Simulation,SNR=[1,2,3,4,5]dB v  .  :: :: :: ::  : :: : : :::: : :  :  —.  . :  :  i1  :  Probability of False Alarm (Q)  Figure 3.4: Complementary ROC performance comparison between different number of SUs at a fixed system average SNR (N = [1, 2, 3, 5], P = 3 dB). Figure 3.5 shows the complementary ROC curves for a fixed number of SUs, N  =  5, at different system average SNRs,  [3, 4, 5] dB. It can be seen that  when N is fixed, the complementary ROC performance improves as Figure 3.6 shows the complementary ROC curves for N  =  5 and I’  increases. =  [3, 4, 5]  dB. However, in this figure, the SNRs for all SUs are assumed to be the same. Once again, both figures show that the performance curves obtained by using the numerical 23  method closely match the simulation results.  100  I  —-U .A.. A V:  :  :NJ:::  N. ::::13..._ I ::::::  :“:..:  O  :  1  :  : :  :  :  110 (1)  ..  •“  o  €  V  10  —2  :: : :: :: :  .  : : ::: ::: : : ::: : ::: :  .  : ::::::: :: : : ::: : :: :  .  .  :: :  Numerical,SNR[1,2,3,4,5)dB Simulation,SNR=[1,2,3,4,5]dB —Numerical, SNR=[2, 3, 4, 5,6)dB U Simulation, SNR [2, 3, 4, 5, 6] dB Numerical, SNR [3, 4, 5, 6,7]dB Simulation, SNR [ 3, 4, 5, 6, 7] dB  :: V : :::;: : ::: :  o —  :.  —  3 1o  10_2  :: :  =  10_i Probability of False Alarm (Qf)  :  :  .:  :‘  10°  Figure 3.5: Complementary ROC performance comparison between different system average SNRs at a fixed number of SUs with each SU having a different SNR (N 5, [3, 4, 5] dB). =  =  Figure 3.7 illustrates a general comparison of the complementary ROC perfor mance among several fusion techniques with N  =  5 and P  =  [1, 2, 2, 6, 7] dB. These  SNR values are selected to show the performance difference among fusion techniques, where two of the SNR values are slightly higher than the rest. In particular, we ex pect to see that WGC obtains better detection performance than EGC. In the figure, the performance of EGC is obtained using (2.17) and (2.18), that of LRT is obtained using (2.19), that of the OR fusion rule is obtained by simulation and that of WGC is obtained by the numerical method. It can be seen that WGC outperforms both 24  10  I  •  .  .  ,.  :  :..  a O do o  : :  1  : :  : :  :  :::  :  .  .  .  .  :  :  •  a)  ............•  o S2  ...  Ic,  >  10  ‘  :  .  .  ::::::  —2  0 — —  D  Ic 10_2  .  Numerical,SNR=[3,3,3,3,3]dB Simulation,SNR=[3,3,3,3,3]dB —Numerical,SNR=[4,4,4,4,4]dB Simulation,SNR=[4,4,4,4,4]dB Numerical,SNR=[5,5,5,5,5]dB Simulation,SNR=[5,5,5,5,5]dB  .  :  ::.:::  V.  :.  :  : :  :  .  :  .  10_i Probability of False Alarm (Qf)  10°  Figure 3.6: Complementary ROC performance comparison between different system average SNRs at a fixed number of SUs with all SUs having the same SNR (N = 5, = [3, 4, 51 dB).  25  OR and EGC and provides performance close to that of LRT.  100  :  I  O •  10  : : : : : ::: :  :  0  ::  :  : :  ::  • : ::: :  :  .  :  : : .:::. : :  .::.: .  .  .  CI)  .  Co C,)  .  .  .  .  .  .  .  .  .  . . .  .  o  .  .  —2 a o10 2  .  * El  OR EGC  o WGC ———LRT  .  . .  .  :  :  io  102  100  Probability of False Alarm (Q)  Figure 3.7: Complementary ROC performance comparison between several fusion techniques (N = 5, 1’ = [1, 2, 2, 6, 7] dB).  26  Chapter 4 Performance Analysis of Imperfect Reporting Channels In this chapter, we investigate a problem in CSS in which the reporting channels are noisy. It is shown that there is a significant performance loss when a final decision regarding the PU’s state made at the FC is based on a set of reconstructed versions of quantized energy measurements that are distorted by imperfect reporting channels during transmission. To reduce this performance loss, we propose to apply unequal power allocation (UPA) on the quantized data bits to protect the more significant bits from channel effects. Simulation results show that UPA effectively improves overall sensing performance.  4.1  Introduction  It is shown in literature [4—6] that using multiple SUs achieves better performance than a single SU. However, most works assume that reporting channels from the SUs to the FC are noiseless. In many situations, this is unrealistic. A problem of CSS with Rayleigh faded reporting channels under AWGN noise is  27  considered in [19]. It was shown that the performance of decision fusion at the FC is limited by reporting channel errors. A clustering method was proposed for improving the performance by considering SNR variations between the cluster heads and the FC. The study is limited to the case where only 1-bit binary decisions are forwarded to the FC. Imperfect AWGN reporting channels without fading are considered in [20]. The authors proposed a Two-Step Detector and an Averaging Detector that can be used at the FC to recover the corrupted binary decisions reported by SUs. It was shown that the overall CSS performance is improved over different reporting channel SNRS. However, this work considers only binary local decisions. For energy fusion based CSS, a commonly used, albeit unrealistic, assumption is that an analog energy measurement at a SU can be received exactly at the FC [4,7]. In [6], it is shown that a 6-bit uniform quantization with a predefined input range can achieve CSS performance close to that with unquantized energy values. However, the question of how to select the quantizer input range is not addressed. In [21], it is found that if the quantizer input range is based on noise only inputs where PU is inactive, a 4-bit uniform quantization is sufficient. In [6, 21], imperfect reporting channels are not considered. In order to protect a transmitted bit vector from channel errors, unequal power protection (UEP) of transmitted symbols through UPA is introduced in [22]. The idea is to allocate different transmission power to individual bits according to their bit error sensitivities. An optimization method which determines a set of bit weights that minimizes the mean square error (MSE) between a transmitted and a received bit vector was also introduced in this work. In this chapter, we investigate the sensing performance of CSS when quantized energy are transmitted over Rayleigh faded reporting channels. To combat transmis  28  sion errors, we propose applying UPA to the quantized energy bits. In Section 4.2, we introduce our system model and in Section 4.3, we present and discuss our simulation results.  4.2  System Model  The system model is illustrated in Figure 4.1. We have a CSS system consisting of N SUs and one FC. We assume that the sensing channels undergo independent Rayleigh fading. For ease of analysis, we assume the average sensing channel SNRs are the 3 same for all SUs in the system which is denoted by P  =  P, i  real-valued energy detector output at SU i denoted by  i  is quantized by a M-bit  =  1,2,.  ..  ,  N. The  uniform quantizer.  Figure 4.1: System model for collaborative spectrum sensing with imperfect sensing and reporting channels We assume that the reporting channels also undergo independent Rayleigh fad ing.  For simplicity, the same average SNR assumption is also true for reporting  channels, which is denoted by Pr  =  Pj, I  =  1, 2,... N. The vector x indicates a par ,  ticular quantization interval in the ith quantizer, which is to be transmitted by BPSK modulation over SU i’s reporting channel. We represent  with natural binary code,  e.g. the 10th quantization interval is represented as 1010 with 4-bit quantization. 29  Due to channel errors, x may be distorted when it arrives at the FC. We denoted i  1, 2,.  ..  ,  ,  N as the distorted version that the FC receives. In the final step, the  FC makes a final decision with EGC based on  4.3  Simulation Results  In this section, we present our results based on a 5 SUs CR system with the number 10. The uniform quantizer is assumed to  of degrees of freedom per SU set at 2TW be 4-bit.  Figure 4.2 shows the complementary ROC curves for quantized energy measure ments transmitted over perfect and imperfect reporting channels. Theoretical perfor mance of EGC with unquantized energy computed by (2.18) and (2.19) is shown for comparison. The sensing channel SNRs and the reporting channel SNRs are both set at 10 dB. It can be observed from the figure that the complementary ROC perfor mance of 4-bit quantized energy measurements over perfect reporting channels closely approximates the theoretical EGC performance, however, a significant performance loss is observed when the reporting channels are imperfect. This is because the re ceived vector I, is distorted by the noisy channel. For instance, at Qm  Q  =  10—2, the  degrades from 3 x 10—2 to 10_i. In order to improve detection performance, we  propose to apply UPA on the transmitted bit vector According to [22], UPA on  can be achieved by multiplying a set of weights to  the BPSK symbols. This set of weights can be obtained according to an optimization criterion that minimizes the MSE between x and M 2  } 2 E{d  ,  expressed as [22]  M 2  P(,) P(.j,Ij,k),  =  .  (4.1)  ,c=i 7 =i  where d,  =  is the probability of occurrence of  and P(±j,xj,ç)  is the transition probability between the transmitted bit vector x and the received 30  10  !..L  .  Perforrnane: loss with ithperfect thanie1s: 101  o  :  : :: :: : : :: : .::::::::::::;:::::::  :::  .  .  •...  ::::: :  :::  :  :  : : ::: :  .  .  . .  C)  o  u) c)  .  .  10  —2  : :::::: : : :::::::.::::::::: ::.::  :  :  :  :  .  :  o  :  :  .::::::::::;:::.:::.::.:  :  :  :••::  >  .  ...  .  .•.  .  S..  .  CO .  ..  10  —3  .  —  —  o O  —  .  .  0  . .  .  .  Unquantized (Perfect reporting channels) Quantized (Perfect reporting channels Quantized (Imperfect reporting channels)  ....  : I  I  10  ..  .  10_i 10_2 Probability of False Alarm (Q)  10°  Figure 4.2: Complementary ROC curves for quantized energy measurements trans 3 = 10 dB, Fr = 10 mitted over perfect and Rayleigh faded reporting channels (F dB).  31  bit vector I. By using the numerical method described in [22], we computed a set of weights for x at average SNR values from —10 dB to 15 dB. These values are listed in Table 4.1 with w 1 denotes the most significant bit (MSB) and w 4 denotes the least significant bit (LSB). Table 4.1: Weight values w at different average SNRs SNR(dB) —10 —9 —8 —7 —6 —5 —4 —3 —2 —1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  w 1 1.8928 1.8754 1.8536 1.8267 1.7946 1.7574 1.7159 1.6707 1.6222 1.5709 1.5174 1.4627 1.4073 1.3526 1.3006 1.2530 1.2108 1.1742 1.1429 1.1166 1.0947 1.0766 1.0617 1.0496 1.0398 1.0319  w 2 0.8370 0.6701 0.7232 0.7820 0.8444 0.9073 0.9669 1.0195 1.0622 1.0933 1.1125 1.1209 1.1197 1.1113 1.0989 1.0851 1.0717 1.0595 1.0489 1.0398 1.0323 1.0260 1.0209 1.0168 1.0134 1.0107  W4  0.1502 0.1780 0.1966 0.2201 0.2499 0.2878 0.3363 0.3974 0.4712 0.5541 0.6384 0.7156 0.7796 0.8290 0.8663 0.8946 0.9164 0.9334 0.9469 0.9577 0.9662 0.9731 0.9785 0.9829 0.9864 0.9891  0.0714 0.0451 0.0499 0.0559 0.0638 0.0742 0.0881 0.1073 0.1344 0.1730 0.2281 0.3037 0.3976 0.4980 0.5919 0.6724 0.7387 0.7921 0.8348 0.8688 0.8958 0.9173 0.9343 0.9478 0.9586 0.9671  The weight values listed in Table 4.1 are plotted in Figure 4.3. It is observed that for reporting channels with low average SNRS, the weight value for the MSB is significantly higher than those for other bits. This is because with natural binary  32  representation, an error made on the MSB represents a more serious problem than that on the LSB. For example, a 4-bit data sample 1111 equals to 15 in decimal. If an error occur in the MSB, the bit pattern becomes 0111, which equals to 7 in decimal. Therefore, in order to protect the MSB from channel error, a higher weight is assigned to it at a cost of lower weights for the rest of the bits. At higher SNR values, the weights for the four bits tend to become closer.  2  I  0 ci  1.8  0 V  wl(MSB). w2 w3 w4(LSB)  1.6 1.4 1.2 1 U)  .  .  .  0.8 0.6 0.4 0.2  o  —10  .  —5  0 5 Average SNR (dB)  10  15  Figure 4.3: Weight values w at different average SNRS. For Rayleigh faded reporting channels under the assumption that the SUs have perfect knowledge regarding the instantaneous channel state, UPA can be applied to X%  by a simple lookup table of weights listed in Table 4.1. For example, if the instan  taneous SNR falls into —4.5  <  F  <  —3.5, then the weight values [w 1 w 2 w 3 w ] 4 33  =  [1.7159 0.9669 0.3363 0.0881] are multiplied to the BPSK symbols. Figure 4.5 shows the complementary ROC curves for quantized energy measure ments transmitted over Rayleigh faded reporting channels with and without UPA. The sensing channel SNRs and the reporting channel SNRs are both set at 10 dB. It can be seen that the performance loss due to imperfect channels can be effectively reduced with UPA. For instance, at  Q  =  10—2, the  m Q  is reduced from 10_i to  7 x 10.  100  H  H  Perfor.mance:1os. with imperfect dTannels.  .:  Performance improvement with UPA 10_i  :.:  .  E  : . ::: :  :;.  : :::.  .:.  ..  : ::: :  :“  : C  o o Cl) 0)  . .  .  :  ?  10  .  ::::.::: .  -  . ...:..  :  :::  . .  %,....  O (a)  .  .  .  :  .:.:::.::.::...::  ..  .•  .%  :  :b...:::  .  ..  .  .  ..:  p .  (Cl  %.  ... ..  S  ...  io  —  —  — —  0  0  S  —  7  Unquantized (Perfect reporting channels) Quantized (Imperfect reporting channels) Quantized with UPA (Imperfect reporting channels)  . . .  :  :  :  I  10 1o  10_2 10_i Probability of False Alarm  10°  Figure 4.4: Complementary ROC curves for quantized energy measurements trans mitted over Rayleigh faded reporting channels with and without UPA (T’ = 10 dB, 10 dB). Figure 4.5 shows the complementary ROC curves for quantized energy mea surements transmitted over Rayleigh faded reporting channels at average SNRs of 34  =  [3, 5, 7, 10, 12] dB. The sensing channel SNRs are F,.  5 dB. The figure  shows that the performance worsens as i,. decreases. For instance, at Q  m Q  degrades from 0.68 at I’,.  =  10 dB to 0.83 at F,.  =  10—2 the  5 dB. This figure also illus  =  trates that despite the sensing channel conditions, the overall detection performance for quantized energy reports is limited by imperfect reporting channels even with the average SNRs on the reporting channels are much higher than that of the sensing channels.  1 00  —F—  .  TF  —  I,. =3dB r’.=5dB  :.  .dB  F1QdB F,.=I2dB  10  —  —  * C  0 V o  102 102  —  Unquantized (Perfect reporting channels) Quantized (Rayleigh reporting channels, SNR Quantized (Rayleigh reporting channels, SNR Quantized (Rayleigh reporting channels, SNR Quantized (Rayleigh reporting channels, SNR Quantized (Rayleigh reporting channels, SNR  = = = = =  3 dB) 5 dB) 7 dB) 10 dB) 12 dB)  \\\\.. ___AU)..  10_i Probability of False Alarm (Q)  100  Figure 4.5: Complementary ROC curves for quantized energy measurements trans mitted over Rayleigh faded reporting channels at different reporting channel average SNRs (F 8 = 5 dB, F,. = [3, 5, 7, 10, 12] dB). In addition to Figure 4.5, Figure 4.6 illustrates the performance improvements when UPA is applied to the quantized energy measurements. It can be seen that the 35  complementary ROC performances improve in both average SNR cases, however, the difference in performance gain is slightly larger at  r  =  5 dB than at  r  =  10 dB.  This is because the more significant bits of a quantized energy measurement are more likely to be protected by large weight values in the low SNR case than in the high SNR case. As a result, there is more opportunity for performance improvement when the SNR value are low.  10  With.TJPA,.F.=5.dB .  :  Wh UPA  l 1odB..  10_i  —  —  C I  V *  —  Unquantized (Perfect reporting channels) Quantized (Rayleigh reporting channels, SNR = 5 dB) Quantized with UPA (Rayleigh reporting channels, SNR Quantized (Rayleigh reporting channels, SNR = 10 dB) Quantized with UPA (Rayleigh reporting channels, SNR  102 10_2  10_I  .  =  5 dB)  =  10 dB)  ‘.  I 100  Probability of False Alarm (Qf)  Figure 4.6: Complementary ROC curves for quantized energy measurements trans mitted over Rayleigh faded reporting channels at different reporting channel average SNRs with and without UPA (P 8 = 5 dB, r = [5, 10] dB). The above results show the performance improvements achievable with uniform quantization and UPA when the reporting channels are noisy. For further study, it would be interesting to investigate the additional gains that may be possible with 36  non-uniform quantization.  37  Chapter 5 Conclusion Two problems in collaborative spectrum sensing (CSS) are studied in this thesis. In Chapter 3, weighted gain combining (WGC) energy fusion considering SUs with different sensing channel SNRs is investigated. By utilizing existing results in the probability density function of the weighted sum of central chi-square random vari ables and the weighted sum of noncentral chi-square random variables, an numerical method was developed to analyze the sensing performance of WGC. Simulation re sults confirmed the viability of the proposed numerical approach. In Chapter 4, the performance degradation resulting from imperfect reporting channels and energy measurement quantization in a CSS system is investigated. Sim ulation results shown that there is a significant performance degradation when the reporting channels are imperfect. To reduce the performance loss, an application of unequal error protection through unequal power allocation (UPA) on the transmit ted symbols of a quantized energy bit vector is investigated. Performance gain is quantified by simulation results.  38  5.1  Main Thesis Contributions  • The application of a recent result from the statistics literature on the distri bution of the sum of weighted noncentral chi-square random variables which allows numerical evaluation of the sensing performance of weighted gain com bining energy fusion in CSS. • An investigation on the performance of CSS with quantized energy measure ments reported over imperfect reporting channels as well as an application of unequal power allocation on quantized energy bits to improve overall CSS sens ing performance.  5.2  Recommendations for Future Work  In the study of imperfect reporting channels in Chapter 4, an uniform quantization model was assumed for simplicity. It would be interesting to consider the performance improvements which may be obtained by a joint optimization of the quantization thresholds and bit weights.  39  Bibliography [1] Federal Communications Commission, “Spectrum Policy Task Force,” Report of ET Docket 02-135, Nov. 2002. [2] J. Mitola and G.  Q. Maguire, “Cognitive radio: making software radios more  personal,” IEEE Personal Communications, vol. 6, no. 4, pp. 13  —  18, Aug.  1999. [3] 5. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 23, no. 2, pp. 201—220, Feb. 2005. [4] F. F. Digham, M. S. Alouini, and M. K. Simon, “On the energy detection of unknown signals over fading channels,” IEEE International Conference on Com munications, vol. 5, pp. 3575—3579, May 2003. [5] A. Ghasemi and E. S. Sousa, “Collaborative spectrum sensing for opportunistic access in fading environments,” IEEE International Symposium on New Frontiers in Dynamic Spectrum Access Networks, pp. 131—136, Nov. 2005. [6] A. Taherpour, Y. Norouzi, M. Nasiri-Kenari, A. Jamshidi, and Z. Z. Yazdi, “Asymptotically optimum detection of primary user in cognitive radio networks,” lET Communications, vol. 1, no. 6, pp. 1138—1145, Dec. 2007.  40  [7] F. E. Visser, C. J. Janssen, and P. Pawelczak, “Multinode spectrum sensing based on energy detection for dynamic spectrum access,” IEEE Vehicular Tech nology Conference, pp. 1394—1398, May 2008. [8] W. Wang, L. Zhang, W. Zou, and Z. Zhou, “On the distributed cooperative spec  trum sensing for cognitive radio,” International Symposium on Communications and Information Technologies, pp. 1496—1501, Oct. 2007. [9] X. Huang, N. Han, G. Zheng, S. Sohn, and J. Kim, “Weighted-collaborative spectrum sensing in cognitive radio,” CHINA COM 2007. 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López-Blázquez, “Distribution of a sum of weighted central chi-square variables,” Communications in Statistics  -  Theory and Meth  ods, vol. 34, pp. 515—524, 2005. [17]  —,  “Distribution of a sum of weighted noncentral chi-square variables,” So  ciedad de Estadistica e Investigacion Operativa Test, vol. 14, no. 2, pp. 397—415, 2005. [18] J. Ma and Y. Li, “Soft combination and detection for cooperative spectrum sens ing in cognitive radio networks,” IEEE Global Telecommunications Conference, pp. 3139—3143, Nov. 2007. [19] C. Sun, W. Zhang, and K. B. Letaief, “Cluster-based cooperative spectrum sens ing in cognitive radio systems,” IEEE International Conference on Communica tions, pp. 2511—2515, Jun. 2007. [20] T. C. Aysal, S. Kandeepan, and R. Piesiewicz, “Cooperative spectrum sensing over imperfect channels,” IEEE GLOBECOM Workshops, pp. 1—5, Dec. 2008. [21] S. Koivu, H. Saarnisaari, and M. 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