Cooperative Wideband Spectrum Sensing Based on Joint Sparsity

Transcription

1 Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2017 Cooperative Wideband Spectrum Sensing Based on Joint Sparsity ghazaleh jowkar Follow this and additional works at: Part o the Electrical and Computer Engineering Commons The Author Downloaded rom This Thesis is brought to you or ree and open access by the Graduate School at VCU Scholars Compass. It has been accepted or inclusion in Theses and Dissertations by an authorized administrator o VCU Scholars Compass. For more inormation, please contact libcompass@vcu.edu.

2 Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2017 Cooperative Wideband Spectrum Sensing Based on Joint Sparsity Ghazaleh Jowkar Follow this and additional works at: Part o the Systems and Communications Commons The Author This Thesis is brought to you or ree and open access by the Graduate School at VCU Scholars Compass. It has been accepted or inclusion in Theses and Dissertations by an authorized administrator o VCU Scholars Compass. For more inormation, please contact libcompass@vcu.edu.

3 Ghazaleh Jowkar 2017 All Rights Reserved

4 Cooperative Wideband Spectrum Sensing Based on Joint Sparsity A thesis submitted in partial ulillment o the requirements or the degree o Master o Science at Virginia Commonwealth University By Ghazaleh Jowkar Adviser: Dr. Ruixin Niu Department o Electrical and Computer Engineering Virginia Commonwealth University Richmond, Virginia July, 2017

5 ACKNOWLEDGMENT I would like to thank my husband or all o his support at every step o my project, none o this would be possible without him. I would like to express my sincere gratitude to my advisor Dr. Ruixin Niu or the continuous support and guidance o my M.S research study, or his patience, directions, and immense knowledge. His guidelines helped me in every step o research and writing o this thesis. Besides my advisor, I would like to thank the rest o my thesis committee members, Dr. Wei Cheng and Dr. Alen Doce or their insightul comments and encouragement. This work was supported in part by the Air Force Research Laboratory Inormation Directorate through its Visiting Faculty Research Program in I thank Air Force Research Laboratory at Rome, NY or giving me the opportunity and the resources to start this work.i have my special thanks to Dr. Lauren Huie or her guidelines and constructive suggestions. I am grateully indebted to her or her very valuable comments on this thesis.

6 TABLE TO CONTENTS Abstract Chapter 1: Introduction Introduction Thesis Structure Chapter 2: Background Cognitive Radio Cognitive Radio Basics Network Structure Spectrum Sensing Analysis Cooperative Sensing Low Rank Matrix Completion Motivation Model Description Jointly Sparse Signals and Mixed Norm Minimization Reconstruction Chapter 3: System Model and Solution Discussion o the Problem Methodology Low-Rank Matrix Completion Based Spectrum Sensing Spectrum Sensing Based on Mixed Norm Minimization Spectrum Sensing Decision Making Numerical Results or Signals in Single Time Frame

7 3.5 Spectrum Sensing over Multiple Time Frames System Model Methodology: Mixed Norm Minimization Based Solution Numerical Results or Signals over Multiple Time Frames Chapter4: Conclusion Reerences:

8 ABSTRACT COOPERATIVE WIDEBAND SPECTRUM SENSING BASED ON JOINT SPARSITY By Ghazaleh Jowkar, Master o Science A thesis submitted in partial ulillment o the requirements or the degree o Master o Science at Virginia Commonwealth University Virginia Commonwealth University 2017 Major Director: Dr. Ruixin Niu, Associate Proessor o Department o Electrical and Computer Engineering In this thesis, the problem o wideband spectrum sensing in cognitive radio (CR) networks using sub-nyquist sampling and sparse signal processing techniques is investigated. To mitigate multipath ading, it is assumed that a group o spatially dispersed SUs collaborate or wideband 4

9 spectrum sensing, to determine whether or not a channel is occupied by a primary user (PU). Due to the underutilization o the spectrum by the PUs, the spectrum matrix has only a small number o non-zero rows. In existing state-o-the-art approaches, the spectrum sensing problem was solved using the low-rank matrix completion technique involving matrix nuclear-norm minimization. Motivated by the act that the spectrum matrix is not only low-rank, but also sparse, a spectrum sensing approach is proposed based on minimizing a mixed-norm o the spectrum matrix instead o low-rank matrix completion to promote the joint sparsity among the column vectors o the spectrum matrix. Simulation results are obtained, which demonstrate that the proposed mixednorm minimization approach outperorms the low-rank matrix completion based approach, in terms o the PU detection perormance. Further we used mixed-norm minimization model in multi time rame detection. Simulation results shows that increasing the number o time rames will increase the detection perormance, however, by increasing the number o time rames ater a number o times the perormance decrease dramatically. 5

10 CHAPTER ONE INTRODUCTION 1.1 INTRODUCTION With an ever-increasing number o wireless users and devices, the radio requency spectrum becomes a more and more scarce resource. On the other hand, a large percentage o spectrum resources are underutilized by the licensed primary users (PUs). Thereore, the cognitive radio (CR) system has the potential to take ull advantage o the underutilized spectrum resources by allowing unlicensed usage o vacant spectrum. For CR systems, spectrum sensing is a key step to detect spectrum holes/vacancies which can be used by secondary users (SUs) without causing any intererence to PUs. We ocused our research on wideband spectrum sensing in CR networks using sub-nyquist sampling and sparse signal processing techniques. To mitigate multi-path ading, we assume that a group o spatially dispersed SUs collaborate or wideband spectrum sensing, to determine the spectrum holes and identiy potential transmission opportunities or SUs. In some state-o-the-art approaches [1,2], multiple spatially dispersed SUs have been used to mitigate wireless ading eects, and the low-rank matrix completion technique involving convex optimization has been applied to reconstruct a low-rank spectrum matrix, and determine whether or not a certain channel has been occupied by a PU. The spectrum is usually under-utilized, and the spectrum matrix has the spectrum vectors at dierent SUs as its columns. As a result, the spectrum matrix has only a small number o non-zero rows, meaning that it is low-rank. To reduce the burden on the analogdigital converter and the sensing cost, sub-nyquist sampling and compressive sensing have been applied. 6

11 Due to the underutilization o the spectrum resource, we ound that the spectrum matrix is not only low-rank, but also sparse. This motivates us to propose a spectrum sensing approach based on minimizing a l2 / l1mixed-norm o the spectrum matrix to promote joint sparsity among the columns o the spectrum matrix, instead o low-rank matrix completion. We investigated the perormance o our model by perorming detection in multiple time rames using mixed-norm minimization model. Experiment results based simulation demonstrate that the proposed new approach outperorms the low-rank matrix completion based approach in higher SNRs, Also, the Detection perormance will increase by increasing the number o time rames through the comparison o the receiver operating characteristic (ROC) curves. 1.2 THESIS STRUCTURE In Chapter Two, we will give general background on cognitive radio networks and spectrum sensing, and an overview on cooperative spectrum sensing. Then an overview o low rank matrix completion model and joint sparse matrix reconstruction will be provided. In Chapter Three we will go through the system model and discussion o the problem and our solutions to the problem using low rank matrix completion and mixed norm matrix reconstruction models. We compare the results o two proposed model and at the end we explain the system model on multi time rame detection using mixed norm minimization. In Chapter our we will give a brie conclusion and we will review our uture work in using Mixed Norm Matrix Completion model based sequential detection, we will discuss our expected results and our current results. 7

12 CHAPTER TWO BACKGROUND 2.1 COGNITIVE RADIO National regulatory bodies such as FCC are in control o usage o radio spectrum resources and the regulation o radio emissions. FCC assigns spectrum to licensed users or primary users on a long-term basis or large geographical regions. However, due to ineicient usage o the limited spectrum, a large portion o the assigned spectrum remains under-utilized. Thereore, the development o dynamic spectrum access techniques is becoming necessary. Dynamic access techniques reer to the case where non-licensed users or the secondary users, are allowed to temporarily use the unused part o the licensed spectrum. Cognitive radio is the next generation communication network solution, also known as dynamic spectrum access (DSA) networks, to make the use o spectrum more eicient in an opportunistic way without interering with the primary users. Cognitive radio is an intelligent wireless communication system which uses its cognitive capability to become aware o its surrounding environment and by learning rom environment can identiy the available spectrum and adapt its internal states to achieve the optimal perormance. Cognitive radio should adaptively modiy its state and spectrum access to assure that primary user reclaims spectrum usage right. In this chapter recent research on cognitive radios will be reviewed. We overview the basics o cognitive radio technology, architecture, and its applications, and we talk about spectrum sensing, types o detection methods, and cooperative spectrum sensing. Finally, we discuss low rank matrix completion and joint sparse matrix reconstruction models as two reconstruction methods or spectrum sensing. 8

13 2.2 COGNITIVE RADIO BASICS: Cognitive radio (CR) is the next generation o communications and networking that can adapt its operating parameters to utilize the limited network resources in a more eicient and lexible way. Two major unctionalities o CRs are cognitive capability and reconigurability. Beore adapting their operating parameters CRs use their cognitive capability to gather inormation about the channel and make a decision accordingly. Cognitive capability is the ability o the cognitive radio transceiver to gather inormation rom radio environment, and accordingly decide which spectrum band(s) to be used and the best transmission method to be adopted. Reconigurability is the use o the inormation rom the radio environment and change o CRs parameters to achieve optimal perormance. A typical duty cycle o CR includes: Spectrum sensing Spectrum sensing is the ability o a CR to measure the activities o the radio transmissions over dierent spectrum bands and to capture the parameters related to such bands (e.g., power levels, user activities, etc.). Spectrum sensing is one o the most critical unctions o a cognitive radio as it provides the awareness o the spectrum usage in the surrounding environment. Existing spectrum sensing techniques ocuses on detecting the activities o the primary users. Such methods are based on matched ilter detection, energy detection, eature detection, and intererence temperature measurement, respectively. 9

14 Spectrum Analysis Spectrum analysis is to iner i a primary user is occupying the band at a certain time and geographic area. Such a deinition covers only three dimensions o the spectrum space: requency, time, and space. Other dimensions o a given spectrum can be exploited. Spectrum Access Decisions The last step o the cognition cycle o a cognitive radio is to decide the set o transmission actions to be taken based on the outcome o the spectrum sensing and analysis procedures. More speciically, a cognitive radio utilizes the inormation gathered regarding the spectrum bands identiied as available spectral opportunities to deine the radio transceiver parameters or the upcoming transmission(s) over such requency bands. The set o transceiver parameters to be decided depends on the underlying transceiver architecture. 2.3 NETWORK STRUCTURE In a CR network architecture, since secondary users who are not authorized with spectrum usage rights can utilize the temporally unused licensed bands owned by the primary users, the components include both a secondary network and a primary network. A secondary network reers to a network composed o a set o secondary users with/without a secondary base station. Secondary users can only access the licensed spectrum when it is not occupied by a primary user. The opportunistic spectrum access o secondary users is usually coordinated by a secondary base station, which is a ixed inrastructure component serving as a hub o the secondary network. Both secondary users and secondary base stations are equipped with CR unctions. 10

15 A primary network is composed o a set o primary users and one or more primary base stations. Primary users are authorized to use certain licensed spectrum bands under the coordination o primary base stations. Their transmission should not be interered by secondary networks. Primary users and primary base stations are in general not equipped with CR unctions. Thereore, i a secondary network share a licensed spectrum band with a primary network, besides detecting the spectrum white space and utilizing the best spectrum band, the secondary network is required to immediately detect the presence o a primary user and direct the secondary transmission to another available band so as to avoid interering with primary transmission. Since CRs are able to sense, detect, and monitor the surrounding RF environment such as intererence and access availability, and reconigure their own operating characteristics to best match outside situations, cognitive communications can increase spectrum eiciency and support higher bandwidth service. Moreover, the capability o real-time autonomous decisions or eicient spectrum sharing also reduces the burdens o centralized spectrum management. As a result, CRs can be employed in many applications. As a CR can recognize spectrum availability and reconigure itsel or much more eicient communication, this provides public saety personnel with dynamic spectrum selectivity and reliable broadband communication to minimize inormation delay. Moreover, CR can acilitate interoperability between various communication systems. Through adapting to the requirements and conditions o another network, the CR devices can support multiple service types, such as voice, data, video, etc. 11

16 2.4 SPECTRUM SENSING ANALYSIS: As mentioned, spectrum sensing detects the primary user s activity based on the local measurements o secondary users. The ollowing are the most common spectrum sensing techniques: 1) Energy Detector: Ease o implementation and no need o any prior knowledge o primary user s signal have made energy detection the most common type o spectrum sensing. H0 : y( t) n( t), H : y( t) hx( t) n( t) 1 (2.1) in which xt () is the primary user s signal received at the local receiver o a secondary user, nt () is the additive white Gaussian noise, h is the channel gain rom the primary user s transmitter to the secondary user s receiver. H 0 is a null hypothesis, meaning there is no primary user present in the band, H 1 means the primary user s presence. The detection statistic o the energy detector is the average (or total) energy o N observed samples, T N t 1 2 N 1 y(t) (2.2) By comparing the detection statistic T, with a predetermined threshold the decision on occupancy o the channel is made. The perormance o the detector is characterized by two probabilities: The probability o alse alarm P F (the probability that the hypothesis test decides H 1 while it is H 0 ) 12

17 P P ( T H ) (2.3) F r 0 The probability o detection P D (the probability that the test correctly decides H1). P P ( T H ) (2.4) D r 1 A good detector should ensure a high detection probability and a low alse alarm, or it should optimize the spectrum usage eiciency. The region o convergence (ROC) curve is typically used to show the relationship between P F and P D. Cognitive radio with more eicient detection will have a ROC curve closer to the uplet corner and urther away rom the 45-degree line. Choosing a right detection approach has an important role in minimizing spectrum sensing error, improving the spectrum utilization, and protecting the PU rom intererence rom the SUs. By utilizing the spectrum sensing error unction an optimal adaptive threshold level can be developed [9-10]. Besides its low computational and implementation complexity and short detection time, there are some challenges in designing a good energy detector. Noise power might change over time and precise measurement o it can be diicult in real time. The detection threshold depends on the noise power and in the cases where the noise power is very high (low signal-to-noise ratio (SNR)), reliable identiication o a primary user is even impossible[8]. 13

18 An energy detector determines primary user s presence only by comparing the received signal energy with a threshold. As a result, it cannot dierentiate the primary user rom other unknown signal sources, a situation that can trigger alse alarm requently. 2) Feature Detector: (cyclostationary eatures) There are speciic eatures associated with the inormation transmission o a primary user. For instance, the statistics o the transmitted signals in many communication paradigms are periodic because o the inherent periodicities such as the modulation rate, carrier requency, etc. Such eatures are usually viewed as the cyclostationary eatures, based on which a detector can distinguish cyclostationary signals rom stationary noise. In a more general sense, eatures can reer to any intrinsic characteristics associated with a primary user s transmission, as well as the cyclostationary eatures. For example, center requencies and bandwidths extracted rom energy detection can also be used as reerence eatures or classiication and determining a primary user s presence. In this section, we will introduce the cyclostationary eature detection ollowed by a generalized eature detection. Cyclostationary eature [6]: as in most communication systems, the transmitted signals are modulated signals coupled with sine wave carriers, pulse trains, hopping sequences, or cyclic preixes, while the additive noise is generally wide-sense stationary (WSS) with no correlation. Cyclostationary eature detectors can be utilized to dierentiate noise rom primary users signal and distinguish among dierent types o transmissions and primary systems. Unlike energy detector which uses time-domain signal energy as test statistics, a cyclostationary eature detector perorms a transormation rom the time-domain into the requency eature domain and then conducts a hypothesis test in the new domain. Cyclic autocorrelation unction (CAF) o the received signal y(t) is deined by, 14

19 R E y t y t e j2t y [ ( ) *( ) ] (2.5) Where E [.] is the expectation operation, * denotes complex conjugation, and is the cyclic requency. Since periodicity is a common property o wireless modulated signals, while noise is WSS, the CAF o the received signal also demonstrates periodicity when the primary signal is present. I we can represent the CAF using its Fourier series expansion, we will have the cyclic spectrum density (CSD) unction, expressed as, y j2 (2.6) S(, ) R ( ) e The CSD unction have peaks when the cyclic requency α equals to the undamental requencies o the transmitted signal xt (), i.e. ( k / T x ), with T x is the period o xt (). Under H 0 the CSD unction does not have any peaks since the noise is non-cyclostationary. A peak detector or a generalized likelihood ratio test can be urther used to distinguish between the two hypotheses. Dierent primary communication systems using dierent air interaces (modulation, multiplexing, coding, etc.) can also be dierentiated by their dierent properties o cyclostationarity. However, when requency-division multiplexing (FDM) becomes the air interace, identiication o dierent systems may become an issue, since the eatures due to the nature o OFDM signaling are likely to be close or even identical. To address this problem, particular eatures need to be introduced to OFDM-based communications. The OFDM signal is conigured beore transmission so that its CAF outputs peaks at certain pre-chosen cycle requencies, and the dierence in these requencies is used to distinguish among several systems under the same OFDM air interace. 15

20 Compared to energy detectors that are prone to high alse alarm probability due to noise uncertainty and unable to detect weak signals in noise, cyclostationary detectors become good alternatives because they can dierentiate noise rom primary users signal and have better detection robustness in low SNR regime. Generalized eature detection reers to detection and classiication that extracts more eature inormation other than the cyclostationarity due to the modulated primary signals, such as the transmission technologies used by a primary user, the amount o energy and its distribution across dierent requencies, channel bandwidth and its shape, power spectrum density, center requency, idle guard interval o OFDM, FFT-type o eature, etc. By matching the eatures extracted rom the received signal to the a priori inormation about primary users transmission characteristics, primary users can be identiied. Location inormation o the primary signal is also an important eature that can be used to distinguish a primary user rom other signal sources. 3) Matched Filtering and Coherent Detection: I secondary users have inormation about a primary user signal a priori, then the optimal detection method is the matched ilter, since a matched ilter can correlate the already known primary signal with the received signal to detect the presence o the primary user and thus maximize the SNR in the presence o additive stochastic noise. The merit o matched iltering is the short time it requires to achieve a certain detection perormance such as low probabilities o missed detection and alse alarm, since a matched ilter needs less received signal samples. However, the required number o signal samples also grows as the received SNR decreases, so there exists a SNR wall or a matched ilter. In addition, its implementation complexity and power consumption is too high, because the matched ilter needs receivers or all types o signals and corresponding receiver algorithms to be executed. 16

21 Matched iltering requires perect knowledge o the primary user s signal, such as the operating requency, bandwidth, modulation type and order, pulse shape, packet ormat, etc. I wrong inormation is used or matched iltering, the detection perormance will be degraded a lot. Even though perect inormation o a primary user s signal may not be attainable, i a certain pattern is known rom the received signals, coherent detection (a.k.a. waveorm-based sensing) can be used to decide whether a primary user is transmitting or not. [16] 4) Other Techniques: There are several other spectrum sensing techniques proposed in recent literature, and some o them are variations inspired by the above-mentioned sensing: Statistical Covariance-Based Sensing: The dierence o statistical covariance matrices o the received signal and noise is used to dierentiate the desired signal component rom background noise [11-12]. Filter-Based Sensing: ilter banks are used or multicarrier communications in CR networks, and spectrum sensing can be perormed by only measuring the signal power at the outputs o subcarrier channels with virtually no computational cost [13]. Fast Sensing: Quickest detection perorms a statistical test to detect the change o distribution in spectrum usage observations as quickly as possible. The unknown parameters ater a primary user appears can be estimated using the proposed successive reinement, which combines both generalized likelihood ratio and parallel cumulative sum tests. Learning/Reasoning-Based Sensing: optimal detection strategy is obtained by solving a Markov decision process (MDP). 17

22 2.5 COOPRATIVE SENSING The perormance o spectrum sensing is limited by noise uncertainty, shadowing, and multipath ading eect. In Low SNR cases, a hidden primary user problem occurs where secondary users cannot detect the primary transmitter, when the primary user is occupying the channel, thereore the primary user will be interered. To solve this issue the advantage o the independent ading channels (i.e., spatial diversity) and multiuser diversity has been considered and cooperative spectrum sensing is proposed to improve the reliability o spectrum sensing, increase the detection probability to better protect a primary user, and reduce alse alarm to utilize the idle spectrum more eiciently. Centralized cooperative spectrum sensing: a central controller, e.g., a secondary base station, collects local observations rom multiple secondary users, decides the available spectrum channels using some decision usion rule, and inorms the secondary users which channels to access. Distributed cooperative spectrum sensing: secondary users exchange their local detection results among themselves without requiring a backbone inrastructure with reduced cost. Relays can also be used in cooperative spectrum sensing, where the cognitive users operating in the same band help each other relay inormation using ampliy-and-orward protocol. Challenges on cooperative spectrum sensing come rom the limitation o the secondary users. Since SRs can be low-cost devices only equipped with a limit amount o power, they cannot employ very complicated detection hardware with high computational complexity. In wideband cooperative sensing, multiple secondary users have to scan a wide range o spectrum channels and share their detection results. This results in a large amount o sensory data exchange, high energy consumption, and an ineicient data throughput. 18

23 1) User Selection: Due to secondary users dierent locations and channel conditions involving all the secondary users in spectrum sensing is not eicient, and cooperating more eicient approach is to select only a group o users who have higher SNR o the received primary signal. Since detecting a primary user costs battery power o secondary users, and shadow ading may be correlated or nearby secondary users, an optimal selection o secondary users or cooperative spectrum sensing is desirable. I a secondary user cannot distinguish between the transmissions o a primary user and another secondary user, it will lose the opportunity to use the spectrum. The presence/absence o possible intererence rom other secondary users is the main reason o the uncertainty in primary user detection, and coordinating with nearby secondary users can greatly reduce the noise uncertainty due to shadowing, ading, and multi-path eects. A good degree o coordination should be chosen based on the channel coherent times, bandwidths, and the complexity o the detectors. 2) Decision Fusion: Dierent decision usion rules or cooperative spectrum sensing have been studied in the literature. An optimal way to combine the received primary signal samples in space and time is to maximize the SNR o local energy detectors. In general, cooperative sensing is coordinated over a separate control channel, so a good cooperation scheme should be able to use a small bandwidth and power or exchanging local detection results while maximizing the detection reliability. An eicient linear cooperation ramework or spectrum sensing is proposed in [7], where the global decision is a linear combination o the local statistics collected rom individual nodes using energy detection. Compared to the likelihood ratio test, the proposed method has lower computational complexity, closed-orm expressions o detection and alse alarm probabilities, and comparable detection perormance. 19

24 3) Eicient Inormation Sharing: In order to coordinate the cooperation in spectrum sensing, a lot o inormation exchange is needed among secondary users, such as their locations, estimation o the primary user s location and power, which users should be clustered into a group, which users should perorm cooperative sensing at a particular time epoch, and so on. Such a large amount o inormation exchange brings a lot o overhead to the secondary users, which necessitates an eicient inormation sharing among the secondary users. In order to reduce the bandwidth required by a large number o secondary users or reporting their sensing results, only users with reliable inormation will send their local observations, i.e., one-bit decision 0 or 1, to the common receiver. 4) Distributed Cooperative Sensing: Cooperative spectrum sensing has been shown to be able to greatly improve the sensing perormance in CR networks. However, i cognitive users belong to dierent service providers, they tend to contribute less in sensing in order to increase their own data throughput. Using replicator dynamics, the evolutionary game modeling provides an excellent means to address the strategic uncertainty that a user may ace by exploring dierent actions, adaptively learning during the strategic interactions, and approaching the best response strategy under changing conditions and environments. 20

25 2.6 LOW RANK MATRIX COMPLETION: In this section, we introduce low rank matrix completion model and the properties o a low rank measurement matrix. In the next chapter, we use the low rank properties o the measurement matrix ormed by measurement vectors rom multiple cooperative CRs. Capitalizing on such a nice property, we then develop a multiple reaction monitoring (MRM)based cooperative support detection algorithm. To perorm cooperative support detection rom multiple measurements we make an important observation that these measurement vectors permit sparse representations due to low spectrum utilization o the primary system, and that these sparse representations jointly possess a desired low-rank property MOTIVATION We have an n 1 by n 2 array o real numbers and that we are interested in knowing the value o each o the n1 n 2 entries in this array. However, we only get to see a small number o the entries so that most o the elements about which we wish inormation are simply missing. Now the question is i we are able to reconstruct the matrix rom the existing entries? This problem is now known as the matrix completion problem. In mathematical terms, the problem may be posed as ollows: n1 n2 We have a data matrix M R which we would like to know as precisely as possible. However, the only inormation available about M is a sampled set o entries M ij,(i, j), where is a subset o the complete set o entries [ n1] [ n2]. (Here and in the sequel, [ n ] denotes the list {1,..., n}.) In order to reconstruct the matrix M rom its partial entries a ew assumptions about the matrix M is needed. 21

26 2.6-2 Model description Here, we are concerned with the theoretical underpinnings o matrix completion and more speciically in quantiying the minimum number o entries needed to recover a matrix o rank r exactly. This number generally depends on the matrix we wish to recover. Let us assume that the unknown rank- r matrix M is n n. Then it is not hard to see that matrix completion is impossible unless the number o samples m is at least 2 2nr r, as a matrix o rank r depends on these many degrees o reedom. The singular value decomposition (SVD), M u v (2.7) k[r] * k k k Where,..., r are the singular values, and the singular vectors,..., n n u1 ur R R and u u R R are two sets o orthonormal vectors, is useul to reveal these degrees o,..., n2 n 1 r reedom. Inormally, the singular values 1... r depend on r degrees o reedom, the let singular vectors u on (n1) (n 2)... (n r) nr r(r 1) / 2 degrees o reedom, and k similarly or the right singular vectors v k. I m nr r 2 2, no matter which entries are available, there can be an ininite number o matrices o rank at most r with exactly the same entries, and so exact matrix completion is impossible. In act, i the observed locations are sampled at random, we will see later that the minimum number o samples is better thought o as being on the order o nr log n rather than nr. nn nn let P : R R be the orthogonal projection onto the subspace o matrices which vanish outside o ( ( i, j) i and only i M is observed) that is, Y P ( X ) is deined as, ij 22

27 Y ij = { X ij, (i, j)εω 0, otherwise (2.8) so that the inormation about M is given by P ( X).The matrix M can be, in principle, recovered rom P ( X) i it is the unique matrix o rank less or equal to r consistent with the data. In other words, i M is the unique solution to, minimize rank( X ) subject to P ( X ) P ( M ) (2.9) Knowing when this happens is a delicate question which shall be addressed later. For the moment, note that attempting recovery via rank minimization is not practical as rank minimization is in general an NP-hard problem or which there are no known algorithms capable o solving problems in practical time once, say, n 10. In [4], it was proved that: 1) matrix completion is not as ill-posed as previously thought. 2) exact matrix completion is possible by convex programming. The author o [4] proposed recovering the unknown matrix by solving the nuclear norm minimization problem, minimize X * subject to P ( X ) P ( M ), (2.10) where the nuclear norm X * o a matrix X is deined as the sum o its singular values, 23

28 X : ( X) (2.11) * i i It is proved that i is sampled uniormly at random among all subset o cardinality m and M obeys a low coherence condition which we will review later, then with a large probability, the unique solution to nuclear norm minimization problem is exactly M, provided that the number o samples obeys, m Cn 6/5 r log n (2.12) (To be completely exact, there is a restriction on the range o values that r can take on). The number o samples per degree o reedom is not logarithmic or polylogarithmic in the dimension, and one would like to know whether better results approaching the nr log n limit are possible. [4] provides a positive answer. In details, this work develops many useul matrix models or which nuclear norm minimization is guaranteed to succeed as soon as the number o entries is o the orm nrpoly log( n ). 2.7 JOINTLY SPARSE SIGNALS AND MIXED NORM MINIMIZATION MATRIX RECONSTRUCTION Over the last ew years, sparsity has emerged as a general principle or signal modeling. Many signals o interest oten have sparse representations, meaning that the signal is well approximated by only a ew nonzero coeicients in a speciic basis. Compressive sensing (CS) has recently emerged as an active research area which aims to recover sparse signals rom measurement data [14-15]. In the basic CS, the unknown sparse signal is recovered rom a single measurement vector, this is reerred to as a single measurement vector (SMV) model. In our study, we consider the 24

29 problem o inding sparse representation o signals rom multiple measurement vectors, which is known as the MMV model. In the MMV model, signals are represented as matrices and are assumed to have the same sparsity structure. Speciically, the entire rows o signal matrix may be 0. Most sparsity based approaches start by expanding signals on a given waveorm amily (basis, rame, dictionary...), and process the coeicients o the expansion individually. Thereore, an assumption on the coeicients independence is implicitly done. Sparse expansion methods explicitly introduce a notion o structured sparsity. Our approach is based on mixed norms, which may be introduced whenever signal expansions on doubly labeled amilies are considered. S is a sparse expansion o signal. S (2.13) i, j ij ij Where { ij } are the waveorms o a given basis or rame. Considering the mixed norm pq, 1/ q q/ p p pq ij (2.14) i j We shall be mainly concerned with the regression problem, 2 min s q i, j 2 ij ij pq (2.15) 25

30 With 0 a ixed parameter. When { ij } is a basis, we give practical estimates or the regression coeicients ij, obtained by generalized sot thresholding. This ormer case is well adapted when the observation o the signal is noisy. 26

31 CAHPTER THREE SYSTEM MODEL AND SOLUTION 3.1 DISCUSSION OF THE PROBLEM When channel state inormation (CSI) rom PU transmitters to CR receivers is available, the CRs can jointly estimate the common transmitted spectrum o the primary system rom their individually received measurement vectors, which is the widely studied cooperative estimation problem. However, when the CSI is unavailable, CRs can only decide the spectrum occupancy o the PU systems, indicated by the nonzero support o the Transmitted spectrum. This becomes a cooperative support detection problem, which is more challenging than cooperative estimation. In the cooperative multiple nodes, the signals received at SUs exhibit a sparsity property that yields a low-rank spectrum matrix o compressed measurements at the usion center. We propose an approach to take advantage the sparsity property o the spectrum matrix at the usion center. With Adopting a system model rom [1], let us assume that a wideband PU system spans over a total o B Hz, and the overall requency band is divided into N non-overlapping bins o equal bandwidth B/N Hz, which are termed as channels and indexed by n [0, 1,..., N 1]. There are J spatially distributed CRs that cooperate during the sensing stage and are indexed by j [1, 2... J]. Each CR senses only a small spectrum segment o bandwidth M(B/N), so that the Nyquist sampling rate per CR is reduced by M/N, compared to that or monitoring the entire wideband spectrum. Further, it is assumed that the J CRs monitors dierent yet overlapping segments o the 27

32 entire spectrum. Namely, the j th CR monitors M channels with channel indices rom ( j 1) to ( j1) M 1, where 0 is an integer denoting the shit between the channel assignments o two adjacent CRs. When1 M, and ( J 1) M N, each channel is guaranteed to be covered by at least one CR. A scenario or 1 and M 4 is illustrated in Figure 1. Figure 1. A cooperative spectrum sensing system with multiple CRs Let s denote the unknown spectrum o the wideband signals transmitted by the PU. The sparsity o the transmitted spectrum is S 0 = I, which is the l0 -norm o the spectrum vector and measures the size o the nonzero support o S. Let us assume that at the j th CR, we have a spectrum vector which is a aded version o S, rj H js (3.1) 28

33 where H j N N C is a diagonal channel matrix, whose diagonal elements are the independent ading coeicients o the corresponding channels. Note that in the cooperative spectrum sensing system, each CR only monitors M out o N channels, and the actual received spectrum ater passing through a selective ilter becomes, r B r (3.2) s, j j j where {0,1} M B N j is the channel selection matrix o the j th CR. B j is obtained rom a N N identity matrix by keeping only those M rows corresponding to the channel subset o the j th CR. When Nyquist-rate sampling is adopted at each CR, the j th CR collects discrete-time sample vector x j in the orm o, 1 x j F r s, j (3.3) where F is the square discrete Fourier transorm (DFT) matrix. When compressive sensing is used, x j can be pre-multiplied by a random sensing matrix Φ j K M C to collect compressive linear projections rom the iltered waveorm x j [3], where K / M is the compression ratio. In the presence o channel noise, the compressed sample vector at the j th CR can be modeled as, x Φ F r w (3.4) 1 j j s, j j x j is a K 1 vector, which corresponds to a sampling rate o ( K / M )( MB / N) KB / N can be generated by an analog sampler [3]., and By deining A Φ F B, (3.4) can be re-written as, 1 j j j 29

34 x A r w (3.5) j j j j Our goal is to iner the binary occupancy state o each channel, deined as a spectrum state vector d {0,1} N, d [ i] 1 when s [] i is nonzero; otherwise, d [ i] 0. Thereore, the goal is to ind the support o the spectrum vector s, when multiple measurements x j j 1 J are available. 3.2 METHODOLOGY LOW-RANK MATRIX COMPLETION BASED SPECTRUM SENSING A low-rank matrix completion based spectrum sensing approach was proposed in [1]. This work was motivated by the observation that only a small percentage o the channels are occupied by the PUs. As a result, i one deines the spectrum matrix as, R r1, r2,..., r J (3.6) Then there will be only a small number o nonzero rows in R, making it a low-rank matrix. First, let us assume that all the measurements j j 1 J x are stacked as a single ( JK) 1 vector T T T J xt [ x1,, x J], and all the measurement noise vectors j j 1 w are stacked as a single ( JK) 1 T T T vector wt [ w1,, w J]. Next, the spectrum matrix R is vectorized column-wise, namely, T T r vec( R ) [ r,, r ] (3.7) T 1 J Further let us deine ~ Α diag{ Α, Α,..., Α }, which is a block diagonal matrix with the 1 2 J diagonal blocks consist o {Α } J j j1. With these notations and considering the low-rank property 30

35 o R, R can be estimated based on all the measurements by solving the ollowing matrix rank minimization problem: ~ t vec min Rank 2 R 2 R x Α R (3.8) The second term in (3.8) penalizes the model itting error, and is the Lagrangian parameter which provides the relative emphases on the low-rank property o R and the tolerance on measurement model errors. Note that ( 3.8) is an intractable optimization problem since the combinatorial nature o the rank o a matrix. Thereore, rank unction can be replaced by its convex surrogate [4], the nuclear norm unction, denoted as. *. The nuclear norm o a matrix is the sum o all the singular values o the matrix. As a result, the optimization problem in (8) becomes * ~ min R x Αvec R (3.9) R t 2 2 In [1], it was shown that a spectrum sensing approach based on the nuclear norm minimization provides very good detection perormance SPECTRUM SENSING BASED ON MIXED-NORM MINIMIZATION Taking a closer look at (3.6), one can ind that R has a small number o non-zero rows, implying that it is not only low-rank but also sparse, which means it has a small number o nonzero elements. More particularly, the columns in R, namely 1 2 r, r,..., J r, share the same support, and are jointly sparse. This motivates us to explore spectrum sensing algorithm based on matrix mixed-norm minimization. 31

36 We are particularly interested in l2 / l 1 norm o a matrix, deined as, A 2 A 2,1 i, j (3.10) i j Which is the sum o the l 2 norms o the rows o matrix A. Minimizing the l 2 / l 1 norm o a matrix will promote the joint-sparsity among its columns. Here, replacing the nuclear norm in (3.9) with the l 2 / l 1 norm, we have the ollowing convex optimization problem, 2,1 ~ min R x Αvec R (3.11) R t SPECTRUM SENSING DECISION MAKING Once the estimated spectrum matrix R ˆ is ound by solving (3.9) or (3.11), the usion center can make a decision on whether or not a particular channel has been occupied by a PU. More speciically, the usion center irst calculates the energy in the i th channel, averaged over J CRs, which is then compared to a threshold to make a decision: 1 ^ J 2 2 d [ i] ˆ j[ i] J r (3.12) j1 3.4 NUMERICAL RESULTS FOR SIGNALS IN SINGLE TIME FRAME In the simulations, we choose the ollowing parameters or the cooperative spectrum sensing network: N = 20, I = 2, J = 20, M = 4, K=5, 1. So as illustrated in Figure. 1, on the average each channel is monitored by 4 CRs. Both the nuclear norm minimization problem as described in (3.9) and the l 2 / l 1 mixed norm minimization problem deined in (3.11) are solved by the CVX 32

37 package [5], with the Lagrangian coeicient being set as 5 in both optimization problems, respectively. The signal-to-noise ratio (SNR) is deined to be the total signal power over the entire spectrum, normalized by the power o the white noise. Given the true channel state vector d, the probabilities o detection and alse alarm are deined as, T ( ˆ ) Pd E d d d T 1d (3.13) P T ( 1d ) ( ˆ d d ) E T N 1d (3.14) Respectively, where 1 denotes an all-one vector. The ROC curves or the two approaches based on nuclear norm minimization and l 2 / l 1 mixed norm minimization are obtained rom 1000 Monte-Carl trials and provided in Figure 2 or dierent SNR values. It is clear that as SNR increases, the perormance or detecting the PUs is improved. Further, the approach based on matrix mixed norm minimization provides a better detection perormance in higher SNR, since it takes advantage o the sparse property o the spectrum matrix, instead o merely its low-rank property. However, as SNR decrease we observe that increasing the noise level has more eect on sparsity o a matrix than its low rank property. As we can see rom the result, by reducing SNR rom 10 db to 0 db, the matrix rank minimization approach has better detection perormance in lower P values, but as P increases the mixed norm approach shows a better result. At lower SNR such as -5 db, both approaches have approximately the same detection perormance. 33

38 Figure 2. ROC curves or spectrum sensing approaches based on nuclear norm minimization and mixed l 2 / l 1 norm minimization. 34

39 3.5 Spectrum Sensing over Multiple Time Frames So ar, we have assumed that the occupied channels are sparse, now we want to add the time dimension to our assumptions and detect the PUs using signals over multiple time-rames. Let us assume that channel occupancy remains the same at each time rame, however the energy o the occupied sub-channels will change over the time. We start with the ollowing assumptions to model our new system. Fading in the channel, is not changing over time over requency it is changing independently the changes over the space are not our concerns With these assumptions, over time the same number o channels are occupied but their energy will change. We will add time index to our model meaning that we study the channel over multiple time rames. 3.6 SYSTEM MODEL Let us add a time index k to the unknown spectrum o the wideband signal transmitted by the PU at time k, namely Sk. Deine the ading coeicient matrix at time k and sensor j as Hjk, then ( 3.1) becomes, rjk = Hjk Sk (4.1) Since ading is not changing over time we can rewrite the equation as ollows, 35

40 r jk H S (4.2) j k At sensor j, using time-invariant channel selection matrix B and random sensing matrix j j, then we have the compressed sample vector at the j th CR as, X F B r W 1 jk j j jk jk A r W j jk jk (4.3) Where W jk is an additive noise at time k and CR j. 3.7 METHODOLOGY: MIXED NORM MINIMIZATION BASED SOLUTION Now i we deine spectrum matrix at each time rame as ollows, R [r,r,,r ] (4.4) k 1k 2k jk Let us assume that we have z number o time rames, let us stack all the spectrum matrices { R } in one large matrix RF. Z k k 1 RF [R,R,,R ] (4.5) 1 2 Z Let us assume that all the measurements {x } J jk j 1 are stacked as a single ( JK) 1 vector X [X,X,,X ] and all the measurement noise vectors {W } J jk j 1 are stacked T T T T tk 1k 2k Jk 36

41 T T T T As a single ( JK) 1vector Wtk [ W1 k, W2k,, W Jk ]. Now we wat to vectorize all the stacked vectors o {x } Z tk k 1. to a single vector o ( JKZ ) 1 and the same or stacked vectors o noise {W } Z tk k 1 XT [X,X,,X ] (4.6) T T T T t1 t2 tz WT [ W, W,,W ] (4.7) T T T T t1 t2 tz Next, we want to vectorize the spectrum matrix, we start by vectorizing { R } Z to a k k 1 ( NJZ ) 1 T T T T vector rk Vec(R k ) [ r1 k, r2k,, rjk ] then we stack all {r } Z k k 1 to a single vector, Vec(RF) [ r, r,, r ] (4.8) T T T T 1 2 Z With these notations and considering its joint-sparse property, RF can be estimated based on all the measurements by solving the ollowing matrix mixed norm minimization problem, ~ 2 RF XT Vec(RF) RF 2,1 2 (4.9) min Where ~ is the diagonal matrix o {{ } J 1 } Z Ajk j k 1. However, we know that A jk does not change over time. Thereore, ~ will be as ollows, ~ Α diag{ Α, Α,..., Α, Α, Α,..., Α,, Α, Α,..., Α } (4.10) 1 2 J 1 2 J 1 2 J k1 k2 kz 37

42 3.8 Numerical Results or Signals over Multiple Time Frames In the simulations, with the same parameters or the cooperative spectrum sensing network as beore, the ROC curves or the approach based on l 2 / l 1 mixed norm minimization are obtained rom 1000 Monte-Carl trials and provided in Figure 3 or dierent number o time rames. It is shown that as number o time rames increases rom one to two, the perormance or detecting the PUs is improved. Figure 3. ROC curves or spectrum sensing based on mixed l 2 / l 1 norm minimization approach. 38

43 CHAPTER 5 Conclusion In this thesis, the problem o wideband spectrum sensing in CR networks using sub- Nyquist sampling and sparse signal processing techniques was investigated. To mitigate multipath ading, we assumed that a group o spatially dispersed SUs collaborate or wideband spectrum sensing, to determine whether or not a channel is occupied by PUs. Due to the underutilization o the spectrum by the PUs, the spectrum matrix has only a small number o non-zero rows. In some existing state-o-the-art approaches, the spectrum sensing problem was solved using the low-rank matrix completion technique involving matrix nuclear norm minimization. Motivated by the observation that the spectrum matrix is not only low-rank, but also sparse, we proposed a spectrum sensing approach based on minimizing the l 2 / l 1 mixed-norm o the spectrum matrix to promote joint sparsity among the spectrum matrix s columns, instead o low-rank matrix completion. Experiment results based simulation showed that the proposed new approach outperorms the lowrank matrix completion based approach, through the comparison o the ROC curves. In practice channels are steady over time. Thereore, by adding time index, we proposed a spectrum sensing approach based on mixed l 2 / l 1 norm minimization over multiple time rames. In our model, we assumed that channel occupancy remains the same at each time rame but the energy o the occupied sub-channels will change over the time. We showed that increasing the number o time rames rom one to two will improve the detection perormance. However, our observation showed that increasing the number o time rames rom two to three will have less eicient detection. This issue guides us to our next step o the study and in our uture works we plan to research and explain this phenomenon. 39