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Collaborative spectrum sensing in a cognitive radio system with non-Gaussian noise Feng, Tan 2008

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COLLABORATIVE SPECTRUM SENSING IN A COGNITIVE RADIO SYSTEM WITH NON-GAUSSIAN NOISE  by TAN FENG B.ENG., Lakehead University, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIRNCE in THE FACULTY OF GRADUATE STUDIES (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2008  ©  Feng Tan, 2008  Abstract With  the deployment of new wireless communication devices and services, the de  inand for radio spectrum continues to grow. Spectrum utilization can he improved using the Cognitive radio, concept which allows secondary users to opportunistically access the unused licensed spectrum bands without causing undue interference to licensed users. Most works on spectrum sensing assume a Gaussian noise model; however, in some situations, an impulsive noise model may be more appropriate. In this thesis, we consider the mixture Gaussian noise and the Laplacian noise model. Approximate closed-form expressions for the probability density functions and cumulative distribution functions of the output of an energy detector with Lapla cian noise were obtained using the Pearson approximation technique. An optimal detection scheme based on the likelihood ratio test (I.RT) for mixture Gaussian and Laplacian noise models was studied. Two sub-optimal algorithms, namely DFC de tection and EFC detection, are also evaluated. The results show that in contrast to the Gaussian noise case, EFC detection does not always outperform DFC detection and 1-out-of-N fusion rule does not always provide the lowest Pm for a given Pj among K-out-of-N rules in a non-Gaussian noise environment. An algorithm, in which large magnitude SU energy measurements are eliminated at the FC, is pro posed to improve the detection performance in impulsive noise, it is shown that substantial detection performance can be achieved. In addition, we study a system model in which the reporting channels between the SUs and the FC, and the chan nels between any two SUs within the cluster experience Rayleigh fading. The results show that in contrast to the Gaussian noise case, the cluster-based schemes do not always outperform the conventional DFC detection.  11  Contents Abstract  ii  Table of Contents  iii  List of Figures  V  List of Abbreviations and Symbols  viii  Acknowledgments 1  2  3  xi 1  Introduction 1. 1  Evolution of ‘Wireless Communicat ion Systems  1  1.2  Motivation and Goals  2  1.3  Contributions  3  1.4  Thesis Organization  4  Background and Literature Review  5  2.1  Cognitive Radio  5  2.2  Spectrum Sensing  6  2.3  Energy Detector  6  2.4  Local Spectrum Sensing  8  2.5  Collaborative Spectrum Sensing  .  9  Single SU Detection of PU state in Mixture Gaussian and Lapla 14  cian Noises 3.1  Non-Gaussian Noises  14  3.2  PDFs of Energy Detector Output  16  111  3.3  4  5  6  3.2.1  PU Inactive  17  3.2.2  PU Active  18  Pearson Approximation  21  3.3.1  Mixture Gaussian Noise Model  21  3.3.2  Laplacian Noise Model  28  Collaborative SU Detection of PU State in a CR System  33  4.1  DFC Detection  33  4.2  EFC Detection and LRT Detection  .  .  40 45  Imperfect Reporting Channels 5.1  Cluster-based Collaborative Spectrum Sensing  45  5.2  Spectrum sensing with mixture Gaussian noise  48  Conclusions and Suggestions for Future Work  55  6.1  Contributions of the Thesis  55  6.2  Future Work  56 57  Bibliography  iv  List of Figures detector  2.1  Block diagram of an  2.2  Collaborative spectrum sensing in a shadowed environment. In this  energy  .  case only the Secondary User 2 may be able to detect the PU 3.1  .  .  .  15 0.05, NVR  19  25  3.2  pdf for PU inactive: a  3.3  pdf for PU active: a  3.4  ROC performance of single SU  3.5  pdf comparison of the output of an energy detector using (3.20) and (3.32): a  0.05, NVR  =  =  19  25  20  24  25 and PU ON  =  0.05, NVR  =  24  25 and PU OFF  =  0.8. NVR  =  25  25 arid PU ON  pdf comparison of the output of an energy detector using (3.15) and (3.26):  3.9  0.05, NVR  =  =  pdf comparison of the output of an energy detector using (3.20) and (3.32): a  3.8  =  =  pdf comparison of the output of an energy detector using (3.15) and (3.26): a  3.7  10  pdf comparison of Gaussian and IVlixture Gaussian noise. All pdf’s have a variance of 1  3.6  .  7  a  =  0.8, NVR  =  25  25 and PU OFF  Pj comparison using (3.27) arid pdf from (3.15): a  =  3.10 Pm comparison using (3.34) and pdf from (3.20): a  0.8, NVJ?  =  0.8, NVR  =  =  3.11 ROC performance comparison using Figure 3.9 arid Figure 3.10: a 0.8, NVR  =  25  26  25 26 =  27  25  3.12 ROC performance comparison of Pearson approximation, Separation method, and the numerical method: a 3.13 a as a function of NVR for  k  =  1  v  =  0.8, NVJ?  =  25  29 29  3.14 pdf comparison of the output of an energy detector using (3.32) arid simulation: Laplacian: noise and PU ON  30  3.15 pdf comparison of the output of an energy detector using (3.26) and 31  simulation: Laplacian noise and PU OFF 3.16 Pf comparison using (3.27) and simulation: Laplacian noise  31  3.17 i comparison using (3.34) and simulation: Laplacian noise  32  3.18 ROC performance comparison of Pearson approximation, Separation 32  method, and numerical method: Laplacian noise 4.1  ROC performance of DFC with K-out-of-N fusion rules using (4.1): mixture Gaussian noise (a  4.2  =  35  3000)  =  0.05, NVI?  =  25)  36  ROC performance of DFC with K-out-of-N fusion rules using (4.1): mixture Gaussian noise (a  4.4  0.05, NVR  HOC performance of DFC with K-out-of-N fusion rules using (4.1): mixture Gaussian noise (a  4.3  =  =  0.08, NVR  =  25)  36  ROC performance of DFC with K-out-of-N fusion rules using (4.1): 37  Laplacian noise 4.5  ROC performance of 1he proposed algorithm using DFC 1-out-of-N fusion rule: mixture Gaussian noise (a  4.6  0,05, NVR  =  38  3000)  ROC performance of the proposed algorithm using DFC 1-out-of-N fusion rule: mixture Gaussian noise (a  4.7  =  =  0.05, NVR  =  25)  39  ROC performance comparison among EFC-EG, LRT and DFC using K-out-of-N fusion rules: Mixture Gaussian noise (a  =  0.05, IVVR  =  41  3000) 4.8  ROC performance comparison among EFC-EG, LRT and DFC using K-out-of-N fusion rules: Mixture Gaussian noise (a  4.9  =  0.05, NVR  =  25) 42  ROC performance comparison among EFC-EG, LRT and DFC using K-out-of-N fusion rules: Mixture Gaussian noise (a  =  0.8, NVR  =  25) 42  4.10 ROC performance comparison among EFC-EG, LRT and DFC using K-out-of-N fusion rules: Laplacian noise  vi  43  4.11 ROC performance of the proposed algorithm using EFC detection: mixture Gaussian noie (a  =  0.05, NVR  =  44  3000)  4.12 ROC performance of the proposed algorithm using EFC detection: mixture Gaussian noise (a  0.05, NVR  =  25)  44  5.1  Cluster-based CCS in CR system  46  5.2  ROC performance comparison of conventional EFC and cluster-EFC detection schemes  5.3  47  ROC performance comparison of cluster-EFC and cluster-DFC de tection schemes with imperfect cluster channels  5.4  48  R.OC performance comparison among conventional DFC, conventional EFC and cluster-based EFC: mixture Gaussian noise (a  =  0.8 and 49  NVR=25) 5.5  Number of additional bits versus total number of SUs: mixture Gaus sian noise (a  5.6  =  0.8, IVVR  25 and a  =  0.05, NVR  51  3000)  Number of additional bits versus average sensing channels SNR: mix ture Gaussian noise (a  5.7  =  =  0.8, NVR  =  25 and a  0.05, NVR  =  3000) 51  ROC performance cotnparison among conventional DFC, conventional EFC and cluster-based EFC: mixture Gaussian noise (a  =  0,05, 52  NVR=25) 5.8  ROC performance comparison among conventional DFC, conventional EFC and cluster-based EFC: mixture Gaussian noise (a  =  0.05, 53  NVR= 3000)  vii  List of Abbreviations and Symbols Acronyms 30  The third generation wireless communications technology  GSM  Global system for mobile communications  B3G  Beyond the third generation wireless communications technology  FCC  Federal communications commission  CR  Cognitive Radio  LU  Licensed user  ULU  Unlicensed user  PU  Priniary User  SU  Secondary User  FC  Fusion center  RF  Radio frecuency  CDS  Collaborative detection scheme  CSS  Collaborative spectrum sensing  WRAN  Wireless regional area networks  TBP  Time bandwidth product  DFC  Data fusion collaboration  EFC  Energy fusion collaboration  EG-EFC  Equal gain energy fusion collaboration  ROC  Receiver operating characteristic  LRT  Likelihood ratio test  AWGN  Additive white gaussian noise  viii  SNR  Signal to noise ratio  PDF  Probability density function  CDF  Cumulative distribution function  lID  Independent and identical distributed  INVR  Noise variance ratio  Symbols s(t)  PU signal  n(t)  Noise process  h(t)  Amplitude gain of the charmel  8 f  Center frequency  iV  One-side noise power spectral density PU signal energy Noise energy  -  SNR a  Impulsive index Central moment  T  Observation time interval  W  Bandwidth Degree of freedom Threshold of an energy detector  fR(r)  PDP of a Rayleigh distribution Probability of detection of a single SU  Pm  Probability of miss detection of a single SU  1 P  Probability of false alarm of a single SU Probability of error of a single SU CR system probability of detection CR. system probability of miss detection  ix  Q  CR system probability of false alarm  0 H  PU OFF hypothesis  1 H  PU ON hypothesis Central chi-square variate with 2TW degrees of freedom  X%ur(2rj)  Non-central chi-square variate with a non-centrality pararrieter 2r  Operators and Notation 1I  Absolute value of a complex number Convolution  {.} [.}*  Expectation Complex conjugate  x  Acknowledgments Undoubtedly, the completion of this thesis came from the support of many individ uals. First and foremost, I would like to take this opportunity to express iriy sincere appreciation to my supervisor, Professor Cyril Leung, who gave me the chance to pursue my master degree in wireless communications and whose patience and deep insights have helped me immeasurably throughout the course of my thesis research. He has taught me the importance of fundamental knowledge and how to solve a problem efficiently, which will be a very important asset for pursuing my career in  the future. I would also like to extend my great thanks to my co-supervisor, Professor ,Julian Cheng, who has always been there to provide me with invaluable guidance, continuous encouragement and financial support in my research. This thesis would never have been written without his assistance. Finall, I would like to thank my parents, Jing Wang  and  Jiaxuan Tan, for  their endless love, constant suØport, immense encouragement and sacrifices over the years. I would not have been able to complete this thesis without them.  FENG TAN  The University of British Columbia Vancouver, Canada December 2008  xi  Chapter 1 Introduction Sections 1.1 and 1.2 provide an overview of the evolution of wireless communication systems and the motivation and goals of the thesis. The thesis contributions are surrimarized in Section 1.3, arid Section 1.4 outlines the thesis organization.  1.1  Evolution of Wireless Communication Systems  The last one hundred years have seen many advances in wireless communication systems, such as radio, TV broadcasting, and emergency services [1]. During the past decade, wireless communications has been one of the fastest growing industry sectors worldwide  arid  it is predicted that this trend will be further accentuated in  the next several years [2]. Three generations of cellular communication systems have been developed so far. First generation (1G) systems for mobile telephony was analog and it only carried voice traffic. An example of second generation (2G) systems is the digital Global System for Mobile Communications (GSM) standard. Third generation (3G) systems offer significantly higher capacity and support variable data transmission rates so that indoor data rates up to 2 Mb/s and mobile data rates up to 144 Kb/s are supported. There are two key objectives for Beyond 3G (B3G) systems: (1.) to provide even higher data rates of up to 100 Mb/s in mobile and 1 Gb/s in stationary environments; (2) to develop handsets that can smoothly operate with  1  (liffererit networks, such as cellular, UMTS, WiFi [2].  1.2  Motivation and Goals  The demand for radio spectrum continues to grow due to the deployment of new wire less communication devices and services. In contrast to the enormous bandwidth available in optical fiber communication systems, spectrum resources in wireless communication systems are lirmted. Today’s cellular wireless networks are regu lated under a fixed spectrum assignment policy in the sense that frequency bands are exclusively licensed to users, and each user has to operate within its allotted frequency band. With most of’the spectrum already allocated, few frequency bands are currently available for deploying future wireless systems. However, recent measurements by the Federal Communications Commission (FCC) have  shown  that most of the licensed spectrum bands are largely under  utilized for significant periods  of time  [3]. This  observation  suggests that it is not  the spectrum shortage, bt the inefficient usage of the licensed spectrum that is the problem. In order to address it, FCC has taken the initiative to open up TV bands for unlicensed access [4] and IEEE has formed a new working group (IEEE 802.. 22) on Wireless regional area networks (WRAN) whose goal is to develop a standard for unlicensed access to TV spectrum on a non-interfering basis [5]. These developments indicate that the traditional approach to using spectrum is no longer sufficient and new techniques need to be developed. Cognitive radio (CR.) [6], has been proposed as a possible approach to improve spectrum utilization. The idea is to develop a low cost arid highly flexible wireless device that enables spectrum holes to be used by unlicensed users (ULUs) without causing significant interference.to the licensed users (LUs). Spectruni hole is defined as a band of frequencies originally assigned to an LU (also known as a  primary user  or PU), but which is unused at a particular time and location. Spectrum efficiency can be improved by allowing an ULU (also known as a secondary user or SU) to access such bands at the right locations and times.  2  In a CR system, the detection of PU activity is critical. A variety of detection methods have been proposed [7], [8], [9], [10], [11], [12], [13], [12]. However, for most of these methods, the SU requires prior knowledge of the PU’s signal format. In practice, this is typically unknown and a popular method is based on energy detection. in a fading environment, a single SU’s detection performance may he degraded because it has to distinguish between an unused spectrum and a deep fade. In order to improve spectrum sensing performance, collaborative spectrum sensing (CSS) has been proposed. One commonly used collaborative detection scheme (CDS) is data fusion collaborative (DFC) detection with K-out-of-N fusion rule. In this method, the decision on the ON/OFF activity of the PU is individually made by each SU first; each of the N binary decisions is then transmitted to a fusion center (FC) where a final decision is made. The final decision is  PU on  “  if and only if K or  more of the binary decisions are ON. Another CDS is energy fusion collaborative (EFC) detection. In this scheme, each SU forwards its measured energy to the FC where a final decision is made by comparing the sum of energies of all collaborating SUs against a threshold. To my knowledge, works on CSS so far assume a Gaussian noise model. However, in practice, certain noises can not he reasonably characterized as Gaussian due to infrequent but high level noise spikes. In such cases, it is important to model these noise spikes more accurately as a poor noise model may lead to a bad detection design. in this thesis, we consider two non-Gaussian noise models; namely mixture Gaussian noise and Laplacian noise.  1.3  Contributions  The main contributions of the thesis are: • Approximate closed-form expressions for the pdfs and CDFs of the output of an energy detector with Laplacian noise were obtained approximation technique.  3  using  the Pearson  • An optimal detection scheme based on the likelihood ratio test (LRT) for mixture Gaussian and Laplacian noise models was studied. Two sub-optimal algorithms, namely DFC detection and EFC detection, are also evaluated. The results show that in contrast to the Gaussian noise case, EFC detection does not always outperform DFC detection and 1-out-of-N fusion rule does not always provide the lowest Pm for a  given  Pf among K-out-of-N fusion  rules in a non-Gaussian noise environment. • An algorithm, in which large magnitude SU energy measurements are elimi nated at the FC, is propoed to improve the detection performance in impulsive noise. It is shown that substantial detection performance can he obtained. • We study a system model in which the reporting channels between the SUs and the FC, and the channels within the cluster experience Rayleigh fading. The results show that in contrast to the Gaussian noise case, the cluster-based detection schemes do nOt always outperform the conventional DFC detection scheme.  1.4  Thesis Organization  The thesis is organized as follows. Chapter 2 provides a review of related works oii CCS with Gaussian noise model. Chapter 3, the problem of single SU detection in mixture Gaussian and Laplacian noise environment is discussed. Detection perfor inance using DFC, EFC, and LRT detection in a CR system is evaluated in Chapter 4. In Chapter 5, a scenario in which the reporting channels between SUs and the FC, and the channels between any two S Us within the same cluster experience Rayleigh fading is studied. The main contributions of the thesis are summarized in Chapter 6, which also includes some topics suitable for future research.  4  Chapter 2 Background and Literature Review In this chapter, related works on CCS are summarized. We introduce the system model and illustrate the performances of various sensing methods.  2.1  Cognitive Radio  As mentioned in section 1, recent study has shown that most of the licensed fre quency bands are poorly utilized [4], resulting in spectrum holes. As defined in [6], a spectrum hole is a band of frequencies assigned to a PU, but at a particular time and specific geographic location, the band is not being utilized by that user. Spectrum utilization can be significantly improved by allowing SUs to access the spectrum  holes. Cognitive radio [14], has been proposed to enable frequency reuse  and improve spectrum efficiency. It is a low cost and highly flexible wireless device that has three key aspects [15]: • Sensing ability to identify spectrum holes. • Ability to change frequency and spectrum shape to fit into the spectrum holes. • Non-interfering transmission to the PU.  5  2.2  Spectrum Sensing  Let 1 (H denote the event that the PU is inactive (active). The received signal 0 H ) at SU  j is [16] 1is(t) +‘n(t), =  ui  (2.1)  Mo where  j  =  1, 2,  ...,  N, N is the total number of SUs in the CR, system, s(t) is the  PU’s transmitted signal, h is the amplitude gain of the channel between the PU and the SU  j, and  n(t) repreents the noise process.  In spectrum sensing, we are interested in minimizing probability of miss detec tion for a given probability of false alarm. A  miss  detection occurs if the PU is  active but the detection decision is that PU is not active, i.e.  Pm  =  JH 0 Pr{H } 1 . A  false alarm occurs when the PU is not active but the detection decision is that it is active, i.e. Pf 1  —  =  Ho;. The probability of detection Pd, which equals to 1 Pr{H  Pm,, indicates the level of interference protection provided to the PU because  whenever a PU is detected as active on a band, an SU may not use that band. The probability of false alarm indicates the fraction of time that a channel which is not used by the PU is also riot used by an SU.  2.3  Energy Detector  In a CR system, it is important to accurately determine if the PU is active or inactive on any channel. A variety of detection methods have been proposed, such as cyclo stationary detection  [71,  [8], [9], energy detection [10], [11], fast Fourier transforiri  detection [12], [13] and correlation detection [12]. Each method has its advantages and disadvantages. For most of these methods, the STJ reqmres prior knowledge of th.e PU’s signal format [8]. In practice, this is typically unknown and in such a case, energy detection is often used. For an energy detector, as shown in Figure 2.1, the input band pass filter first selects the center frequency  8 and the bandwidth of interest W to remove the out f  6  ()2  BPF  j zF  Jr  DideHorH  ThrèshidDevic9 Figure 2.1: Block diagram of an energy detector of band noise.  5 rfj  filter is followed by a squaring device to measure the received  energy and an integrator over an observation interval, 7. Finally, the output is compared against a threshold to decide the presence or absence of the PU. Each sample of the Gaussian noise process can be expressed in the form [17] n()  =  >  rij5hl’t.  ‘t)  (2.2)  i=-co  w 1 ore  ui —  When the PU is inactive, over the time interval (O,T), the noise energy can be approximated by a finite sum of 2 terms as [17]  =  where u  =  f  Q)dt 2 n  =  (2.3)  TW is the degree of freedom of a bandlirnited noise signal over a finite  observation period I. If we dfine  n.  =  n,i W 0 N  /  2.4  where N 0 is the one-side noise power spectral density and the decision statistic Y can be written as [17]  (2.5)  =  which is the sum of the square of 2’u independent Gaussian variates, each with zero mean and unit variance. Then, Y follows a central chi-square distribution with 2u degrees of freedom. When the PU is active, the decision statistic Y has a non-central chi-square distribution with 2i. degree of freedom  7  and a non centrality parameter  2r, where r is the channel’s average SNR [17]. Thus, we can describe the decision statistic as [16]  I  x (2r), xL,  1 H  U  (2.6)  ho  the pdf of Y can then he written as [1.6]  I  fy(y) =  (/), H 1 exp(-)f, 1  (2.7)  0 H  UF(Uyexp( ) 2 ,  where F(.) is the gamma function and I(.) is the vth order modified Bessel function of the first kind [18].  2.4  Local Spectrum Sensing  In a non-fading environment where h is non-time-varying, the probability of detec tion arid the probability of false alarm are given by [161 Pd  =  Pr(Y > A I H ) 1  I  Pf =Pr(Y > A  /)  =  ) 0 H  =  (2.8)  (2.9)  where P(x, y) is the incomplete gamma function, and Q,L(a, b) is the generalized Marcurn-Q function given by [16] Co  QU(a,b) =  I  (i)  exp  2±a (_x ) 2  I_i(ax)dx.  (2.10)  0 As expected, Pf is independent of SNR since there is no PU present under H hypothesis. When h is varying due to fading, the probability of detection can be obtained by averaging the SNR-coriditioned Pd over the fading statistics as  f  )f(x) dx  8  (2.11)  where f(x) is pdf of the instantaneous SNR. If the signal amplitude follows a Rayleigh distribution, then the SNR follows an exponential pdf as  j(r) where  =  exp  (-)  (2.12)  E[r]. In this case, a closed-form expression for Pd can be obtained as [16] u—2  /  —  1  /  i  Pd=exP-)Z--,)  x  2.5  =  exp  / 2(1+))  n  / —\ tl+r  u—i  +  u1N/____ 2 / u_ -exp  .!)  (2.13)  2(1+))  Collaborative Spectrum Sensing  In order to improve spectrum sensing performance, SUs may collaborate with each other by sharing their ohservtions. This is done by combining information from the SUs and making a final decision at the FC. Collaboration has many advantages: for example, as shown in Figure 2.2, when an SU experiences a deep fade, it may not he able to sense the PU’s presence, and may decide incorrectly that a frequency band is free. However, by combining sensing information from a number of SUs, the probability of missed detection. may be reduced. Depending on the cost of the reporting channels between the SUs and the FC, each SU can either transmit a one-bit decision or measured energy to the FC. In most cases, in order to minimize communication overhead, an SU may transmit a one-bit PU ON/PU OFF decision to the FC. If the K-out-of-N rule [16], [19], [20] is used, the FC declares the PU to he present if and only if at least K SUs have detected a PU signal. For K  =  1, we have the OR fusion rule and for K  =  N. we have the AND fusion rule. Such a collaborative scheme is known as DFC  detection and was originally used for radar detection applications in the 1940s and more recently in wireless sensor networks. When communication overhead is not concern, an SU can transmit its measured energy to the FC. By comparing the sum of energies of all collaborating SUs against a threshold, the FC makes an overall decision. This scheme is known as the EFC detection.  9  SnyIJr2  Ur  SendaW tI, 1  Figure 2.2: Collaborative spectrum sensing in a shadowed environment. In this case only the Secondary User 2 may be able to detect the PU Among the K-out-of-N fusion rules, the 1-out-of-N rule has been shown to provide the lowest Pm for a given Pf in AWGN when all SUs experience the same pass loss effect,  and  the performance degrades by increasing K [20]. Under this  assumption, the probability of detection and false alarm can be calculated because different SUs will have equal probability of detection and false alarm, resulting in an equal weighting of the individual decisions. The overall probability of detection arid false alarm are then given:by [16] =  =  () ()  P(1  P(1  —  -  p)fl_i  (2.14)  P  (2.15)  where Pd and Pf are the individual SU probability of detection and probability of false alarm, respectively. It is observed from the receiver operating characteristic (ROC) diagram in [20] that fusing the decisions from different SUs reduces the negative impact of fading. The performance of collaborative sensing scheme improves as we increase the number of collaborating SUs. This is because as the number of SUs increase, so does the probability that at least one user will have a channel better than the AWGN channel.  10  In [211, the authors note that when SUs have different SNR levels, collaborating all SUs in the network does not always achieve the optimum performance. Instead, optimal performance is achievd by collaborating certain number of SUs that have high SNR levels. In [22], a two-threshold detection scheme is proposed to reduce the bandwidths needed for reporting t;he collaborating SUs’ binary decisions when the number of SUs is large. In this scheme, only SUs with measured energy above an upper threshold or below a lower threshold send their one-bit decisions to the FC. It is observed that this scheme can achieve a performance almost identical to that of the conventional one-threshold detection scheme. Instead of transmitting a one-hit decision to the FC, a soft DFC detection scheme is proposed in [23] that transmits two bits to the FC. The detection performance is shown to he only slightly inferior to the EFC detection scheme. Many authors have shown that EFC detection outperforms DFC detection scheme in AWGN. Among linear diversity combining schemes, it is well-known that the maximal-ratio combining (MRC) provides the best performance [24]. However, MRC requires the prior knowledge of fading channel gains. With the same average SNR levels, equal gain combining (GC) performs only slightly inferior to MRC hut does not require knowledge of the channel gains. For simplicity, the FC often uses equal gain (EG) EFC detection scheme. With EG-EFC detection, the sum of 2u independent chi-square variates is just another chi-square variate with the same degree of freedom and non-centrality pa ramnieter equal to the sum of the 2u individual non-centrality pararnieters [17]. As a results, the decision variable can be described as [22]  {  X(2r),  1 H  N?), 2 X  . 0 H  (2.16)  The probability of detection and the probability of false alarm are  Qd =  Prob(Y  >  Iiii)  11  = QN?J(V’,  %/)  given  by [22]  (2.17)  =  Prob(Y  >  \IHo)  ( M fQ  (2.18)  f(Nu)  =  If the SNR fbllows the gamma distribution as  =  )  F(’V’u  fR(r)  =  N,.(jj)  exp(—), then  )fR(r)dr.  (2.19)  In [25], the authors propose a detection scheme in which the credibility of the observation is taken into account. In their problem, credibility is a function of the channel gain  Ii  and the distance between the SUs and PU. This algorithm leads to  a better detection perforrriance than EGC. The optimal spectrum sensing detection algorithm is based on LRT [26], in which the measured energies from all collaborating SUs are used. if we denote the measured energies by a vector: Y, then the likelihood ratio test function is fy(yi, Y2,  ,  yHj)  ,  y11o)  —  fy(yi, Y2, y,.  .  .  .  2.20  in the case that y are independent and identical distributed (lID), (2.20) can be rewritten as  =  where L(y)  =  fi L(y)  (2.21)  f (vjlHi) f(IHo)  In order to minimize collaborative spectrum sensing time, a detection scheme is proposed in [27] in which the SUs with higher SNRs are functioned as relays to forward their observations to the SUs with low SNRs. This method can effectively reduce the probability of miss detection and the detection time. So far, most works have focused on the scenario that observations are reported to the FC through perfect channels. However, in practice, the channels between collaborating SUs and the FC may be subject to fading as well. In [28], the reporting channels between SUs and th FC are assumed to undergo Rayleigh fading, and a cluster-based CCS scheme is proposed to improve the detection performance. The SUs with the largest reportmg channel gains collect the observations from all other  12  SUs within the sarrie cluster to make a cluster decision, arid then forward the decision to the FC. By employing this algorithm, the reporting error due to the fading channel can he greatly reduced and the detection performance can be significantly improved compared with the conventional algorithm which all collaborating SUs forward their observations to the FC to make a final decision.  13  Chapter 3 Single SU Detection of PU State in Mixture Gaussian and Laplacian NOises 3.1  Non-Gaussian Noises  For characterizing man-made iadio frequency (RF) noise and low frequency atmo spheric noise, the Gaussian noise model is not appropriate due to infrequent but  high level noise spikes. Commonly used non—Gaussian noise models include Lapla— cian noise, Cauchy noise, and mixture noise [29]. Relative to Gaussian noise, the tail of a non-Gaussian noise pdf typically decays at a lower rate. In this chapter, we assess the detection performance using mixture Gaussian arid Laplacian noise models. The Laplacian pdf is given by [29]  p(n)  1 =  exp  —/ nj  where u 2 is the noise variance. The pdf of mixture noise is [29]  14  ,  —cc <m < cc  (3.1)  x  ———  20  x  •  — —.  Gaussian impulsive impulsive impulsive  noise index index index  V  V  V  0.05 Ratio = 3000 0.05 Ratio = 25 0.8 Ratio = 25  = = =  V  V  15  10  Co C 0 t3 0  -  V  V  V V  Co  0  5 z  VVVVVV  XX  0  3.5  4.5  4  5.5  5 Data  Figure 3.1: pdf comparison of Gaussian and Mixture Gaussian noise. All pdf’s have a variance of 1  p(n) where a  e [0, 1],  Ti  =  (1  —  a)(n) + aF(n)  (3.2)  is a Gaussian pdf, and F is some other pdf with a heavier tail.  When F is Gaussian, p(n) is the pdf of mixture Gaussian noise (n2  p(n) = (1- a) where  +a  exp  exp  (-a)  (3.3)  is the variance of (n) and u- is the variance of F(n). The total noise  variance is (1  —  a)a + ao-.  Laplacian noise model has been used in ultra-wideband receiver design and in modeling impulsive noise. The mixture Gaussian model has been found to provide a good fit to empirical noise data, and the typical values for the noise variance ratio (NVR)  and a are usually in the range [20, 10000] and [0.01,0.33], respectively  [29]. The Gaussian and iriixture Gaussian pclfs for different values of NVR and a  15  are shown in Figure 3.1. Thè main difference is a heavier-tailed characteristic for the mixture Gaussian pdf due to noise samples that are large in magnitude. It is also observed that a smaller a and a larger N VR value correspond to a heavier tail. Normally, a heavier tailed noise has a larger degree of non-Gaussianity, also known as excess kurtosis, defined as  V  For a Gaussian pdf,  xi =  =  (3.4)  -3.  0, while the Laplacian pdf has  ii =  3. For the mixture  Gaussian pdf, we have  —  3a(i—a)[o—u] 2 ((1 a)o + auf-)  (3.o)  —  2_  =  F  The  xi  1 (1—a)+a*NVR  36  NVR (1—a)+a*NVR  (3.7)  values for several (a, NV R) values are Shown in the following table a  0.05 NRV xi  3.2  =  =  3000  56.25  NRV xi  a  0.05  =  =  25  16.96  =  NRV xl =  =  0.8 25  0.678.  PDFs of Energy Detector Output  In this section, we derive the pdfs of the output of an energy detector assuming mixture Gaussian noise under hypotheses ]-1 and H . 1 Let X denotes tile input of an energy detector, then random variable Y  =  2 X  has the CDF Fy(y)  =  Prob(Y  =  Prob  y)  =  2 Prob(X  /)  (IXI  16  y) (3.8)  Differentiating with respect to y, we can obtain the pdf of Y as  —3.2.1  Px(/)  px(m/)  +  (3.9)  2/  PU Inactive  When the PU is not active, the pdf of Y is py(y)  The CDF  (1-a)  =  of  Y  =  exp  (-)  exp  +  (-h)  (3.10)  is  /  exp  u  (—)  du +  a  1  f  exp  u  (—)  du.  (3.11)  This CDF function cannot he written the characteristic function of Y =  can  in a  simple, closed expression; l-iowever,  he determined as 1—a .  +  (1— J2l.’o)2  a (1—  (3.12)  2 j2voj-)2  If we define S as the sum of 2TW lID RVs 2TW  S  (3.1:3)  =  The characteristic function of 5, at the output of the energy detector, is s(jv)  =  i—a a + (1 —j2vo)t (1 —j2iu)r  2TW .  (3.14)  The inverse Fourier transform (3.14) yields the pdf  P8(Y) =  /[  2TW  exp(-jvy) di’.  +  17  (3.15)  3.2.2  PU Active  When the PU is active, the pdf of Y conditioned on the mean rn is py(y)  =  (1  1 2 a  +  (  1 exp J2iry  a)  —  x—m\ — 22  )  exp(’_x_m 2a  (3.16)  )  and the corresponding characteristic function is ‘y(jv)  1  —  a  exp  =  (1 —j2vo) -  (1  ‘i —j2i’a,  /  a  +  1’  )  jrni  (3.17) ‘\  2 xP ). e 2 . 1  2  j21’uF)  —  Proceeding in a similar way as in section 3.2.1, we can write the characteristic function of S as sCv)  1 =  —  a  I1  exp  ‘‘  j2L/) (1— j2vu jmzi N a exp 2 \1 —J l)UF/ 2 (1 —j2vu-)r —  (  +  (3.18) 2TW  and the pdf of S as 00  1—a =  ./(1 -j2uu)2  exp  7  jmv  “  _j2va,)  (3.19)  -00  +  a exp  (1 —j2u)  -  (  jrriv  N  } ) 2 T WexP(_jvY)dI/.  1  If the SI\R follows an exponential distribution, the pdf of S,  psf(y),  can he  calculated as 00  PSf(Y)  =  /Ps(Y)fR(r)dr  (3.20)  1)  where fJ?(’r)  =  2 r exp(—).  Since closed-forni expressions for (3.15) and (3.20) are not achievable, we eval uate the pdfs numerically. The results are plotted, along with simulation results, in  18  0. 0.14 0.12 0.1 0.08 0.06 0.04 0.02  20  40  60 Data  Figure 3.2: pdf for PU inactive: a  =  0.05, NVR  =  25  Data  Figure 3.3: pdf for PU active: a  19  =  0.05, NVR  =  25.  SV  O  U1?t{’J .IJp3UIS ig JOj aSlOiW  ‘kJiU?TSStfl?Q-UOU JO OOJ tJ S?91OU1 O  AN  Ofl? osiou mnoJcFwI ij& 2uou  ‘(  =  -j4pj  Rd  pajiod  Jo JJq?qoid oq’ ‘uopp uj  o.irii ui oou’uuoj.iod  ‘o  =  v) oiou  01{4 Ac[  DI 044  (oou)  pu /g pm Pj upuodsauoo  ?OUOUIUU  aSJ)UT pm?  v  UWIIIOJJ9CI  S9iO)p )M  001119 1rfflSSfl?Q 04  ktUT3iSSfli?Q-UOU JO oap iiUItS  oinb  1f1iM $OOUI?llUOJIOd 1101400 [Op 01.{4 ‘1I)0$  044  3oTd  poid  usn  1000  oq’j  pu’r  ots  $utoq  onl4x!m  tIIfli’400d$  U00M49  Rd 044  f 044  soououodxo  ui3Jodoi ot p ‘uTp?J opw uisuos oq sjg oj2us  ooa2 94Jfl90.I U014131flU119 p119  ouis jo  M404  U  pmnqo 2uiaodo  19uu?P JOAXU  ‘T4 wilru ii’’-’ o  044 ‘p oodxo sy  0.LJ Pu?  oom?uuopod DOll  !!qeqOJ 4 weie esi o 1 _0[  001.  = • .:  •.  000  =  elAN oo elAN 00  = =  9S!OU UB!OeIdel 1fl!M u!SSe 9JflXiW e ‘.fl!M U!SSe) 8JflXiW  —  —  —  9=eiAN 9Q=eqpMUe!SSeeJnXiW.—.—. OS!OU uelssneo  V  x .  V  \•  .  .  .  .  .  V  V  .  V.  “  ::::::‘%c.’  :  ::::::  :.  \  :  .  :  :  :  :  g  %%,• V  V..  V  V  ::::  3.3  Pearson Approximation  The motivation to use the Pearson approxi nation is to obtain approximate closed form expressions for Pm and Pf. In section 3.3.1 and section 3.3.2, we consider the mixture Gaussian and Laplacian noise models, respectively.  3.3.1  Mixture Gaussian Noise Model  According to the procedure described in [30], [31], a selection parameter  ii  is cal  culated based on the first four central moments as determined from the empirical data. Depending on the value of  ic,  the approximate closed-form expression is se is calculated using  lected among seven different possibilities. The value of i(/2+3)2  31)(22  .4(42  where  and  =  /32  =  —  3i  —  (391)  6)  are the skewness and kurtosis, respectively, in (3.21),  PN and UN denote the Nth central and noncentral moments, respectively.  Since the output of an energy detector is the sum of 2TW lID RVs X, where  j  =  1, 2,  .  .  ,  2TW. We calculate  /N  and  UN  as follows  2TW  N (X_v ) =E 1  (  =  N ,K 1 k . 2  .  ,  k2T7+1  )  E[(X1]E[(X2]  .  .  .  (3.22)  =zE (X) 1 ij (3.23)  E[(X)j + E[(X)] + k] 2 where E[X  =  f  kfx(x)dX. 2 X  21  ...  + E[(X,)]  ‘When the PU is inactive, X  where ‘a denotes the noise process, and we  = ‘ny,  have  1  L[X 9k  1  2k  a  —  a  ± exp  ‘  2 (—n  dn (3.24)  (1  With a  =  —  a)  0.05 and NVR  —29. Since —oo <  =  2k!: 2kk! ! (°) + 1  t  = 25, 112 =  (CF)  1.9e + 2,  le + 4,  =  =  9.5e + 5, and  <0, e use a Type one approximation [31] C(1 + ‘(1  fz(z) =  Z)62  —  1 b  z  (3.25)  0, where b 1 s  =  ()  = sign(/L ), 3  t = 6(4  2 =b b 0 (),  Oi  =  0.5[t  —  =  0 2 + 16(r+ 1), b i(r+2)  2— s*t + 2)j, 02  — i)(6 + 3 — 22)  —  — b1, 0  =  0.5[1;  and G  =  2 + s*t(t + 2),j,  02  Oi ,  —  =  ) (b i 2 +o2) ((oj  After equating the mean of (3.25) to that of Z, the resulting pdf is given by  fz(z) where 6  =  ii  1 +b  G(1 +  )ei(i —  =  J,  z  (3.26)  —  The CDF of Z can be shown as [31] 0, Fz(z) =  blGopU21_(O1+l)B(Z)(0l +  +S 1 z<—b 1,02 + 1),  1+ö < —b  1,  /b and 1 b , 2  x(z) =  (3.27)  2± b  + 2 z>b  where B()(.,.) is the incomplete beta function, with 1 p’ =  z <  = p’/p, p =  1 + 2 /b 1 (b ) ,  /(1 +  When the PU is active, through a fading channel, X denotes a Rayleigh pdf, and we have  22  =  (h +  2  where h  k] 2 E[X  =  kj 2 E[(h + n)  =  E  (2k)  (3.28)  (2k)  =  where [32]  1.3.  I 1 I  E[n]  With a i  =  ]E[nJ 2 E[h  -  E[h]  and  h2mnh]  =  (1  —  a)  q! 2)!  •pa’•  p= 2rn+1  , 2m.m!.u m 2  p=2m  (u) 1  +a  g 2(  (-)  0.05 and iVVR  6. Since 1  =  25,  =  .  .  .  2m  =  /2  6.7e+002, ,u  =  3.6e+004, p.  =  4.48e±006  cc, we use a Type SIX approximation [31]  <i <  0.5, 6  2,4.  (3.30)  I  Go(z  a)°2z°1,  —  a  r(r±2)  z  <  cc  (3.31)  otherwise  0, =  =  q=1,3...,2m+1.  0,  .tz(z)  where a  q  .  (3.29)  +(_l)(2), i  =  1,2, r  =  62—uli—1)(6+3l—  1))i, and=i3i(r+2) _1)F(O 2 =aO1_O2iP(8i)(F(0i_0 282)_1, 0 +16(r+1). 2 o +  After equating the mean of (3.30) to that of Z, the resulting pdf is given by fz(z)  = fz(z  —  (3.32)  ö)  where (3.33) The CDF of Z can he shown as [31]  Fz(z)  =  1 1  1—  CoaG2°1B(0i  —09—1,02 + 1). -  0,  a+ < z <cc  —  (3.34)  z<a+8.  We adopt the same system model described in section 3.2. Figure 3.5 plots the pdf of the output of an energy detector for PU ON using (3.20) and (3.32)  23  0.03  0.025  IL 0  0  20  40, 60  80  100 120 Threshold  140  160  180  Figure 3.5: pdf comparison of the output of an energy detector using (3.20) and (3.32): a  =  0.05, NVR  =  25 and PU ON  0.2 0.18 0.16 0.14 0.12 Li  O0.1 0  0.08 0.06 0.04  20  40 Threshold  Figure 3.6: pdf comparison of the output of an energy detector using (3.15) and (3.26): a  =  0.05, NVR  =  25 and PU OFF.  24  0.035 —  e  —  Pearson Method Numerical Method  0.03  0.025 0.02 U 0  0.015 0.01  0.005  100 Threshold  50  Figure 3.7: pdf (3.32): a  =  comparison of the output of  0.8, NVR  =  150  an energy detector using (3.20) and  25 and.PU ON  0.09 Pearson Approximation Numerical Method  0.08 0.07 0.06  LI.  U 0  0.05 0.04 0.03 0.02 0.01 n ‘O  Figure 3.8: pdt cornpanson (3.26): a  =  0.8, NVR  mmmmxmxc 100 40 60 80 Threshold  20  =  of  the output  25 and PT.] OFF.  25  of  an energy  (ietector  using (3.1.5) arid  100  _::::::::I::::::I ‘:::::::::::::::::::::::::.:  10  ::: ::::.:::::  E  ::  ‘—O—NumericalMethod Approximation  .::  CO CO Cl)  2  10  ...  >  CO  2 i0  io  :::  0  20  .::::.  40  ::::::  60  .  .:  :  :..  80  100  Threshold  Figure 3.9: Pf comparison using (3.27) and pdf from (3.15): a  10  =  0.8, NVI?  =  25  =  25.  0 .  Numerical Method Approximation  I 10  7 10  Figure 3.10:  i  .3  .  .  10  20  Pm  .  I  —  0  30  40  50 60 Threshold  70  80  comparison using (3.34) and pdf from (3.20): a  26  90  =  100  0.8, NVR  10° ‘—0— Numerical Method Approximation  :‘  i  10l  10  2  102  10_i  10°  Probability of false alarm  Figure 3.11: ROC performance comparison using Figure 3.9 and Figure 3.10: a = 0.8, NVR =25 with mixture Gaussian noise (a = 0.05, AVR = 25). It is observed that when the PU is active, the Pearson method yields a reasonable pdf approximation; however, when the PU is inactive, using (3.15) and (3.26), Figure 3.6 shows that the Pearson method is ineffective. The reason is because when the PU is inactive, with this set of mixture Gaussian noise parameters, the first four central moments cannot provide enough information to lead us to the correct type of approximation. When a is increased to 0,8, mixture Gaussian noise becomes less impulsive. Figure 3.7 plots the pdf of the output of an energy detector for PU ON using (3.20) and (3.32) with mixture Gaussian noise (a = 0.8, NVR = 25). We can see that when the PU is active, the Pearson method is not very accurate for small threshold value. This is because the Pearson method does not guaranteed the approximation start from origin  [311. Figure 3:8 plots the pdf of the output of an energy detector for  PU OFF using (3.15) and (3.26) with mixture Gaussian noise (a = 0.8, NVR = 25). It is observed that when the PU is inactive, the Pearson method yields a good pdf approximation.  27  Figure 3.9 plot the Pf using (3.27) and the pdf from (3.15). We can see Pf by the Pearson method starts to lose accuracy for the threshold value above 40. Figure 3.10 plots the Pm using (3.34) and the pdf from (3.20). it is observed that  ‘m  by  the Pearson method performances bad for threshold values smaller than 10. Figure 3.11 plots the ROC performance using Figure 3.9 and Figure 3.10. As seen, the detection performance by the Pearson approximation works well for P smaller than 0.1. In order to improve the detection performance for Pf greater than 0.1, we conic up with a separation method as described in the following. The approximation of the pdfs using (3.26) and (3.32) are divided into two parts from a threshold value P of an energy detector. The right hand side is approximated using Pearson and the left hand side is approximated by a polynomial function. Further study suggests that good approximation requires a polynomial of at least a degree of four. However, a general expression for such polynomial is difficult to obtain; therefore, we use a line to simplify the problem. We set P = Pp+Pr where 5 and P are the threshold values corresponding to the pdf peak using = Pearson and numerical method, respectively. Figure 3.12 plots the ROC performance —  using the separation method, the Pearson method, and the numerical method. It is observed that the detection performance can he significantly improved. In order to better illustrate the usefulness of Pearson approximation with mix ture Gaussian noise, in Figure 3.13, we plot the impulsive index a as a function of noise variance ratio NVR for  ic  equals to 1 when PU is inactive. As seen, Pearson  is applicable for a greater than 0.7.  3.3.2  Laplacian Noise Model  Following the same procedures, in this section, we plot the pdfs and CDFs of the output of an energy detector with Laplacian noise model. Figure 3.14 plots the pdf of the output of an energy dtector for PU ON using (3.32) and simulation. We can see that when the PU is active, the Pearson method with Laplacian noise has the same accuracy issue as mentioned in Figure 3.7. Figure 3.15 plots the pdf of the output of an energy detector for PU OFF using (3.26) and simulation. It is  28  10° •  —0— Numerical Method Pearson Approximation + Separation Method  (I)  .  a)  .  .  .  .  1101  :::: 10  —.  10_I Probability of false alarm  102  10°  Figure 3.12: ROC performance comparison of Pearson approximation, Separation method, and the numerical method: a  =  0.8, NV]?  1t0  =  25  I  900 800 700 600 >  500  :  400 300  :  . .  •  200  100  •:  0 0  0.1  0.2  I  I  I  I  I  I  0.3  0.4  0.5 a  0.6  0.7  0.8  Figure 3.13: a as a function of NVR for  29  , =  0.9  1.  U  Threshold  Figure 3.14: pdf comparison of the output of an energy detector using (3.32) and simulation: Laplacian noise and PU ON observed that when the PU is inactive, the Pearson method with Laplacian noise yields a good pdf approximation. Figure 3.16 plots the Pf using (3.27) and simulation. Figure 3.17 plots the using (3.34) and simulation. As seen, the Pf and P 7 with Laplacian noise model exhibit similar qualitative performance as the mixture Gaussian noise model (us cussed in Figure 3.9 and Figure 3.10. Figure 3.18 plots the ROC performance using the Pearson method, Separation method, and the numerical method. it is observed that the overall detection performance using Pearson approximation works well for Pf smaller than 0.1. After applying the separation method, detection performance for Pf greater than 0.1 can be greatly improved.  30  0.09 Pearson approximation Simulation results  0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01, 0.0  20  40  60  80  100  Threshold  Figure 3.15: pdf comparison of the output of an energy detector using (3.26) and simulation: Laplacian noise and PU OFF 100  Pearson approximation —0— Numerical method  10_i  :.:::::::::;.  CO CO  G)  .  0 Co  o 10  :  -2  .  :::::.::::::::::::::::::  ::::.:: ..  CO  .  ::: .  .  2 1o  io  :::::::..::;:::::  0  20  40  60  80  100  Threshold  Figure 3.16: Pf comparison using (3.27) and simulation: Laplacian noise.  31  .: 10  i I 3 0 _ 0  ::::  :  :  50  100 Threshold  150  200  Figure 3.17: Pm comparison using (3.34) and simulation: Laplacian noise  102  10_a  io  102  100  Probability of false alarm  Figure 3.18: ROC performance comparison of Pearson approximation, Separation method,  arid numerical method: Laplacian noise.  32  Chapter 4 Collaborative SU Detection of PU State in a CR System 4.1  DFC Detection  As mentioned in chapter 2, CSS has been proposed to improve the detection per formance [16]. Among the K-out-of-N fusion rules, the 1-out-of-N rule has been shown to provide the lowest Pm, for a given Pf in AWGN when all SUs experience the same pass loss effect, and the performance degrades by increasing K; there fore, the FC should always declare PU ON if any SU detects the PU activity [20]. However, the i-out-of-N rule ‘does not always provide the lowest P, for a given Pf among K-out-of-N fusion rules in a non-Gaussian noise environment. In this section,  we evaluate the performance of DFC detection in mixture Gaussian and Laplacian noise. In addition, since non-Gaussian noises exhibit spikes with large magnitude, it would be expected that the detection of the PU state should take those spikes into consideration so that the, Pf is not excessive. We will show that by eliminating large magnitude SU energy measurements at the FC, and then make decision on the PU state using the K-out-of-N fusion rules, the detection performance can he significantly improved for heavier tailed noises. Let Qd and  Q.  denote the CR system probability of detection and probability  of false alarm, respectively. When all SUs experience the same pass loss effect, we  33  have [16]  ()  Qd=  P(l  (‘Y)  =  (4.1)  -  -  Pf)  (4.2)  i=K  where N is the total number of collaborating SUs, Pd and Pf are the individual SU probability of detection and probability of false alarm, respectively. We consider a CR system with one PU, seven collaborating SUs and a FC. The sensing channels between the PU and the SUs experience Rayleigh fading and have an average SNR of 10dB. The reporting channels between the SUs and the PC are assumed to be perfect. Figure 4.1 plots the ROC performance using (4.1) and (4.2) with mixture Gaus sian noise (a  =  0.05, NRV  =  3000). It is observed that in contrast to the Gaussian  noise case, the 1-out-of-N rule does not provide the lowest P for a given Pf, it actually provide the highest  rn 3 ‘  We also observe crossovers when using K-out-of-N  fusion rules. The reason is because the Gaussian noise samples are usually within a few standard deviations of the mean; thus, energy measurements which are sig nificantly differ from the mean can be interpreted as resulting from the PU signal. However, with this set of non-Gaussian noise parameters, large energy measurements occur that are entirely due to the noise spikes, leading to excessive Pf when using the 1-out-of-N rule. As K increases, the K-out-of-N fusion rules start to reduce the effect of the noise spikes, and the ROC performance is improved. Figure 4.2 plots the ROC performance using (4.1) and (4.2) with mixture Gaus sian noise (a  =  005, NRV  =  25). When NRV decreases, the noise becomes less  impulsive. It is observed that the K-out-of-N fusion rules for this set of noise pa rarneters also result in crossovers. We can see that as the Pf decreases, the best fusion rule gradually changes from 1-out-of-N to N-out-of-N. This can be explained intuitively that when the noise becomes less impulsive, the effect of the noise spikes decreases; therefore, leading tO a decrease in  34  PTr,  for a given Pf.  10  ••••  5  — —  10  —  —  io’ io  —  \  DFCloütof7rule :::: DFC3outof7rule DFC4outof7 rule DFC5outof7rule DF 6 :jt 7 rule DFC2outof7rule DFC7outof7rule  ! :  cf  3 iO_  :  :::::  ::•:::•:•  :‘%  .::  ..  .1;.:,!,  10_i  10  10°  Figure 4.1: ROC performance of DFC with K-out-of-N fusion rules using (4.1): mixture Gaussian noise (a  =  0.05, NVR  =  3000)  Figure 4.3 plots the ROC performance using (4.1) and (4.2) with mixture Gaus sian noise (a  =  0.8, NRV  =  25). With this set of noise parameters, the noise  becomes even less impulsive. It is observed from the ROC performance that sim ilar to the Gaussian noise case, the 1-out-of-N rule start to outperform  the rest  K-out-of-N fusion rules, and the performance degrades by increasing K. Figure 4.4 plots the HOC performance using (4.1) and (4.2) with Laplacian noise. Since Laplacian noise has a degree of non-Gaussianity between mixture Gaussian (a  =  0.8, NRV  =  25) and mixture Gaussian (a  =  0,05, NRV  =  25), it would  be expected that the K-out-of-N fusion rules should result in crossovers. As seen, 2-out-of-N provides the lowest Pm for Pf smaller than 0.1, and 3-out-of-N provides the lowest P, for Pf bigger than 0.1. In the following, we propose  an  algorithm in which large magnitude SU energy  measurements are eliminated at tim FC first, and then the decision of the PU state is  made using DFC it-out—of—N fusion rules. Since all rules exhibit similar qualitative  35  :°  5)  10 ::‘,  o  Co Cl)  o  .  10  .  ::  >, • .  O  0  DFC with 7 out of 7 rule  r  p  10  5  —  —  —  — —  DFC DFC DFC DFC DFC  with with with with with  10 1o 4  1  5outof7 4 out of 7 3 out of 7’ 2 outof7 1 out of 7  rule rule rule :: rule :;: rule ,.  •::  .:.::,  : :::  : :  102 Probability of false alarm  10_i  : :  100  Figure 4.2: ROC performance of DFC with K-out-of-N fusion rules using (4.1): 0.05, NVR  mixture Gaussian noise (a  =  25)  100  ‘.,  210 —  — —  DFCwith1outof7rule:::::::::::::::::::;:’;:::::: DFCwitb2outof7 rule : : ::::::::::: DFC with 3 out of 7 rule • DFC with 4 out of 7 rule “ \ : : : : : DFCwith5outof7 rule DFC with 6 out of 7 rule : DFCwith7outof7 rule •  — —  0 I  \.  10_i 102 Probability of false alarm  10°  Figure 4.3: ROC performance of DFC with K-out-of-N fusion rules using (4.1): mixture Gaussian noise (a  =  0.08, NVR  =  36  25).  10  . .‘.  ,•..•..  ..,  ..,.•.,.,.  _i 10  102 B  —  3 io 10  DEC lout of7 rule —DFC2outof7rule DFC3outof7rule — . DFC4outof7rule 0 DFC5outof7rule DFC6outof7rule DEC7outof7rule ::::  —  :  :  :::::  : :  .  10  1 02 Probability of false alarm  io  100  Figure 4.4: ROC performance of DFC with K-out-of-N fusion rules using (4.1): Laplacian noise performance, we use the 1-out-of- N rule as an illustrative example. Figure 4.5 plots the ROC performance when eliminates the largest m energy measurements with mixture Gaussian noise (a 1, 2.  . . . ,  N  —  =  0.05, NVR  =  3000), where rn  =  K. It is observed that substantial performance improvements can be  achieved by increasing m. This is because with this set of noise parameters, the effect of the noise spikes are tremendous; therefore, Pm decreases as rn increases, leading to a better detection performance. Figure 4.6 plots the R.OC. performance when eliminates the largest rn energy measurements with mixture Gaussian noise (a 1,2,.  . . ,  =  0.05, NVR  =  25), where m  =  N— K. We can see that with this set of noise parameters, the best detection  performance is gradually obtained from no eliminate to eliminate N  —  K energy  measurements. The reason is because when the noise becomes less impulsive, the effect of the noise spikes decreases; therefore, eliminating more energy measurements may increase the possibility of eliminating a PU signal, leading to an increase in  37  m  2  —  10  .  —  I outof7 rule (Drop Ilargest term) : 1 out of 7 rule (Drop 2 largest terms) —1 out of7 rule (Drop 3largestterms):::::. o 1 out of7 rule (Drop4largestterms)::::: 1 n 1 out of 7 rule (Drop 5 largest terms) I out of 7 rule (Drop 6 largest terms).:: :: DFCloutof7rule :::::  —  —  -  10  io  102 Probability of false alarm  .  :.:.i.. .  : I :::  ::  10_i  10°  Figure 4.5: ROC performance of the proposed algorithm, using DFC i-out-of-N fusion rule: mixture Gaussian noise (a  =  0.05, NVR  =  3000)  for a given Pf. With mixture Gaussian noise (a  =  0.8, ATVR  =  25), we found that the proposed  algorithm is not applicable. The reason is because with this set of noise parame ters, the noise becomes Gaussian-like; therefore, eliminates energy measurements is equivalent to reduce the number of collaborating SUs in the CR system. We find that eliminating rn largest energy measurements using K-out-of-N fu sion rule gives almost identical performance as eliminating in  —  1 largest energy  measurements using K + 1-out-of- N fusion rule. For example, eliminating 2 largest energy measurements using 1-out-of-? fusion rule yields almost the same detection performance as eliminating 1 largest energy measurement using the 2-out-of-7 fusion rule. This is because one of the two measurements dropped by the 1-out-of-? fusion rule is almost always above the threshold value on the 2-out-of-7 fusion rule.  38  100  ----—-..  .-.  .-.Q--.  ‘,...  .  s  .  .  -.  .  .  .  :  \  —‘.  \:  :  \  :  :  .  \  o  :  -1  .  :  :  . .  .  ‘..  .  :  \: \ :  .  .  :  E  (0 •  210  :  ‘V  ci)  :::::::  .  .  —2  —1 outof7 1 out of 7 — —. 1 out of 7 0 1 out of 7 a 1 out of 7 1 out of 7 DFC with  —  —  rule (Drop 1 largestterm) rule (Drop 2 largest terms) .:: rule (Drop 3 largest terms) rule (Drop 4 largest terms) . :• rule (Drop 5 largest terms) rule (Drop 6 largest terms) : 1 out of 7 rule  \...  \  -  .  ..  : ...  \  —‘  \  ‘\  \.  ..  102 io Probability of false alarm  100  Figure 4.6: ROC performance of the proposed algorithm using DFC 1-out-of-N fusion rule: mixture Gaussian noise (a  =  0.05, NVR  39  =  25).  4.2  EFC Detection and LRT Detection  Let O derìote the energy measured by SU  {  j, and w denote the corresponding  weighing factor. The objective is to decide between the following two hypotheses  B  1, 0,  By comparing the  weighted  3 0 7 Zw  A,  1 if (4.3)  j=1  =  otherwise,  H)  sum energy of all collaborating SUs against a threshold  A, the FC makes an overall decision on the ON/OFF activity of the PU. When all weighing factors ‘w are equal, this scheme is known as EG-EFC detection. Many authors have shown that EFC detection outperforms DFC detection scheme, and the optimal detection scheme in AWGN is based on the LRT. Our simulation results show that in a non Gaussian environment, EFC detection does not always outper form DFC detection, hut the LRT is still the optimal. We also study the detection performance of the proposed eliminating algorithm in Section 4.1. Figure 4.7 plots the ROC performance using EFC and LRT detection schemes with mixture Gaussian noise (a  =  0.05, NRV  =  3000). It is observed that in  contrast to the Gaussian noise case, the EFC detection does not outperform DFC detection, and the performance difference between the EFC detection and LRT detection is substantial. The reason is because for the Gaussian noise case, the sum energy of all collaborating SUs average out the effect of the Gaussian noise; however, a single large magnitude non-Gaussian noise sample can dominate the sum energy of all collaborating SUs. Therefore increasing the number of collaborating SUs in the CR system  can lead to  an  increase in the probability that a large magnitude  noise sample is observed. Figure 4.8 plots the ROC performance using EFC and LRT detection schemes  with mixture Gaussian noise (a  =  0.05, NRV  =  25). With this set of noise pa  rameters, we can see that the performance difference between the EFC detection and LH.T detection is getting smaller. This is because when the noise becomes less impulsive, the effect of the noise samples also decreases.  40  10  ‘r-,-  —  DFCloutof7rule : :: —DFC2outof7 rule :: DFC3outof7rule s5 DFC4outof7 rule 0 DFC5outof7rule::::: a DFC6outof7rule Sf I EFC :: Likelihood Ratio DFC7outof7 rule :.:.::  ‘  ::: s.:  —  102  io  :  :.:•  :.:V.,:.....  ::::::  :: :.  10_i  102  10°  Probability of false alarm  Figure 4.7: ROC performance comparison among EFC—EG, LRT and DFC using K-out-of-N fusion rules: Mixture Gaussian noise (a  41  =  0.05, NVR  =  3000).  2  :  DFC with 1 out of 7 rule DFC with 2 out of 7 rule DFC with 3 out of7 rule DFC with 4 out of 7 rule —8--— DFC with 5 out of 7 rule —e--—DFCwith6outof7rule 2 10 • —“— DFC with 7 out of7 rule —  —  •  —  I  —  :  V  —  V  V  -  V  io  100  10 Probability of false alarm  Figure 4.8: R.OC performance comparison among EFC-EG, LRT and DFC using K-out-of-N fusion rules: Mixture Gaussian noise (a  =  0.05, NVR  =  25)  100  VV:VVVVVV  ::.:....:.  —  —  VV.V...VVVVV..V. :V  V:..::  .  V4  —  .  0 V  :  : : : ::: : : :  : :.  :..  with I out of7 rule —DEC with2outof7rule DFCwith3outof7rule DEC with 4 out of 7 rule DEC with 5 out of 7 rule DEC with 6 out of 7 rule DEC with 7 out of 7 rule Likelihood Ratio  V  V  —.  .— —  1 0i  102 10_i Probability of false alarm  100  Figure 4.9: ROC performance comparison among EFC-EG, LRT and DFC using K-out-of-IV fusion rules: I\’lixt;ure Gaussian noise (a  42  =  0.8, NVR  =  25).  10  .  o  ,.  :-..  : : :  : :  ‘::.:  ..-...  102  :  : :  • •  : :  L DFCloutof7rule ———DFC2outof7rule DFC3outof7rule DFC4outof7 rule 0 DFC5outof7rule DFC 6 out of 7 rule “ DFC7outof7 rule ‘ EFC LikelihoodRatio  10_i Probability of false alarm  10°  Figure 4.10: ROC performance comparison among EFC-EG, LRT and DFC using K-out-of-N fusion rules: Laplacian noise Figure 4.9 plots the ROC performance using EFC and LRT detection schemes with mixture Gaussian noise (a  =  0.8, NRV  =  25). With this set of noise pa  rameters, it is observed that the detection performance is similar to Gaussian noise case, where the EFC detection outperforms the DFC detection, and the difference between the EFC detection and the LRT is small. Figure 4.10 plots the ROC performance when eliminates the largest rn energy measurements with mixture Gaussian noise (a 1, 2  N  =  0.05, NVR  =  1, 2,.  .  .  3000), where m  =  K. Figure 4.11 plots the ROC performance when eliminates the largest  .—  m energy ineasurerrierits with mixture Gaussian noise (a m  =  ,  N  —  =  0.05, NVR  =  25), where  K. As seen, they exhibit similar qualitative performance as shown  in Figure 4,5 and Figure 4.6.  43  2 —  — —  — —  0  •  Energy Energy Energy Energy Energy Energy Energy  Fusion Fusion Fusion Fusion Fusion Fusion Fusion  .:  (Drop (Drop (Drop (Drop (Drop (Drop  io  :  :::  ilargest term) 2 largest terms) 3 largest terms) 4 largest terms) 5 largest terms) 6 largest terms)  .1  :  :.::: :1:  .:  .  -  10  102  10  100  1  Probability of false alarm  Figure 4.11: ROC performance of the proposed algorithm using EFC detection: mixture Gaussian noise (a  0.05, N VR  =  =  3000)  100 .  ,..., ..  : :-c--.  io  ...  -.  ::::  .:.  .  ..::::::::::::...:::::::,:\x .E::%.s  a:.  •\  •••\ \  ..  .  0  :-.  .  ‘\  \.  —.:  .  .  ‘<•. .0  210  .‘  ::.:  ‘4. -2  .  :  . .  Energy — — — Energy Energy • • — —. Energy —e--— Energy — e — Energy Energy  1o  Fusion Fusion Fusion Fusion Fusion Fusion Fusion  .  .  .  (Drop (Drop (Drop (Drop (Drop (Drop  1 2 3 4 5 6  . :.  .  largest term) largest terms) largest terms) largest terms) largest terms) largest terms)  10_2  -...  .  .  ,  :. .  ....  -..  ‘  •  .  ..  -.“.  .  :  :  10_i  .  10°  Probability of false alarm  Figure 4.12: ROC performance of the proposed algorithm using EFC detection: mixture Gaussian noise (a  =  0.05, NVR  =  44  25).  Chapter 5 Imperfect Reporting Channels In this chapter, we investigate the ROC performance of sensing schemes with im perfect reporting channels between the SUs and the FC and among SUs. A clusterbased collaborative spectrum sensing scheme is introduced in Section 5.1, arid the performance in mixture Gaussian noise is discussed in Section 5.2.  5.1  Cluster-based Collaborative Spectrum Sens ing  Most works assume that the reporting channels from the SUs to the FC are perfect, i.e. noise-free and unlimited bi rate. However, in practice, the channels between the SUs and the FC may be subject to fading and noise as well. In [28], the reporting channels between the SUs and the FC are assumed to undergo Rayleigh fading, and a cluster-based CCS scheme is proposed to improve the HOC performance. Within each cluster, the SU with the largest reporting channel gain, hereafter referred to as a cluster head, collects measurements from other SUs in the cluster through perfect channels to make a cluster decision, and then forwards the decision to the FC. If each SU within the same cluster sends a binary decision to the cluster head and the T fusion rule, we have cluster-based DEC. cluster decision is made using a K-out-of-A If the cluster decision is based on the sum energy of all collaborating SUs within the  45  Figure 5.1: Cluster-based CCS in CR system same cluster, we have cluster-based EFC. We assume that clustering has been done using the algorithm proposed in [33], [34] and a system block diagram is shown in Figure 5.1. The  Qj  and  Q,.,  can be computed as [28]  Qj =1-111(1- Qj.j)(1 Qrn =  fl[m,(i  —  Qe,i)  -  Qe,i) + Qf,iQe,i]  + (1  where K is the number of clusters, Qf,j and  —  Qrn,i  (5.1)  (5.2)  Qrn,i)Qe,i]  denote the probability of false  alarm and probability of miss detection of the cluster head in cluster i,  Q  denotes  the probability that the cluster decision is incorrectly decoded at the FC due to the imperfect reporting channels. The performances of cluster-based DFC, cluster-based EFC, and conventional  DFC in which each SU forwards its binary decision to the FC and the overall de tection decision is made using a K-out-of-N fusion rules, are compared in [28]. The results indicate that by employing time cluster-based algorithms, the reporting error due to the fading channel can be greatly reduced; therefore, leading to a better ROC  46  10  ::::: 1::::: : :1  —  :  .2 lO  —  —  Cluster—based EFC: Conventional EFC  : : : :  :  a)  :  .:  V C’)  E  ::::::::: : : :  :•::  :  : : ::: : :::  :  : s  ...,  ‘__..•.%  ....  •.•.•2’%5.%  ..55  .........5  :1:  5  .. .•.  o >,  :::.:  102 :.  .  ..........  .  3 1o 1  :::::::.  :::::\  :  .  :  .\\.•  ..  :  :  ;::.:,:  1  102 Probability of false alarm  10_i  100  Figure 5.2: ROC performance comparison of conventional EFC arid cluster-EFC detection schemes performance. However, the best ROC performance is given by conventional EFC where all collaborating SUs forward their measured energies to the FC. A disadvantage of such a scheme is the high energy consumption compared with the cluster-based scheme in which oniy one SU within each cluster transmits during the reporting transmission. The large bandwidths required for the reporting channels is another issue. We adopt the same system model in [28], and Figure 5.2 plots the R,OC performance using cluster-based EFC and conventional EFC. As expected, it is observed that conventional EFC provides a lower P, for a given Pf. The assumption of perfect channels between SUs within the same cluster may he unrealistic; therefore, we now consider a model in which channels between any two SUs within the same cluster experience fading and study the performance with cluster-based DFC and cluster-based EFC. Figure 5.3 plots the ROC perforrriance with imperfect cluster channels using  47  100  ,,J..  .i.  I  : :  C  -.‘-  o C)  10  .  I,,  .  .  .  V  :  -1• .—.-  ..  ci  ......  ‘:..  V...;.  -o  Cl)  ..  E  :  ::.::  >  V..  VVVV V  :  .::::  .V.V. V VVç  V  .\‘  10  —2  . o  0-  .  .  Cluster—based DFC 5db cluster channels Cluster—based DFC1Odb cluster channels Cluster—based DFC 15db cluster channels Conventional DFC Cluster—based EFC 10db cluster channels Cluster—based EFC 5db cluster channels Cluster—based EFC 15db cluster channels :  :N.::  c  ‘  .  •  V•V  —  •  — —  —  —,  10  io  .  .  :  ::‘  : :  102 io Probability of false alarm  100  Figure 5.3: ROC performance comparison of cluster-EFC and cluster-DFC detection schemes with imperfect cluster channels cluster-based EFC and cluster-based DFC. We can see that the performance using cluster-based DFC can he significantly affected by the conditions of the cluster channels; however, the performance using cluster-based EFC tends to insensitive to the channel conditions. Also, it is observed that the perforrriance difference between cluster-based DFC and cluster-based EFC decreases as the average cluster channels SNR increases. The reason is because the sum energy of SUs can average out the effect of the Gaussian noise even in a low SNR detection environment.  5.2  Spectrum Sensing with Mixture Gaussian Noise  We adopt the same system model as described in [28], and the performance using conventional EFC, conventional DFC, and cluster-based EFC are studied assuming a mixture Gaussian noise model for three different set of noise parameters: (1) a 0.05 and NVR =3000; (2) a  =  0.05 and NVR =25; (3) a  48  =  0.8 and NVR =25.  =  10°  .  .  .,.  ,.  .•.  I : Conventional EFC ‘,\ c -\ Cluster—basedEFC :::\: Conventional DFC with 1 outof7rule:::’:::. :: :.:: \ Conventional DFC with 3 out of 7 rule \.,.;..:.:..:. ConventionalDFCwith4outof7rule\ :• Conventional DEC with 5 out of 7 rule : Conventional DEC with 6 out of 7 rule: Conventional DEC with 7 out of 7 rule : : Conventional DEC with 2 out of 7 rule .  -  10  -2  :  ..  I  o  \:  o I —  10  —  I  \\  —  102  10_i Probability of false alarm  10°  Figure 5.4: ROC performance comparison among conventional DFC, conventional EFC and cluster-based EFC: mixture Gaussian noise (a  49  =  0.8 and N VR  =  25).  Figure 5.4 plots the ROC performance using conventional EFC, conventional DFC and cluster-based EFC with mixture Gaussian noise (a  =  0.8, NVR  =  25). It  can be seen that similar to the Gaussian noise case, with this set of noise parameters, the cluster-EFC outperforms the conventional DFC with K-out-of-N fusion rules, and the best performance is given by the conventional EFC. Figure 5.5 plots the number of additional bits required for reporting transmission using cluster-based EFC as a function of the total number of the collaborating SUs with mixture Gaussian noise.(a  =  0.8, NVR  =  25 and a  =  0.05, NVR  =  3000).  Figure 5.6 plots the number of additional bits required for reporting transmission using cluster-based EFC as a function of the average reporting channels SNR with mixture Gaussian noise (a  =  0.8, NVR  =  25 and a  =  0.05, NVR  =  3000). It  is observed that the additional bandwidths required increase with the number of SUs and the average sensing channels SNR. This is because when the number of SUs and the average sensing channels SNR increase, so does the sum energy. We also observe that the additional bandwidths required increase as the noise becomes more impulsive. The reason is because as the noise becomes more impulsive, the magnitude of the noise spikes also becomes larger. Figure 5.7 plots the ROC performance using conventional EFC, conventional DFC, and cluster-based EFC with mixture Gaussian noise (ci.  =  005, NVR  =  25).  When a decreases, the noise becomes more impulsive. With this set of noise param eters, it is observed that the cluster-based EFC does not outperform conventional DFC with K-out-of-Al fusion rules, and the conventional EFC does not provide the lowest P, for a given Pf. The reason is because when the noise becomes more impulsive, the effect of the lare magnitude noise spikes also increases. Figure 5.8 plots the ROC performance using conventional EFC, conventional DFC, and cluster-based EFC with mixture Gaussian noise (a  =  0.05, NVR  =  3000).  With this set of noise parameters, the noise becomes even more impulsive. We can see that in contrast to the Gaussian noise case, conventional DFC with i-out-of-N fusion rule does not provide the lowest P,. for a given Pf, and cluster-EFC, along with conventional-EFC perform quite worse because a single large magnitude noise  50  rnIr19OOOQOOOO a=0.O5NVR=25 a=0.8NVR=25  11 U  0 10.5  •  .:  ‘  10  0  DCUU  0 -  00000000  V  V  V  V  V  9.5  9  poop  .  8.5  . :  .  8 p 0  10  5  :  .  15 NumberofSU  20  25  30  Figure 5.5: Number of additional bits versus total number of SUs: mixture Gaussian noise (a = 0.8,  NVR  =  25  NVR  and a = 0.05,  =  3000)  16 15  0  ‘  /  a0.O5andNVR=3000 a0.8andNVR25  14 Cl)  .  V  V  V  .  13 0  V  V V  _12  .  V  V  V  D0  oil  /  E10uoDUDDDDDp V  z  ,.  V  V  V V  ........pg  9  ‘QQQQ  8 7  0  5  Figure 5.6: Number Gaussian noise (a  =  of  .  10 15 20 25 Average sensing channels SNR in dB  V  30  additional bits versus average sensing channels SNR:  0.8, NVR  =  25 and a  51  =  0.05, NVR  =  3000).  mixture  —  ‘-e.  --  .%  ‘—  —_‘s,___. .  0  a, a,  .\.\ :\:  :::::: : .::‘—ç  io  .  .  \.•  .  ::.\  : :  :::  :\:  .:.ç\:.;.: ..\..:...:.:: .  ::::::  :‘.  :::\:::-  ‘S  10  2 —  —  —.  0  •  -  3 1o 3 i0  —  —.  ConventionalEFC Conventional DFC with Conventional DFC with Conventional DFC with Conventional DFC with Conventional DFC with Conventional DEC with Conventional DEC with Cluster—basedEEC  :.:“  1 2 3 4 5 6 7  out out out out out out out  of 7 of 7 of 7 of 7 of 7 of 7 of 7  rule rule ::: rule:: ;: rule rule: •: rule: : rule: : :  “  :  .  .  :: :;  :.  :::  : ::  :  1  : :‘  :-  :  :  :  \ :  —:  102 10_i Probability of false alarm  10°  Figure 5.7: ROC performance comparison among conventional DFC, conventional EFC and cluster-based EFC: inixture Gaussian noise (a = 0.05, NVR  52  =  25).  —  Conventional DFC with I out of 7 rule : —ConventionalDFCwith2outof7rule Conventional DFC with 3 out of 7 rule Conventional DFC with 4 outof7 rule:4:: :::::::. 0 Conventional DEC with 5outof7 wle :::.::::.: :.:i o Conventional DFC with 6 out of 7 rule Conventional DFC with 7 out of 7 rule :. Conventional EEC :: Cluster—based EFC  z  ...  —  10  -2 .  —  .  :  .  :  .  .  -  ‘l: .  .  .  ..  :  :  .  lo  102 Probability of false alarm  10_i  10°  Figure 5.8: ROC performance comparison among conventional DFC, conventional EFC and cluster-based EFC: mixture Gaussian noise (a  53  =  0.05, NVR  =  3000).  spikes can dominate the sum energy of all collaborating SUs.  54  Chapter 6 Conclusions and Suggestions for Future Work in this section, we summarize the main contributions of the thesis and identify some topics for possible future research.  6.1  Contributions of the Thesis  • Approximate closed-form expressions (3.26, 3,27, 3.32, 3.34) for the pdfs and CDFs of the output of an energy detector with Laplacian noise were obtained using the Pearson approxiiriation technique. • An optimal detection scheme based on the likelihood ratio test (LRT) for mixture Gaussian and Laplacian noise models was studied. Two sub-optimal algorithms, namely DFC detection and EFC detection, are also evaluated. The results show that in contrast to the Gaussian noise case, EFC detection does not always outperform  DFC detection and i-out-of-N fusion rule does  not always provide the lowest  Pm  for a given .Pf among K-out-of-N fusion  rules in a non-Gaussian noise environment. • An algorithm, in which large magnitude S U energy measurements are elimi nated at the FC, is proposed to improve the detection performance in impulsive  55  noise. It is shown that substantial detection performance  can  be obtained.  • We study a system model in which the reporting channels between the SUs and the FC, and the channels within the cluster experience Rayleigh fading. The results show that in contrast to the Gaussian noise case, the cluster-based detection schemes do not always outperform the conventional DFC detection scheme.  6.2  Future Work  • The detection schemes are based on the assumption that communication chan nels are time-invariant within every spectrum sensing period.  Fast fading  would be an interesting extension to the present work. • In the thesis, the locations of the PU and SUs are assumed fixed. Mobility issue would be another interesting extension.  56  Bibliography. [1] T. S. Rappaport. Wireless communications: principles and practice, 2nd ed. Prentice-Hall, 2001. [2] M. Steer. Beyond 3G. IE.E Microwave Magazine, vol. 8:76—82, February 2007. [3] Federal Communications Commission. Spectrum policy task force. Report of ET Docket 02-135, November 2002. [4] Federal Communications Commission. 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