PERFORMANCE METRICS, SAMPLING SCHEMES, AND DETECTION ALGORITHMS FOR WIDEBAND SPECTRUM SENSING. A Dissertation. Submitted to the Graduate School

Transcription

1 PERFORMANCE METRICS, SAMPLING SCHEMES, AND DETECTION ALGORITHMS FOR WIDEBAND SPECTRUM SENSING A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Zhanwei Sun J. Nicholas Laneman, Director Graduate Program in Electrical Engineering Notre Dame, Indiana December 2013

2 c Copyright by Zhanwei Sun 2013 All Rights Reserved

3 PERFORMANCE METRICS, SAMPLING SCHEMES, AND DETECTION ALGORITHMS FOR WIDEBAND SPECTRUM SENSING Abstract by Zhanwei Sun This dissertation studies the problem of wideband spectrum sensing for cognitive radio by partitioning it nto four fundamental elements: system modeling, performance metrics, sampling schemes, and detection algorithms. Each element can potentially couple individual channels, and appropriate designs of wideband spectrum sensing should consider the four elements jointly. We propose a p-sparse model to characterize the primary occupancy in a band of channels as a Bernoulli process, and suggest a pair of new performance metrics more appropriate for wideband spectrum sensing, specifically, the probability of insufficient spectrum opportunities P ISO and the probability of excessive interference opportunities P EIO. We suggest two narrower band Nyquist sampling schemes with correspondingly much lower rates than wideband Nyquist rate, i.e., partial-band Nyquist sampling (PBNS) and sequential narrow band Nyquist sampling (SNNS), and establish a unified sub-nyquist sampling structure, within which we study several important sub-nyquist sampling schemes in literature. We investigate the aliasing patterns inherent in sub-nyquist sampling and identify two extremes, specifically, uniform aliasing and periodic aliasing, and develop corresponding detection algorithms that allow tradeoffs between primary protection and secondary opportunities relevant to the goal of channel detection characterized P m, the probability of missed detection,

4 Zhanwei Sun and P f, the probability of false alarm, as well as the goal of wideband detection characterized by P ISO and P EIO. For performance metrics that couple individual channels, multi-channel detection algorithms have an advantage over channel-by-channel detection algorithms even for Nyquist sampling that give independent observations across channels. Most importantly, integer undersampling (IU), which corresponds to the simplest sub-nyquist sampling scheme, exhibits the best observed sensing performance in the regime of better protection for the primary system, i.e., the regime of low P M and high P F, or the regime of low P EIO and high P ISO, for moderate and high signal-to-noise ratio (SNR 0 db); on the other hand, SNNS exhibits globally best performance for low SNR (< 0 db) for the cases studied. These observations discourage studies on the design of more sophisticated sub-nyquist sampling schemes and development of more advanced sparse reconstruction algorithms to the problem of wideband spectrum sensing, since their performance is inferior to either IU or SNNS depending on the system parameters and the detection regime considered.

5 To my parents. ii

6 CONTENTS FIGURES vi TABLES x ACKNOWLEDGMENTS xi CHAPTER 1: INTRODUCTION Wideband Spectrum Sensing for Cognitive Radio Open Research Challenges Contributions of the Dissertation Outline of the Dissertation CHAPTER 2: BACKGROUND Cognitive Radio and Dynamic Spectrum Access Wideband Sampling Schemes Alias-Free Sampling Compressed Sensing Sparsity and Compressibility Compressive Measurements Sparse Reconstruction Sub-Nyquist Sampling Multi-Branch Sub-Nyquist Sampling Single-Branch Sub-Nyquist Sampling Spectrum Sensing Narrowband Spectrum Sensing Wideband Spectrum Sensing Primary System Model Performance Metrics Sampling Schemes Detection Algorithms Summary iii

7 CHAPTER 3: ELEMENTS OF WIDEBAND SPECTRUM SENSING System Model Primary Channelization Model Primary Occupancy Model: p-sparse Primary Signal Model Secondary Model New Performance Metrics: Probability of Excessive Interference Opportunities and Probability of Insufficient Spectrum Opportunities Sampling Schemes Nyquist Sampling Partial-Band Nyquist Sampling Sequential Narrowband Nyquist Sampling Sub-Nyquist Sampling A Unified Model for Sub-Nyquist Sampling Aliasing Effect of Sub-Nyquist Sampling Integer Undersampling Multi-Coset Sampling Generalized Co-Prime Sampling Random Sampling on A Grid Detection Algorithms Channel-by-Channel Bayesian Detection Wideband Bayesian Detection Summary CHAPTER 4: NYQUIST SAMPLING SCHEMES Channel Detection with (P M, P F ) Wideband Detection with (P ISO, P EIO ) Channel-by-Channel Energy Detection Ranked Energy Detection Ranked Posterior Belief Detection Numerical Results Channel-by-Channel Energy Detection Ranked Channel Detections Summary CHAPTER 5: SUB-NYQUIST SAMPLING SCHEME PART I - PERIODIC ALIASING Detection Statistic for Integer Undersampling Channel Detection with (P M, P F ) Wideband Detection with (P ISO, P EIO ) Bin-by-Bin Energy Detection Ranked Bin Posterior Belief Detection Ranked Bin Posterior Belief-Stochastic Channel Detection.. 94 iv

8 5.3.4 Numerical Results Summary CHAPTER 6: SUB-NYQUIST SAMPLING SCHEME PART II - UNIFORM ALIASING Lomb-Scargle Periodogram An Approximation Approach for Uniform Aliasing Channel Detection with (P M, P F ) Belief Propagation based Channel-by-Channel Detection Channel-by-Channel Energy Detection Orthogonal Matching Pursuit Approximating the Declared Primary Occupancy Level within the Bayesian Detection Framework Orthogonal Matching Pursuit Detection within the Bayesian Framework Ranked Energy Detection with an Estimated Number of Declared ON Channels Wideband Detection with (P ISO, P EIO ) Numerical Results Sub-Nyquist Sampling Schemes with Uniform Aliasing Detection with an Estimated Number of Declared ON Channels VS. Detection with a Fixed Number of Declared ON Channels More Results for Channel Detection with (P M, P F ) Wideband Detection with (P ISO, P EIO ) Summary CHAPTER 7: COMPARISON OF DIFFERENT SAMPLING SCHEMES General Comparison of Wideband Sampling Schemes Channel Detection with (P M, P F ) Wideband Detection with (P ISO, P EIO ) Which Sampling Scheme is Most Favorable? Extensibility to More General System Models Summary CHAPTER 8: CONCLUSIONS AND FUTURE WORK APPENDIX A: A.1 Facts for Circularly Symmetric Complex Gaussian Random Variables 142 A.2 Covariance and Variance of the LSP A.3 Conditional Mean and Variance of T BIBLIOGRAPHY v

9 FIGURES 1.1 A cognitive radio network with spectrum sensing Cognitive cycle of cognitive radio system Effective bandwidth and Landau rate: Landau rate is the same for the three example spectra, but the Nyquist rate increases from (a) to (b) to (c) Sub-Nyquist sampling: (a) General multi-branch sub-nyquist sampling; (b) Random sampling; (c) Random demodulation Primary channelization models: (a) Equal bandwidths, equal primary signal power levels, and known channel locations; (b) Equal bandwidth, unequal primary signal power levels, and known channel locations; (c) Unequal bandwidths, unequal primary signal power levels, and unknown channel locations Wideband spectrum sensing Equivalent baseband received signals y(t) Illustration of partial-band Nyquist sampling (PBNS) Implementation of partial-band Nyquist sampling (PBNS): (a) Equivalent baseband implementation; (b) Real implementation A unified model for sub-nyquist sampling Integer undersampling (IU) causes periodic aliasing of a set of frequencies. (a) Complex analog signal in the wideband of interest with 10 channels. (b) Aliasing effect of IU. (c) Aliased signal: each bin is a complete overlap of 2 channels Comparsion of (a) sequential narrowband Nyquist sampling (SNNS) and (b) integer undersampling (IU) in time and frequency Implementation of generalized co-prime sampling Illustrative example of several sub-nyquist sampling schemes with a sub-sampling factor of 2. (a) wideband Nyquist sampling (WBNS); (b) integer undersampling (IU); (c) multi-coset sampling (MCS) with: D = 6, A = {0, 2, 3}; (d) co-prime sampling (CPS) with: D 1 = 4, d 1 = 0, D 2 = 3, d 2 = 0; (e) random sampling on a grid (RSG) vi

10 3.10 Relationship between different sub-nyquist sampling schemes and their aliasing patterns Detection regions for ranked posterior belief detection for partial-band Nyquist sampling with L = 2 bands, and the desired number of spectrum opportunities S d = 1 for: (a): C II < 1: (b) 1 C II 1/β; (c) C II > 1/β Detection regions for partial-band Nyquist sampling with L = 2 bands, and the desired number of spectrum opportunities S d = 2: (a) ranked posterior belief detection for C II < 1 ; (b) ranked energy detection β 2 +2β for τ > 0; (c) channel-by-channel energy detection for τ > Probability of excessive interference opportunities (vertical axes) and probability of insufficient spectrum opportunities (horizontal axes) for partial-band Nyquist sampling with channel-by-channel energy detection. M = 16 channels, S d = 1 spectrum opportunities sought, and I d = 0 interference opportunities allowed Probability of excessive interference opportunities (vertical axes) and probability of insufficient spectrum opportunities (horizontal axes) for partial-band Nyquist sampling with parameter L = 16: M = 16 channels, SNR = 0 db, primary occupancy probability p = 0.1, number of frequency samples per channel N 0 = 20, I d = 0 interference opportunities allowed Probability of excessive interference opportunities (vertical axes) and probability of insufficient spectrum opportunities (horizontal axes) for partial-band Nyquist sampling with parameter L = 16: M = 16 channels, SNR = 0 db, number of frequency samples per channel N 0 = 20, S d = 8 spectrum opportunities sought, I d = 0 interference opportunities allowed Probability of excessive interference opportunities (vertical axes) and probability of insufficient spectrum opportunities (horizontal axes) for partial-band Nyquist sampling with parameter L = 16: M = 16 channels, SNR = 10 db, primary occupancy probability p = 0.1, number of frequency samples per channel N 0 = 8, I d = 0 interference opportunities allowed Probability of excessive interference opportunities (vertical axes) and probability of insufficient spectrum opportunities (horizontal axes) for partial-band Nyquist sampling with parameter L = 16: M = 16 channels, SNR = 10 db, number of frequency samples per channel N 0 = 8, S d = 2 spectrum opportunities sought, I d = 0 interference opportunities allowed vii

11 5.1 Probability of excessive interference opportunities (vertical axes) and probability of insufficient spectrum opportunities (horizontal axes) for integer undersampling: M = 16 channels, SNR = 10 db, number of frequency samples per channel N 0 = 4, sub-sampling factor ρ = 4, I d = 0 interference opportunities allowed Different sub-nyquist sampling schemes with uniform aliasing, with channel-by-channel energy detection: Number of channels M = 16, sub-sampling factor ρ = 4, SNR=15 db and number of samples per channel N 0 = Detection with an estimated number of declared ON channels versus detection with a fixed number of declared ON channels for generalized co-prime sampling: number of channels M = 16, sub-sampling factor ρ = 4, SNR=15 db and number of samples per channel N 0 = Comparison of different detection algorithms for generalized co-prime sampling (GCPS): number of channels M = 16, sub-sampling factor ρ = 4, and number of samples per channel N 0 = 8. Similar results are observed if random sampling on a grid with fixed number of taps (RSG-FN) or minimum aliasing variance (MAV) sampling is used Comparison of different detection algorithms for generalized co-prime sampling (GCPS): number of channels M = 16, SNR=0 db, subsampling factor ρ = 4, and number of samples per channel N 0 = 8. Similar results are observed if random sampling on a grid with fixed number of taps (RSG-FN) or minimum aliasing variance (MAV) sampling is used Comparison of different detection algorithms for generalized co-prime sampling (GCPS): number of channels M = 16, SNR=20 db, subsampling factor ρ = 2, and number of samples per channel N 0 = 4. Similar results are observed if random sampling on a grid with fixed number of taps (RSG-FN) or minimum aliasing variance (MAV) sampling is used Wideband sampling schemes in the three-dimensional space subject to the same overall sensing window duration and the overall sampling rate Probability of missed detection (vertical axes) and probability of false alarm (horizontal axes) for different sampling schemes using channelby-channel energy detection. M = 16 channels, N 0 = 8 frequency samples per channel, primary occupancy probability p = 0.1, and subsampling factor ρ = viii

12 7.3 Probability of insufficient spectrum opportunities (vertical axes) and probability of excessive interference opportunities (horizontal axes) of different sampling schemes with ranked posterior belief detection. M = 16 channels, SNR = 10 db, N 0 = 8 frequency samples per channel, primary occupancy probability p = 0.1, and sub-sampling factor ρ = Probability of insufficient spectrum opportunities (vertical axes) and probability of excessive interference opportunities (horizontal axes) of different sampling schemes with corresponding most favorable detection algorithms. M = 16 channels, SNR = 10 db, N 0 = 4 frequency samples per channel, primary occupancy probability p = 0.1, and subsampling factor ρ = Probability of insufficient spectrum opportunities (vertical axes) and probability of excessive interference opportunities (horizontal axes) of different sampling schemes with corresponding most favorable detection algorithms. M = 16 channels, SNR = -10 db, N 0 = 20 frequency samples per channel, primary occupancy probability p = 0.1, and subsampling factor ρ = Preferable operating regimes for different wideband sampling schemes and the developed detection algorithms, for moderate / high SNR levels and moderate / large desired number of spectrum opportunities Probability of missed detection (vertical axes) and probability of false alarm (horizontal axes) of multi-coset sampling compared to integer undersampling and minimum aliasing variance sampling with channelby-channel energy detection: M = 16 channels Probability of excessive interference opportunities (vertical axes) and insufficient spectrum opportunity (horizontal axes) of multi-coset sampling compared to integer undersampling and minimum aliasing variance sampling with ranked energy detection: M = 16 channels, subsampling factor ρ = ix

13 TABLES 3.1 ILLUSTRATIVE EXAMPLES OF PERIODIC ALIASING AND UNI- FORM ALIASING DEPENDENCIES OF ELEMENTS OF WIDEBAND SPECTRUM SENSING ACROSS CHANNELS WIDEBAND SPECTRUM SENSING VERSUS SPARSE RECON- STRUCTION x

14 ACKNOWLEDGMENTS Foremost, I would like to express my sincere gratitude and appreciation to my advisor Dr. J. Nicholas Laneman, for his long-time support and guidance in research, and especially for his patience and understanding. He is so knowledgable, insightful, and energetic. I have benefited a lot from discussions with him, and there is always more I can learn from him. Thank Prof. Yih-Fang Huang, Prof. Thomas Fuja and Prof. Martin Haenggi for their willingness and time to serve on my committee. I have received a lot of help and suggestions from them in pursuing my PhD. I would like to thank Glenn Bradford for his continuous help in lab and demonstration, and all the great experience we share on and off campus, inside and outside the country. Many thanks also go to the previous and current group members, Michael Dickens, Brian Dunn, Matthieu Bloch, Ebrahim Molavianjazi, Utsaw Kumar, Mostafa Khoshnevisan and Peyman Hesami, for all the great insights and fruitful discussions from you. Thank all my loved friends. They form an integral part of my Notre Dame life. Above all, thank my parents for their endless love, support and understanding. Without them, this dissertation would be simply impossible. xi

15 CHAPTER 1 INTRODUCTION Wireless communication systems, from point-to-point links to networks connecting millions of radios, depend upon the radio frequency (RF) spectrum as their propagation environment. The growth in users, applications, and bandwidth of modern wireless communication systems has resulted in the RF spectrum becoming increasingly crowded. Each radio transmitter creates a spectrum footprint in time, frequency, and location depending upon its circuitry, transmission formats, power levels, and regulatory constraints. The spectrum footprints of multiple transmitters can overlap and lead to harmful interference at radio receivers, which has led to a need for both technical and regulatory approaches to manage interference on the one hand and to more effectively utilize the spectrum on the other hand. One of the important future directions for addressing these challenges is dynamic spectrum management, or spectrum sharing, among compatible applications. Instead of the traditional fixed frequency allocations, dynamic spectrum management allows radios to identify the footprints of other radios in situ and basically fill in the gaps, which are called spectrum opportunities, or white or gray space. Dynamic spectrum management is being enabled by emerging technologies like cognitive radio and dynamic spectrum access, which in turn rely on spectrum sensing and database approaches. This dissertation develops a framework for wideband spectrum sensing that includes basic models as well as new performance metrics and sensing algorithms. The remainder of this chapter motivates the problem in more detail, summarizes the main 1

16 contributions of the work, and provides an outline for the dissertation. 1.1 Wideband Spectrum Sensing for Cognitive Radio Cognitive radio (CR) [29, 44] and dynamic spectrum access (DSA) [77] are promising approaches to the spectrum scarcity problem because they allow unlicensed, secondary users (SUs) to identify and opportunistically access the unused or underutilized spectrum, provided they do not cause harmful interference to the licensed, primary users (PUs). Fig. 1.1 illustrates an example of a typical CR system. Spectrum sensing and database queries have been identified as key functionalities for CR. Although it was removed as an FCC requirement in the TV white spaces [21], spectrum sensing remains critical to CR technology for several reasons. First, it is preferable for low power mobile devices since it only depends on the SUs to identify spectrum opportunities. Second, database queries may not be appropriate in frequency bands other than the TV bands in which the primary activities can be changing dynamically on the order of a few milliseconds or seconds. Last but not least, spectrum sensing can augment any database solution, for example, to enhance the efficiency of the database queries, or to update the database information. In wideband CR networks, the SUs are often required to be capable of sensing multiple frequency bands at a time. Traditional signal sampling approaches follow Shannon s well-known sampling theorem, which states that, after appropriate downconversion, the sampling rate must be at least twice the highest frequency present in the downconverted signal, which is referred to as the Nyquist sampling rate [47]. Thus wideband spectrum sensing conventionally requires very high sampling rates, which could be prohibitively expensive to implement. Rather than first sampling the downconverted wideband signal at the Nyquist rate and then compressing all the samples, compressed sampling, or often called compressive sampling or compressed sensing [5, 11, 16], can be used to convert the wideband signal directly to a smaller 2

17 Primary System Spectrum Sensing Interference Oppourtunities Secondary System Figure 1.1. A cognitive radio network with spectrum sensing. 3

18 set of compressed measurements in the digital domain. Sub-Nyquist sampling [42, 66] is an important class of compressed sampling algorithms that can be easy to implement. The average sampling rate of a sub-nyquist sampling scheme can go far below the Nyquist sampling rate. Conceptually, sub-nyquist sampling combines the analog to digital conversion (ADC) and the digital domain compression into one step. 1.2 Open Research Challenges The motivations and studies of sub-nyquist sampling date back to [25, 37, 39, 50, 51, 72], and significant growth has been witnessed in recent years with the booming studies in compressed sensing. Traditional approaches in sub-nyquist sampling often aim at estimating the power spectral density (PSD) [22, 25, 39, 66], and recent work on sub-nyquist sampling for wideband spectrum sensing motivated by theorems from the field of compressed sensing often aim at reconstructing the original signal with corresponding sparse reconstruction algorithms that minimize the mean-square estimation error (MSE) [26 28, 59, 69, 70, 74, 79]. Although this work motivates potential use of sub-nyquist sampling to reduce the stringent hardware requirements for wideband spectrum sensing, it lacks insightful discussions relevant to the ultimate goal of spectrum sensing for CR: to find spectrum opportunities for the SUs under certain constraints on primary protection. As a matter of fact, low MSE is neither necessary nor sufficient in fulfilling the goal. Some other work uses the traditional single channel detection performance metrics, i.e., the probability of missed detection P M, and the probability of false alarm P F, or their average across channels, to characterize the wideband sensing performance [2, 3, 46, 48, 70]. However, the goal of wideband spectrum sensing for CR can be inherently different from that of single channel detection. Specifically, wideband spectrum sensing can tolerate much higher P F, since the SUs may not be interested in finding all of the spectrum opportunities. Instead, finding a fraction of them could be sufficient [55]. 4

19 There are three issues in applying the compressed sensing approaches directly to the problem of wideband spectrum sensing, because in essence, sparse reconstruction is an estimation problem, and wideband spectrum sensing is a detection problem. First, sparse reconstruction algorithms for compressed sensing often require exact knowledge of the sparsity, which is defined as the number of non-zero elements of the target vector, and aim to reconstruct the target signal with the same number of non-zero elements from the compressed measurements. However, the sparsity of the target primary activities is time-varying and unknown to the SU. Second, sparse reconstruction algorithms do not allow tradeoffs between primary protection and secondary opportunities. Third, most theoretical analysis and well-known results for compressed sensing are based upon sparse signal, i.e., if only a very small fraction of its entries are non-zero, or at least, compressible signal, i.e., if only a very small fraction of its entries are significant and most of its entries are relatively small [23]. However, the wideband signal in a real-world environment is neither sparse nor compressible in the frequency domain. It is sparse only if it is noise-free, or if the signal-to-noise ratio (SNR) is impractically high. In fact, in a wideband system in which there are multiple PUs, only the primary occupancies across channels are sparse. The wideband signal received at the SU is often contaminated by noise and potentially other interference, and hence is not strictly sparse. Furthermore, relevant to the existing sophisticated sub-nyquist sampling schemes in the literature, a unified structure for sub-nyquist sampling, which allows a comprehensive study and comparison between different schemes, would be appealing [56]. A common framework for comparing sensing schemes and obtaining insights for system design is sorely needed. 1.3 Contributions of the Dissertation The main contributions of this dissertation are: 5

20 1. We suggest a general framework for wideband spectrum sensing by breaking down the system design into four elements, specifically, primary system modeling, performance metrics, sampling schemes, and detection algorithms, each of which can potentially couple the individual channels. We provide examples that couple and do not couple the individual channels for each element and comment that appropriate designs of wideband spectrum sensing should consider the four elements jointly. 2. We propose a p-sparse model to characterize the band of channels as a Bernoulli process, which contrasts the widely used K-sparse model in which the number of ON channels is fixed and known a priori to the SU. 3. We suggest a pair of new system performance metrics more appropriate for wideband spectrum sensing, specifically, the probability of insufficient spectrum opportunities P ISO and the probability of excessive interference opportunities P EIO. 4. We propose narrower band Nyquist sampling schemes with much lower sampling rate than the wideband Nqyuist rate, specifically, partial-band Nyquist sampling (PBNS) that monitors a fraction of the entire bandwidth relevant to the goal of finding a desired number of spectrum opportunities, and sequential narrowband Nyquist sampling (SNNS) that sacrifices sensing accuracy per channel. 5. We establish a unified structure for sub-nyquist sampling that allows us to compare several sub-nyquist sampling schemes, i.e., multi-coset sampling (MCS) [30], co-prime sampling (CPS) [65], and random sampling on a grid (RSG) [1]. The unified model also allows us to readily generalize co-prime sampling from two branches to multiple branches. 6. We identify two extreme aliasing patterns for sub-nyquist sampling, i.e., uniform aliasing of generalized co-prime sampling (GCPS) and RSG, and periodic aliasing of integer undersampling (IU), and discuss their effect on wideband spectrum sensing. We demonstrate that although GCPS can achieve similar performance to RSG, it is much easier to implement practically. 7. We develop energy based and posterior belief based detection algorithms tailored for different sampling schemes and identify the most favorable ones with respect to the channel receiver operating characteristic (ROC) that characterizes tradeoffs between P M and P F, as well as the system ROC that characterizes tradeoffs between P ISO and P EIO. 8. We propose an approach to estimate the number of declared ON channels within the Bayesian detection framework, which allows sparse reconstruction algorithms to tradeoff between P M and P F, for sub-nyquist sampling schemes with uniform aliasing. 9. We compare the best observed sensing performance of different sampling schemes. We observe that IU, which corresponds to the simplest sub-nyquist sampling 6

21 scheme, achieves the best observed performance for moderate and high SNR (SNR 0 db) in the regime of better protection for the primary system. Furthermore, SNNS exhibits globally best observed performance for low SNR (SNR<0 db) for the cases studied. On the other hand, the more sophisticated sampling schemes that often lead to uniform aliasing, are inferior to either IU or SNNS depending on the system parameter settings and the detection regime considered. 10. We discuss the extensibility of the detection algorithms to more general primary system models. 1.4 Outline of the Dissertation The remainder of this dissertation is organized as follows. In Chapter 2, we provide a background survey on cognitive radio and spectrum sensing, with an emphasis on wideband spectrum sensing relevant to the scope of this dissertation. In Chapter 3, we elaborate on the fundamental elements for wideband spectrum sensing. Specifically, we introduce the primary system model, suggest new performance metrics relevant to the goal of wideband spectrum sensing, establish a unified sub-nyquist sampling structure, and discuss the general aspects of various detection algorithms. In Chapter 4, we develop optimal detection algorithms for Nyquist sampling schemes relevant to the goal of channel detection as well as the goal of wideband detection. In Chapter 5, we develop several detection algorithms for IU and identify the best among the group. In Chapter 6, we propose an approximation approach for sub- Nyquist sampling with uniform aliasing and develop several detection algorithms based upon the Lomb-Scargle periodogram. We also suggest an approach that allows sparse reconstruction algorithms to tradeoff between primary protection and secondary opportunities within the primary system model studied. In Chapter 7, we compare the best observed performance of various sampling schemes and identify the most favorable designs. We also discuss extensibility of the main detection algorithms to more general primary system models. In Chapter 8, we conclude the dissertation and discuss directions for future research. 7

22 CHAPTER 2 BACKGROUND This chapter provides a survey of the relevant literature and techniques in the areas of cognitive radio (CR) and spectrum sensing. Specifically, Section 2.1 introduces the definition of cognitive radio and dynamic spectrum access (DSA). Section 2.2 provides a short survey on non-uniform sampling, which is often considered in sampling wideband signals. Section 2.3 provides a thorough exploration of spectrum sensing for CR, especially of wideband spectrum sensing from four fundamental elements that are relevant to the scope of the dissertation. Section 2.4 summarizes the background survey and points to open research challenges. 2.1 Cognitive Radio and Dynamic Spectrum Access It is widely acknowledged that the traditional static and centralized allocation of wireless spectrum by governmental regulators is inefficient due to spatial, spectral, and temporal variation in utilization by licensed primary users (PUs) and due to the pace of wireless innovation and penetration. Recent spectrum measurements [53] suggest that certain frequency bands are almost never used in some regions. Cognitive radio (CR) and dynamic spectrum access (DSA) techniques are new communication paradigms that can offer new ways of exploiting the underutilized spectrum. Cognitive radio, which is coined by Mitola [44], corresponds to a broader paradigm than DSA. CR is a context-aware intelligent radio potentially capable of autonomous reconfiguration by learning from and adapting to the communication environment. DSA, however, commonly refers to a specific application of cognitive radio in which 8

23 secondary users (SUs), also referred to as unlicensed users, may access licensed spectrum bands as long as they do not cause harmful interference to primary users (PUs), also referred to as licensed users. TV broadcasters, public safety users, cellular operators and point-to-point microwave links are examples of primary users. Figure 2.1 shows the tasks required for cognitive radio in open spectrum. It is referred to as the cognitive cycle [29]. In the spectrum analysis, modeling and learning step, the cognitive radio measures the spectrum, estimates the PU s transmission parameters and models the PU s transmission structure through observations over a long time period. This information is then used to formulate the threshold in the spectrum sensing, channel predication step. Finally, in the spectrum management, cognitive transmission step, the cognitive radio adapts itself to transmit in the open band, potentially changing its carrier frequency, transmit power, modulation type and packet length. If multiple SUs exist, they must share the spectrum according to some channel access protocol. In March 2010, the US Federal Communications Commission (FCC) released Connecting America: The National Broadband Plan [20]. It calls for the Federal government to identify 500 MHz of wireless spectrum that can be freed up for wireless broadband service over the next 10 years. The plan aims to stimulate economic growth, spur job creation, and boost capabilities in energy, healthcare, education, government, and other areas. A key issue in spectrum allocation, whether static or dynamic, centralized or decentralized, is identifying available spectrum without causing excessive interference that causes unacceptable degradation to incumbent wireless services and devices. After approximately 80 years of spectrum regulation by the Federal Communications Commission (FCC) and over a 10 years of research and development of CR and DSA, it seems clear that defining available spectrum and excessive interference for different applications are significant challenges to regulators and engineers. 9

24 Radio environment: Primary users and other secondary users Channel allocation, power and packet length control RF stimuli RF stimuli Spectrum management, cognitive transmission Spectrum holes and noise statistics Spectrum analysis modeling, and learning channel capacity Spectrum holes and noise statistics information Spectrum sensing, channel prediction Figure 2.1. Cognitive cycle of cognitive radio system. 10

25 Two classes of solutions for addressing these challenges within the context of DSA are being considered from both engineering and regulatory viewpoints: Spectrum Sensing. The first class of solutions, broadly called spectrum sensing, has a secondary device measure signal levels across time and frequency around its location in order to identify available spectrum as well as potential adjacent primary users. The FCC shared a Notice of Proposed Rule Making (NPRM) in May 2004 to allow wireless devices to use vacated TV channels provided no harmful interference is caused to incumbent services, including wireless microphones. Later in November 2004, the IEEE Working Group was formed to develop a standard for a CR-based physical layer (PHY) / media access control layer (MAC) for secondary operation in the TV bands, and spectrum sensing was an initial ingredient in the standards discussion, requiring sensitivity of -116 dbm over 6 MHz for TV channels and -107 dbm over a 200 khz bandwidth for wireless microphones [14]. Database Queries. A more recent class of solutions, broadly called databases queries, has a secondary device cross-reference its location, obtained from GPS for example, with a special database of emitter locations and secondary access policies. In a September 2010 Opinion and Order [21], the FCC officially freed up the TV channels but did not require spectrum sensing for devices that incorporate a geo-location database. The geo-location database could instruct secondary users to vacate channels when interfering with broadcast TV signals as well as help prevent devices from interfering with each other. It remains to be seen if a viable database solution and corresponding TV white spaces ecosystem will emerge. Although it has been removed as an FCC requirement in the TV white spaces, spectrum sensing remains critical to CR and DSA technology. Indeed, the recent FCC order states that regulators believe that spectrum sensing will continue to develop and improve, and some form of spectrum sensing may very well be included in TV band devices [21]. Spectrum sensing can of course augment any database solution, but the importance of spectrum sensing can be further appreciated by considering limitations of database-only solutions. First, access to the geo-location database generally requires additional infrastructure for internet connectivity outside the band utilized on a secondary basis. Therefore, database solutions could be feasible for larger devices such as base stations; for low power mobile devices, spectrum sensing may remain preferable, since it only depends on the CR itself to identify the channel 11

26 opportunities. Second, although a database solution may be appropriate for the TV bands in which the overwhelming majority of primary users operate from fixed locations, mobile devices in other frequency bands will be particularly challenging to maintain in a database. In July 2012, President s Council of Advisors on Science and Technology (PCAST) released a report suggesting spectrum sharing between government users and commercial users in order to meet the nation s growing demand for spectrum resources and identified 1 GHz of spectrum to be initially made available for dynamic spectrum sharing, including 50 MHz in 4.9 GHz public safety band and 75 MHz in 5.9 intelligent transportation system GHz band [18]. Thus wideband spectrum sensing techniques can be potentially applied. 2.2 Wideband Sampling Schemes In this section we provide a short survey on wideband sampling schemes that are often seen in the context of wideband spectrum sensing for CR. We start our discussion from the basics for spectrum analysis. A stochastic process x = {x(t), < t < } is wide-sense stationary (WSS) if its mean µ x (t) is constant and its autocorrelation r xx (s, t) is a function of the time difference [40], specifically, ( ) µ x (t) E x(t) = µ x < +, (2.1) ( ) r xx (s, t) E x (s)x(t) = r xx (s t) < +. (2.2) where x is the conjugate of the complex signal x, and E( ) is the expectation operator. The power spectral density (PSD) of a WSS random process is S(f) = + r xx (τ)e j2πft dτ, (2.3) 12

27 which is often estimated by the periodogram from the finite discrete x[n] = x(nt ), where T > 0 is the sampling period and 1/T is the sampling frequency[47], the periodogram is defined as P x (f) = 1 N where N is the number of samples taken. N 1 n=0 x[n]e j2πft 2, (2.4) Alias-Free Sampling The concept of alias-free sampling is first introduced in [52]. A sampling scheme {t n } for WSS random processes x is said to be alias free relative to a family of spectra if any spectrum of the family can be recovered by a linear operation on the correlation / interpolation sequence. Later in [7] the authors provide a general formulation of the problem of alias-free sampling and present criteria for alias-free sampling relative to various classes of spectral distributions without strict definition of alias-free in a general context. A trivial alias-free sampling approach is Nyquist sampling, which refers to a uniform sampling at a rate equal to or greater than the Nyquist rate, i.e., twice the highest frequency present in the downconverted signal. General alias-free sampling is often achieved by random sampling, for which the sampling scheme {t n } is independent of x and constitutes a stationary stochastic point process on the real line [38]. One important random sampling scheme that has been shown to be alias-free is Poisson sampling [39], in which the sampling instants are generated by a Poisson point processes, i.e., t 0 = 0, (2.5) t n = t n 1 + α n, n = 1, 2,, (2.6) 13

28 where α n are independent identically-distributed (i.i.d) positive random variables with a common exponential distribution with cumulative distribution function (CDF) F (x) = 1 e βx, where β is the average sampling rate. Note that the actual realizations of sampling times t n need not to be known to effectively recover the spectrum of x. Random sampling corresponds to one of the most important non-uniform sampling schemes. The concept of non-uniform sampling was first introduced decades ago in [72]. Later Lomb [37] and Scargle [50, 51] extend the definition of the classical periodogram for uniformly spaced time series to unevenly spaced time series, which is now known as Lomb-Scargle periodogram (LSP), for the purpose of detection of a periodic signal observed in noise. The LSP is often used to estimate the power spectral density (PSD) directly without interpolation or reconstruction. The generalized LSP is defined as [37] ( ( (x(tn Px ls (f) 1 ) x ) cos ( 2πf(t n τ) )) 2 σ 2 ( N 1 n=0 cos2 2πf(t n τ) ) + ( (x(tn ) x ) sin ( 2πf(t n τ) )) 2 ) N 1 n=0 sin2 ( 2πf(t n τ) ), (2.7) where x and σ 2 are the mean and variance of the data, and τ is defined by tan(4πfτ) N 1 n=0 sin ( 4πf(t n τ) ) N 1 n=0 cos ( ). (2.8) 4πf(t n τ) Specifically, the LSP of Nyquist samples with missing data is simply the classic periodogram in (2.4) with missing data set to zero [68]. More details can be found in Section Compressed Sensing Compressed sensing (CS), also known as compressive sensing, has recently emerged as a very powerful approach to signal processing, enabling the acquisition of signals 14

29 at rates much lower than Nyquist rate [6, 11, 12, 16, 23, 49]. CS exploits the sparsity inherent in many natural and man-made signals. Sparsity characterizes the property that the information rate of a continuous time signal may be much smaller than suggested by its bandwidth, or that a discrete-time signal depends on a number of degrees of freedom that is comparably much smaller than its length [11] Sparsity and Compressibility There has been some inconsistency in the definition of K-sparse in the field of compressed sensing. A vector is defined to be K-sparse if the number of non-zero elements is at most K in [10, 42, 45], or if the number of non-zero elements is exactly K in [5, 15, 28]. In this dissertation, we use the latter definition, i.e., a vector x C N is said to be K-sparse if exactly K N elements of x are non-zero. The set of indices corresponding to the non-zero elements are called the support of x, which is denoted by supp(x). For k {1, 2,..., N}, denote the set of all K-sparse vectors as Σ K {x C N : supp(x) K} (2.9) Many natural and man-made signals are not strictly sparse, but can be approximated by a K-sparse representation. Such signals are said to be compressible. Specifically, define the best K-term approximation error of a vector x C N in l p norm as σ K (x) p = inf z Σ K x z p. (2.10) where the l p norm of the vector x is defined as usual by ( N ) 1/p x p = x i p, 0 < p <. (2.11) i=1 If σ K (x) p decays quickly in K, e.g., σ K (x) p K r for some r > 0, then x is called 15

30 compressible Compressive Measurements CS provides a framework for acquisition of an N 1 discrete-time signal vector x that is sparse in some N N sparsity basis matrix Ψ, i.e., x = Ψα and supp(α) K. ( ) CS theory demonstrates that x can be recovered using M = KO log( N ) nonadaptive linear projection measurements on to an M N basis matrix Φ that is K incoherent with Ψ [35], where O is the commonly used Bachmann-Landau notation describing the rate of growth in complexity of a function when the argument goes to infinity, i.e., suppose f(x) and g(x) are two functions defined on some subset of the real numbers, then f(x) = O ( g(x) ) (2.12) if and only if here exist constants N and C such that f(x) Ng(x), x > N. (2.13) Note that the incoherent condition means that the rows φ j of the matrix Φ cannot sparsely represent the elements of the sparsity-inducing basis ψ i, and vice versa [49]. The measurement (projections) y can be expressed as y = Φx = ΦΨα. (2.14) Sparse Reconstruction One might expect that the approach for a recovery procedure would be to search for the sparsest vector x which is consistent with the measurement vector y = Φx. This is an l 0 -minimization problem, which is NP-hard in general [23]. Therefore, 16

31 two practical and tractable alternatives have been proposed, i.e., convex relaxation leading to l 1 -minimization, and greedy algorithms [23]. Recovery via l 1 optimization is ˆx = arg min x x 1, s.t. y = Φx. (2.15) In the real-valued case, the optimization problem in (2.15) is equivalent to a linear program, and in the complex-value case, it is equivalent to a second-order cone program. Standard and efficient algorithms apply to both problems [23]. One of the most important sparse reconstruction algorithms is orthogonal matching pursuit (OMP) [61, 62]. Sparse reconstruction algorithms require the perfect knowledge of the sparsity level of the target signal and outputs a reconstructed signal with the same sparsity level. However, the sparsity of the target primary activities is time-varying and unknown to the SU in CR systems. Also, sparse reconstruction algorithms are estimation approaches and do not allow tradeoffs between primary protection and secondary opportunities Sub-Nyquist Sampling Traditional signal sampling approaches follow Shannon s well-known theorem, which states that the sampling rate must be at least twice the highest frequency present in the signal, and the signals can be reconstructed by sinc interpolation. Nyquist sampling could be prohibitively expensive to implement for a wideband signal. Although CS provides a promising approach to recover the target signal from compressive measurements, it is for discrete-time signals only. The concept of analogy-to-information converter (AIC) [35, 36] is proposed to combine the analog to digital conversion (ADC) and the digital domain compression into one step. Specifically, AIC samples and reconstructs a class of multiband signals with spectral support F, at rates arbitrarily close to the Landau minimum rate [34], 17

32 f (a) f (b) f (c) Figure 2.2. Effective bandwidth and Landau rate: Landau rate is the same for the three example spectra, but the Nyquist rate increases from (a) to (b) to (c). which equals the Lebesgue measure of F. Fig. 2.2 illustrates an example of effective bandwidth with the Landau rate much lower than the corresponding Nyquist rate. Once again, random sampling is a common way to implement AIC. Another analogous term to AIC is sub-nyquist sampling. The literature lacks a strict definition of sub-nyquist sampling. In broad terms, it refers to a sampling scheme for which the overall sampling rate is below the Nyquist rate. In some cases, it can be interpreted as decimation on wideband Nyquist samples, for which the time between any two consecutive sampling instants must be an integer multiple of the Nyquist sampling period. Most existing work on sub-nyquist sampling for wideband spectrum sensing, including this dissertation, focuses on the case of decimation from wideband Nyquist samples. In particular, we develop a unified model for sub-nyquist sampling of decimation on wideband Nyquist samples. More discussions can be found in Section From the implementation point of view, sub-nyquist sampling can be classified 18

33 as multi-branch sub-nyquist sampling and single-branch sub-nyquist sampling Multi-Branch Sub-Nyquist Sampling For a multi-branch sub-nyquist sampling approach, the sampling procedure is implemented by a set of uniform samplers working at much lower, yet not necessarily identical, rates. Schemes falling into this category include: Multi-coset sampling. In [22, 30], the authors propose a set of uniform samplers working at the same rate that is much smaller than the Nyquist rate, yet different phase offsets / time delays to sample the band-limited signal at an overall sampling rate much lower than Nyquist rate. The sampling scheme is known as multi-coset sampling (MCS), or non-uniform periodic sampling and has been widely studied in the field of wideband spectrum sensing for CR [3, 4, 33, 41, 43, 48, 66]. Co-prime sampling. Co-prime sampling (CPS) is a sub-nyquist sampling approach recently studied in [65], and its relationship to MCS is discussed in [4]. CPS uses two uniform low-rate samplers with co-prime decimation factors relative to Nyquist. MCS is the most well-known multi-branch sub-nyquist sampling approach Single-Branch Sub-Nyquist Sampling On the other hand, only one sampler is required for single-branch sub-nyquist sampling, which always relies on a certain form of randomization. The two most important single-branch sub-nyquist sampling schemes are: Random sampling. Studies of random sampling date back to [25] and [39] and have been revisited from time to time. As discussed in Section 2.2.1, the sampling instants {t n } form a stochastic point process on the real line. Note that general random sampling approaches do not have any constraints on the stochastic point process. Recently, the authors in [2] propose a sub-nyquist sampling scheme called random sampling on a grid (RSG), which in essence is decimation on wideband Nyquist samples. Random demodulation. An alternative approach is to move the randomization process one step forward before the ADC, which is referred to as random demodulation [43, 63]. Specifically, the signal is first demodulated by (multiplied 19

34 with) a high-rate pseudo noise sequence to smear any frequency tones across the entire spectrum, and then passed through a lowpass anti-aliasing filter, and finally sampled with a uniform sampler at a relatively low rate. Implementations of multi-branch sub-nyquist sampling and single-branch sub-nyquist sampling are illustrated in Fig. 2.3, in which Y(t) is the downconverted signal. 2.3 Spectrum Sensing Traditionally, spectrum sensing is understood as measuring the spectrum to decide whether PUs are active or not, but if the ultimate cognitive radio [44] is considered, it is a more general term that may involve obtaining the spectrum usage characteristics across multiple dimensions such as time, space, frequency, and code, as well as determining what type of signals are occupying the spectrum, i.e., modulation scheme, waveform, bandwidth, carrier frequency, and so forth. This will of course require more powerful signal analysis techniques with additional computational complexity. For the purpose of this dissertation, we only consider spectrum sensing in the traditional sense. Spectrum sensing is best addressed as a cross layer problem. Specifically, PHY layer sensing focuses on how to detect the presence of primary signals rapidly and robustly. It is accomplished by using or not using the parameters of the primary signals such as transmission power, waveform, modulation schemes. The most well-known PHY layer sensing schemes include energy detection (power detection, periodogram detection), coherent detection (matched filter detection) [8] and feature detection (cyclostationary detection) [24, 58]. Other PHY layer detection algorithms include but are not limited to: sequence detection [57], eigenvalue based detection [76], etc. MAC layer sensing concentrates on how to schedule sensing for efficient discovery of spectrum opportunities, especially in the case of multiple channels and multiple SUs. Important issues associated with MAC layer sensing in DSA include how often 20

35 e sτ 1 Uniform sampler 1 D 1 z 1 F 1 = 2MB 0 D 1 y(t) e sτ 2 Uniform sampler 2 F 2 = 2MB 0 D 2 D2 z 2 v[n] e sτ q Uniform sampler q D q z q F q = 2MB 0 D q (a) y(t) Random sampler F s = 2MB 0 ρ (b) v[n] Uniform y(t) sampler v[n] p(t) (c) F s = 2MB 0 ρ Figure 2.3. Sub-Nyquist sampling: (a) General multi-branch sub-nyquist sampling; (b) Random sampling; (c) Random demodulation. 21

36 to sense the availability of licensed channels, in which order to sense, and how long a sensing period should be. Recently, significant effort has been devoted to the field of MAC layer sensing and scheduling [13, 31, 32, 54, 67, 78] Narrowband Spectrum Sensing In simplest form, spectrum sensing of a single channel is a binary hypothesis testing problem. Specifically, H 0 : y(n) = z(n), n = 0, 1,..., N 1, (2.16) H 1 : y(n) = x(n) + z(n), n = 0, 1,..., N 1, (2.17) where x(n) is the signal to be detected, z(n) is the additive white Gaussian noise (AWGN), and n is the sample index. For simplicity, let 0 and 1 denote the two hypotheses, let the random variable H denote the state of the signal, and let the random variable Ĥ denote the sensing decision. Thus, the probability of missed detection and the probability of false alarm are defined as P M Pr { Ĥ = 0 H = 1 }, (2.18) P F Pr { Ĥ = 1 H = 0 }. (2.19) Small P F is necessary in order to provide possible high throughput in dynamic spectrum access networks, since a false alarm wastes a spectrum opportunity. On the other hand, small P M is necessary in order to limit the interference to PUs. A detection algorithm can seek tradeoffs between P M and P F by varying the detection threshold. Energy detection has been widely applied since it requires limited a priori knowledge of the primary signals to be detected. It is also one of the lowest complexity 22

37 schemes. Specifically, T = N 1 n=0 y(n) 2 Ĥ=1 τ, (2.20) where T is the detection statistic, N is the number of samples in the sensing window, and τ is some pre-determined threshold. The noise is often modeled as AWGN, thus Ĥ=0 T under hypothesis 0 is a scaled central chi-square random variable. Specifically, T H 0 σ2 noi 2N χ2 2N, (2.21) where χ 2 k denotes a central chi-square random variable with k degrees of freedom and σnoi 2 is the noise power. Note that the factor of 2 follows from the signals being complex-valued. Depending on the modeling of the primary signal, T under hypothesis 1 can either be a scaled central chi-square random variable if the primary signal is modeled by AWGN [75], or a scaled non-central chi-square random variable if the primary signal is modeled by some unknown deterministic signal [64] Wideband Spectrum Sensing The problem of wideband spectrum sensing can be considered in the following four aspects: primary system model, performance metrics, sampling schemes, and detection algorithms. The following subsections summarizes relevant literature on each of these aspects Primary System Model Existing work on wideband spectrum sensing for CR uses different channelization models for the primary system along with several performance metrics. Some of the most commonly used models in the literature include, but are not limited to: Equal bandwidths, equal primary signal power levels, and known channel locations, which are used in [2, 3, 43, 48]; 23

38 Equal bandwidths, unequal primary signal power levels, and known channel locations, which are used in [70]; Unequal bandwidths, unequal primary signal power levels, and unknown channel locations, which are used in [26, 33, 46, 59, 79]. Fig. 2.4 illustrates the channelization models mentioned above. The K-sparse model is commonly used to characterize the primary occupancies across channels, in which the number of non-zero elements is known a priori to the SU [2, 3, 26, 28, 33, 43, 46, 48, 65, 70]. Thus, the simulation results are often generated based upon only one static realization of the primary occupancies, with averaging over only the primary signals and noise. On the contrary, we propose a Bernoulli model called the p-sparse model and the simulation results in this dissertation average over primary occupancies as well as the primary signals and noise. Ergodicity in the primary occupancies is assumed Performance Metrics Most work focusing on primary signal reconstruction uses MSE as the performance metric [4, 17, 26 28, 41, 59, 69, 70, 74, 79]. However, the ultimate goal of spectrum sensing for CR is to find spectrum opportunities for secondary use. Low MSE is neither sufficient nor necessary in fulfilling this goal. A significant amount of work uses the performance metrics for single channel detection, i.e., the probability of missed detection P M and the probability of false alarm P F, or their averages over channels, for wideband spectrum sensing. Specifically, [46] studies the detection performance in the form of probability of detection / missed detection versus the sub-sampling factor, and [2, 3, 70] study tradeoffs between the probability of missed detection and the probability of false alarm in the form of receiver operating characteristic (ROC) for a given primary occupancy level and sub-sampling factor. 24

39 M channels f (a) M channels (b) f f (c) Figure 2.4. Primary channelization models: (a) Equal bandwidths, equal primary signal power levels, and known channel locations; (b) Equal bandwidth, unequal primary signal power levels, and known channel locations; (c) Unequal bandwidths, unequal primary signal power levels, and unknown channel locations. 25

40 Other performance metrics proposed in the literature include, but are not limited to: the detection probability of wideband, which is defined as the probability that all ON channels are correctly detected, and the false alarm probability of wideband, which is defined as the probability that any of the OFF channels are falsely detected as ON [73], the (empirical) probability of detecting a given number of ON channels [48], and so forth. Despite the above discussions on performance metrics, there are other performance metrics that make more sense for wideband spectrum sensing, as we will introduce in Section Sampling Schemes All the approaches and discussions in Section 2.2 are relevant to sampling design for wideband spectrum sensing Detection Algorithms The work focusing on primary signal reconstruction often relies on reconstruction algorithms for compressed sensing such as orthogonal matching pursuit (OMP) [4, 17, 28, 41, 59, 69, 74]. Some go one step further in making a hard decision on whether the PUs are active or not based upon the reconstructed signals. This type of detection algorithm is referred to as recover-and-detect in this dissertation. A stringent requirement on compressed sensing is that the target signal be sparse. However, the wideband signal in a real-world environment is neither strictly sparse nor compressible in the frequency domain since the signal received at the SU is often contaminated by noise and other possible interference. Another approach is to estimate the auto-correlation sequence, or equivalently, the PSD of the primary signal directly without any attempts of signal reconstruction under the assumption that the number of active PUs is fixed and known [65]. 26

41 Note that the above discussions categorize detection algorithms by their way of generating the detection statistics from the samples. In Section 3.4, we discuss two classes of detection algorithms based upon how decisions are made across channels. Specifically, channel-by-channel detection makes decisions on each channel independently, and multi-channel detection makes decisions across channels jointly. We conclude that the design of detection algorithms depends not only on the sampling schemes, but also on the specific performance metrics being used. For performance metrics that couple the individual channels, e.g., P ISO and P EIO, multi-channel detection is advantageous over channel-by-channel even for wideband Nyquist samples with conditionally independent observations across channels. 2.4 Summary Recent years have witnessed a rapid growth in wideband spectrum sensing for CR motivated by theorems and results from CS, with accordingly more attention paid to design of sophisticated sub-nyquist sampling schemes and correspondingly advanced reconstruction algorithms. However, the problem of wideband spectrum sensing needs to be addressed from first principles. Specifically, since the goal of wideband spectrum sensing for CR can be inherently different from that of single channel detection and finding a fraction of the spectrum opportunities could be sufficient than finding them all, new performance metrics more relevant to this goal should be considered, which will shape the way of the corresponding wideband spectrum sensing methods in both sampling and detection. Moreover, a comprehensive study and comparison between different wideband sampling schemes is needed, and developed in the following chapters. 27

42 CHAPTER 3 ELEMENTS OF WIDEBAND SPECTRUM SENSING This chapter discusses the general components and design for wideband spectrum sensing and establishes the framework for later chapters. Specifically, a wideband spectrum sensing system needs to consider the following four elements jointly: System modeling; Performance metrics; Sampling schemes; Detection algorithms. Each element can lead to dependencies across channels, thus complicating the design for wideband spectrum sensing. Given the system and the performance metrics, a wideband spectrum sensing scheme can be partitioned into wideband sampling and wideband detection as illustrated in Fig The remainder of the chapter is organized as follows. In Section 3.1, we introduce the system model. In Section 3.2, we motivate new performance metrics relevant to the goal of wideband spectrum sensing. In Section 3.3, we introduce Nyquist y(t) Wideband sampler v[n] Wideband detector Ĥ Figure 3.1. Wideband spectrum sensing. 28

43 sampling schemes, as well as a unified sub-nyquist sampling model, and discuss several important sub-nyquist sampling schemes within that model. We also discuss the aliasing patterns inherent in sub-nyquist sampling and identify two extreme cases. In Section 3.4, we briefly discuss the categorization of detection algorithms. In Section 3.5, we present some remarks on system design for wideband spectrum sensing. 3.1 System Model We introduce three aspects of the primary system model: primary channelization model, primary occupancy model, and primary signal model Primary Channelization Model The wideband signals of interest are received in a collection of M consecutive narrow-band frequency channels. Each channel has equal bandwidth B 0 and the entire bandwidth is B = MB 0 and equal primary power levels. This model corresponds to Fig. 2.4 (a) and similar channelization models have been used in the literature in [2 4, 28, 46, 48]. Let f m denote the center frequency of the m th channel, m = 1, 2,..., M. The signals are subject to additive white Gaussian noise (AWGN) across channels. Let σnoi 2 and σsig 2 denote the power of the noise and the signal in a channel, respectively Primary Occupancy Model: p-sparse Let H m denote the state of of the m-th channel, with H m = 1 and H m = 0 corresponding to the PU in the m-th channel being active and inactive, respectively. Let H [H 1, H 2,..., H M ]. As discussed in Section 2.3.2, the K-sparse model is commonly used to characterize the primary occupancies across channels, in which the number of ON channels 29

44 is fixed and known a priori to the SU [2, 3, 26, 43, 48, 70]. Note that the K-sparse model inherently couples the activities of the individual channels. On the contrary, we propose a Bernoulli model to characterize the primary occupancy of each channel. Specifically, the states of the channels are modeled by i.i.d Bernoulli random variables that are independent across time and frequency, i.e., H m Bernoulli(p), (3.1) where Bernoulli(p) denotes the Bernoulli distribution with mean p, i.e., the random variable being 1 with probability p and 0 with probability 1 p, 0 p 1. We refer to the parameter p as the primary occupancy probability. Define the primary occupancy level K as the number of ON channels. With independent occupancies across channels, K Binomial(M, p), (3.2) where Binomial(M, p) is the binomial distribution with parameters M and p. Note that the p-sparse model differs with the commonly used K-sparse model in that: 1) the state of each channel is random; and 2) it is unknown to the SU at any given time. A fundamental assumption is that no changes occur in the primary occupancies during a sensing window. This assumption is reasonable as long as the duration of the sensing window is much smaller than that of a primary ON / OFF period [57]. We restrict our discussion to one sensing window with fixed duration of T win = NT nyq in time, where F nyq = 1/T nyq = 2B is the wideband Nyquist sampling frequency and the positive integer number N is the number of wideband Nyquist samples per channel in a sensing window. If other sampling schemes with different sampling rates are utilized, we assume the same sensing window duration to ensure fair comparisons. 30

45 y p (t) y(t) 2MB 0 MB 0 f c Figure 3.2. Equivalent baseband received signals y(t) Primary Signal Model If present, we use an AWGN model for the primary signal within each channel [75]. The main reason for assuming symmetric channels is to simplify the theoretical analysis. However, most analysis and detection algorithms to be discussed in this dissertation do not rely on the symmetry property. Also, discussion of more general primary system models with asymmetric channel power levels can be found in Section 7.5. Let σnoi 2 and σsig 2 denote the average received power of the noise and the signal in a channel, respectively. The received signal r(t) at the SU is first down-converted to baseband before further signal processing and decision making. Without loss of generality, we consider only the equivalent baseband received signals throughout this dissertation as shown in Fig. 3.2 Let x(t), z(t) and y(t) be the received wideband primary signal, the AWGN, and the received signal in the continuous-time domain at the SU, respectively. Let x[n], z[n] and y[n] denote the corresponding wideband Nyquist samples, i.e., x[n] x(nt nyq ), y[n] y(nt nyq ), and z[n] z(nt nyq ). (3.3) Let Y[k] denote the normalized discrete Fourier transform (DFT) of y[k] in a 31

46 sensing window, i.e., for 0 k < N, where Y[k]= 1 N 1 y[n]e j2πkn 1 N = N n=0 N 1 N n=0 y[n]ω kn N, (3.4) ω N e j 2π N. (3.5) Thus, the system model described above is Y[k] = H m(k) X[k] + Z[k], (3.6) where X[k] and Z[k] are zero mean, independent Gaussian random sequences with variances σ 2 sig and σ 2 noi, respectively. Y[k] are also zero mean, conditionally independent Gaussian random variables given H, with variances Λ Y [k] H m(k) σ 2 sig + σ 2 noi. (3.7) Here k m(k) = N 0 (3.8) is the index of the channel into which the k-th frequency sample falls, where the positive integer N 0 is the number of samples per channel, and x is the ceiling function that maps x to the smallest integer not less than x. Specifically, if only one sample is taken for each channel, then m(k) = k. Define I m as the set of frequency indices that fall into channel m, i.e., I m { k : (m 1)N k mn 0 }. (3.9) 32

47 We can write (3.4) in matrix form as Y = Fy, (3.10) where y [ y[0], y[1],..., y[n 1] ], and Y [ Y[0], Y[1],..., Y[N 1] ] and F is the unitary DFT matrix defined by F = 1 N ωn 1 1 ωn ω 1 (N 1) N 1 ωn 2 1 ωn ω 2 (N 1) N ω (N 1) 1 N ω (N 1) 2 N... ω (N 1) (N 1) N. (3.11) Secondary Model Throughout this dissertation, we assume a single SU capable of wideband spectrum sensing. Aside from knowing the SNR and the fact that the primary occupancies of the M channels is p-sparse, no further prior information on the PU activities is assumed. Let Ĥm denote the sensing decision for the m-th channel, i.e., Ĥm = 1 and Ĥ m = 0 if the SU declares the m-th channel to be ON and OFF, respectively. Let ˆK denote the declared numbers of ON channels, so that M ˆK represents the declared number of OFF channels. 3.2 New Performance Metrics: Probability of Excessive Interference Opportunities and Probability of Insufficient Spectrum Opportunities The detection performance for a single channel is often characterized by the pair (P M, P F ), where P M is the probability of missed detection and P F is the probability of false alarm as introduced in Section For multi-channel spectrum sensing, a possible extension of these performance metrics would be P M = 33

48 {P M,1, P M,2,..., P M,M } and P F = {P F,1, P F,2,..., P F,M }, i.e., vectors of individual probabilities of missed detection and individual probabilities of false alarm. However, a natural goal of wideband spectrum sensing is to find a sufficient amount of spectrum opportunities for secondary use under certain protection requirements for the primary system, not necessarily all spectrum opportunities. Thus, we propose new performance metrics based upon this goal and discuss their connections with the existing performance metrics used in the literature. Let S denote the number of spectrum opportunities, i.e., the number of correctly identified OFF channels. Let S d denote the desired number of spectrum opportunities for the SU. The probability of insufficient spectrum opportunities (ISO) is defined as the probability that the number of spectrum opportunities falls below the desired level S d, i.e., P ISO (S d ) Pr{S < S d }. (3.12) For one SU, a natural choice of the desired number of spectrum opportunities could be S d = 1, i.e., the SU requires at least one spectrum opportunity to make its transmission. Wider bandwidth transmissions and/or multiple SU transmissions can be captured by S d > 1. If an ON channel is declared to be OFF, it leads to an interference opportunity since the SU may decide to transmit based upon the decision. Let I denote the number of interference opportunities, i.e., the number of missed ON channels. For the purpose of minimizing the possible interference to the PUs, a maximum desired number of interference opportunities, denoted by I d, is required by the primary system. The probability of excessive interference opportunities (EIO) is defined as the probability 34

49 that the number of interference opportunities is greater than the desired level I d, i.e., P EIO (S d ) Pr{I > I d }. (3.13) One example for the desired number of interference opportunities is I d = 0, i.e., the primary system prefers no interference opportunity to any channel. In this case, P EIO is exactly the probability of missed detection for wideband defined in [73]. Thus, P ISO and P EIO correspond to a broader range of performance metrics for wideband spectrum sensing. In this dissertation, we focus on the special case of I d = 0. The probability mass functions (PMFs) of the number of spectrum opportunities and the number of interference opportunities, P ISO and P EIO are given by P ISO (S d ) = P EIO (I d ) = S d 1 k=0 M k=i d +1 Pr { S = k } = F S (S d 1), (3.14) Pr { I = k } = 1 F I (I d ), (3.15) respectively, where F S ( ) and F I ( ) are the corresponding cumulative distribution functions (CDFs). Note that both pairs of performance metrics, i.e., (P M, P F ), and (P ISO, P EIO ), are general performance metrics independent of the sampling schemes and the detection algorithms used by the SU. There are two major differences between the two pairs of performance metrics: P M,m and P F,m are defined conditionally on the real states of channels, while P ISO and P EIO are defined unconditionally; P M,m and P F,m are constant performance metrics across symmetric channels, while P ISO and P EIO are performance metrics for the entire bandwidth and inherently introduce coupling across channels. For illustration, if we consider one channel only, i.e., M = 1, S d = 1 and I d = 0, the 35

50 relationship between these two pairs of performance metrics are P ISO = p + (1 p)p F, (3.16) P EIO = pp M. (3.17) In essence, (P ISO, P EIO ) takes the primary utilization rate into account in defining the spectrum opportunities. This inclusion makes more sense, as a channel might be of little interest to the SU even with small (P M, P F ), if the primary occupancy of that channel is high. This dissertation considers both pairs of performance metrics. Also note that the discussions on (P M, P F ) that follow are based on the individual channels only. For symmetric channels with identical primary occupancy probabilities and identical SNR levels, (P M,m, P F,m ) share the same receiver operating characteristic (ROC) across channels. 3.3 Sampling Schemes Sampling schemes for wideband spectrum sensing fall into two categories: Nyquist sampling and sub-nyquist sampling Nyquist Sampling Nyquist sampling leads to conditionally independent observations across channels, which makes channel detection much easier compared to sub-nyquist sampling that couples observations across channels. However, if the spectrum of interest is relatively wide, the rate required for wideband Nyquist sampling (WBNS) could be very high and costly to implement. As alternatives, we consider two types of Nyquist based sampling schemes that monitor much narrower frequency bands with corresponding Nyquist rate. 36

51 partial-band for spectrum sensing wideband of interest M channels f Figure 3.3. Illustration of partial-band Nyquist sampling (PBNS) Partial-Band Nyquist Sampling As discussed in Section 3.2, the SU may only be interested in finding a desired number of spectrum opportunities rather than finding all of them in a wideband CR network. This motivates the idea of partial-band Nyquist sampling (PBNS), which samples only a fraction of the entire bandwidth at the corresponding Nyquist rate and ignores the remaining part, as illustrated in Fig To implement PBNS, an analog bandpass filter with corresponding bandwidth and center frequency, as well as a mixer that down-converts the signal to baseband, are needed at the receiver front end before the sampler, as illustrated in Fig Again, we only consider the baseband equivalent signal for PBNS. PBNS is characterized by the number of channels in the partial band. Specifically, PBNS with positive integer parameter L samples L channels of the entire bandwidth at the corresponding Nyquist sampling rate of 2LB 0. Since all channels are symmetric, it makes no difference which part of the spectrum is monitored by the SU. Thus, we assume the partial band contains the first L channels for simplicity. Note that WBNS can be viewed as a special case of PBNS with L = M. The number of samples taken for each channel is N 0 for PBNS and the number of samples in a sensing window is N = T win 2LB 0 = NL/M. Let V[k] denote the frequency samples 37

52 y(t) LB 0 Uniform sampler v[n] (a) F s = 2LB 0 2LB 0 LB 0 v PB (t) Uniform sampler v[n] f c (b) F s = 2LB 0 Figure 3.4. Implementation of partial-band Nyquist sampling (PBNS): (a) Equivalent baseband implementation; (b) Real implementation. 38

53 for PBNS. Thus, Y[k], 0 k LN 0 1, V[k] =, (3.18), LN 0 k N 1 where we use to indicate that no observation is available. Compared to more sophisticated sub-nyquist sampling schemes that follow, the advantages of PBNS lie in two major aspects. First, the sampler is straightforward to implement. It is simply the traditional Nyquist sampler over a smaller bandwidth. Second, channel-by-channel detection algorithms, which are optimal for performance metrics that do not couple the channels, such as P M and P F, are much easier to implement and more computationally efficient. The disadvantage is that the number of channels PBNS can monitor is limited to L, and no a posteriori information for the remaining channels is available Sequential Narrowband Nyquist Sampling One way to monitor all channels while sampling at a much lower rate than F nyq is to divide the entire bandwidth in frequency as well as the entire sensing window in time into several parts, and sense each sub-band sequentially in sub-windows. We refer to this approach as sequential narrowband Nyquist sampling (SNNS). SNNS is characterized by the sub-sampling factor, which is defined as the ratio of the Nyquist rate to the actual sampling rate, i.e., ρ = F nyq F s. (3.19) To implement SNNS, the entire bandwidth is split into ρ sub-bands, with each subband containing M/ρ channels and a total bandwidth of B/ρ. At the receiver front end, ρ analog passband filters, or one analog passband filter along with a tunable 39

54 mixer, are needed. Narrowband Nyquist sampling is implemented sequentially in sub-windows for each sub-band, with the entire sensing window T win split into ρ sub-windows. Thus, the number of samples collected for each channel is N 0 /ρ. Compared to PBNS that monitors only a fraction of the wideband in a sensing window, SNNS is able to get posterior information for all channels. However, this is achieved by providing less information per channel, which then reduces the accuracy of the corresponding sensing result. Another disadvantage of SNNS is that it is more complicated to implement than PBNS and most of the sub-nyquist sampling schemes that follow, which contradicts the original goal of reducing the complexity of the hardware implementation Sub-Nyquist Sampling Sub-Nyquist sampling schemes, which sample the entire bandwidth all at once at an average sampling rate much lower than the wideband Nyquist rate, are often much easier to implement and have attracted significant research attention. However, as we will see, sub-nyquist sampling couples the observation across channels, which in principle requires more sophisticated detection algorithms that are more computationally complex A Unified Model for Sub-Nyquist Sampling We develop a unified model for sub-nyquist sampling schemes using multirate filter banks analogous to multi-coset sampling introduced in [22], as illustrated in Fig Specifically, a sub-nyquist sampling scheme is characterized by parameters { D, a }, in which D > 1 is the down-sampling factor, a[l] {0, 1}, 0 l < D, is a sampling branch indicator, with a[l] = 1 and a[l] = 0 corresponding to an active and inactive sampling branch, respectively. 40

55 y[n] a[0] z 0 D D z 0 a[1] z 1 D D z 1 v[n] a[d 1] z (D 1) D D z D 1 Figure 3.5. A unified model for sub-nyquist sampling. Let q D 1 l=0 a[l], A {l : a[l] = 1, 0 l < D}. Note that a and A are two equivalent representations of the active sampling branches. The sub-sampling factor ρ is defined as the ratio of the down-sampling factor to the number of active sampling branches, i.e., ρ D q. (3.20) Note that the unified sub-nyquist sampling model in Fig. 3.5 differs with the multibranch sub-nyquist sampling in Fig. 2.3 (a) in that: (1) the former corresponds to decimation on wideband Nyquist samples, and thus defined in the digital domain, whereas the latter is defined in the continous-time domain; (2) the down-sampling factors of all branches are the same in the former and may be different in the latter. Also note that WBNS can be viewed as an extreme case within the unified model with q = D, i.e., all branches active. Based on the above model, the sub-nyquist samples v[n] in the time domain are modeled as v[n] = ã[n]y[n], 0 n < N, (3.21) 41

56 where ã[n] a [ (n) D ], (3.22) and () D is the mod-d operator. Taking the finite duration of a sensing window into account, a wideband spectrum sensing scheme based on sub-nyquist sampling can be characterized by parameters { N, D, a }. Let à denote the set of active Nyquist samples of a sub-nyquist sampling in a sensing window: à { n : 0 n < N, ã[n] = 1 }. (3.23) Let Q à (3.24) denote the number of active wideband Nyquist samples in a sensing window. For simplicity, we only consider N = κd, where κ is a positive integer. Thus, Q = κq and ã[n] is periodic in the sensing window if κ 2. Let V and à denote the normalized DFT of v and ã in a sensing window, respectively. Since multiplication in the time domain corresponds to circular convolution in the frequency domain according to the duality property of the DFT, the counterpart of (3.21) in the frequency domain is V[n] = NÃ[n] Y[n], (3.25) where the operator denotes N-point circular convolution. Expressed in matrix form, V = Fv = NΦY, (3.26) 42

57 where Ã[0] Ã[N 1] Ã[N 2]... Ã[1] Ã[1] Ã[0] Ã[N 1]... Ã[2] Φ Ã[2] Ã[1] Ã[0]... Ã[3] Ã[N 1] Ã[N 2] Ã[N 3]... Ã[0] (3.27) is called the aliasing matrix. Φ is Hermitian, i.e., Ã[N k] = Ã [k], since ã[n] is real, where z is the complex conjugate of z. Let φ kn denote the kn-th entry of Φ, i.e., φ kn = Ã[ (k n) N ]. (3.28) As we will see, the aliasing matrix determines the aliasing pattern of a specific sub- Nyquist sampling scheme, and affects the performance of various detection algorithms Aliasing Effect of Sub-Nyquist Sampling It is clear from (3.25) and (3.27) that aliasing in the frequency domain is inevitable for sub-nyquist sampling, i.e., V[k] is a weighted sum of all Y[n], 0 n < N, with the aliasing weight for each frequency sample being Nφ kn. Different sub-nyquist sampling schemes lead to different combinations of φ kn, 0 k, n < N. Nevertheless, all sub-nyquist sampling schemes with the same sub-sampling factor are subject to the following two constraints: N 1 n=0 N 1 φkn 2 = Ã[k] 2 = k=0 n=0 N 1 n=0 φkk 2 N 1 = Ã(0) 2 1 = ã[n] N ã[n] 2 = Q, (3.29) 2 = Q2 N, (3.30) 43

58 with (3.29) resulting directly from Parseval s Theorem. To study the different aliasing effects of the sampling schemes, we define the aliasing variance of a sub-nyquist sampling scheme with the set of active Nyquist samples à as where 2 1 σa(ã) N 1 0<k<N ( Ã[k] ) 2 2, µ( Ã) (3.31) µ(ã) 0 k<n Ã[k] 2 Ã[0] 2 N 1 = N Q ρ(n 1) (3.32) 2 is constant for a fixed sub-sampling factor. Note that σa(ã) can be computed by substituting (3.22), (3.23) and (3.32) into (3.31) for a given sub-nyquist sampling scheme. We study two extreme aliasing patterns that lead to the maximum and minimum aliasing variances, respectively. Maximum aliasing variance Periodicity in the observation window leads to large aliasing variance. To see this, suppose N = κd, where κ is a positive integer. Thus, ã[id + l] = ã[l] = a[l], for 1 i < κ, 0 l < D, and Ã[k] = 1 N 1 ã[n]ω kn = 1 κ 1 D 1 ã[id + l]ω k(id+l) = 1 D 1 N N N n=0 i=0 l=0 l=0 κ 1 ã[l]ω kl ω kid κ D 1 N l=0 = a[l]ωkl 0, k = l κ, 0 l < D,. (3.33) 0, otherwise i=0 44

59 The last step follows from κ 1 ( ) k = l ω kd i = i=0 κ, κ, 0 l < D,. (3.34) 0, otherwise Thus, N D elements of Ã[k] are zero and the D non-zero elements occur periodically. Note that smaller period D leads to larger aliasing variance. Therefore, the maximum possible aliasing variance is obtained when D = ρ, for which Q N, k = lq, 0 l < ρ, Ã[k] =. (3.35) 0, otherwise The corresponding sub-nyquist sampling scheme corresponds to integer undersampling (IU) from WBNS. As a result of circular convolution, IU causes periodic aliasing (PA) of a set of frequencies. Minimum aliasing variance The sub-nyquist sampling schemes that exhibit more randomness in the sensing window often give smaller aliasing variance. One 2 could imagine the ideal minimum aliasing variance to be σa(ã) = 0, or equivalently, Ã[k] 2 = Q 2 N, k = 0,. (3.36) µ(ã), 1 k < N The choice corresponds to an aliasing pattern that spreads the power of any frequency evenly over the entire bandwidth, which we refer to as uniform aliasing (UA). Although no sub-nyquist sampling scheme can actually achieve zero aliasing variance, as we will see in later sections, there are several sub-nyquist sampling schemes that can achieve very small aliasing variance. Moreover, for a given duration of a sens- 45

60 ing window N and a specific sub-sampling factor ρ, the sampling pattern with the minimum aliasing variance can be obtained by exhaustive search over all possible sub-sampling patterns, which we refer to as the minimum aliasing variance (MAV) sampling hereafter. An illustrative example of the difference between periodic aliasing and uniform aliasing with N = 8, ρ = 2 is shown in Table 3.1, where for uniform aliasing the off-diagonal elements s i are comparably much smaller than the diagonal elements l, 1 i Integer Undersampling The simplest sub-nyquist sampling scheme is to sample the wideband uniformly at a lower sampling rate than the wideband Nyquist rate. We focus on the uniform sub-nyquist sampling scheme in which the sampling rate is an integer fraction of the wideband Nyquist rate, i.e., integer undersampling (IU) that causes periodic aliasing. To the best of our knowledge, existing work on sub-nyquist sampling focuses exclusively on non-uniform sampling and IU has not attracted any attention in the field of wideband spectrum sensing for CR. As we will see in Chapter 7, IU exhibits appealing performance in the regime of better protections for the primary system. IU can be modeled by one active branch within the unified sub-nyquist sampling model in Fig. 3.5 as D = ρ, and q = 1, (3.37) i.e., only one value of a[n] is non-zero, 0 n < ρ, where ρ is the sub-sampling factor. Note that IU corresponds to one branch of decimation from Nyquist samples, but would most likely be implemented as a low rate analog-to-digital converter. We refer to the aliased channels after periodic aliasing as bins, which are illus- 46

61 TABLE 3.1 ILLUSTRATIVE EXAMPLES OF PERIODIC ALIASING AND UNIFORM ALIASING Periodic aliasing Uniform aliasing ã[n] Ã[n] Φ l s 1 s 2 s 3... s 30 s 31 s 31 l s 1 s 2... s 29 s 30 s 30 s 31 l s 1... s 28 s s 2 s 3 s 4 s 5... l s 1 s 1 s 2 s 3 s 4... s 31 l trated in Fig Note that for IU: Each bin contains ρ channels, which overlap completely and are indistinguishable. The M bins are periodic with period M/ρ. Thus, the signals in the first M/ρ bins are sufficient observations for spectrum sensing. Note that both SNNS and IU monitor the entire bandwidth with uniform sampling at a rate of F nyq /ρ. However, they are quite different in implementation in the timefrequency dimension as illustrated in Fig

62 5B 0 4B 0 3B 0 2B 0 replica B 0 F s = F nyq 2 0 B 0 2B 0 3B 0 4B 0 5B 0 f (a) replica (b) F MBins (c) F Figure 3.6. Integer undersampling (IU) causes periodic aliasing of a set of frequencies. (a) Complex analog signal in the wideband of interest with 10 channels. (b) Aliasing effect of IU. (c) Aliased signal: each bin is a complete overlap of 2 channels. 48

63 frequency frequency wideband of interest wideband of interest a sensing window time a sensing window time (a) (b) Figure 3.7. Comparsion of (a) sequential narrowband Nyquist sampling (SNNS) and (b) integer undersampling (IU) in time and frequency Multi-Coset Sampling Multi-coset sampling (MCS) introduced in Section fits directly into the unified sub-nyquist sampling model in Fig 3.5 with a given down-sampling factor D and a set of active sampling branches A. Note that MCS has been specified to contain at least two active branches. If the definition can be extended to any number of active branches, IU can be viewed as a special case of MCS with D = ρ, q = 1. Moreover, if D N, MCS is non-periodic in the sensing window Generalized Co-Prime Sampling We extend co-prime sampling introduced in Section from two branches to a general number of branches and refer to it as generalized co-prime sampling (GCPS). Specifically, as illustrated in Fig. 3.8, a J-GCPS with J sampling branches 49

64 z d 1 D 1 D 1 z d 1 y[n] z d 2 D 2 D 2 z d 2 v[n] z d J D J D J z d J Figure 3.8. Implementation of generalized co-prime sampling. is characterized by a vector of down-sampling factors D [D 1, D 2,..., D J ] and a vector of delays d [d 1, d 1,..., d J ], with gcd(d i, D j ) = 1, 1 i, j J, i j, where gcd(n 1, n 2 ) is the greatest common divisor of two positive integers n 1 and n 2, and 0 d j < D j, 1 j J. Note that GCPS is inherently periodic, and for every GCPS implementation in Fig. 3.8, there is an equivalent implementation in Fig. 3.5, with J 1, if n J j=1 D = D j, and a[n] = A j, j=1 0, otherwise,, (3.38) where A j { } n : n = d j + kd j, 0 k < D/D j, 1 j J. However, if D > N, the sampling is non-periodic in the sensing window and it causes uniform aliasing. Within the unified structure, the original CPS proposed in [65] can thus be viewed as a special case of J = 2. We note that the analysis of CPS in [65] is limited to the time domain, i.e., the authors show that the auto-correlation sequence can be reconstructed for any value of time lag, which is then used to estimate the power spectral density (PSD). As an alternative approach, we use the Lomb-Scargle periodogram to 50

65 study the performance of GCPS in the frequency domain directly Random Sampling on A Grid Within the unified sub-nyquist sampling model in Fig. 3.5, random sampling on a grid (RSG) in a sensing window can be modeled by ( ) 1 D = N, and a[n] Bernoulli, (3.39) ρ where ρ is the sub-sampling factor and a[n] are i.i.d. Practically, RSG exhibits small aliasing variance with a high probability. However, it also has the possibility to give the maximum aliasing variance for sub-nyquist sampling, because a realization of RSG reduces to IU with probability Pr { RSG reduces to IU } ( ) N ( ( ) 1 ρ = ρ 1 1 ) N 1 1 ρ, ρ ρ which is so small that the possibility is negligible for large N and ρ 2. It is worth noting that the active branches of RSG are time-varying, and the number of active branches is a binomial random variable. This leads to a time-varying sub-sampling factor in contrast to the fixed sub-sampling factor for IU, MCS, and GCPS. Thus, we also consider random sampling on a grid with fixed number of taps (RSG-FN), for which the number of active branches is fixed, yet the set of active branches are still time-varying. In Fig. 3.9, we provide an illustrative example of different sub-nyquist sampling schemes compared to WBNS. In Fig. 3.10, we summarize the connection between different sub-nyquist sampling schemes discussed above with regard to the aliasing patterns. 51

66 t t WBNS IU t MCS t t CPS RSG Figure 3.9. Illustrative example of several sub-nyquist sampling schemes with a sub-sampling factor of 2. (a) wideband Nyquist sampling (WBNS); (b) integer undersampling (IU); (c) multi-coset sampling (MCS) with: D = 6, A = {0, 2, 3}; (d) co-prime sampling (CPS) with: D 1 = 4, d 1 = 0, D 2 = 3, d 2 = 0; (e) random sampling on a grid (RSG). 52

67 Increase D Increase J MCS GCPS Periodic Sampling IU MAV RSG, RSG-FN Uniform Aliasing Periodic Aliasing Figure Relationship between different sub-nyquist sampling schemes and their aliasing patterns. 3.4 Detection Algorithms We consider two classes of detection algorithms based upon how decisions are made across channels. Specifically, Channel-by-channel detection makes decisions on each channel independently, Multi-channel detection makes decisions for all channels jointly. Along with the other three elements discussed in the preceding sections, dependencies across channels for wideband spectrum sensing arise as listed in Table 3.2 with illustrative examples for each case. Note that partial-band Nyquist sampling is sub-wideband Nyquist Nyquist sampling. However, in essence it is Nyquist sampling with corresponding narrower bandwidth. For the simplest case in which the first three elements are all independent across channels, channel-by-channel detection gives the optimal performance. Otherwise, we might expect multi-channel detection algorithms to perform better. We 53

68 TABLE 3.2 DEPENDENCIES OF ELEMENTS OF WIDEBAND SPECTRUM SENSING ACROSS CHANNELS Independent Dependent Primary occupancies p-sparse K-sparse Performance metrics (P M, P F ) (P ISO, P EIO ), MSE Sampling schemes Nyquist sampling sub-nyquist sampling Detection algorithms channel-by-channel detection multi-channel detection will see some examples that confirm this intuition in the sections that follow. In this section, we introduce channel-by-channel Bayesian detection and wideband Bayesian detection for (P M, P F ) and (P ISO, P EIO ), respectively, and general sampling schemes based upon the p-sparse model for primary occupancies. Within the Bayesian detection framework, the goal of wideband spectrum sensing for CR is to recover the primary occupancy that minimizes the detection risk. Specifically, costs are assigned to each outcome for each true state and the associated detection risk is defined as the expected cost. For a sequence of M channels, the primary occupancy has 2 M possibilities. Thus, wideband spectrum sensing is essentially a 2 M -hypothesis Bayesian detection problem Channel-by-Channel Bayesian Detection First consider the case of single channel detection. Let C ij be the cost of declaring state i if the true state is j, i, j {0, 1}. Specifically, C 01 and C 10 are the costs of missed detections and false alarms for one channel, respectively. It is common to assume that the cost of a correct outcome is zero, i.e., C 00 = C 11 = 0. We derive the Bayesian detection rule in a slightly different way from the classic 54

69 Bayesian detection model [60]. The conclusions are the same since they are essentially identical. Define the detection risk of declaring a particular state i for channel m given the observation as R I,m (i v) = C ij Pr { H m = j v }, (3.40) Specifically, j=0,1 R I,m (0 v) = C 01 Pr { H m = 1 v }, R I,m (1 v) = C 10 Pr { H m = 0 v }, for the model considered. Note that we use subscript I to denote the channelby-channel risk associated with P M and P F, which is to be differentiated with the wideband risk associated with P ISO and P EIO that follows. The Bayesian detection rule is Ĥ m = arg min R I,m (i v). (3.41) i={0,1} Now consider wideband spectrum sensing if (P M, P F ) are used as performance metrics. We assume additive costs across channels, which is reasonable if different channels exhibit independent primary activities and are equally attractive to the SU. Specifically, given the true state H, the vector cost of declaring the state Ĥ is defined as C(Ĥ,H) = C 10 1{Ĥm =1, H m =0} + C 01 1{Ĥm =0, H m =1}, (3.42) 1 m M 1 m M Ĥ, H H, where H is the set of all 2 M possible states, H m and Ĥm are the m-th element of vector H and Ĥ, respectively, and 1{statement} is the indicator function, 55

70 which is defined as a function of an event that evaluates to 1 if the event occurs and 0 if the event does not occur. Define the detection risk R I (Ĥ v) as the expected cost of declaring the primary occupancy Ĥ given the measurement v. Specifically, (C(Ĥ, ) R (Ĥ v) = E I H H) v = C(Ĥ, H)Pr{ H = H v }, (3.43) H H where E X ( ) is the expectation operator with regard to the random variable X and Pr { H = H v } is the conditional probability of H given v. Thus, the Bayesian detection rule that minimizes the detection risk is, Ĥ = arg min R I (Ĥ v). (3.44) Ĥ H It is worth noting that the posterior belief of the primary occupancy of the entire bandwidth Pr { H v } plays a key role in the Bayesian detection model. In the following a few steps we will derive Pr { H v } within the unified sub-nyquist sampling model. We start from exploiting the properties of the aliasing matrix. Specifically, Lemma 1. rank(φ) = Q. Proof. Substituting Ã[k] = 1 N 0 n<n ã[n] into (3.27), Φ = FΨ, (3.45) 56

71 where Ψ is defined as Ψ 1 N ã[0] ã[0]... ã[0] ã[1] ã[1]ω 1... ã[1]ω (N 1) ã[2] ã[2]ω ã[2]ω 2 (N 1). (3.46) ã[n 1] ã[n 1]ω (N 1) 1... ã[n 1]ω (N 1) (N 1) Thus, rank(φ) = rank(ψ) = Q, since the DFT matrix F has full rank. It is clearly seen from (3.46) that rank(ψ) = Q, as only Q elements of ã[k] take value 1, and the remaining N Q elements take value 0. This makes Ψ a derived matrix of the inverse discrete Fourier transform (IDFT) matrix by deleting N Q rows and keeping only Q rows. The remaining rows are linearly independent as all rows of the original IDFT matrix are linearly independent. Substituting (3.45) into (3.26) and then taking the IDFT on both sides, then v = ΨY. (3.47) For ease of exposition, let ṽ ( ), (, (v )Ã and Ψ (Ψ )Ã) where X I represents a sub-matrix whose columns are withdrawn from X according to the set of column coordinates I. Thus, (3.47) is equivalent to ṽ = ΨY. (3.48) 57

72 From (3.45) and (3.46), we can also deduce that Φ = Ψ Ψ = Ψ Ψ. (3.49) Based on (3.6), there exists a probabilistic model of observing Y given a primary occupancy H. Specifically, Y has a zero mean, multivariate Gaussian distribution, with conditionally independent components whose variances are given in (3.7). Let Σ Y denote the covariance matrix of Y, then ( ) Σ Y = diag {Λ Y [k], 0 k < N, (3.50) where diag( ) defines a diagonal matrix with the diagonal entries specified by the input arguments and all the off-diagonal entries zero. Thus, ṽ has a zero mean, multivariate Gaussian distribution with covariance matrix Σṽ = ΨΣ Y Ψ, (3.51) and the posterior belief Pr { H v } in (3.43) can be computed from fṽ H (ṽ H) by Bayes theorem. Specifically, Pr { H ṽ } = fṽ H (ṽ H)Pr { H } H H f ṽ H(ṽ H )Pr { H }. (3.52) Note that PBNS can be viewed as a special case of sub-nyquist sampling with corresponding parameter settings, i.e., Q = N, Σṽ = Σ Y = diag ( {Λ Y [k] ), and the measurements are uncorrelated. In this case, the vector detection problem reduces to a channel-by-channel detection problem, and the classical detection algorithms, such as energy detection, can be readily applied. For a more general sub-nyquist sampling scheme in which Q < N, the detection 58

73 problem in (3.44) can be, at worst, solved by an exhaustive search in H, with the computational complexity increasing exponentially with M, which is computationally prohibitive for large M. In the chapters that follow, we develop some efficient algorithms that seek tradeoffs between the computational complexity and the sensing performance following the Bayesian detection rule for different aliasing patterns of sub-nyquist sampling Wideband Bayesian Detection A similar Bayesian detection framework can be established if (P ISO, P EIO ) are used as performance metrics. Specifically, define C ISO and C EIO as costs of insufficient spectrum opportunities and excessive interference opportunities, respectively. The wideband risk R II of declaring a particular primary occupancy is R (Ĥ v) = C II ISOP (Ĥ v) + C ISO EIOP EIO (Ĥ v), (3.53) in which P ISO (Ĥ v) and PEIO(Ĥ v) are conditional probabilities of insufficient spectrum opportunity and excessive interference opportunity of declaring primary occupancy Ĥ = Ĥ given the observation v. The corresponding detection rule is Ĥ = arg min R II Ĥ H (Ĥ v) = arg min Ĥ H R II (Ĥ V). (3.54) P ISO (Ĥ v) and PEIO(Ĥ v) involve computations of the likelihood of each state in H, which is computationally prohibitive. In the chapters that follow, we develop ranked posterior belief detection algorithms that recover the channel states sequentially based upon the sorted posterior beliefs of each channel being in an ON state following the Bayesian rule in (3.54). The above discussions establish the general Bayesian detection frameworks based upon the raw observation v. However, the computation of the likelihood, which is 59

74 required for Bayesian detection, often involves inversion of matrices with large dimension. To reduce the computational complexity, we use some deterministic function of the raw observations, specifically, the average power per channel, to generate the alternative detection statistics from the raw observation. More details can be found in later chapters. Note that realizations of the same Bayesian detection rule could be different for different sampling schemes. Also, detection algorithms specialized to a particular sampling scheme may exist as well, such as orthogonal matching pursuit (OMP) for sub-nyquist sampling with uniform aliasing. A comprehensive study of several detection algorithms can be found in Chapter 4, Chapter 5, and Chapter 6, for Nyquist sampling, sub-nyquist sampling with periodic aliasing, and sub-nyquist sampling with uniform aliasing, respectively. When comparing different detection algorithms for the same sampling scheme, or the same detection algorithm for different sampling schemes, we are not only interested in a particular operating point, but also particularly interested in full tradeoffs between the primary protection and the secondary opportunities in the forms of channel receiver operating characteristic (ROC), which shows tradeoffs between P M and P F, as well as wideband ROC, which shows tradeoffs between P ISO and P EIO. As we will see, ROCs can be obtained by varying the detection threshold in energy based detection, or varying the cost ratios in Bayesian detection. 3.5 Summary In this chapter, we discuss the four elements for wideband spectrum sensing that could couple the individual channels, based upon which we establish general frameworks for later chapters. We introduce the system model for the dissertation and propose a p-sparse model to characterize the primary occupancies across channels. We also motivate new wideband performance metrics relevant to the goal of wide- 60

75 band spectrum sensing. We discuss structures and classification of sampling schemes as well as detection algorithms, which will be expanded upon in later chapters. 61

76 CHAPTER 4 NYQUIST SAMPLING SCHEMES In this chapter, we develop several detection algorithms for Nyquist based sampling schemes, specifically, channel-by-channel energy detection, ranked energy detection, and ranked posterior belief detection. We analyze and compare the detection regions and sensing performance in the form of the system receiver operating characteristic (ROC), which corresponds to the best tradeoff between P ISO and P EIO, for different detection algorithms. We provide some illustrative examples as well as a much broader range of numerical results. The results suggest that for wideband performance metrics (P ISO, P EIO ) that couple the individual channels, multi-channel detection outperform channel-by-channel detection even for Nyquist sampling that gives conditionally independent observations across channels. The remainder of the chapter is organized as follows. In Section 4.1, we introduce the channel-by-channel energy detection algorithm which is optimal in minimizing the channel-by-channel risk associated with (P M, P F ). In Section 4.2, we develop multi-channel detection algorithms exclusively for wideband performance metrics (P ISO, P EIO ). In Section 4.2.4, we provide several numerical results and compare the performance of different detection algorithms. In Section 4.3, we conclude the chapter. 4.1 Channel Detection with (P M, P F ) The study in this section is primarily based on PBNS with parameter L. The results for WBNS and SNNS can be deduced from PBNS with different parameter 62

77 settings. Define the detection statistic of channel m as the average energy of all samples in the channel, i.e., T m = 1 N 0 V[k] 2, (4.1) k I m for 1 m L. Since no real-world observations are available for any channel with an index greater than L, the SU simply assumes those channels are always unavailable, or equivalently, always declare those channels to be ON. For the sake of reducing computational complexity in matrix inversion, we consider non-coherent, energy based detection algorithms, and use T {T 1, T 2,..., T M } to replace the raw observation V, and equivalently v, in the Bayesian detection structure introduced in Section 3.4. Note that T is not a sufficient statistic compared to the original data sequence given that the samples are conditionally independent Gaussians. However, it is widely used in the non-coherent energy detection. First, it is much more computationally efficient. Second, it applies to signal models beyond the AWGN model. Nyquist sampling leads to conditionally independent observations across frequencies and channels. Thus, the detection rule in (3.41) can be simplified to Ĥ m = arg min i={0,1} = arg min i={0,1} R I,m (i T m ) = arg min i={0,1} j C ij Pr { } H m = j T m j C ij q m (j), (4.2) where we use q m (j) to denote the posterior belief that channel m is in state j given the observations, i.e., q m (j) Pr { H m = j T m }. (4.3) 63

78 For the primary system model considered, T m is a scaled chi-square random variable with 2N 0 degrees of freedom under both hypotheses. Specifically, T m H m σ2 noi + H m σ 2 sig 2N 0 χ 2 2N 0, (4.4) for 1 m L. The likelihood ratio LR m for channel m is defined as LR m f T m H m (t m 1) f Tm H m (t m 0) = e γn 0 1+γ tm (1 + γ), (4.5) N 0 where γ σ 2 sig/σ 2 noi is the SNR level. The posterior belief can be deduced by Bayes theorem from the likelihood ratio. Specifically, f Tm Hm (t m 1)Pr { H m = 1 } q m (1) = i=0,1 f T m H m (t m i)pr { H m = i } = LR m 1 p, (4.6) + LR p m 1 q m (0) = 1 q m (1) = 1 + p LR, (4.7) 1 p m where f Y X (y x) is the conditional probability density function (PDF) of random variable Y given X. Note that LR m, and q m (i) are both functions of the detection statistic T m. Substituting (4.5) into (4.2), the detection rule can be simplified to Ĥ m=1 T m τ = 1 + γ ( ) pc I log Ĥ m=0 γn 0 (1 p) N 0 log(1 + γ), (4.8) where C I C 01 C 10, (4.9) is the first type cost ratio. Note that detection rule in (4.8) corresponds to the traditional channel-by-channel energy detection (CED) with the threshold determined 64

79 by the cost ratio given all other system parameters. For a given detection threshold τ, the probability of missed detection and the probability of false alarm are given by P M,m (τ) = F χ 2 ( ) 2N0 τ 1 + γ ; 2N 0, (4.10) P F,m (τ) = 1 F χ 2( 2N0 τ; 2N 0 ), (4.11) respectively, where F χ 2(x; k) denotes the CDF of the central chi-square distribution with k degrees of freedom. By varying the detection threshold τ, we can obtain the receiver operating characteristic (ROC). For symmetric channels with identical detection threshold, P M,m (τ) and P F,m (τ) in (4.10) and (4.11) are the same for all channels in the partial-band. Thus, we omit the subscript m in further discussion when there is no ambiguity. 4.2 Wideband Detection with (P ISO, P EIO ) In this section, we develop several multi-channel detection algorithms for system design with the wideband performance metrics (P ISO, P EIO ) Channel-by-Channel Energy Detection We first study the performance of CED for (P ISO, P EIO ). For PBNS with parameter L and CED with a given detection threshold τ, the PMFs of the number of spectrum opportunities and the number of interference opportunities are given by Pr { S = s, τ } ( ) L ( s ( ρ s, = P S (τ)) 1 P S (τ)) (4.12) s Pr { I = i, τ } ( ) L ( i ( ρ i, = P I (τ)) 1 P I (τ)) (4.13) i 65

80 respectively, where ( ) ) P S (τ) (1 p) 1 P F (τ) = (1 p)f χ 2( 2N0 τ; 2N 0, (4.14) ( ) 2N0 τ P I (τ) pp M (τ) = pf χ γ ; 2N 0 (4.15) are the probability of a spectrum opportunity and the probability of an interference opportunity per channel with a given detection threshold τ, respectively. Thus, P ISO and the P EIO can be obtained by substituting (4.12) and (4.13) into (3.14) and (3.15), respectively. For SNNS, since the entire sensing window T win is split into ρ sub-windows, and the entire bandwidth is split into ρ sub-bands with narrowband Nyquist applied to each sub-band, the number of samples available for each channel is N 0 /ρ. Thus, P M and P F per channel for SNNS can be obtained by replacing N 0 by N 0 /ρ in (4.10) and (4.11), and P ISO and P EIO can be obtained by replacing L with M in (4.12) and (4.13), respectively Ranked Energy Detection Channel-by-channel detection is optimal in minimizing the channel-by-channel risk, which is defined based upon the individual P M,m and P F,m, if the primary occupancies and the secondary observations are independent across channels. For the wideband performance metrics (P ISO, P EIO ) that couple the individual channels, it is expected that channel-by-channel detection could be worse than other more sophisticated detection algorithms. With the goal of finding a desired number S d of spectrum opportunities restricting to no interference opportunities to the primary system, a proper multi-channel detection algorithm should first choose channels with larger posterior beliefs of being in the OFF state as candidates for spectrum opportunities, until a certain number 66

81 of channels have been found. For symmetric channels, a channel with a smaller detection statistic in (4.1) has a larger posterior belief of being in an OFF state. Motivated by the above discussion, we propose a ranked energy detection (RED) algorithm that compares channels in the partial band and makes decisions only for a certain number of top candidates. Algorithm 1. Ranked energy detection with parameters ( K, τ), where S d K L: 1. Sort the channels according to T m in ascending order. 2. Make decisions for the first K channels using CED with the given detection threshold τ according to (4.8) and ignore the remaining M K channels. 3. If the total number of declared OFF channels of the top K candidates is less than S d, disregard the decision results and declare all channels to be ON instead. The last step is reasonable because the performance metric P ISO does not give partial credit for finding spectrum opportunities less than the desired amount. Also note that K is a parameter that controls the number of channels on which the detection algorithm would make decisions based upon the real-world observations, and the decisions on the remaining M K channels are ON regardless of the observations. Ranked energy detection essentially separates the number of monitored channels and the number of channels on which decisions are made. General theoretical analysis for ranked energy detection is rather involved. As an illustrative example, we study the simplest case in which K = S d = 1 and I d = 0 for L = 2. The detection algorithm can thus be specialized to T min = min m={1,2} { Tm }, mmin = arg min m={1,2} 1, T min > τ, Ĥ mmax = 1, and Ĥm min =. 0, T min τ { } { } Tm, and mmax = arg max Tm, m={1,2} There are three different cases for the primary occupancies of the two channels 67

82 with respect to the number of ON channels, i.e., both channels are OFF, only one channel is ON, and both channels are ON. In any case, both channels are declared ON if T min > τ, leading to neither spectrum opportunity nor interference opportunity. Both channels are OFF: P S ( τ H = [0, 0] ) = Pr { Tmin < τ } = F χ 2( 2N0 τ; 2N 0 ), (4.16) P I ( τ H = [0, 0] ) = 0. (4.17) Both channels are ON: ( ) P S τ H = [1, 1] = 0, (4.18) ( ) { P I τ H = [1, 1] = Pr Tmin < τ } ( ) 2N0 τ = F χ γ ; 2N 0. (4.19) Only one channel is ON: Without loss of generality, we assume channel 1 is OFF and channel 2 is ON. Then ( ) { P S τ H = [0, 1] = Pr T1 < T 2, T 1 < τ } τ ( + ) = f T1 (t 1 ) f T2 (t 2 )dt 2 dt 1 0 t 1 τ ( ( ) + ( ) 2N0 t 2 = f χ 2 2N0 t 1, 2N 0 f χ 2 0 t γ, 2N 0 )dt 2 dt 1, (4.20) ( ) { P I τ H = [0, 1] = Pr T2 < T 1, T 2 < τ } τ ( + ) = f T2 (t 2 ) f T1 (t 1 )dt 1 dt 2 0 t 2 = τ 0 ( ( 2N0 t ) + 2 f χ γ, 2N 0 f χ 2 t 2 ( 2N0 t 1, 2N 0 ) dt1 ) dt 2. (4.21) where f χ 2(x; k) is the PDF of central chi-square distribution with k degrees of freedom. An analogous result can be obtained if channel 1 is ON and channel 2 is OFF. Thus, the unconditional probability of a spectrum opportunity and the probability 68

83 of an interference opportunity given the detection threshold τ are P S (τ) = H H Pr { H = H } P S ( τ H = H ) ( ) ( ) ( ) = (1 p) 2 P S τ H = [0, 0] + 2p(1 p)ps τ H = [0, 1] + p 2 P S τ H = [1, 1], (4.22) P I (τ) Pr { H = H } ( ) P I τ H = H H H = (1 p) 2 P I ( τ H = [0, 0] ) + 2p(1 p)pi ( τ H = [0, 1] ) + p 2 P I ( τ H = [1, 1] ), (4.23) respectively, and P ISO (τ) = Pr { S < S d } = 1 PS (τ), (4.24) P EIO (τ) = Pr { I > I d } = PI (τ). (4.25) In principle, a similar procedure can be applied for general S d, I d, K and L to complete the theoretical analysis for ranked energy detection. However, the expressions are quite complicated if any parameter is much larger than 1. In Section 4.2.4, we provide several results for RED based on simulation, and we also verify the theoretical analysis in this section by comparing it with the simulation results for the special parameters. As we will see, RED outperforms CED in terms of ROC defined by (P ISO, P EIO ) Ranked Posterior Belief Detection Note that RED needs to take the number of channels to make decisions on as an explicit input. Intuitively, reasonable detection algorithms should first choose channels with larger posterior beliefs of being in an OFF state as candidates for spectrum opportunities. The energy based ranked channel detection (RCD) algorithm proposed in [55] is one illustrative example of such algorithms, since smaller energy 69

84 implies higher posterior belief of being in an OFF state for symmetric channels with equal noise and primary signal power levels. If we rank the channels by the posterior beliefs, we obtain a more general algorithm following the Bayesian rule in (3.54) that also applies for asymmetric channel models. The idea is to start from an empty candidate channel set and add one channel with the largest posterior belief of being in an OFF state to the candidate channel set each step. Each newly added channel leads to potential increase in both the spectrum opportunity and the interference opportunity, and affects the detection risk accordingly. Once all M channels are added to the candidate channel set, the number of declared OFF channels is chosen to minimize the detection risk and the corresponding number of channels are actually declared OFF. Algorithm 2. Ranked posterior belief detection (RPBD) for partial-band Nyquist sampling (RPBD-PBNS): 1. Initialize: Compute the posterior beliefs of each channel in the OFF state q i (0) and rank them in descending order with respect to q i (0), i.e., q J(1) (0) q J(2) (0) q J(ρ) (0), where J(i) is the index mapping of the sorted channels. For initialization, declare all channels to be ON. Set P ISO,0 = 1, and P EIO,0 = 0, (4.26) R II,0 = C ISO P ISO,0 + C EIO P EIO,0 = C ISO, (4.27) since there are no spectrum opportunities nor interference opportunities if all channels are declared ON. Also, set P S,0 (0) = 1, P S,0 (s) = 0 for s 1. Set the iteration counter to i = Update: If i < L, update P S,i (S d ), P ISO,i, P EIO,i, and R II,i according to (4.28), (4.29), (4.30), and (4.31), respectively. P S,i (S d ) = P S,i 1 (S d 1) q J(i) (0) + P S,i 1 (S d ) q J(i) (1), (4.28) P ISO,i = P S,i 1 (S d 1) q J(i) (0) + (1 P ISO,i 1 ) q J(i) (1), (4.29) i 1 P EIO,i = 1 q J(i) (1), (4.30) k=0 R II,i = C ISO P ISO,i + C EIO P EIO,i. (4.31) 3. Finalize: After the iteration terminates at i = L, choose the number of declared 70

85 OFF channels that minimizes the detection risk, i.e., ˆK = arg min R II,i. (4.32) 0 i L Thus, channels J(1), J(2),..., J(ˆK) are declared as OFF, and the remaining L ˆK channels are declared as ON. Note that RPBD either declares all channels to be ON, or declares at least S d channels to be OFF, since R II,i = C ISO + C EIO P EIO,i > C ISO = R II,0, 0 < i < S d, (4.33) where the inequality follows from P ISO,i = 1 and P EIO,i > 0 for 0 < i < S d. General theoretical analysis for RPBD is rather complicated and beyond the scope of this dissertation. As an alternative, we study two illustrative examples with L = 2. Without loss of generality, we assume channel 1 and channel 2 are considered. L = 2, and S d = 1: Without loss of generality, assume q 1 (0) > q 1 (1), or equivalently, t 1 < t 2. Therefore, P ISO,1 = P EIO,1 = Pr { H 1 = 1 T 1 = t } = q 1 (1), (4.34) P ISO,2 = Pr { H 1 = 1 T 1 = t 1 } Pr { H2 = 1 T 2 = t 2 } = q1 (1)q 2 (1), (4.35) P EIO,2 = 1 Pr { H 1 = 0 T 1 = t 1 } Pr { H2 = 0 T 2 = t 2 } = 1 q1 (0)q 2 (0). (4.36) Along with (4.27), the risks of declaring a specific number of OFF channels are R II,1 = (C ISO + C EIO )q 1 (1), (4.37) R II,2 = C ISO q 1 (1)q 2 (1) + C EIO ( 1 q1 (0)q 2 (0) ). (4.38) 71

86 By comparing the detection risks, the detection rule can be specialized by the followings three inequalities jointly: ˆK=0 t 1 1 ˆK=1 α log(βc II), (4.39) ˆK=1 t 2 ˆK=2 βe αt 1 t 1 1 α log(c II), (4.40) ( ) (C II 1 + βe αt 2 ) ˆK=0 1 (1 C II ) ˆK=2 ( 1 + βe αt 2 ), (4.41) where α γn γ, (4.42) β p ( ) N0 1, (4.43) 1 p 1 + γ C II C EIO C ISO. (4.44) Again, the above results are generated for t 1 < t 2 only. Thus, several facts are worth nothing: For t 1 < t 2, ˆK = 0, ˆK = 1 and ˆK = 2 are equivalent to Ĥ = [1, 1], Ĥ = [0, 1] and Ĥ = [0, 0], respectively. If C II > 1, it is always better to declare one OFF channel than declaring two regardless of the observation according to (4.40), since the left-hand side is always greater than the right-hand side. If C II > 1/β, it is always better to declare all channels to be ON than declaring one OFF channel regardless of the observation according to (4.39), since the left-hand side is always greater than the right-hand side. The same analysis applies for t 1 > t 2 and similar results can be obtained. The detection regions of RPBD are illustrated in Fig. 4.1 for different values of system cost ratios. Note that the shape of the detection region of RED with parameter L d = S d = 1 is identical to Fig. 4.1 (b), with the boundaries of the region determined by certain values of the detection threshold τ. In both cases, Fig. 4.1 (c) 72

87 can be viewed as special cases of Fig. 4.1 (b) with corresponding parameters. Thus, we can expect that the wideband ROCs of the two detection algorithms differ only in the regime of small system cost ratio C II, or equivalently, the regime of large detection threshold, which corresponds to a regime of low P ISO and high P EIO. On the other hand, in the preferable regime that better protects the primary system, i.e., the regime of low P EIO and high P ISO, the two detection algorithms exhibit exactly the same performance. L = 2, and S d = 2: Since P ISO,2 = P EIO,2 = 1 Pr { H 1 = 0 T 1 } Pr { H2 = 0 T 2 }, = 1 q1 (0)q 2 (0), (4.45) the risk of declaring two OFF channels is R II,2 = (C ISO + C EIO ) ( 1 q 1 (0)q 2 (0) ). (4.46) Thus, by comparing R II,0 and R II,2, the detection rule can be specialized to ( )( βe αt βe αt ) ˆK=0 ˆK= C II, (4.47) which is illustrated in Fig. 4.2 (a) for C II < 1/(β 2 + 2β). For C II 1/(β 2 + 2β), the detection region reduces to Fig. 4.1 (c), i.e., all channels are declared OFF regardless of the observations since the left-hand side of (4.47) is always greater than the right-hand side. Also, the detection regions for RED and CED are also provided as comparisons in Fig. 4.2 (b) and Fig. 4.2 (c). Note that unlike the case of S d = 1 < L = 2 in which ranked posterior belief detection and ranked energy detection with parameter L d = S d = 1 have partially overlapping ROCs, the ROCs here share no overlap since there is no one-to-one correspondence in the detection regions. 73

88 t 2 t 2 Ĥ=[0, 1] Ĥ=[1, 1] Ĥ=[0, 1] Ĥ=[1, 1] Ĥ=[0, 0] Ĥ=[1, 0] Ĥ=[1, 0] 0 (a) t 1 0 (b) t 1 t 2 Ĥ=[1, 1] 0 (c) t 1 Figure 4.1. Detection regions for ranked posterior belief detection for partial-band Nyquist sampling with L = 2 bands, and the desired number of spectrum opportunities S d = 1 for: (a): C II < 1: (b) 1 C II 1/β; (c) C II > 1/β. 74

89 t 2 t 2 Ĥ=[1, 1] τ Ĥ=[1, 1] Ĥ=[0, 0] Ĥ=[0, 0] (a) t 1 t 1 τ (b) t 2 Ĥ=[0, 1] Ĥ=[1, 1] τ Ĥ=[0, 0] Ĥ=[1, 0] τ (c) t 1 Figure 4.2. Detection regions for partial-band Nyquist sampling with L = 2 bands, and the desired number of spectrum opportunities S d = 2: (a) ranked posterior belief detection for C II < 1 ; (b) ranked energy β 2 +2β detection for τ > 0; (c) channel-by-channel energy detection for τ > 0. 75

90 Let R L,Sd (Ĥ) denote the detection region of Ĥ if the number of partial bands is L and the target number of spectrum opportunities is S d. Thus, the conditional P ISO (H) and P EIO (H) given H = H can be calculated in theory. P ISO (H) = Ĥ H P EIO (H) = Ĥ H (Ĥ ) P ISO = Ĥ, H = H ), R L,Sd (Ĥ (Ĥ ) P EIO = Ĥ, H = H ), (4.48) R L,Sd (Ĥ where P ISO (Ĥ = Ĥ, H = H) and P EIO(Ĥ = Ĥ, H = H) can be obtained given Ĥ and H. Once again, note that we fix I d = 0 throughout this thesis and it is not reflected in the definition of the detection region. Based upon the conditional P ISO and P EIO, the unconditional P ISO and P EIO can be obtained. Specifically, ) P ISO = E H (P ISO (H) = P ISO (H)Pr { H = H }, (4.49) H H ) P EIO = E H (P EIO (H) = P EIO (H)Pr { H = H }. (4.50) H H Numerical Results This section provides several numerical results to compare the performance of different detection algorithms Channel-by-Channel Energy Detection In Figure 4.3, we compare the wideband ROC with channel-by-channel energy detection in theory based upon (3.14), (3.15), (4.12) and (4.13) by varying the the detection threshold τ, for S d = 1 and I d = 0. Several scenarios that differ in the primary occupancy probabilities, the number of samples per channel, and the SNR levels are provided. Note that the x-axis is logarithmic in scale for better illustration of the results. Also note that the logarithmic scale in either the x-axis or the y-axisis 76

91 is widely used for better illustration when necessary depending on the system setups. First, note that the end point of each curve corresponds to the case in which P M = 1 and P F = 0, i.e., all channels are declared OFF regardless of the observations, or equivalently, τ = +. This can be easily verified by substituting P M = 1 and P F = 0 into (4.14) and (4.15), respectively. Second, the lower left corner corresponds to the regime of better protection for the primary system. We notice that with independent channel-by-channel detection, it is not always advantageous to sample at a higher sampling rate. Specifically, PBNS with the least number of channels in the partialband performs best in its achievable regime for the case that satisfies the following conditions concurrently: Small primary occupancy probability; Small number of samples per channel; Small SNR levels. The reason is that as we increase the number of channels in PBNS, we increase the possibilities of finding more spectrum opportunities and more interference opportunities at the same time. Because the sensing window is not sufficiently long, the amount of increased interference opportunities overwhelms the amount of increased spectrum opportunities, thus leading to worse overall performance for increased number of channels in the partial-band. The results indicate that the observations in the partial-band are not efficiently utilized. In the next section, we will observe one way for making better use of the observations available Ranked Channel Detections In this part we compare the performance of RED and RPBD with that of CED. The simulation results are averaged over 10 3 to 10 6 Monte Carlo trials for different SNR levels and number of samples per channel, with the primary occupancies, the primary signals, and the noise signals drawn independently in each trial. 77

92 SNR= 10 db, N 0 =20, p=0.1 SNR= 10 db, N 0 =20, p= P EIO L=1 L=2 0.1 L=4 L= P ISO (a) P EIO L=1 0.2 L=2 L=4 L= P ISO (b) SNR= 10 db, N 0 =40, p=0.1 SNR=0 db, N 0 =20, p= L=1 L=2 L=4 L=8 P EIO 0.3 P EIO L=1 L=2 0.1 L=4 L= P ISO (c) P ISO (d) Figure 4.3. Probability of excessive interference opportunities (vertical axes) and probability of insufficient spectrum opportunities (horizontal axes) for partial-band Nyquist sampling with channel-by-channel energy detection. M = 16 channels, S d = 1 spectrum opportunities sought, and I d = 0 interference opportunities allowed. 78

93 Results for several scenarios that differ in the system parameters are illustrated in Fig , based upon which we can draw the following conclusions: As expected, CED has the worst wideband performance, since it is designed to optimize the channel goal; For RED, which parameter L d gives the best performance depends on which part of the ROC regime about which we are concerned. For example, increasing the number of channels to make decisions would degenerate the wideband performance in the regime of better protection for the primary system, which results from the fact that the amount of increased interference opportunities overwhelms the amount of increased spectrum opportunities RPBD exhibits the best wideband performance, which can be viewed as an envelope of all RED with parameters ranging from L d = S d to L d = L. However, in the regime of better protection for the primary system, it exhibits the same performance as that of RED with parameter L d = S d, for reasons addressed in Section Overall, the advantage of RPBD over RED with parameter L d = S d is observed to be better for the regime of low P ISO and high P EIO, or the regime of high P ISO and low P EIO with S d close to L. 4.3 Summary In this chapter, we develop several detection algorithms for partial-band Nyquist sampling with wideband wideband performance metrics (P ISO, P EIO ), specifically, channelby-channel energy detection (CED), ranked energy detection (RED) and ranked posterior belief detection (RPBD). We provide complete analysis for CED and discuss the performance of RED and RPBD in theory with simple, illustrative examples. The results illustrate that with the performance metrics coupling the individual channels, more sophisticated multi-channel detection algorithms outperform the channel-bychannel detection algorithms, even for Nyquist sampling that give conditionally independent observations across channels. Also, we identify RPBD as the best detection 79

94 P EIO = Pr{I > 0} CED RPBD RED (L d =2) RED (L d =3) RED (L d =4) P = Pr{S < 2} ISO (a) P EIO = Pr{I > 0} CED RPBD RED (L d =4) RED (L d =5) RED (L d =6) P = Pr{S < 4} ISO (b) P EIO = Pr{I > 0} CED RPBD RED (L d =8) RED (L d =9) RED (L d =10) P ISO = Pr{S < 8} (c) P EIO = Pr{I > 0} CED RPBD RED (L d =12) RED (L d =13) RED (L d =14) P ISO = Pr{S < 12} (d) Figure 4.4. Probability of excessive interference opportunities (vertical axes) and probability of insufficient spectrum opportunities (horizontal axes) for partial-band Nyquist sampling with parameter L = 16: M = 16 channels, SNR = 0 db, primary occupancy probability p = 0.1, number of frequency samples per channel N 0 = 20, I d = 0 interference opportunities allowed. 80

95 10 1 p= p=0.2 P EIO = Pr{I > 0} CED RPBD RED (L d =8) RED (L d =9) RED (L d =10) P = Pr{S < 8} ISO (a) P EIO = Pr{I > 0} CED RPBD RED (L d =8) RED (L d =9) RED (L d =10) P = Pr{S < 8} ISO (b) 10 0 p= p=0.5 P EIO = Pr{I > 0} CED RPBD RED (L d =8) RED (L d =9) RED (L d =10) P ISO = Pr{S < 8} (c) P EIO = Pr{I > 0} CED RPBD RED (L d =8) RED (L d =9) RED (L d =10) P ISO = Pr{S < 8} (d) Figure 4.5. Probability of excessive interference opportunities (vertical axes) and probability of insufficient spectrum opportunities (horizontal axes) for partial-band Nyquist sampling with parameter L = 16: M = 16 channels, SNR = 0 db, number of frequency samples per channel N 0 = 20, S d = 8 spectrum opportunities sought, I d = 0 interference opportunities allowed. 81

96 P EIO = Pr{I > 0} CED RPBD RED (L d =1) RED (L d =2) RED (L d =3) P EIO = Pr{I > 0} CED RPBD RED (L d =2) RED (L d =3) RED (L d =4) P = Pr{S < 1} ISO (a) P = Pr{S < 2} ISO (b) P EIO = Pr{I > 0} CED RPBD RED (L d =3) RED (L d =4) P EIO = Pr{I > 0} CED RPBD RED (L d =4) P ISO = Pr{S < 3} (c) P ISO = Pr{S < 4} (d) Figure 4.6. Probability of excessive interference opportunities (vertical axes) and probability of insufficient spectrum opportunities (horizontal axes) for partial-band Nyquist sampling with parameter L = 16: M = 16 channels, SNR = 10 db, primary occupancy probability p = 0.1, number of frequency samples per channel N 0 = 8, I d = 0 interference opportunities allowed. 82

97 P EIO = Pr{I > 0} p=0.1 RPBD RED (L d =2) RED (L d =3) RED (L d =4) P EIO = Pr{I > 0} p=0.2 RPBD RED (L d =2) RED (L d =3) RED (L d =4) P ISO = Pr{S < 2} (a) P ISO = Pr{S < 2} (b) P EIO = Pr{I > 0} p=0.4 RPBD RED (L d =2) RED (L d =3) RED (L d =4) P EIO = Pr{I > 0} p=0.5 RPBD RED (L d =2) RED (L d =3) RED (L d =4) P ISO = Pr{S < 2} (c) P ISO = Pr{S < 2} (d) Figure 4.7. Probability of excessive interference opportunities (vertical axes) and probability of insufficient spectrum opportunities (horizontal axes) for partial-band Nyquist sampling with parameter L = 16: M = 16 channels, SNR = 10 db, number of frequency samples per channel N 0 = 8, S d = 2 spectrum opportunities sought, I d = 0 interference opportunities allowed. 83

98 algorithms for (P ISO, P EIO ). Most of the detection algorithms in this chapter can be applied to sub-nyquist sampling schemes, as we will see in the following chapters. 84

99 CHAPTER 5 SUB-NYQUIST SAMPLING SCHEME PART I - PERIODIC ALIASING For integer undersampling (IU) that causes periodic aliasing, each bin contains multiple channels that are indistinguishable from the observations. Thus, detection algorithms for IU can be classified into two types based upon how decisions on different channels in the same bin are correlated. Specifically, Bin detection makes the same decisions for all channels in the same bin. Channel detection makes independent decisions for channels in the same bin. In this chapter, we study both bin detection and channel detection algorithms for IU. We derive the optimal detection algorithm for single channel detection that exhibits the most favorable channel ROC, and develop several detection algorithms for wideband detection with (P ISO, P EIO ) and compare their performance from the perspective wideband ROCs. The remainder of the chapter is organized as follows. In Section 5.1, we define the detection statistic for IU. In Section 5.2, we propose stochastic channel detection that allows independent decisions for channels in the same bin and illustrate that it reduces to bin-by-bin energy detection with a fixed detection threshold in minimizing the channel-by-channel risk. In Section 5.3, we study the performance of bin-by-bin energy detection and develop bin detection based as well as channel detection based multi-channel detection algorithms for wideband performance metrics (P ISO, P EIO ). In Section 5.3.4, we provide numerical results and identify the most favorable detection algorithm from the perspective of wideband ROC. In Section 5.4, we conclude the chapter. 85

100 5.1 Detection Statistic for Integer Undersampling IU corresponds to the simplest sub-nyquist sampling that causes periodic aliasing of frequencies and channels. Each bin, which is defined as the aliased channel, is a periodic aliasing of ρ channels as discussed in Section Specifically, ρ 1 V[k] = l=0 [ Y k + l N 0 M ] ρ 1 = ρ l=0 [ H m(k)+l M X k + l N 0 M ] [ + Z k + l N 0 M ] ρ ρ ρ (5.1) for 0 k < N/ρ, or equivalently, 1 m(k) M/ρ, which can be easily verified by substituting (3.35) into (3.26). The bins are periodic with period M/ρ. The bin detection statistic is defined as the bin energy, which is defined in exactly the same way as channel energy for PBNS in (4.1). Specifically, T b = 1 N 0 k I b V[k] 2. (5.2) However, unlike one channel that has only two states, each bin has 2 ρ states with respect to vector of the individual channel states, and ρ+1 states with respect to the number of active channels in the bin. Let the random variable K b denote the number of active channels in the b-th bin. Thus, T b (K) T b K b = K 1 ( ) Kσ 2 2N sig + ρσnoi 2 χ 2 2N0. (5.3) 0 Note that K b H m Binomial(ρ, p), (5.4) m M b under the p-sparse model, where M b is the set of channels that fall into the b-th bin 86

101 for IU, i.e., M b {m : m = b + (k 1) Mρ, k = 1,..., ρ }, (5.5) 1 b M/ρ. Essentially, bin detection algorithms neglect all the 2 ρ 2 possibilities of vector states and choose only between a vector of all 0s and a vector of all 1s. 5.2 Channel Detection with (P M, P F ) Nevertheless, We start from a more general stochastic channel detection algorithm that allows different decision outcomes for channels in the same bin and will later show that it reduces to bin-by-bin energy detection, the most straightforward detection algorithm with a fixed detection threshold analogous to CED in (4.8). Specifically, Algorithm 3. Stochastic channel detection (SCD) for integer undersampling: given the bin detection statistic T b = t, declare each channel in the bin as ON independently with an alarm probability θ(t). Specifically, 1, with probability θ(t), Ĥ m = 0, with probability 1 θ(t), m M b, (5.6) where the alarm probability θ = θ(t) is defined as a function of the bin detection statistic. SCD has different realizations with different choices of alarm probability functions. Specifically, bin-by-bin energy detection can be viewed as a special case of SCD with the alarm probability function chosen as a step function that takes value 0 if the argument is less than a threshold and 1 otherwise. Note that the likelihood of each state with respect to the number of active channels 87

102 in the bin is f Tb K b (t k) = 2N ( ) 0 γk + ρ f 2N0 t χ 2 γk + ρ ; 2N 0. (5.7) In a similar way to that of PBNS, the posterior belief of the number of active channels in a bin is q b (k) = f Tb K b (t k)pr { K b = k } k=0,1,...,ρ f T b K b (t k)pr { K b = k } 1 η(t) ( ) N0 2N0 e N 0 t kγ+ρ kγ + ρ ( ) ρ p k (1 p) ρ k k ζ(k, t)e N 0 t kγ+ρ, (5.8) k = 0, 1,..., ρ, where η(t) ρ k=0 ( ) N0 ( ) 2N0 e N 0 t kγ+ρ ρ p k (1 p) ρ k (5.9) kγ + ρ k is a constant for a given observation T b = t, and ζ(k, t) 1 ( ) N0 ( ) 2N0 ρ p k (1 p) ρ k. (5.10) η(t) kγ + ρ k Let R B I,b(θ K, T b ) denote the channel-by-channel risk of the b-th bin conditioned on the number of active channels in the bin and given the observation. Specifically, R B I,b(θ K b, T b ) = C 01 K b (1 θ) + C 10 (ρ K b )θ. (5.11) The result follows from the fact that the expected number of missed ON channels and false alarmed OFF channels are K b (1 θ) and (ρ K b )θ, respectively, given the number of active channels K b, respectively. Let RI,b(θ T B b ) denote the channel-by-channel risk of the b-th bin for a particular 88

103 alarm probability. Then, ) RI,b(θ T B b ) = E Kb (RI,b(θ K B b, T b ) = ρ q b (k) ( C 01 K b (1 θ) + C 10 (ρ K b )θ ). (5.12) k=0 We would like to find the optimal alarm probability function that minimizes the channel-by-channel risk. From (5.12), the derivative of the bin risk function with respect to the alarm probability θ for a given T b = t is dr B I,b(θ T b = t) dθ ρ = C 10 q b (k) ( ρ (1 + C I )k ) k=0 = C 10 ρ C 10 k=0 ρ k=0 ζ(k, t) ( ρ (1 + C I )k ) e N 0 t kγ+ρ ζ (k, t)e N 0 t kγ+ρ. (5.13) where q b (k) > 0 and ζ(k) > 0 are determined by all parameter settings as defined in (5.8) and (5.10), respectively, and ζ (k, t) ζ(k, t) ( ρ (1 + C I )k ). (5.14) Fact 1. For any cost ratio C I > 0 and observation T b = t, there exists a K 0, 0 < K 0 < ρ, such that 0, 0 k K 0, ζ (k). (5.15) < 0, K 0 < k ρ The result is clear since ζ (k, t) is a monotonically decreasing function with respect to k and ζ (0, t) > 0, ζ (ρ, t) < 0. Lemma 2. For any cost ratio C I > 0 and observation T b = t, there exists a τ = 89

104 τ ( C I, N 0, γ, p ), such that dr B I,b(θ T b = t) dθ = C 10 ρ k=0 ζ (k, t)e N 0 t kγ+ρ > 0, t < τ, = 0, t = τ,. (5.16) < 0, t > τ Proof. First, it is easy to verify that ρ k=0 ρ k=0 ζ (k, )e N 0 t kγ+ρ > 0, (5.17) ζ (k, + )e N 0 t kγ+ρ < 0. (5.18) Thus, there must be at least one point t = τ such that ρ k=0 ζ (k, τ)e N 0τ kγ+ρ = 0. Next, for t > τ, ρ K 0 ζ (k, t)e N 0 t kγ+ρ = ζ (k, t)e N 0 t kγ+ρ + k=0 k=0 k=0 ρ k=k 0 +1 K 0 = ζ (k, t)e N 0 (t τ) kγ+ρ e N 0 τ kγ+ρ + K 0 < ζ (k, t)e N 0 (t τ) K 0 γ+ρ e N 0 τ kγ+ρ + k=0 = e N 0 (t τ) K 0 γ+ρ ( ρ k=0 ζ (k, τ)e N 0 τ kγ+ρ ζ (k, t)e N 0 t kγ+ρ ρ k=k 0 +1 ρ k=k 0 +1 ) ζ (k, t)e N 0 (t τ) kγ+ρ e N 0 τ kγ+ρ ζ (k, t)e N 0 (t τ) K 0 γ+ρ e N 0 τ kγ+ρ = 0, (5.19) where the inequality follows from that e N 0 t kγ+ρ of k. Similarly, it can be shown that ρ k=0 ζ (k, t)e N 0t kγ+ρ can be shown in a similar way that ρ k=0 ζ (k, t)e N 0t kγ+ρ is a monotonically increasing function > 0 for t < τ. Actually, it is a monotonically decreasing function of t by taking the derivative. Thus, we complete the proof for Lemma 2. 90

105 Lemma 2 implies that the bin-by-bin risk is monotonically increasing with respect to the alarm probability θ if t < τ, and monotonically decreasing if t > τ. Thus, the optimal stochastic channel detection that minimizes the channel-by-channel risk is 0, t < τ, θ = θ(t) =, (5.20) 1, t τ which is equivalent to T ON τ, (5.21) OFF i.e., BED with the corresponding threshold determined by the cost ratio and other parameters as well. Note that it is possible that τ = τ ( C I, N 0, γ, p ) < 0 for very large value of cost ratio C I. In this case, channels in the bin are declared ON regardless of the detection statistic. Given a detection threshold τ, the probability of missed detection and the probability of false alarm are P M,m (τ) = P F,m (τ) = τ 0 k=1 ρ 1 ( ρ 1 = k k=0 + ρ 1 τ k=0 ρ Pr { H m = 1, K b = k } f T Kb (t k + 1)dt ) F χ 2 ( ) 2N 0 τ ρ + (k + 1)γ ; 2N 0, (5.22) Pr { H m = 0, K b = k } f T Kb (t k)dt k=0 ρ 1 ( ) ( ( ) ) ρ 1 2N0 τ = 1 F χ2 k ρ + kγ ; 2N 0, (5.23) respectively, where b M/ρ. ( ) N k is the usual binomial coefficient, for every m M b, 1 91

106 5.3 Wideband Detection with (P ISO, P EIO ) In this section we discuss detection algorithms for the wideband performance metrics (P ISO, P EIO ) Bin-by-Bin Energy Detection We first study the performance of BED in terms of the wideband ROC. Let S b and I b denote the number of spectrum opportunities and the number of interference opportunities in the b-th bin, and let P Sb (s) = Pr { S b = s } and P Ib (i) = Pr { I b = i } denote their PMFs, respectively. For a given detection threshold τ, the conditional PMFs of S b and I b are Pr { K b = ρ } + ρ k=1 Pr{ K b = ρ k } Pr { T b (ρ k) τ }, s = 0, P Sb (s τ) = Pr { K b = ρ s } Pr { T b (ρ s) < τ }, 1 s ρ,, 0, others (5.24) Pr { K b = 0 } + ρ k=1 Pr{ K b = k } Pr { T b (k) τ }, i = 0, P Ib (i τ) = Pr { K b = i } Pr { T b (i) < τ }, 1 i ρ,, (5.25) 0, others respectively, for 1 b M/ρ. Since M/ρ S = S b, (5.26) b=1 M/ρ I = I b, (5.27) b=1 the unconditional PMFs of the number of spectrum opportunities and the number of 92

107 interference opportunities of the entire bandwidth are P S (s) = P S1 (s) P S2 (s) P SM/ρ (s), (5.28) P I (i) = P I1 (i) P I2 (s) P IM/ρ (i), (5.29) respectively, where is the convolution operator. P ISO and P EIO can be achieved by substituting (5.28) and (5.29) into (3.14) and (3.15), respectively. For the same reason as addressed in Section 4.2.4, it is expected that other more sophisticated detection algorithms could outperform BED Ranked Bin Posterior Belief Detection Analogous to RPBD for PBNS, we propose ranked posterior belief detection for IU within the Bayesian detection structure, which is a bin detection algorithm and differs from Algorithm 1, i.e., RPBD for PBNS only in the way of updating P ISO, P EIO, and R II at each iteration. Algorithm 4. Ranked bin posterior belief detection (RBPBD) for integer undersampling: 1. Initialize: Compute q i (k), 1 i ρ, the posterior beliefs of the number of ON channels in bins and rank the bins in descending order with respect to q i (0), i.e., q J(1) (0) q J(2) (0) q J(ρ) (0), where J(i) is the index mapping for sorted bins. Set P ISO,0 = 1, P EIO,0 = 0, R II,0 = C ISO, P S,0 (0) = 1, and P S,0 (s) = 0 for s 1. Set the iteration counter to i = Update: If i < ρ, update P S,i, P ISO,i, P EIO,i, and R II,i according to (5.30), (5.31), 93

108 (5.32), and (5.33), respectively. P S,i (s) = P S,i 1 (s) q J(i) (ρ s), (5.30) P ISO,i = S d 1 s=0 P S,i (s), (5.31) i 1 ( P EIO,i = 1 1 qj(i) (0) ), (5.32) k=0 R II,i = C ISO P ISO,i + C EIO P EIO,i. (5.33) 3. Finalize: After the iteration terminates at i = ρ, choose the number of declared OFF bins that minimizes the detection risk, i.e., B off = arg min R II,i. (5.34) 0 i ρ Bins J(1), J(2),..., J(B off ) are then declared as OFF, i.e., all channels in these bins are declared OFF, and the remaining M/ρ B off bins are declared as ON, i.e., all channels in these bins are declared ON Ranked Bin Posterior Belief-Stochastic Channel Detection Although similar analysis for stochastic channel detection for wideband performance metrics (P ISO, P EIO ) is possible within the Bayesian detection structure, the optimal detection rule that minimizes the wideband risk often involves multidimensional optimization, which is too complicated and beyond the scope of this dissertation. An alternative approach that can achieve channel detection while still following the Bayesian rule is to randomly pick one channel from the appropriate bin to declare as OFF at each iteration, which updates the posterior beliefs of P ISO and P EIO, as well as the wideband risk, and the optimal number of declared OFF channels is chosen in the end. Unlike RPBD for PBNS which always chooses the next channel with the smallest posterior belief of being in an ON state from the remaining channels at each iteration, it is not clear from which bin we should pick the next channel depending on the iteration number without an exhaustive search over all possibilities. Thus, we consider a simple approach, i.e., all channels in a bin with a higher posterior belief 94

109 of no active channels in the bin are always chosen before any channel in a bin with lower posterior belief of no active channels in the bin. Since the number of spectrum opportunities in a bin cannot be greater than the true number of OFF channels in that bin, or greater than the number of declared OFF channels, then 0, s > max(ρ k, n), Pr { S b = s K b = k, ˆK b = n } ( )( ) ρ k k =, (5.35) s ( ) n s, otherwise ρ n where S b and ˆK b are the number of spectrum opportunities and the number of declared OFF channels for the b-th bin, respectively. Similarly, since the number of interference opportunities in a bin cannot be greater than the true number of ON channels in that bin, or greater than the number of declared OFF channels, then 0, i > max(k, n), Pr { I b = i K b = k, ˆK b = n } ( )( ) k ρ k =, (5.36) i ( n ) i, otherwise ρ n where I b is the number of interference opportunities for the b-th bin. Thus, P Sb ˆK b (s n) = P Ib ˆK b (s n) = ρ Pr { S b = s K b = k, ˆK b = n } q b (k), (5.37) k=0 ρ Pr { I b = s K b = k, ˆK b = n } q b (k). (5.38) k=0 Algorithm 5. Ranked posterior belief detection for integer undersampling (IU-RPBD): 1. Initialize: Compute q b (k), 1 b ρ, the posterior beliefs of the number of ON channels in bins and rank the bins in descending order with respect to q b (0), i.e., 95

110 q J(1) (0) q J(2) (0) q J(ρ) (0), where J(b) is the index mapping for sorted bins. Set the iteration counter to b = 1. Set P ISO,(1,0) = 1, P EIO,(1,0) = 0, R II,(1,0) = C ISO, P S,(1,0) (0) = 1, and P S,(1,0) (s) = 0 for s Update: If b M/ρ, for each 1 n ρ, compute P S,(b,n) and P I,(b,n) according to P S,(b,n) (s) = P S,(b 1,ρ) (s) P SJ(b) ˆK J(b) (s n), (5.39) P I,(b,n) (i) = P S,(b 1,ρ) (i) P SJ(b) ˆK J(b) (i n), (5.40) respectively. Update P ISO,(b,n), P ISO,(b,n), and R II,(b,n) accordingly. 3. Finalize: The iteration terminates at b = ρ and the detection decision is (B off, ˆK b ) = arg min R II,(b,n). (5.41) (b,n):1 b M/ρ, 0 n ρ Specifically, the first B off bins are declared as OFF, i.e., all channels in these bins are declared OFF, ˆK b channels in the ˆK-th bin are randomly declared OFF, and all the remaining channels are declared as ON. Note that the cases of (b, 0) are undefined for b 2, since they are equivalent scenarios to (b 1, ρ). Also, it is worth noting that It is expected that IU-RPBD can perform no worse than IU-RBPBD, since the former chooses the best from a much broader range of possibilities which is a superset of the latter. The difference between IU-RPBD and PBNS-RPBD is that the update functions are recursive for the former, but non-recursive for the latter at each iteration. Due to its complexity, theoretical analysis of RBPBD and RPBD for IU is beyond the scope of this dissertation. Thus, the results and discussions that follow in Section are based upon simulations Numerical Results In Fig. 5.1, we compare several detection algorithms discussed in this section for wideband performance metrics (P ISO, P EIO ) with different desired number of spectrum opportunities. The simulation results are averaged over 10 5 Monte Carlo trials, with 96

111 the primary occupancies, the primary signals, and the noise signals drawn independently in each trial. We can draw the following conclusions analogous to those of PBNS: Although BED is optimal in minimizing the bin and channel risk, it has the worst performance for the system design with (P ISO, P EIO ). RPBD, which incorporates both ranked bin detection and stochastic channel detection, exhibits better performance than RBPBD as expected. Although we provide only one specific scenario with certain parameter settings, the above conclusions apply for general cases and do not depend on the system parameters. 5.4 Summary In this chapter, we derive the optimal stochastic channel detection algorithm that minimizes the channel-by-channel risk, which appears to be bin-by-bin energy detection with a fixed detection threshold. We also develop multi-channel detection algorithms for wideband performance metrics (P ISO, P EIO ) and identify the most favorable algorithm, i.e., ranked posterior belief detection, from the perspective of the wideband ROC. 97

112 P EIO = Pr{I > 0} CED RPB BD RPB SCD P EIO = Pr{I > 0} CED RPB BD RPB SCD P = Pr{S < 2} ISO (a) P = Pr{S < 4} ISO (b) CED RPB BD RPB SCD CED RPB BD RPB SCD P EIO = Pr{I > 0} P EIO = Pr{I > 0} P ISO = Pr{S < 8} (c) P ISO = Pr{S < 12} (d) Figure 5.1. Probability of excessive interference opportunities (vertical axes) and probability of insufficient spectrum opportunities (horizontal axes) for integer undersampling: M = 16 channels, SNR = 10 db, number of frequency samples per channel N 0 = 4, sub-sampling factor ρ = 4, I d = 0 interference opportunities allowed. 98

113 CHAPTER 6 SUB-NYQUIST SAMPLING SCHEME PART II - UNIFORM ALIASING This chapter focuses on detection algorithms for sub-nyquist sampling with uniform aliasing based upon the Lomb-Scargle periodogram (LSP). We propose a reasonably good approximation approach, together with the belief propagation algorithm [71], to estimate the posterior beliefs of the states of each channel, based upon which we develop similar detection algorithms to those for PBNS and IU. We also introduce an approximation method to estimate the number of declared ON channels within the Bayesian detection framework, based upon which both the previously proposed ranked energy detection (RED), and the traditional sparse reconstruction algorithms, such as orthogonal matching pursuit (OMP), can be applied and tradeoffs between the probability of missed detection and the probability of false alarm for the individual channels are enabled. In contrast to the detection algorithms that estimate the number of ON channels within the Bayesian framework, we also consider alternative algorithms that declare a fixed number of ON channels based upon parameter settings. To summarize, five detection algorithms will be studied in this chapter: RED with an estimated number of ON channels; RED with a fixed number of ON channels; OMP with an estimated number of ON channels; OMP with a fixed number of ON channels; ranked posterior belief detection (RPBD). The remainder of the chapter is organized as follows. In Section 6.1, we introduce the Lomb-Scargle periodogram, based upon which we define the energy type detection 99

114 statistics. In Section 6.2, we propose an approximation approach to characterize the relationship between detection statistics across channels. In Section 6.3, we develop detection algorithms for channel detection with performance metrics (P M, P F ). In Section 6.4, we develop detection algorithms for wideband detection with performance metrics (P ISO, P EIO ). In Section 6.5, we compare the detection algorithms through numerical results. In Section 6.6, we conclude the chapter. 6.1 Lomb-Scargle Periodogram The Lomb-Scargle Periodogram (LSP), first proposed in [37], is an estimate of the PSD with missing data, which in the simplest form, is essentially the classical periodogram with the missing data set to zero. Thus, the LSP of y[n], denoted by P ls Y (f), is the classical periodogram of v[n] in (3.21), denoted by P V(f), i.e., P ls Y(f) P V (f) = 1 N N 1 n=0 Samples of the LSP can be computed via the FFT, i.e., ( ) P ls Y[k] P ls Y k Fnyq = V[k] 2 =N N ã[n]y[n]e j2πfn 2. (6.1) N 1 n=0 2 φ kn Y[n], (6.2) where φ kn is defined in (3.28). Since Y[n] are independent circularly symmetric complex Gaussian random variables, V[k], 0 k < N, are circularly symmetric complex Gaussian random variables as well (see Appendix). Thus, Fact 2. Given H, or equivalently, Λ Y as defined in (3.7), P ls Y [k] is a scaled chi-square random variable with 2 degrees of freedom, i.e., P ls Y[k] µ ls H k (H) χ 2 2 2, (6.3) 100

115 where µ ls k (H) E (P lsy[k] ) N 1 H = N φ kn 2 Λ Y [n]. (6.4) Since the variance of a central chi-square random variable with k degrees of freedom is 2k, the variance of the LSP is V ar (P lsy[k] ) H = ( µ ls k (H) ) 2 4 n=0 4 = ( µ ls k (H)) 2, (6.5) which can also be obtained through an alternative approach as shown in the Appendix. Before exploring correlation among the LSP samples, we note that P ls Y [n] resembles Y[n] 2 as they are both central chi-square random variables with 2 degrees of freedom, which motivates the development of the energy detection algorithms that do not require signal reconstruction. As a result, the detection algorithms and the corresponding analysis are analogous to those of the non-coherent energy detection in [9, 64]. 6.2 An Approximation Approach for Uniform Aliasing We develop an approximation approach for sub-nyquist sampling with uniform aliasing. Consider the ideal uniform aliasing, i.e., φ kn = µ(ã) as defined in (3.32). Thus, (6.4) reduces to ( N 1 µ ls k (H) = N = n=0 φ kn 2 σ 2 noi + N 1 n=0 H m(n) φ kn 2 σ 2 sig ( 1 + (ρ 1)(N ) 0 1) H m(k) σsig 2 + Λ m(k). (6.6) N 1 ) 101

116 Here the aliased noise and interference power for channel m, which is a deterministic function of the states of channels other than m, is Λ m ρσnoi 2 + δσsig 2 H n, (6.7) where n M,n m δ (ρ 1)N 0 N 1. (6.8) As developed in the Appendix, the covariance of LSP samples conditioned on the primary occupancy is cov (P lsy[k 1 ], P lsy[k 2 ] N 1 2 H )=N 2 φ k1 nφ k 2 nλ Y [n]. (6.9) Though we have not obtained a complete and rigorous analysis, it appears that the correlation coefficients between the LSP samples are often small. Define the detection statistic for the m-th channel as n=0 T m 1 P ls N Y [k]. (6.10) 0 k I m Thus, it appears reasonable to approximate the detection statistics T m as independent for uniform aliasing, T ls m H 1 ( ) H m σ 2 2N sig+δσsig 2 H n +ρσnoi 2 χ 2 2N 0, (6.11) 0 n m and T m are conditionally independent for 1 m M given the primary occupancy H. Note that we use the symbol to denote approximation in distribution. Also note that the traditional periodogram can be viewed as a special case of the 102

117 LSP and the detection statistics for PBNS and IU in (4.1) and (5.2) are basically special cases of (6.10) with corresponding parameter settings. Thus, the log likelihood ratio (LLR) of a channel can be approximated by ( γδ n m LLR m log(lr m ) N 0 log H n + ρ γ ( 1 + δ n m H ) )+ n + ρ γn 0 t ( m γδ n m H n + ρ )( γ(1 + δ n m H ). (6.12) n) + ρ If we replace H n in (6.12) with the posterior beliefs of the channels being in the ON state, i.e., ( γδ n m LLR m N 0 log q ) n(1) + ρ γ ( 1 + δ n m q n(1) ) + + ρ γn 0 t ( m γδ )( n m q n(1) + ρ γ ( 1 + δ n m q n(1) ) ), (6.13) + ρ we can thus use the belief propagation algorithm to estimate the posterior belief that each channel in an ON state recursively. Algorithm 6. Belief propagation for uniform aliasing: 1. Initialize: Set the initial probabilities q m,0 (1) = p, 1 m M. Set the iteration counter to i = Update: Update the LLR m,i in (6.13), where the subscript i is the iteration counter. Update the posterior beliefs q m,i (1) by Set i i + 1. q m,i (1) = ellr m,i 1 + e LLR m,i. (6.14) 3. Finalize: If the termination condition is reached, i.e., the iteration counter i exceeds a pre-determined threshold, or q i q i 1 2 falls below a pre-determined threshold, where q i = [ q 1,i (1), q 2,i (1),..., q M,i (1) ], terminate the iteration and set q m (1) = q m,i (1). Analogous to PBNS, the posterior beliefs can be used in the Bayesian detection 103

118 algorithms that aim to minimize the associated detection risk, as will be discussed later. 6.3 Channel Detection with (P M, P F ) Belief Propagation based Channel-by-Channel Detection Given the posterior beliefs of a channel in the ON state derived from the belief propagation algorithm, the Bayesian detection rule in(3.41) can be simplified to Ĥ m=1 1 q m (1), (6.15) Ĥ m=0 1 + C I which we refer to as belief propagation based channel-by-channel posterior belief detection (BP-CPBD). Note that we abuse the term channel-by-channel a little bit, since the decision on one channel also relies on the observations from other channels in estimating the posterior beliefs using belief propagation. The detection rule is channel-by-channel after obtaining the posterior beliefs Channel-by-Channel Energy Detection Analogous to CED for PBNS in (4.8) and BED for IU in (5.21), CED also applies to sub-nyquist sampling with uniform aliasing. According to (6.11), the conditional P M,m and P F,m for a fixed detection threshold τ are approximately P M,m (τ ) ( 2N 0 τ H n F χ 2 σ 2 n m sig + Λ ; 2N 0 ), (6.16) m P F,m (τ ) ( ) 2N0 τ H n 1 F χ 2 ; 2N 0, (6.17) Λ m n m 104

119 respectively, where Λ m is a deterministic function of n m H n as defined in (6.7). For the p-sparse model, Thus, { Pr n m H n Binomial(M 1, p). (6.18) n m } H n = k = ( M 1 k ) p k (1 p) M 1 k, (6.19) for k = 0, 1,..., M 1. Therefore, the probability of missed detection and the probability of false alarm for channel m can be approximated finally by taking expectation over n m H n Orthogonal Matching Pursuit All the detection algorithms discussed thus far make sensing decisions directly based upon the aliased signal without any attempt in recovering the original signal. On the other hand, numerous sparse reconstruction algorithms in the field of compressed sensing that aim to recover the original signal with a priori knowledge of its sparsity level have been applied in the field of sub-nyquist sampling. Orthogonal matching pursuit (OMP) is a well-known, greedy sparse reconstruction algorithm that can recover a sparse signal with high probability. This algorithm expects the columns of the measurement matrix Ψ to be approximately orthonormal, which makes Ψ Ψ in a loose sense close to an identity matrix, and thus Ψ ṽ = Ψ ΨY a locally rough approximation for Y, where A is the Hermitian transpose of matrix A. Therefore, the largest coordinate of Ψ ṽ corresponds a nonzero element of Y with high probability and one coordinate for the support of the signal Y is found. After transmission, the contribution of the corresponding basis in Ψ is subtracted 105

120 from the measurement vector Y. The algorithm repeats until s, which is the sparsity level of the original signal that is known a priori, nonzero elements are found, or a termination condition is reached [62]. Algorithm 7. Orthogonal matching pursuit (OMP): 1. Initialize: Initialize the support set J 0 =, the residual r 0 = ṽ, and the iteration counter to i = Identify: Select the largest coordinate λ of Ψ r i 1 in absolute value. 3. Update: Add λ to the support set J i J i 1 {λ}. Estimate the support value by solving the least squares problem Ŷ i = arg min ṽ Ψ Ji u 2, (6.20) u and update the residual: r i = ṽ Ψ Ji Ŷ i. (6.21) Set i i + 1. If i s, goto step 2. The algorithm terminates after s iterations. The final estimation is Ŷ = Ŷs and the support set is J s. by Note that 2 is the l 2 norm operator. The least square problem can be solved Ŷ i = Ψ J i ṽ, (6.22) where X (X X) 1 X (6.23) is the pseudo-inverse of the matrix X. Note that sparse reconstruction algorithms such as OMP cannot be applied directly to wideband spectrum sensing, since they require the a priori knowledge of the sparsity level of the original signal, which is often unavailable. Additional high-level comparisons between approaches can be found in Table

121 TABLE 6.1 WIDEBAND SPECTRUM SENSING VERSUS SPARSE RECONSTRUCTION Problem Type Inputs Wideband Spectrum Sensing Detection Measurement / aliasing matrix Observations Costs / detection threshold Sparse Reconstruction Estimation Measurement / aliasing matrix Observations Sparsity level Output Hard decisions of channel states Estimate of original signals Approximating the Declared Primary Occupancy Level within the Bayesian Detection Framework In order to apply the sparse reconstruction algorithms to wideband spectrum sensing, an estimate of the sparsity level of the primary occupancy is required. However, the declared primary occupancy level does not necessarily need to be identical to the true primary occupancy level for the sake of trading off between P M and P F. Intuitively, the declared primary occupancy level should be be larger than the true primary occupancy level if small P M is required. At the extreme, if the declared primary occupancy level is M, all channels are declared ON and no missed detections would be made. On the other hand, the declared primary occupancy level could be smaller than the true primary occupancy level if small P F is required. At the other extreme, if the declared primary occupancy level is 0, all channels are declared OFF and no false alarms would be made. 107

122 We propose an approximation approach to obtain the optimal declared primary occupancy level within the Bayesian detection framework, which relies on the symmetry property. For ease of exposition, denote C K ( ˆK, K) as the cost of declaring ˆK ON channels if the true number of ON channels is K. The estimate of the number of declared ON channels within the Bayesian detection framework is where ˆK = arg min 0 ˆK M = arg min 0 ˆK M M C K ( ˆK, K)Pr { K = K ṽ } K=0 M C K ( ˆK, K)Pr { ṽ K = K } Pr { K = K }, (6.24) K=0 Pr { K = K } = ( ) M p K (1 p) M K. (6.25) K since K is a binomial random variable for the p-sparse model. Computation of C K ( ˆK, K) and Pr { ṽ K } are both highly involved. Thus, we are interested in approximating them. Approximation of Pr { ṽ K } : There is no direct probability model between ṽ and K. Rather, there exists a conditional probability model of ṽ given a specific H, as indicated in (3.7), (3.50) and (3.51). Thus, in order to treat all vector states of the same weight as one entity, we need to bypass the difference between them and identify the common attributes. The most straightforward way is to look into the total / average power of the entire bandwidth, which clearly depends on the number of ON channels, but does not distinguish between two different vector states with the same weight. Specifically, let T be the average of T m, i.e., T = 1 M m M T m = 1 N 0 k<n P ls Y[k]. (6.26) 108

123 As developed in the Appendix, the mean and the variance of T given the number of ON channels K are E ( T K ) = N 2 var ( T K ) (N) 2 (1 + ρ) 2ρ 2 ρ ( σnoi 2 + K ) M σ2 sig, (6.27) ( σnoi 2 + K ) 2 N(2N ρ 1) M σ2 sig + 2ρ 2 (N 1) K(M K) σ M noi. 4 2 (6.28) To calculate the precise conditional distribution of T given K is complex. Instead, we use the Gaussian approximation, which is appropriate since N is likely to be large for wideband spectrum sensing. Hence, the conditional distribution of T given K is approximately, ( T K N E ( T K ) ), var ( T K ), (6.29) where N (µ, σ 2 ) denotes a Gaussian random variable with mean µ and variance σ 2. Since T is a deterministic function of ṽ, we then further approximate Pr { ṽ K } in (6.24) by Pr { T K }. Therefore, the optimization problem in (6.24) is replaced by ˆK = arg min 0 ˆK M = arg min 0 ˆK M M C K ( ˆK, K)Pr { K = K T } K=0 M C K ( ˆK, K)Pr { T } { } K = K Pr K = K. (6.30) K=0 Approximation of C K ( ˆK, K): We approximate C K ( ˆK, K) from the costs for the individual channels based upon two fundamental assumptions. To elaborate, define H l as the set of all possible vector states with l ON channels, i.e., { H l H H n M } H n = l. (6.31) 109

124 For H l, we have the following three facts: H l = ( ) M, l H l1 H l2 =, 0 l 1, l 2 M, l 1 l 2, 0 l M H l = H. Let Ĥl be the declared vector state if the number of declared ON channels is l, and ˆK l the corresponding declared ON channel set, i.e., Ĥ l = arg min R I (H ṽ), (6.32) H H l which is defined analogously to the rule in (3.44). Assumption 1. Ĥ l arg max Pr { ṽ H = H }. (6.33) H H l The approximation in Assumption 1 is made by pairwise comparison between all possible vector states in H l. Consider choosing between two vector states H (1) and H (2). Analogous to (3.43), we have R I (H (i) v) = H H l C(H (i), H)Pr { H = H v }. (6.34) Thus, R I (H (1) v) R I (H (2) v) = (C(H (1), H) C(H (2), H) )Pr { H = H v } H H l 1 } H {H (1),H (2) ( C(H (1), H) C(H (2), H) )Pr { H = H v } 2 = Pr { H = H (2) v } Pr { H = H (1) v }, (6.35) 110

125 where 2 is because C(H (1), H (2) ) = C(H (2), H (1) ), H (1), H (2) H l, H (1) H (2), 1 l M, (6.36) since by declaring H (2) if the true state is H (1), the number of false alarm bits must be identical as the number of missed detection bits, and vice versa, and 1 is the key approximation we make to obtain Assumption 1. It is a greedy approximation in that when comparing the detection risks of vector states pairwise, the detection risks are computed by neglecting all possibilities other than the two vector states in consideration. Thus, Assumption 1 can be obtained by substituting (6.35) into (6.32). Assumption 2. ˆK l+1 = ˆK l + {n}, for some n M ˆK l, (6.37) where M {1, 2,..., M}. Assumption 2 states that Ĥl and Ĥl+1 differ only in one bit, i.e., Ĥ l+1 can be viewed as generated by flipping one 0 bit to 1 from Ĥl. Assumption 2 is an implicit assumption for most sparse reconstruction methods that find and reconstruct the non-zero elements of a target signal element by element. A direct consequence of (6.37) is (K 1 K 2 )C 10, K 1 K 2, C K (K 1, K 2 ) C(ĤK 1, ĤK 2 )=. (6.38) (K 1 K 2 )C 01, K 1 < K 2 To summarize, the declared number of ON channels can be approximated by substituting (6.25), (6.27), (6.28), (6.29) and (6.38) into (6.30). The result is sub-optimal due to the approximations. Nevertheless, the simulation results in Section 6.5 suggest 111

126 that these assumptions and approximations are useful by showing improvement in detection performance. Again, we stress that the procedure for obtaining the declared number of ON channels within the Bayesian framework relies on the symmetric channel model Orthogonal Matching Pursuit Detection within the Bayesian Framework Given the estimate of the number of ON channels to declare, OMP can be used to reconstruct the signal and find the corresponding support set. For the general case in which N 0 > 1, we expect the s = ˆKN 0 supports of Ŷ to fall into exactly ˆK channels, which is, unfortunately, rarely true even for a small N 0. One solution is to modify the original OMP to ensure channel recovery that reconstructs N 0 samples in a channel concurrently at each iteration. The corresponding algorithm, which estimates the number of ON channels within the Bayesian framework, is described below. Algorithm 8. Orthogonal matching pursuit detection with an estimated number of declared ON channels (OMP-AN): 1. Initialize: Estimate the declared number of ON channels ˆK via (6.30). Initialize the recovered ON channel set ˆK 0 =, the support set J 0 = and the residual r 0 = ṽ. Set the iteration counter to i = Identify: Select the coordinate λ according to λ = arg max ( Ψ r i 1 ) 2 Im, (6.39) m where 2 is the l 2 norm of a vector, i.e., x 2 = ( n i=1 x2 i ) Update: Add λ to the recovered ON channel set ˆK i : ˆKi ˆK i 1 {λ}. Add I λ to the support set J i : J i J i 1 I λ. Estimate the support value Ŷi and update the residual r i in the same way as in (6.20) and (6.21). Set i i + 1. If i ˆK, goto step 2); otherwise, the algorithm terminates after ˆK iterations. The final estimation of the signal is Ŷ = Ŷ ˆK and the declared ON channel set is ˆK = ˆKˆK. 112

127 On the other hand, orthogonal matching pursuit detection with a fixed number of declared ON channels (OMP-FN) differs from Algorithm 8 only in the declared number of ON channels. The fixed number of declared ON channels can be viewed as a pre-determined threshold depending on parameter settings including the primary occupancy probability and the requirements on the probability of false alarm or the probability of missed detection, and so forth. As we will see in Section 6.5, OMP- AN exhibits better performance compared to OMP-FN, which illustrates that the approach in approximating the optimal number of declared ON channels is useful Ranked Energy Detection with an Estimated Number of Declared ON Channels In contrast to OMP based algorithms that recover the target signal before making decision, we also propose a ranked channel energy detection based upon the estimate of the number of declared ON channels within the Bayesian framework, which does not require signal reconstruction compared to OMP-AN. Algorithm 9. Ranked channel energy detection with an estimated number of declared ON channels (RCED-AN): 1. Estimate the declared number of ON channels ˆK via (6.30). 2. Sort the channels according to T m in descending order. 3. Declare the first ˆK channels ON and the remaining M ˆK OFF. Analogously, we also consider ranked channel energy detection with a fixed number of declared ON channels (RCED-FN), which differs from RCED-AN only in the first step, i.e., the number of declared ON channels is fixed and given as a pre-determined parameter. It is worth noting that the LSP is used implicitly in the OMP based algorithm, which indicates a strong connection between OMP-AN and RCED-AN. To see this, 113

128 substitute (3.49) into ( Ψ r i 1 ) Im 2 in (6.39) and rewrite it for the first iteration, ( Ψ ṽ) 2 Im = (ΦY)Im 2 = V I m 2 1 = N M T m. (6.40) Substituting (6.40) back into (6.39), we notice that the first declared ON channels for both algorithms are identical, since the same criterion is used. However, the two algorithms diverge in seeking the second ON channel. Specifically, OMP-AN corrects the residual by subtracting the contribution of the corresponding basis / bases every time a new channel is declared. On the contrary, RCED-AN doe not estimate the hidden signal at all, thus making no correction at each step. OMP based detection algorithms have a strong limitation that they can only declare at most M/ρ ON channels, since Ψ JM/ρ reaches a square matrix of size Q Q, where Q is the number of active wideband Nyquist samples in a sensing window defined in (3.24). More basis vectors added to Ψ JM/ρ make the columns correlated and the estimate of the corresponding samples zero [45]. This suggests that by reconstruction, the OMP based algorithms fold all the primary signals from the ON channels as well as all the noise in the entire wideband into at most M/ρ channels. Other than randomly choosing from the remaining channels, one way to work around the limitation on the number of declared ON channels is to choose the (M/ρ + 1)-th (and beyond) channel based upon the ordering of ( Ψ r i 1 ) 2 Im, m M, at some intermediate stage i. Two simple cases are i = 1 and i = M/ρ. If the intermediate result in the first iteration is used, the sensing result for ˆK > M/ρ is expected to be similar to the energy based detection algorithms. Since we would like to see if reconstruction could help in improving the performance compared to the algorithms that do not require signal reconstruction at all, we use the ordering in the M/ρ-th iteration to declare channels beyond M/ρ for OMP based detection algorithms. Note that OMP based algorithms are more computationally complex than the 114

129 energy based detection algorithms, since they require matrix inversion at each iteration. 6.4 Wideband Detection with (P ISO, P EIO ) Note that CED and RED (Algorithm 1) for PBNS can be readily applied to sub- Nyquist sampling with uniform aliasing. RPBD (Algorithm 2) for uniform aliasing is also possible based upon the posterior beliefs derived from the belief propagation algorithm, which we refer to as belief propagation based ranked posterior belief detection (BP-RPBD). The rest algorithms mentioned in this chapter are developed exclusively within the Bayesian detection framework in which P M and P F are the performance metrics, and thus will not be applied to this scenario. 6.5 Numerical Results Similar to Section and Section 5.3.4, we generate simulation results averaged over 10 3 to 10 6 Monte Carlo trials for different SNR levels and number of samples per channel, with the primary occupancies, the primary signals, and the noise signals drawn independently in each trial. As for random sampling, RSG-FN is used for fair comparisons for reasons addressed in Chapter Sub-Nyquist Sampling Schemes with Uniform Aliasing As discussed in Section , the sub-nyquist sampling schemes exhibiting more randomness in the sensing window have smaller aliasing variances, which leads to uniform aliasing. Specifically, GCPS introduced in Section with a period larger than the sensing window length, and RSG-FN introduced in Section leads, with a high probability, to uniform aliasing. In Fig. 6.1 we provide an illustrative example to compare the performance of different sub-nyquist sampling schemes as 115

130 10 0 p = 0.125, SNR=0 db 10 0 p = 0.125, SNR=10 db P m GCPS 5 GCPS RSG FN MAV Approx P f (a) 10 0 p = 0.125, SNR=20 db P m GCPS 5 GCPS RSG FN MAV Approx P f (b) 10 0 p = 0.250, SNR=10 db P m GCPS 5 GCPS RSG FN MAV Approx P f (c) P m GCPS 5 GCPS RSG FN MAV Approx P f (d) Figure 6.1. Different sub-nyquist sampling schemes with uniform aliasing, with channel-by-channel energy detection: Number of channels M = 16, sub-sampling factor ρ = 4, SNR=15 db and number of samples per channel N 0 =

131 well as the approximation for ideal uniform aliasing as discussed in Section Although only four scenarios differing in the SNR levels and the primary occupancy probabilities are provided, the following conclusions apply for a much broader range of parameter settings in not only the SNR levels and the primary occupancy probabilities, but also the number of samples per channel and the sub-sampling factors. The approximation approach in Section 6.2 appears to be reasonably good for sub-nyquist sampling with uniform aliasing. GCPS achieves remarkably small aliasing variance compared to MAV, which corresponds to the sub-nyquist sampling scheme with the minimum possible aliasing variance obtained by exhaustive search. We notice a small divergence for RSG-FN from the ideal uniform aliasing. The result is reasonable, since unlike GCPS and MAV that are pre-fixed, RSG- FN is realized at runtime. It has a high probability to cause uniform aliasing, yet non-zero probability to cause periodic aliasing as discussed in Section , as well as more intermediate aliasing patterns between the two extremes. We believe that the similarity in sensing performance among GCPS, RSG-FN and MAV lies in the fact that they all cause uniform aliasing. This observation appears to apply regardless of system parameter settings and different performance metrics. That said, GCPS is much easier to implement practically Detection with an Estimated Number of Declared ON Channels VS. Detection with a Fixed Number of Declared ON Channels In Fig. 6.2, we compare the sensing performance of the detection algorithms that estimate the number of ON channels within the Bayesian detection framework with those that do not estimate the number of ON channels, with comparisons for both the K-sparse model and the p-sparse model for the primary occupancies. We stress that the results for GCPS, RSG-FN and MAV are very similar and we illustrate the results for GCPS only. The primary occupancy probability in the p-sparse model is chosen in such a way that the expected number of ON channels is the same as the 117

132 10 0 K sparse: K = p sparse: p = P m RCED AN RCED FN OMP AN OMP FN P f (a) P m RCED AN RCED FN OMP AN OMP FN P f (b) Figure 6.2. Detection with an estimated number of declared ON channels versus detection with a fixed number of declared ON channels for generalized co-prime sampling: number of channels M = 16, sub-sampling factor ρ = 4, SNR=15 db and number of samples per channel N 0 = 8. number of ON channels in the K-sparse model. For the K-sparse model in which the number of ON channels remains the same for each trial, the detection algorithms that estimate the number of ON channels are worse than the corresponding algorithms that declare a fixed number of ON channels. The gaps between the two algorithm classes characterize the losses in the approximations in obtaining the number of declared ON channels in Section On the other hand, for the p-sparse model in which the number of ON channels is varying, we observe a substantial improvement of the detection algorithms that approximate the number of declared ON channels. Thus, the assumptions and estimation approaches in estimating the number of declared ON channels within the Bayesian detection framework are useful despite the inherent approximation losses, and the detection al- 118

133 gorithms that estimate the number of ON channels are more robust for the scenario in which the primary occupancy level is uncertain, or time varying More Results for Channel Detection with (P M, P F ) In Fig. 6.3 we compare the performance of different detection algorithms for several different scenarios. Again, the results are for GCPS only and very similar results are observed for RSG-FN and MAV. Also, similar results apply for a much broader range of parameter settings than provided here. For low SNR ( 0 db), different detection algorithms exhibit relatively close performance, with OMP based detection algorithms being the worst. For moderate and high SNR ( 10 db), OMP-AN and BP-CPBD outperform the other detection algorithms. OMP-AN can potentially outperform BP-CPBD in the regime of worse protection for the primary system, i.e., the regime of high P M and low P F, for high SNR and small primary occupancy level. Since the opposite regime with low P M and high P F is more desirable for wideband spectrum sensing, the value of sparse reconstruction algorithms is questionable for wideband spectrum sensing based upon sub-nyquist sampling. Among the several detection algorithms considered, and for the several scenarios studied, BP-CPBD exhibits the most favorable performance in the regime of better protection for the primary system, therefore making it what we believe to be the most favorable detection algorithm for sub-nyquist sampling with uniform aliasing Wideband Detection with (P ISO, P EIO ) In Fig. 6.4 and Fig. 6.5 we compare several detection algorithms relevant to the goal of wideband detection characterized by (P ISO, P EIO ) for different system settings. As expected, similar results and conclusions can be drawn to those of PBNS. CED exhibits the worst performance. BP-RPBD is the most favorable detection algorithm of all the multi-channel detection algorithms studied. 119

134 10 0 SNR=0 db, p= SNR=10 db, p= P m CED CPBD RED AN OMP AN P f (a) P m CED CPBD RED AN OMP AN P f (b) SNR=20 db, p=0.125 CED CPBD RED AN OMP AN SNR=10 db, p=0.25 P m P f (c) P m CED RPBD RED AN OMP AN P f (d) Figure 6.3. Comparison of different detection algorithms for generalized co-prime sampling (GCPS): number of channels M = 16, sub-sampling factor ρ = 4, and number of samples per channel N 0 = 8. Similar results are observed if random sampling on a grid with fixed number of taps (RSG-FN) or minimum aliasing variance (MAV) sampling is used. 120

135 P EIO = Pr{I > 0} P EIO = Pr{I > 0} CED RED BP RPBD P ISO = Pr{S < 2} (a) CED RED BP RPBD P ISO = Pr{S < 4} (b) P EIO = Pr{I > 0} P EIO = Pr{I > 0} CED RED BP RPBD P ISO = Pr{S < 8} (c) CED RED BP RPBD P ISO = Pr{S < 12} (d) Figure 6.4. Comparison of different detection algorithms for generalized co-prime sampling (GCPS): number of channels M = 16, SNR=0 db, sub-sampling factor ρ = 4, and number of samples per channel N 0 = 8. Similar results are observed if random sampling on a grid with fixed number of taps (RSG-FN) or minimum aliasing variance (MAV) sampling is used. 121

136 P EIO = Pr{I > 0} CED RED BP RPBD P ISO = Pr{S < 2} (a) P EIO = Pr{I > 0} CED RED BP RPBD P ISO = Pr{S < 4} (b) P EIO = Pr{I > 0} P EIO = Pr{I > 0} CED RED BP RPBD P ISO = Pr{S < 8} (c) CED RED BP RPBD P ISO = Pr{S < 12} (d) Figure 6.5. Comparison of different detection algorithms for generalized co-prime sampling (GCPS): number of channels M = 16, SNR=20 db, sub-sampling factor ρ = 2, and number of samples per channel N 0 = 4. Similar results are observed if random sampling on a grid with fixed number of taps (RSG-FN) or minimum aliasing variance (MAV) sampling is used. 122

137 6.6 Summary In this chapter, we focus on two elements for wideband spectrum sensing, specifically, sampling schemes and corresponding detection algorithms, based upon the system model and the performance metrics introduced in Chapter 3. We develop a number of detection algorithms for sub-nyquist sampling with uniform aliasing and either relevant to the goal of channel detection characterized by performance metrics (P M, P F ), or relevant to the goal of wideband detection characterized by performance metrics (P ISO, P EIO ). We introduce an approximation method to estimate the number of declared ON channels within the Bayesian detection framework, which allows the traditional sparse reconstruction algorithms such as orthogonal matching pursuit to be applied to the problem of multi-channel detection and to tradeoff between probability of missed detection and probability of false alarm. The results illustrate that sparse reconstruction algorithms, which are often more computationally complex, can potentially improve the sensing performance compared to other much easier detection algorithms, but only for very high SNR ( 20 db) and in the regime of less interest for wideband spectrum sensing for CR, i.e., the regime of high P M and low P F. Furthermore, we show that GCPS with a sufficiently small number of co-prime branches can give very small aliasing variance, and with same detection algorithm being used, GCPS exhibits similar performance to that of MAV, which corresponds to the sub-nyquist sampling that gives the minimum possible aliasing variance. Moreover, GCPS is much easier to implement practically compared to RSG. 123

138 CHAPTER 7 COMPARISON OF DIFFERENT SAMPLING SCHEMES In this chapter we compare the performance of all wideband sampling schemes discussed in the dissertation for channel detection with performance metrics (P M, P F ), and for wideband detection with performance metrics (P ISO, P EIO ). We identify the most favorable sampling scheme, which appears to be IU in most cases, and discuss the extensibility of some important detection algorithms to more general primary system models with respect to three aspects, i.e., primary channel model, primary signal model, and primary occupancy model, for different sampling schemes. The remainder of the chapter is organized as follows. In Section 7.1, we summarize the wideband sampling schemes to be compared and establish the basis for comparison. In Section 7.2, we compare the best performance of each sampling scheme for channel detection with performance metrics (P M, P F ), and in Section 7.3, we compare the best performance of each sampling scheme for wideband detection with performance metrics (P ISO, P EIO ). In Section 7.4, we draw some conclusions on the most favorable wideband sampling schemes. In Section 7.5, we discuss the extensibility of certain detection algorithms to more general primary system models. In Section 7.6, we conclude the chapter. 7.1 General Comparison of Wideband Sampling Schemes Based upon the insights from previous chapters, we are particularly interested in comparison of five wideband sampling schemes in this section. 124

139 Partial-band Nyquist sampling (PBNS); Sequential narrowband Nyquist sampling (SNNS); Integer undersampling (IU); Generalized co-prime sampling with 3 branches (3-GCPS), which corresponds to a typical sub-nyquist sampling with uniform aliasing that has the simplest practical implementation as discussed in Section and Section 6.5.1; Multi-coset sampling (MCS) with non-singular parameter D. For fair comparisons, the following two parameters are held constant for different sampling schemes: The sensing window duration for the entire bandwidth; The overall sampling rate. Subject to these constraints, the abovementioned wideband sampling schemes can be differentiated along three dimensions: The total bandwidth monitored during the sensing window, which is necessarily equal to the entire bandwidth; The normalized sampling rate, which is defined as the actual overall sampling rate divided by the corresponding Nyquist rate; The effective number of samples per channel, which is defined as the total number of available frequency samples divided by the number of channels monitored, regardless of the correlations among frequency samples. Comparisons among the wideband sampling schemes along these three axes are illustrated in Fig Channel Detection with (P M, P F ) In Figure 7.2, we compare the channel ROCs for SNNS with CED of Section 4.2.1, IU with BED of Section 5.2, and 3-GCPS with BP-CPBD of Section 6.3 for several scenarios that differ in the primary occupancies, the SNR levels and the sub-sampling 125

140 Effective number of frequency samples per channel Sub-Nyquist sampling PBNS SNNS Total bandwidth monitored Normalized sampling rate Figure 7.1. Wideband sampling schemes in the three-dimensional space subject to the same overall sensing window duration and the overall sampling rate. 126

141 factors. Note that the results for SNNS and IU are generated in theory, while the results for 3-GCPS are generated by simulation. We make the following observations based upon the numerical results for low primary occupancy level: IU achieves the most favorable performance in the regime of better protection for the primary system; SNNS achieves the most favorable performance in the regime of worse protection for the primary system but better opportunities for the secondary system, with the crossover point to IU determined by the specific system parameters. Sub-Nyquist sampling with uniform aliasing is either dominated by IU or dominated by SNNS. Since the regime of better protection for the primary system is preferable for wideband spectrum sensing for CR, IU with BED from Section 5.2 appears to be the most favorable wideband sensing scheme. It is important to stress, however, that, the performance of IU relies on the primary occupancy level. Specifically, the lower the primary occupancy probability, the better that IU can perform. This is because high primary occupancy probability leads to more ON channels, thus more primary signal aliasing for IU. On the contrary, the results of SNNS are independent of the primary occupancies, since the channels are not coupled. Finally, as the sub-sampling factor increases, the detection performances for IU and SNNS both degrade significantly, since larger sub-sampling factor means more aliasing in the signal for IU, and less number of samples per channel for SNNS. 7.3 Wideband Detection with (P ISO, P EIO ) In Fig Fig. 7.5, we provide several illustrative examples in which we compare the best sensing performance that a sampling scheme can achieve with ranked posterior belief detection. Several conclusions can be drawn from the results: 127

142 10 0 SNR=0 db, N 0 = SNR=20 db, N 0 = P m P m GCPS IU SNNS P f (a) GCPS IU SNNS P f (b) Figure 7.2. Probability of missed detection (vertical axes) and probability of false alarm (horizontal axes) for different sampling schemes using channel-by-channel energy detection. M = 16 channels, N 0 = 8 frequency samples per channel, primary occupancy probability p = 0.1, and sub-sampling factor ρ =

143 Generally, SNNS exhibits the best performance in the regime of worse protection for the primary system; PBNS works best, especially in the regime of better protection for the primary system, for S d M and small primary occupancy probability. However, if the desired number of spectrum opportunities is far less than the total number of channels in the wideband, it is essentially meaningless to scan the entire bandwidth given the limited sensing window duration and overall sampling rate. For low SNR (< 0 db) scenarios, and relatively moderate and high number of spectrum opportunities sought, SNNS exhibits the most favorable overall performance, though the gaps between different sampling schemes are relatively small. For moderate and high SNR (SNR > 0 db), and relatively moderate and high number of spectrum opportunities sought, IU exhibits the most favorable performance in the regime that better protects the primary system. 7.4 Which Sampling Scheme is Most Favorable? Based upon the discussions in Section 7.2 and Section 7.3, we notice that the conclusions are consistent whether channel detection or wideband detection are considered: for moderate and high SNR levels, and moderate and large number of spectrum opportunities sought, IU appears to be the most favorable sampling scheme in the regime that better protects the primary system. Specifically, we summarize the most favorable detection regimes of different wideband sampling schemes in Fig. 7.6, for moderate / high SNR levels, and moderate / large number of spectrum opportunities sought. Note that the highlighted regime corresponds to the preferable regime of wideband spectrum sensing that better protects the primary system. Also, for very small number of spectrum opportunities sought, PBNS achieves the most favorable performance in the preferable regime, and for very low SNR level, SNNS achieves the most favorable performance in the preferable regime. Another important observation is that, for all cases studied, sub-nyquist sampling schemes with uniform aliasing, which are often more sophisticated and complex to 129

144 P EIO = Pr{I > 0} PBNS SNNS IU 3 GCPS P EIO = Pr{I > 0} PBNS SNNS IU 3 GCPS P = Pr{S < 2} ISO (a) P = Pr{S < 4} ISO (b) P EIO = Pr{I > 0} PBNS SNNS IU 3 GCPS P ISO = Pr{S < 6} (c) P EIO = Pr{I > 0} PBNS SNNS IU 3 GCPS P ISO = Pr{S < 8} (d) Figure 7.3. Probability of insufficient spectrum opportunities (vertical axes) and probability of excessive interference opportunities (horizontal axes) of different sampling schemes with ranked posterior belief detection. M = 16 channels, SNR = 10 db, N 0 = 8 frequency samples per channel, primary occupancy probability p = 0.1, and sub-sampling factor ρ =

145 P EIO = Pr{I > 0} S d =2, p= P = Pr{S < 2} ISO (a) PBNS SNNS IU 3 GCPS P EIO = Pr{I > 0} PBNS SNNS IU 3 GCPS S d =4, p= P = Pr{S < 4} ISO (b) 10 0 S d =6, p= S d =4, p= P EIO = Pr{I > 0} PBNS SNNS IU 3 GCPS P ISO = Pr{S < 6} (c) P EIO = Pr{I > 0} PBNS SNNS IU 3 GCPS P ISO = Pr{S < 4} (d) Figure 7.4. Probability of insufficient spectrum opportunities (vertical axes) and probability of excessive interference opportunities (horizontal axes) of different sampling schemes with corresponding most favorable detection algorithms. M = 16 channels, SNR = 10 db, N 0 = 4 frequency samples per channel, primary occupancy probability p = 0.1, and sub-sampling factor ρ =

146 P EIO = Pr{I > 0} PBNS SNNS 0.05 IU 3 GCPS P = Pr{S < 2} ISO (a) P EIO = Pr{I > 0} PBNS SNNS 0.1 IU 3 GCPS P = Pr{S < 4} ISO (b) P EIO = Pr{I > 0} PBNS SNNS 0.1 IU 3 GCPS P = Pr{S < 6} ISO (c) P EIO = Pr{I > 0} PBNS 0.2 SNNS 0.1 IU 3 GCPS P = Pr{S < 8} ISO (d) Figure 7.5. Probability of insufficient spectrum opportunities (vertical axes) and probability of excessive interference opportunities (horizontal axes) of different sampling schemes with corresponding most favorable detection algorithms. M = 16 channels, SNR = -10 db, N 0 = 20 frequency samples per channel, primary occupancy probability p = 0.1, and sub-sampling factor ρ =

147 Primary protections: PM/PEIO Better Worse Better SNNS Sub-Nyquist sampling with uniform aliasing IU Worse Secondary opportunities: P F /P ISO Figure 7.6. Preferable operating regimes for different wideband sampling schemes and the developed detection algorithms, for moderate / high SNR levels and moderate / large desired number of spectrum opportunities. 133