Compressive Wideband Spectrum SensingBased on Random Matrix Theory

CAO Kaitian（曹開(kāi)田），DAI Linyan（戴林燕），HANG Yiling（杭燚靈），ZHANG Lei（張 蕾），GU Kaidong（顧凱冬）

1 Key Lab of Broadband Wireless Communication and Sensor Network Technology，Ministry of Education，Nanjing University of Posts and Telecommunications，Nanjing 210023，China

2 College of Overseas Education，Nanjing University of Posts and Telecommunications，Nanjing 210023，China

Introduction

Cognitive radio（CR）has been proposed as a promising and effective technology to solve the conflict between wireless spectrum scarcity and underutilization，which can greatly improve the spectrum usage efficiency［1］by allowing secondary users（SUs）to share the licensed spectrum with primary users（PUs）.To access the licensed spectrum，SUs firstly need to sense a wide frequency range to find the available spectrum holes and avoid interfering with the operations of PUs.Therefore，spectrum sensing is a fundamental technique and plays an essential role in a cognitive radio network（CRN）.

To sense a wideband spectrum，the traditional spectrum sensing schemes require very high-rate analog-to-digital converters（ADCs）according to Nyquist-Shannon sampling criterion.However，in practice，the requirement could not be met because such a high-speed sampling rate moves inevitably toward a physical barrier［2］.Fortunately，compressive sampling （CS）theory has emerged as a framework that can significantly reduce the front-end data acquisition burden.Thus，in recent years，the spectrum sensing techniques based on CS have received considerable attention.However，most of the existing spectrum sensing schemes based on CS firstly need to recover the signal before spectrum sensing， which leads to the terribly disadvantageous influence on the computational complexity and spectrum sensing time.In fact，signal recovery is not necessary in signal detection because we are just interested in whether the spectrum is occupied by PU or not，not PU signals itself.Therefore，spectrum sensing approaches directly without requiring signal reconstruction［2-4］have been discussed.However， these approaches need priori knowledge of the noise variance，channel gain and PU signal.

Eigenvalue-based spectrum sensing algorithms［5］the are recently proposed as blind detection methods.In these eigenvalue-based sensing algorithms， the maximum eigenvalue of the received signal sample covariance matrix is usually used and supposed to converge to Tracy-Widom distribution［6］according to random matrix theory （RMT）under PU signal is absent，and the decision threshold is derived based on the limiting Tracy-Widom distribution.However，the drawbacks existed in the eigenvalue-based sensing algorithms proposed in Ref.［5］limit their applications in the real world：（1）requiring the infinite number of receivers and signal samples，otherwise，resulting in sensing performance decrease；（2）sampling the signal with at least Nyquist rate is a big challenge for wideband spectrum sensing； （3）requiring a look-up table to approximately calculate the decision threshold since probability density function （PDF） of Tracy-Widom distribution has no closed form expression.Kortun et al.found the closed form PDF［7］to estimate the exact decision threshold with finite number of samples and receivers.However，drawbacks（1）and（2）described above cannot be solved yet.Moreover，the sensing performance of the method in Ref.［7］is quite sensitive to the number of samples.

In order to overcome the aforementioned problems，an eigenvalue-based compressive wideband spectrum sensing（ECWSS）algorithm based on CS is proposed in this paper.In the proposed ECWSS algorithm，the exact decision threshold can be derived by the closed form PDF of the extreme eigenvalues，and the compressive measurements that are far fewer than Nyquist samples are directly utilized to sense the wideband spectrum without reconstructing PU signal.In addition，to mitigate the data acquisition overhead of SUs，a sensor-assisted cooperative spectrum sensing framework is also addressed in which the sensor nodes（SNs）around SU aim at receiving the compressive samples instead of SU and fusion center（FC）fuses the binary decisions made by SUs and makes the final decision.

1 Compressed Sensing Theory

In this section，we firstly give a brief introduction to the CS theory.Given a real-valued signal s′∈RN，which can be viewed as an N ×1column vector in RN.Suppose that the signal s′can be represented in terms of a group orthogonal basis vectors，then the signal s′can be expressed as s′is the N×Northogonal basis vector matrix，θ＝［θ1θ2… θN］Tis an N×1column vector of weighting coefficients.The signal s′isρ-sparse if onlyρcoefficients in vectorθare nonzero or large andρ ?Nis satisfied.

Let y′＝Φs′＝ΦΨθbe the M ×1measurement vector（M?N）whereΦis the M ×N measurement matrix.CS theory demonstrates that as long as the signal s′isρ-sparse in Ψ domain and all these above matrices satisfy restricted isometry property （RIP），the original signal s′can be recovered by M compressive measurements y′ with overwhelming probability［8］.A related condition，referred to as incoherence，requires that the rows ofΦcannot sparely represent the columns ofΨ，and vice versa［9］.

However，signal recovery is not necessary in signal detection applications.This paper aims to detect the PU signal directly with compressive samples far fewer than Nyquist samples without reconstructing the PU signal.Baraniuk［9］indicates that both RIP and incoherence can be satisfied with high probability by selectingΦ as a random matrix.Therefore，it is crucial to construct the measurement matrixΦfor the signal recovery.Without loss of generality，the random matrixΦcan be constructed as follows：given M and N，let the entriesφi，jof Φ be independent and identically distributed（i.i.d）random variables with EIn addition，we require that the distribution ofφi，jis a sub-Gaussian distribution［2］.

2 Sensor-Assisted Compressive Wideband Spectrum Sensing Scenario and System Model

One of the reasons why the traditional spectrum sensing techniques cannot be directly used for performing wideband spectrum sensing is because they make a single binary decision for the whole spectrum and thus cannot identify individual spectrum holes that lie within the wideband spectrum.Meanwhile，in practice，PUs only occupy apart of spectral range that are the sub-channels or sub-bands rather than the whole wide frequency range at any time.Therefore，for convenience，the wideband spectrum ［0，fL］of the input wideband signal s（t）can be supposed to be divided into L non-overlapping sub-bands with their frequency boundaries located at 0＜f1＜f2＜… ＜fL，where fLis the maximum frequency in s（t）.

In the sensor-assisted cooperative spectrum sensing paradigm shown in Fig.1，some SNs are closely placed around the SU to periodically sample the signal within a subband of interest at a sampling rate far lower than Nyquist frequency，and forward their samples to the nearby SU over a common channel，which can significantly alleviate the communication and data acquisition overhead of SU.SUs just focus on making hard decisions based on these samples，and FC fuses the hard decisions and decides whether the spectrum band is occupied by PU.Due to far fewer compressive samples than Nyquist samples，we can save energy and prolong the lifetime of the sensor-assisted CRN.

Fig.1 Sensor-assisted cooperative spectrum sensing scenario

We suppose that there are K SNs around each SU and M consecutive compressive measurements are sampled by each SN，then for any sub-band signal of interest，the compressive measurements observed by SNk（k ＝1，2，…，K）under two hypotheses H0and H1are given as follows，

where s is the PU signal，yk＝（yk（1） yk（2） … yk（M））Tis the M ×1 compressive measurement vector at SNk，hkdenotes the channel fading coefficient from PU to SNk， xk＝（xk（1） xk（2） … xk（N））Tis the received PU signal vector by SNk，and wk＝（wk（1） wk（2） … wk（N））T～N（0，σ2IN）denotes the i.i.d Gaussian noise.Hypotheses H0and H1represent absence and presence of PU，respectively.Suppose that the signal xkis independent of the noise wk，therefore，the compressive measurements yk（m）（m ＝1，2，…，M）are the sub-Gaussian variables［2］.Φis the known M×Nrandom measurement matrix，and for any xk∈ RN， Φ satisfies the following expression with probability at least 1-2e-cMδ2［2］，

where 0＜δ ＜1and c＞0is a small constant.

3 ECWSS Scheme and Performance Analysis

Based on Eq.（1），we can obtain the CS matrix at each SU as follows

For all K SNs，Eq.（1）can be generally rewritten as

where x ＝［x1x2… xK］and w ＝［w1w2… wK］are the N×K matrices.Note that the received PU signal xkis independent of the Gaussian noise wkandthus we can deduce the compressive sample covariance matrix under the two hypotheses as follows

In this paper，the proposed ECWSS decision rule based on the ratio of maximum to minimum eigenvalue（MME）can be expressed as follows

where Tis the test statistic andγis the decision threshold.The probabilities of detection Pdand false alarm Pfcan be defined as

where f1and f0denote the PDFs of the test statistic Tunder H1and H0，respectively.

In practice，since we do not know whether the PU signal exists or not，whereas the noise definitely exists.Therefore，it is reasonable to estimate the decision threshold in terms of the predefined Pf.In order to compute Pfgiven in Eq.（8），we should know the PDF of the test statistic T under H0.The authors in Ref.［10］gives the following theorem.

Theorem Let an M×Kcomplex Gaussian random matrix A be distributed as A～CN（0，σ2IM?IK）.Then the complex central Wishart matrix and its distribution are denoted by W＝AHA ～CWK（K，σ2IK）.The condition number of a Wishart matrix Wis defined as cond（W）and the density of xis given by

where fλmax，xis the joint density ofλmaxand x.fλmax，xis still complex，as seen in Refs.［7］and［10］.Edelman derives the following proposition in his Ph.D.dissertation［11］.

Proposition If the number of cooperative receivers（K）and the number of samples（M）are approximately equal（i.e.，K＝M）and both K and M are large，then the density of Tunder hypothesis H0can be approximated as

where f0（T）denotes the probability density function of T under H0.Therefore，the probability of false alarm can be derived as

Thus，from Eq.（11）we can obtain

Thus，we can calculate the probability of detection Pdbased on Eqs.（7）and（12）at each SU.FC combines the binary decisions transmitted by SUs with“OR”rule and makes the final hard decision.

In this paper，we also consider the ECWSS scheme using the generalized orthogonal matching pursuit （gOMP）［12］reconstruction algorithm and MME scheme with Nyquist samples in Ref.［5］as the benchmark for comparison.For our propose ECWSS scheme，the calculation ofneeds K×M multiplications and additions，and computation of the maximum （or minimum） eigenvalue needs O （K3）multiplications and additions.Therefore， the total computational complexity（multiplications and additions）of ECWSS is K×M ＋O（K3）.For MME scheme，its total computational complexity is K×N＋O（K3）［5］.Compared with the proposed ECWSS scheme，the ECWSS with gOMP recovery algorithm has additional computational complexity O（ρMN）［12］.Therefore，the proposed ECWSS in this paper has a considerably computational advantage over the other two methods.

4 Simulation Results

In this section，randomly generated signals are used and simulations are written in Matlab.Simulation results are averaged over 5 000Monte-Carlo realizations to demonstrate the detection performance of the proposed ECWSS without reconstruction compared with the MME scheme based on Nyquist samples in Ref.［5］and the ECWSS with gOMP algorithm.We define signal to noise ratio，N＝100，σ2＝1and the number of SUs is 20.To test the robustness of all the three schemes to the noise uncertainty，we let the estimated noise variance beand define the noise uncertainty factor as（dB）.

Figure 2illustrates the receiver operating characteristic（ROC）curves of the three methods underη＝1 dB.The detection performance of both the two ECWSS schemes rises dramatically as the compression ratioincreases.Furthermore，we have also observed that MME scheme detection performance remains constant regardless of M because MME adopts Nyquist samples not the compressive measurements at all.The figure reveals that the proposed ECWSS method approximates the MME detection performance when noise uncertainty exists（η＝1dB）.

Fig.2 ROC curves（SNR＝-10dB，K＝40，η＝1dB）

The probability of detection versus SNR is shown in Fig.3.The detection performance of all the three schemes greatly ascends with the increase of SNR.The larger Kis，the better the detection performance of the three schemes is.This is because more number of cooperative SNs can lead to the better sensing performance gain.The simulation results illustrate that the detection performance of the proposed ECWSS without signal reconstruction approximates that of MME algorithm with almost no performance degrade；meanwhile，the complexity of ECWSS algorithm is much lower than those of the other two schemes.Therefore，in terms of the computational complexity，the proposed ECWSS scheme is actually the optimal algorithm among all the three schemes.

Fig.3 Pdvs.SNR

5 Conclusions

This paper presents a wideband spectrum sensing approach based on the extreme eigenvalues of CS covariance matrix of the received signal.The sensor-assisted cooperative sensing paradigm is also addressed in this paper to alleviate the load of the SUs，and makes them aim at operations over the available spectrum rather than data acquisition.In addition，the exact and simple closed form PDF of the ratio of MME is used to set threshold.The theoretical analyses and simulation results reveal that the proposed ECWSS scheme has the lowest computational complexity among the three schemes and approximates the optimal MME algorithm.

［1］Fragkiadakis A G，Tragos E Z，Askoxylakis I G.A Survey on Security Threats and Detection Techniques in Cognitive Radio Networks［J］.IEEE Communications Surveys ＆Tutorials，2013，15（1）：428-445.

［2］Davenport M A，Boufounos P T，Wakin M B，et al.Signal Processing with Compressive Measurements［J］.IEEE Journal of Selected Topics in Signal Processing，2010，4（2）：445-460.

［3］Najafabadi D M，Tadaion A A，Sahaf M R A.Wideband Spectrum Sensing by Compressed Measurements［C］.IEEE Symposium on Computers and Communications，Cappadocia，Turkey，2012：667-671.

［4］Wimalajeewa T，Chen H，Varshney P K.Performance Analysis of Stochastic Signal Detection with Compressive Measurements ［C］.The 44th Asilomar Conference on Signals，Systems and Computers，Monterey，CA，USA，2010：813-817.

［5］Zeng Y H，Liang Y C.Eigenvalue-Based Spectrum Sensing Algorithms for Cognitive Radio ［J］.IEEE Transactions on Communications，2009，57（6）：1784-1793.

［6］Johnstone I M.On the Distribution of the Largest Eigenvalue in Principle Components Analysis［J］.Annals of Statistics，2001，29（2）：295-327.

［7］Kortun A，Ratnarajah T，Sellathurai M，et al.On the Performance of Eigenvalue-Based Cooperative Spectrum Sensing for Cognitive Radio ［J］.IEEE Journal of Selected Topics in Signal Processing，2011，5（1）：49-55.

［8］Candes E J，Wakin M B.An Introduction to Compressive Sampling［J］.IEEE Signal Processing Magazine，2008，25（2）：21-30.

［9］Baraniuk R G.Compressive Sensing ［J］.IEEE Signal Processing Magazine，2007，24（4）：118-121.

［10］Ratnarajah T，Vaillancourt R，Alvo M.Eigenvalues and Condition Numbers of Complex Random Matrices［J］.Society for Industrial and Applied Mathematics，2005，26（2）：441-456.

［11］Edelman A.Eigenvalues and Condition Numbers of Random Matrices ［D］.Cambridge： Massachusetts Institute of Technology，1989.

［12］Wang J，Kwon S，Shim B.Generalized Orthogonal Matching Pursuit［J］.IEEE Transactions on Signal Processing，2012，60（12）：6202-6216.