Detection based on eigenvalues of received signal covariance matrix is currently one of the most effective solution for spectrum sensing problem in cognitive radios. However, the results of these schemes often depend on asymptotic assumptions since the distribution of ratio of extreme eigenvalues is exceptionally mathematically complex to compute in practice. In this paper, a new approach to determine the distribution of ratio of the largest and the smallest eigenvalues is introduced to calculate the decision threshold and sense the spectrum. In this context, we derive a simple and analytically tractable expression for the distribution of the ratio of the largest and the smallest eigenvalues based on upper bound on the joint probability density function (PDF) of the largest and the smallest eigenvalues of the received covariance matrix. The performance analysis of proposed approach is compared with the empirical results. The decision threshold as a function of a given probability of false alarm is calculated to illustrate the effectiveness of the proposed approach

Alouini, Mohamed-Slim

Imran, Muhammad Ali

Shakir, Muhammad Zeeshan

Tang, Wuchen

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      Shakir, M. Z., Tang, W., Imran, M. A., and Alouini, M.-S. (2011) Collaborative Spectrum Sensing Based on Upper Bound on Joint PDF of Exreme Eigenvalues. In: 2011 19th European Signal Processing Conference, Barcelona, Spain, 29 Aug - 02 Sep 2011, pp. 1214-1218.     There may be differences between this version and the published version. You are advised to consult the publisher’s version if you wish to cite from it.    http://eprints.gla.ac.uk/136982/            Deposited on: 15 February 2017                       Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk COLLABORATIVE SPECTRUM SENSING BASED ON UPPER BOUND ONJOINT PDF OF EXTREME EIGENVALUESMuhammad Zeeshan Shakir#, Wuchen Tang†, Muhammad Ali Imran† and Mohamed-Slim Alouini##Division of Physical Sciences and Engineering, King Abdullah University of Science and TechnologyThuwal 23955-6900, Makkah Province, Saudi Arabiaphone: +(966) 2 808 0288, fax: +(966) 2 802 0143, Email: {muhammad.shakir, slim.alouini}@kaust.edu.sa†Centre for Communication Systems Research (CCSR), University of SurreyGuildford GU1 7XH, Surrey, United Kingdomphone: +(44) 148 368 6003, fax: +(44) 148 368 6011, Email: {w.tang, m.imran}@surrey.ac.ukABSTRACTDetection based on eigenvalues of received signal covari-ance matrix is currently one of the most effective solu-tion for spectrum sensing problem in cognitive radios.However, the results of these schemes always dependon asymptotic assumptions since the distribution of ra-tio of extreme eigenvalues is exceptionally mathemati-cally complex to compute in practice. In this paper, anew approach to determine the distribution of ratio ofthe largest and the smallest eigenvalues is introducedto calculate the decision threshold and sense the spec-trum. In this context, we derive a simple and analyti-cally tractable expression for the distribution of the ra-tio of the largest and the smallest eigenvalues based onupper bound on the joint probability density function(PDF) of the largest and the smallest eigenvalues of thereceived covariance matrix. The performance analysisof proposed approach is compared with the empiricalresults. The decision threshold as a function of a givenprobability of false alarm is calculated to illustrate theeffectiveness of the proposed approach.1. INTRODUCTIONIn cognitive radio networks, the spectrum resources areavailable for a secondary user (SU) only when they arenot occupied by the primary user (PU), which aimsat avoidance of intolerable interference. Thus, the SUshould be able to detect the presence of the PU. In thiscontext, high probability of accurate detection in spec-trum sensing becomes extremely important in the im-plementation of cognitive radio networks.The detection algorithms have their mathematicalfoundations in the random matrix theory. One of thesealgorithms is the eigenvalue ratio (ER) detector [1],which uses the ratio of the largest eigenvalue to thesmallest eigenvalue as a test statistic and then givesthe decision threshold. ER based detection has beenconsidered as an effective and efficient solution to thisproblem [1–5]. The key advantage of this detectionmethod lies in the fact that there is no need to obtain aprior information of the PU. The main idea is to decidewhether the spectrum resources are being used by a PUor not. In other words, these eigenvalues behave differ-ently, which depends on the presence or absence of PU.Due to these properties, the eigenvalue ratio based de-tection outperforms the energy detector (ED), which isa traditional approach for spectrum sensing, especiallywhen the noise uncertainty increases larger [5, 6].The decision threshold is precalculated, which is de-termined by the distribution of the test statistics TN .However, the exact distribution of the test statisticsof the ER detector is generally a mathematically in-tractable function. Some semi-analytical approaches forthe distribution are presented in [5, 7] where the com-putational complexity becomes intractable with the in-crease in number of SUs K and received samples N .The exact expression for this ratio has also been derivedin [3], however the distribution can only be evaluated nu-merically. The complexity of the exact expression maybecome computationally cumbersome with the increasein K and N . As a consequence, a Gaussian approxima-tion is introduced in [8] to derive the analytical distri-bution of TN such that the decision threshold γ can becalculated. The derived decision threshold is based onthe asymptotic Gaussianity of the extreme eigenvaluedistribution obtained by fitting asymptotic moments ofTracy-Widom distribution of order 2 [9]. Despite thesimplicity of the decision threshold, the proposed ap-proximation is only valid under the assumption thatthe distribution of the largest and the smallest eigenval-ues converges to the Trace-Widom distribution of ordertwo [9]. It has been also shown that such convergenceonly occurs when K →∞, N →∞ and KN→ c ∈ (0, 1).However, the resulting cumulative distribution function(CDF) of the Trace-Widom random variable involve ma-trix determinants with function entries that are diffi-cult to evaluate when K and N are larger. Therefore,the most of the previous approaches not only based onasymptotic assumptions but also leads to mathemati-cally complicated solutions. In this paper, our mainconcern is to develop a mathematically tractable solu-tion to solve the problem of sensing the presence or ab-sence of the PU with the collaboration of finite numberof SUs. However, we consider a case when few SUs arecollaborating to sense the presence PU.The contributions of this paper are described as fol-lows: we introduce a mathematically tractable approachto derive the distribution of ratio of the largest and thesmallest eigenvalues. The derived distribution is basedon the upper bound on the joint probability densityfunction (PDF) of the largest and the smallest eigenval-ues. The proposed approach gives useful results whichare applicable to the scenarios when few SUs are collab-orating to sense the presence or absence of the PU in agiven sensing time.The rest of this paper is organized as follows. Section2 defines the system model to explain detection problemof PU; section 3 introduces the upper bound on the jointPDF of the largest and the smallest eigenvalues. Nextin section 4, we derive the distribution of the ratio ofextreme eigenvalues based on the upper bound. Thisis followed by the calculation of decision threshold fora given probability of false alarm. Later, we presentnumerical results to validate the simulation and analyt-ical results and illustrate the usefulness of the proposedupper bound approach for realistic cognitive scenarios.Finally in section 5, we conclude the results.2. EIGENVALUE RATIO DETECTIONPROBLEMConsider there are K collaborating SUs such that eachuser collects N samples during the sensing time to de-tect a PU. The SUs may be considered as a K receiveantennas in one secondary terminal or K secondary ter-minals each with single antenna, or any combinationsof these. The collected samples from K collaboratingSUs will be forwarded to a fusion center for combinedprocessing [9].The aim of the SU cognitive phase is to construct andanalyze tests associated with the following hypothesistesting problem:H0 : y(n) = w(n) (1)H1 : y(n) = h(n) s(n) +w(n) (2)where y(n) = [y1(n), · · · , yK(n)]Tis the K×1 observedcomplex time series, w(n) represents a K × 1 circularlysymmetric complex Gaussian (CSCG) noise with zeromean and variance σ2w. In (2), the vector h(n) ∈ CK×1typically represents the propagation channel betweenthe PU and K collaborating SUs and the signal s(n) de-notes a standard scalar i.i.d circular complex Gaussianprocess w.r.t samples n = 1, 2, · · · , N and stands for thesource signal to be detected with E[s2(n)]= σ2s 6= 0.We stack the observed data into K ×N data matrix Ywhich may be expressed asY =y1(1) y1(2) · · · y1(N)y2(1) y2(2) · · · y2(N)....... . ....yK(1) yK(2) · · · yK(N) (3)As the sample number N → ∞, the sample covariancematrix, R(N) = 1NY Y ∗, converges to E [yy∗], wherey is the column vector of N samples collected by asingle receiver. From the eigenvalues of R(N), it ispossible to infer the absence or presence of the pri-mary signal. Denote the normalized covariance matrixas RN =Nσ2wR(N) = 1σ2wY Y ∗. Under the hypothesisH0, RN is a complex white Wishart matrix subject toCWK(N,IK), where IK is a K × K identity matrix,while it turns out to be the class of spiked populationmodels under the hypothesis H1 [10].Suppose the ordered eigenvalues of R(N) and RNare 0 < lK < lK−1 < · · · < l1 and 0 < λK < λK−1 <· · · < λ1 respectively, with the relationshipλk =σ2wNlk (4)where k = 1, 2, · · · ,K.Let us define the test statistics for the for ER baseddetection asTN =λ1λK(5)and denoting γ as the decision threshold employed bythe detector such thatTNH1≷H0γ (6)to decide if the target spectrum resource is occupied ornot.3. UPPER BOUND ON JOINT PDF OFEXTREME EIGENVALUESIn this section, we present the upper bound on the jointPDF of the largest and the smallest eigenvalues. Ifgλ1,λK (x, y) denotes the joint PDF of the largest and thesmallest eigenvalues such that gλ1,λK (x, y) satisfies [11]:gλ1,λK (x, y) ≤ fλ1,λK (x, y)= CK,Ne− 12(x+y) x12(N+K−3) y12(N−K−1)(7)where CK,N =14Γ(K−1)Γ(N−K+1) with Γ (·) is theGamma function [12]. The associated marginal PDFof the largest and the smallest eigenvalues can be re-spectively calculated asfλ1(x) =∫ ∞0fλ1,λK (x, y)dy= CK,N 212(N−K+1) Γ(12(N −K + 1))× e−x2 x12(N+K−3) (8)fλK (y) =∫ ∞0fλ1,λK (x, y)dx= CK,N 212(N+K−1) Γ(12(N +K − 1))× e−y2 y12(N−K−1) (9)Also, the associated marginal CDFs of the largestand the smallest eigenvalues can be respectively calcu-lated asFλ1(x) =∫ x0fλ1(x′)dx′= CK,N 2N Γ(12(N −K + 1))(Γ(12(N +K − 1))− Γ(12(N +K − 1),x2))(10)0 20 40 60 80 100 120 140 160 180 200 22000.10.20.30.40.50.60.70.80.91Largest Eigenvalue (λ1)CDF(Fλ1(x))(a)SimulationUpperbound approach0 20 40 60 80 100 120 140 160 180 200 22000.10.20.30.40.50.60.70.80.91Smallest Eigenvalue (λK)CDF(FλK(y))(b)SimulationUpperbound approachFig. 1: CDFs of the largest and the smallest eigenvalues for K = 2and N = 150.FλK (y) =∫ y0fλK (y′)dy′= CK,N 2N Γ(12(N +K − 1))(Γ(12(N −K + 1))− Γ(12(N −K + 1),y2))(11)where Γ (, ) is the incomplete Gamma function [12].Fig. 1(a) and Fig. 1(b) shows the CDFs of the largesteigenvalue and the smallest eigenvalue using (10) and(11) respectively for K = 2 and N = 150. The an-alytical CDFs are compared with the empirical CDFs(compare the solid curve with the dashed curve). It canbe seen that the marginal CDFs of the largest and thesmallest eigenvalues are behaving same as the empiri-cal CDFs. However, the known difference between thetwo CDFs is due to the obvious reason that the ana-lytical CDFs are derived from the upper bound on the0 50 100 150 200 250 30000.10.20.30.40.50.60.70.80.91Largest Eigenvalue (λ1)CDF(Fλ1(x))(a)SimulationUpperbound approach0 50 100 150 200 250 30000.10.20.30.40.50.60.70.80.91Smallest Eigenvalue (λK)CDF(FλK(y))(b)SimulationUpperbound approachFig. 2: CDFs of the largest and the smallest eigenvalues for K = 4and N = 150.joint density or distribution. It is to note that the re-sults are still useful to calculate the decision thresholdnumerically as a function of probability of false alarm.Similarly, in Fig. 2(a) and Fig. 2(b) the CDFs of thelargest and the smallest are calculated for K = 4 andN = 150. It is observed that the intrinsic differencebetween the empirical and upper bound CDFs (due toupper bound on joint density or distribution) increaseswith the increase in number of collaborating SUs. How-ever, a tighter upper bound results is expected to bereported at a later date which is calculated using [13].Despite the fact that our results are based on the upperbound, the decision threshold can be easily calculatednumerically to decide the occupancy of the spectrum.4. DISTRIBUTION OF EIGENVALUERATIO (TN)In this section, we introduce a novel application of up-per bound on the joint PDF of the largest and smallestTable 1: Numerical values of (15) for selected values of K and NF = FZ(z) F = 0.1 F = 0.2 F = 0.3 F = 0.4 F = 0.5 F = 0.6 F = 0.7 F = 0.8 F = 0.9K = 2, N = 150 0.8200 0.8800 0.9300 0.9700 1.0100 1.0600 1.1000 1.1600 1.2500K = 4, N = 100 0.8200 0.9000 0.9600 1.0100 1.0600 1.1200 1.1800 1.2600 1.3700K = 4, N = 150 0.8400 0.9100 0.9600 1.0000 1.0400 1.0900 1.1300 1.1900 1.2800K = 8, N = 100 0.8900 0.9700 1.0400 1.0900 1.1500 1.2100 1.2800 1.3600 1.49000 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.500.511.522.53False Alarm Probability (Pfa)DecisionThreshold(γ)SimulationUpperbound approachFig. 3: Decision threshold as a function of probability of falsealarm for K = 2 and N = 150eigenvalues to calculate the decision threshold as a func-tion of probability of false alarm.Using (7), we first derive PDF of the ratio of thelargest and the smallest eigenvalues TN and then deriveCDF of the ratio. The CDF of the ratio between λ1 andλK i.e., z =xycan be expressed asFZ(z) = P{xy≤ z} = P{x ≤ yz, y > 0} (12)=∫ ∞0∫ yz−∞fλ1,λK (x, y)dxdy (13)Using (13), the PDF of the ratio can be calculated asfZ(z) =ddzFZ(z) =∫ ∞0yfλ1,λK (yz, y)dy=∫ ∞0yfλ1,λK (yz, y)dy= CK,N∫ ∞0ye−y2(z+1)z12(N+K−3)yN−2 dy= CK,N z12(N+K−3)∫ ∞0e−y2(z+1)yN−1 dyFinally, we arrived atfz(z) = CK,NΓ(N) 2N z12(N+K−3)(1 + z)−N (14)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.500.511.522.53False Alarm Probability (Pfa)DecisionThreshold(γ)SimulationUpperbound aproachFig. 4: Decision threshold as a function of probability of falsealarm for K = 4 and N = 150The CDF of the ratio TN can be calculated asFZ(z) = CK,N Γ(N) 2N∫ z0z′12(N+K−3)(1 + z′)−Ndz′= CK,N 2N(Γ(12(N −K + 1))× Γ(12(N +K − 1))+2z12(K−N−1)Γ(N)(K −N − 1)× 2F1(N,12(N −K + 1);12(N −K + 3);−1z))(15)where 2F1(a, b; c; z) is the Hypergeometric Function [12].Using (15), the numerical values of CDFs of ratioof the largest and the smallest eigenvalues for selectedvalues of K and N are available in Table. 1. As desired,the resulting decision threshold γ for a given probabilityof false alarm (Pfa) is calculated numerically by solvingPfa = 1− F (γ). (16)It is to note that in (16), F (γ) is solved numericallyin Matlab using the trapezoidal integration method [14].For a given target Pfa, the corresponding the decisionthreshold for K = 2 and N = 150 is shown in Fig. 3.The analytically calculated decision threshold is com-pared with the simulation based approach (compare the10−410−310−210−110010−410−310−210−1100False Alarm Probability (Pfa)MissedDetectionProbability(Pmd)Energy DetectorER Detector - based on simulationER Detector - based on UpperboundFig. 5: Receiver operating characteristic curves for K = 20 andN = 50.solid curve with the dashed curve). It can be seenthat the results of our proposed approach is accept-able for low probability of false alarm. However, thereis a trade off between the acceptable performance andmathematical tractability with the calculation of deci-sion threshold. Similarly, Fig. 4 shows the calculateddecision threshold as a function of range of probabilityof false alarm for K = 4 and N = 150. It can be seenthat decision threshold based on our approach is use-ful for reasonably small number of collaborating SUs todetect the occupancy of spectrum.Finally, the receiver operating characteristics (ROC)curves are given in Fig. 3 which shows achieved probabil-ity of missed detection under H0 versus the probabilityof false alarm under H1. In this figure, we assume aconstant modulation transmitted signal with K = 20collaborating SUs and N = 50 samples during the givensensing time. The SNR is set to be −10 dB, while thenoise uncertainty is set to 1/2 dB. It is apparent that ERdetector behaves better than energy detector. However,our approach provides a closed lower bound to the sim-ulation result (compare the solid curve with the dashedcurve).5. CONCLUSIONSIn this paper, we used upper bound on joint PDF ofthe largest and the smallest eigenvalues of the receivedcovariance matrix to derive a simple and analyticaltractable expression for distribution of ratio of eigen-values. It has been shown that the upper bound can beexploited effectively to approximate the decision thresh-old for a given probability of false alarm. The proposedapproach is useful specially for reasonably small numberof collaborating SUs. However, a trade off is required tobe defined between acceptable performance and mathe-matical tractability of the detection problem.REFERENCES[1] Y. Zeng and Y. C. Liang, “Eigenvalue based spec-trum sensing algorithms for cognitive radios,” inIEEE Trans. Communs., vol. 57, no. 6, pp. 1784–1793, Jun. 2009.[2] F. Penna, R. Garello and M. A. Spirito, “Coopera-tive spectrum sensing based on the limiting eigen-value ratio distribution in Wishart matrices,” inIEEE Communs. Letters, vol. 13, no. 7, pp. 507–509, Jul. 2009.[3] F. 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Collaborative Spectrum Sensing Based on Upper Bound on Joint PDF of Exreme Eigenvalues

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Collaborative Spectrum Sensing Based on Upper Bound on Joint PDF of Exreme Eigenvalues

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