Spectrum Sensing in Cognitive Radio withSubspace MatchingShujie Hou, Robert Qiu, James P. Browning, Michael WicksCognitive Radio Institute, Department of Electrical and Computer Engineering,Center for Manufacturing Research, Tennessee Technological University,Cookeville, TN 38505, USAEmail:shou42@students.tntech.edu, rqiu@tntech.eduAir Force Research Laboratory, Wright Paterson AFB,OH 45433, USAEmail:James.Browning@wpafb.af.milSensor Systems Division, University of Dayton Research Institute,300 College Park, Dayton, OH 45469Email:Michael.Wicks@udri.udayton.eduAbstractSpectrumsensinghas beenput forwardtomakemore efcient use of scarce radio frequency spectrum. The lead-ing eigenvector of the sample covariance matrix has been appliedtospectrumsensingundertheframeworksofPCAandkernelPCA. Inthispaper, spectrumsensingwithsubspacematchingis proposed. The subspace is comprised of the eigenvectorscorrespondingtodominantnon-zeroeigenvaluesofthesamplecovariancematrix. That is, several eigenvectorsareappliedtospectrumsensingotherthantheonlyuseof leadingone. ThedistancebetweenthesubspacesismeasuredbytheprojectionFrobenius norm. The simulations are done based on the simulatedand captured DTV signals.I. INTRODUCTIONSpectrum sensing is a cornerstone in cognitive radio whichdetectstheavailabilityofradiofrequencybandsforpossibleusebysecondaryuser without interferencetoprimaryuser.Some traditional techniques proposedfor spectrumsensingare energy detection, matched lter detection, cyclostationaryfeature detection, covariance-based detection and feature baseddetection [1][9].Secondaryuser receives a ddimensional columnvectorx. Basedonthereceivedsignal, therearetwohypotheses:the hypothesis H0that is primary signal is absent opposite tothehypothesis H1thatisprimarysignalispresent.Thetwohypotheses H0and H1areH0: x = nH1: x = s +n(1)in which s and n are assumed to be independent. s is primarysignal andniszero-meanwhiteGaussiannoise. Secondaryuserdetectstheprimarysignal betweentwohypotheses H0and H1.Two quantities used most often to measure the performanceof spectrum sensing algorithm are detection probability Pd andfalse alarm probabilityPfwhich are dened asPd= prob(detect H1|x = s +n)Pf= prob(detect H1|x = n)(2)in whichprob represents probability.Theleadingeigenvector(eigenvectorcorrespondingtothelargest eigenvalue) of the sample covariance matrix has beenproved stable for non-white wide-sense stationary (WSS) sig-nal. Spectrum sensing with leading eigenvector of the samplecovariance matrix is proposed and hardware demonstratedsuccessfully [9] under the framework of principal componentanalysis (PCA). Ontheother hand, spectrumsensingwithleading eigenvector has been generalized to the framework ofkernel PCA [10] withkernel trick. However, theprevioustechniques have been based on the leading eigenvector ofthe sample covariance matrix. In this paper, spectrum sensingwith subspace matching is proposed under both the PCA andkernel PCAframeworks. Thesubspaceiscomprisedof theeigenvectors corresponding to dominant non-zero eigenvaluesof the sample covariance matrix.Spectrum sensing with subspace matching proposed in thispaper is the straightforwardextensionof that withleadingeigenvectors. In practice, subspace matching for spectrumsensing can be implemented by two means. First, if thetrainingsamples of theprimaryusers signal areavailable,the subspace of the primary users signal can be learnedpriori which is taken as the template. Then the subspacelearned from the received signal by the secondary user shouldembodylargesimilaritywiththetemplateunder hypothesisH1. Second, the received signal can be divided into twosegments. Sincethesubspacefor theprimaryusers signalis stable, the subspaces of these two segments should embodylarge similarity whenever the primary users signal is presentinthereceivedsignal. Inthispaper, thetrainingsamplesofthe primary users signal are assumed to be available.II. SPECTRUM SENSING WITH LEADING EIGENVECTORUNDER THE FRAMEWORKS OF PCA AND KERNEL PCAInthissection, spectrumsensingwithleadingeigenvectorunder the frameworks of PCA[9] and kernel PCA[10] willbe simply reviewed. The training vectors of the primary signalare assumed to be available priori which ares1, s2, ..., sM.Assumingthereceivedvectorsbythesecondaryuser arex1, x2, ..., xM, spectrumsensing with leading eigenvectorunder the framework of PCA can be summarized as follows,1) The sample covariance matrix of the training set iscomputed byRs=1MM

i=1(si us)(si us)T(3)in which Tdenotes transpose and us is the mean vectorof the training samples.2) The leading eigenvector v1 of Rs is obtained by eigen-decomposition.Rs= VVT(4)in which is a diagonal matrix with ithdiagonalelementi.1 2 d 0 are eigenvalues ofRs. V = (v1, v2, .., vd) in whichviis the eigenvectorofRscorresponding toi.3) Likewise, the leading eigenvector v1for the samplecovariance matrix ofx1, x2, ..., xMis derived too.4) The primary signal is detected if the similarity which ismeasured by inner-product betweenv1and v1is largerthan a pre-dened threshold,v1, v1 > Tpca(5)where denotes inner-product.Inthis paper, theinner-product is usedas ameasureofsimilarity. In [9], the autocorrelation is used instead which is = maxl=0,1,...,dd

k=1v1[k] v1[k + l] > Tpca. (6)The threshold value is determined by the simulation in whichPf= 10%.The leading eigenvector detection mentioned above isbasedonthedataintheoriginal space. In [10], thelead-ingeigenvector detectionbasedon (s1), (s2), ..., (sM),(x1), (x2), ..., (xM) is proposed for spectrumsensingwith the use of kernel function. is the mapping which mapsthe original space datas andx to the feature space data(s)and(x).Assuming (s1), (s2), ..., (sM) is zero mean, the samplecovariance matrix of the training set in the feature space isR(s)=1MM

i=1(si)(si)T. (7)Similarly, the sample covariance matrix R(x)of(x1), (x2), ..., (xM) can also be obtained in which(x1), (x2), ..., (xM) are also assumed to have zero mean.The leading eigenvectors of R(s)and R(x)have beenemployed for detection.A kernel function which just relies on the inner-product offeature space data is dened as [11]k(xi, xj) = (xi), (xj) (8)to implicitly map the original space data xinto a higherdimensional feature space F. With the use of kernel function,the mapping need not know explicitly.The detection algorithm with leading eigenvectors of R(s)andR(x)can be summarized as follows,1) Provided a kernel functionk. The kernel matrixKij=k(si, sj), i, j =1, ..., M(Grammatrix) basedonthetraining samples is obtained. K is positive semi-denite.2) Eigen-decomposing the kernel matrix Kto get theeigenvectors1, 2, ..., Mcorrespondingtoeigenval-ues1 2 M 0.3) AnotherkernelmatrixKij=k(xi, xj)canbederivedbased on the received vectorsx1, x2, ..., xM.4) Likewise, theeigenvectors1, 2, ..., Mcorrespond-ingtoeigenvalues 1 2 M0areobtained by eigen-decomposingK.5) Eigenvectors forR(s)andR(x)can be expressed as(vf1, ..., vfM) = ((s1), (s2), ..., (sM))(1, ..., M),(9)( vf1, ..., vfM) = ((x1), (x2), ..., (xM))(1, ..., M).(10)6) vfiand vfi , i = 1, 2, ..., Mare normalized through i=i/iandi=i/ i.7) The primary signal is detected if the similarity betweenvf1and vf1is larger than a pre-dened threshold_vf1, vf1_ = T1Kt1> Tkpca, (11)in which Tkpca is the derived threshold value and Ktij=k(si, xj).III. SPECTRUM SENSING WITH SUBSPACE MATCHINGUNDER THE FRAMEWORKS OF PCA AND KERNEL PCAIntheprevioussection, onlytheleadingeigenvectors v1and v1, vf1and vf1of the sample covariance matrix are usedfor spectrumsensingintheoriginal andfeaturespaces. Inthissection, spectrumsensingwithleadingeigenvector willbe generalized with the use of several dominant eigenvectors.Under theframeworkof PCA, eigenvectors v1, v2, ..., vpof the sample covariance matrixRsand v1, v2, ..., vpof thesample covariance matrix Rx are used.p is a parameter to bedetermined.The two linear subspaces which consist of vectorsv1, v2, ..., vpand v1, v2, ..., vparerepresentedbymatricesA andB for simplicity,A = (v1, v2, ..., vp), (12)B = ( v1, v2, ..., vp). (13)Similar with leading eigenvector detection, under hypothesisH1, the subspace Bshould embody large similarity withthesubspaceAlearnedfromthetrainingvectorsofprimarysignal. Therefore, under hypothesis H1, the distance betweensubspacesA andB should be small.Thedistancebetweentwosubspacesismeasuredbypro-jection Frobenius norm[12] in this paper,d(A, B) =___AATBBT___2F(14)whereFis the Frobenius normof the matrix. In thispaper, onlythe case that Aand Bhave equal dimensionpisconsidered. Undersuchcondition, projectionFrobeniusnormequals2 sin 22, whereisapdimensional vectorof principal angles betweenA andB and 2isl2norm.Theprimarysignal isclaimedtobedetectedbythesec-ondary user ifd(A, B) =___AATBBT___2F< Tsubpca, (15)whereTsubpcais the threshold value for subspace matchingunder the framework of PCA.Likewise, in the feature space, the eigenvectorsvf1, vf2, ..., vfpfand vf1, vf2, ..., vfpfof R(s)and R(x)are used.The subspaces which consist of vf1, vf2, ..., vfpfand vf1, vf2, ..., vfpfare represented by matricesC andD respec-tively. C andD are real matrices considered here.The distance between Cand Dcan be calculated as follows,d(C, D) =___CCTDDT___2F= trace__CCTDDT_T _CCTDDT__= trace__CCTDDT__CCTDDT__= trace_CCTCCT+DDTDDTCCTDDTDDTCCT_= trace_CCT+DDTCCTDDTDDTCCT_= trace(CTC) + trace(DTD) 2trace(DTCCTD)= 2pf 2trace__CTD_ _CTD_T_(16)whereCTD = (vf1, vf2, ..., vfpf)T( vf1, vf2, ..., vfpf)=_((s1), (s2), ..., (sM))(1, ..., pf)_T((x1), (x2), ..., (xM))(1, ..., pf)= (1, ..., pf)TKt(1, ..., pf).(17)Theprimarysignalisdetectedifd(C, D) +c)de, c 0, (21)wherede is the order of the polynomial.IV. EXPERIMENTSIn this section, the training vectors are derived from discretesampless(n), s(n + 1), ..., s(L + n 1) bys1= (s(n), s(n + 1), ..., s(n + d 1)),s2= (s(n + 1), s(n + 2), ..., s(n + d)),......sM= (s(L + n d), ..., s(L + n 1)),(22)where M = L d+1. Likewise, the received vectorsx1, x2, ..., xMare derived from discrete samplesx(n), x(n +1), ..., x(L + n 1)inthesameway.Intheexperiment,thelength withL = 500 is used.A. Experiments on Simulated SignalThe primary signal is assumed to be the sumof threesinusoidal functions with unit amplitude of each. Polynomialkerneloforder2withc=0isused. Basedon1000MonteCarlosimulations, thedetectionprobabilitywithdimensionsd = 32 and d = 128 are shown in Fig. 1 and Fig. 2,respectively. In Fig. 2, the results are shown with both inner-product andautocorrelationasthemeasureofsimilarityforleading eigenvector detection under the framework of PCA .From Fig. 1 and Fig. 2, it can be seen that the performanceofsubspacematchingundertheframeworkofPCAismuchbetter thanthecorrespondingleadingeigenvector detection.However, theperformanceof subspacematchingunder theframework of kernel PCA is only comparable with the corre-sponding leading eigenvector detection.25 20 15 10 5 00.10.20.30.40.50.60.70.80.91SNR in dBPdDetection Probability with Pf =0.1 Kernel PCA with leading eigenvectorKernel PCA with subspacePCA with leading eigenvectorPCA with subspaceFig. 1. The detection probability withPf= 10% for the simulated signalThe calculated threshold values withPf= 10% are showninFig. 3. Thethresholdvalues arenormalizedbydividing25 20 15 10 5 00.10.20.30.40.50.60.70.80.91SNR in dBPdDetection Probability with Pf =0.1 Kernel PCA with leading eigenvectorKernel PCA with subspacePCA with leading eigenvectorPCA with subspacePCA with leading eigenvector measured by correlationFig. 2. The detection probability withPf= 10% for the simulated signal25 20 15 10 5 00.50.550.60.650.70.750.80.850.90.951SNR in dBNormalized threshold value TkpcaTsubkpcaTpcaTsubpcaFig. 3. Normalized threshold valuestheir corresponding maximal values. All of these methodshavenonoiseuncertaintyproblem. Thethresholdvaluesforsubspace methods are more robust than the correspondingleading eigenvector detection.B. Experiments on Captured DTV SignalDTVsignal [14] capturedinWashingtonD.C. will beemployed to the experiment of spectrumsensing in thissection. Polynomial kernel of order 3 with c = 0 is used. Basedon1000MonteCarlosimulations, thedetectionprobabilitywithd=32isshowninFig. 4withfalsealarmprobability10%. ThenormalizedthresholdvalueswithPf=10%areshown in Fig. 5.From Fig. 4, it can be seen that the performance of subspacematchingundertheframeworkofkernel PCAisbetterthanthe corresponding leading eigenvector detection. However, theperformanceof subspacematchingunder theframeworkofPCA is worse the corresponding leading eigenvector detection.V. CONCLUSIONSpectrum sensing with subspace matching is proposed underthe frameworks of PCA and kernel PCA. The experimental re-sults are compared with the corresponding leading eigenvectordetection. Fromthesimulation, it canbeseenthat subspace25 20 15 10 5 00.10.20.30.40.50.60.70.80.91SNR in dBPdDetection Probability with Pf =0.1 Kernel PCA with leading eigenvectorKernel PCA with subspacePCA with leading eigenvectorPCA with subspaceFig. 4. The detection probability withPf= 10% for DTV signal25 20 15 10 5 00.50.550.60.650.70.750.80.850.90.951SNR in dBNormalized threshold value TkpcaTsubkpcaTpcaTsubpcaFig. 5. Normalized threshold valuesmatching method proposed in this paper is applied to spectrumsensingsuccessfully. Theperformanceofsubspacematchingmethod under the framework of PCAis better than thecorresponding leading eigenvector detection for the simulatedsignal. However, for the DTVsignal, the performance ofsubspace matchingmethodunder the frameworkof kernelPCAis better than the corresponding leading eigenvectordetection.ACKNOWLEDGMENTThis work is funded by National Science Foundation,through two grants (ECCS-0901420 and ECCS-0821658), andOfce of Naval Research, through two grants (N00010-10-1-0810 and N00014-11-1-0006).REFERENCES[1] S. Haykin, D. Thomson, and J. Reed, Spectrum sensing for cognitiveradio, Proceedings of the IEEE, vol. 97, no. 5, pp. 849877, 2009.[2] J. Ma, G. Li, andB. 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