A Unified Framework for GLRT-Based Spectrum Sensing of Signals with Covariance Matrices with Known Eigenvalue Multiplicities

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1 A Unified Framework for GLRT-Based Sectrum Sensing of Signals with Covariance Matrices with Known Eigenvalue Multilicities Erik Axell and Erik G. Larsson Linköing University Post Print N.B.: When citing this work, cite the original article. 20 IEEE. Personal use of this material is ermitted. However, ermission to rerint/reublish this material for advertising or romotional uroses or for creating new collective works for resale or redistribution to servers or lists, or to reuse any coyrighted comonent of this work in other works must be obtained from the IEEE. Erik Axell and Erik G. Larsson, A Unified Framework for GLRT-Based Sectrum Sensing of Signals with Covariance Matrices with Known Eigenvalue Multilicities, 20, Proceedings of the IEEE International Conference on Acoustics, Seech and SignalProcessing ICASSP, htt://dx.doi.org/0.09/icassp Postrint available at: Linköing University Electronic Press htt://urn.kb.se/resolve?urnurn:nbn:se:liu:diva-64320

2 A UNIFIED FRAMEWORK FOR GLRT-BASED SPECTRUM SENSING OF SIGNALS WITH COVARIANCE MATRICES WITH KNOWN EIGENVALUE MULTIPLICITIES Erik Axell and Erik G. Larsson Deartment of Electrical Engineering ISY, Linköing University, Linköing, Sweden ABSTRACT In this aer, we create a unified framework for sectrum sensing of signals which have covariance matrices with known eigenvalue multilicities. We derive the generalized likelihood-ratio test GLRT for this roblem, with arbitrary eigenvalue multilicities under both hyotheses. We also show a number of alications to sectrum sensing for cognitive radio and show that the GLRT for these alications, of which some are already known, are secial cases of the general result.. INTRODUCTION One of the most essential arts of cognitive radio is sectrum sensing. An erroneous decision results in either increased interference for the rimary users missed detection, or underutilized sectrum false alarm. Therefore, it is imortant to design good detectors, that exloit most of the available knowledge about the signal to be detected. All man-made signals have some structure, which is intentionally introduced for examle by the channel coding, the modulation and by the use of sace-time codes. Usually, some of these roerties of the signal are known from standards. In this work, we consider a discrete-time model, and the structure of the signal is then inherent in the covariance matrix of the signal if the signal is stationary. Such structures incurs that some of the eigenvalues of the signal covariance matrix are larger than others, even though the exact eigenvalues or their multilicities may not be known. Detection of correlated signals, exloiting features with unknown arameters is often referred to as blind detection. Blind detectors based on functions of eigenvalues of the samle covariance matrix were roosed and analyzed e.g. in [, 2]. These detectors are blind in the sense that they do not exloit any knowledge of the eigenvalues nor their multilicities. In this work, however, we consider the eigenvalues of the signal covariance matrix to have known multilicities. This can occur, for examle, when a single signal is received by multile antennas SIMO [2, 3, 4, 5], when the signal is encoded with an orthogonal sace-time block code OSTBC [6], or if the signal is an OFDM signal [7]. A related roblem was considered in [8], also dealing with covariance matrices with known eigenvalue multilicities. The roblem of [8] was not only to detect the resence or absence of a signal, but rather to detect the number of signal sources embedded in noise. The aer [8] assumed that each signal source gives rise to a distinct eigenvalue, and that the remaining eigenvalues are equal to the noise ower. This is a secial case of the roblem we consider in this The research leading to these results has received funding from the Euroean Community s Seventh Framework Programme FP7/ under grant agreement no This work was also suorted in art by the Swedish Research Council VR, the Swedish Foundation for Strategic Research SSF and the ELLIIT. E. Larsson is a Royal Swedish Academy of Sciences KVA Research Fellow suorted by a grant from the Knut and Alice Wallenberg Foundation. aer, which allows arbitrary eigenvalue multilicities. The work of [8] made use of the results on rincial comonent analysis in [9]. In this work, we also use the results of [9], in articular for the maximum-likelihood estimation of the covariance matrices. Contributions: We derive the generalized likelihood-ratio test GLRT when the covariance matrices have arbitrary and known eigenvalue multilicities under both hyotheses. We show that this is a unifying framework for some alications of sectrum sensing, which are secial cases of the general result. In articular, we show that the GLRT of [2, 3, 4, 5], for detection of a single signal using multile antennas, is a secial case of the roblem. Furthermore, we show that two of the detectors roosed in [6], for signals encoded with an OSTBC, are equivalent to the GLRT. We also derive the eigenvalues and their multilicities of a synchronized OFDM signal in an AWGN channel, using the model in [7]. From this, we derive the GLRT for detection of an OFDM signal in AWGN. 2. MODEL AND PROBLEM FORMULATION Let y k, k,...,k, be the observe-length column vectors. We wish to discriminate between the two hyotheses : y k N0,Q 0, i.i.d. k,...,k H : y k N0,Q, i.i.d. k,...,k, where Q i has r i distinct eigenvalues λ,i > λ 2,i >... > λ ri,i, with known multilicities q,i,...,q ri,i resectively, and y k R N. Then, r i jqj,i N. Note that the model is real-valued. This is not a restriction in most cases, since a comlex valued model can be slit into its real and imaginary arts. Let Y [y y 2... y K] R N K, and denote by R the samle covariance matrix R K K y k yk T K YYT. 2 k Then the likelihood functions of Y, under the two hyotheses can be written Y Q i ex K2 2π NK/2 K/2 detq tr Q R i. i 3 3. DETECTION In general, the covariance matrices Q i are unknown. A standard technique to deal with unknown arameters, that usually erforms well, is the generalized likelihood-ratio test GLRT: Y H, Q Y, Q H η, 4 0

3 where Q i is the maximum-likelihood ML estimate of Q i, and the multilicitiesq,i,...,q ri,i of the eigenvalues ofq i are assumed to be known. As already mentioned, this can be the case for examle in a SIMO transmission [2, 3, 4, 5], if the signal is encoded with an orthogonal sace-time block code [6], or if the signal is OFDM modulated as we will show in Section 4.4. This also includes the secial case of [0] when the structure of the transmitted signal is assumed to be comletely unknown, so that all eigenvalues of the covariance matrix are assumed to have multilicity one. 3.. ML-Estimation of the Covariance Matrices In this subsection, we will show the maximum-likelihood estimates that are required for the GLRT. The main work in deriving the ML estimates ofq i was done in [9]. We will then use the result of [9] to derive the likelihood functions and the GLRT in 4. Letu,i,...,u N,i denote the eigenvectors ofq i, normalized so that u j,i, j,i. Define the set of indices S k,i k k q j,i +,..., q l,i 5 j r i k S k,i,...,n. For examle, if there are two distinct eigenvalues with multilicities q, and q 2, N q, resectively under hyothesis H, then S,,...,q, and S 2, q, +,...,N. The covariance matrix Q i is comletely defined by its eigenvalues and eigenvectors, and can be written Q i r i k l λ k,i u j,iu T j,i. Denote by d k and v k, k,...,n, the eigenvalues sorted in descending order, and the corresonding normalized eigenvectors resectively of the samle covariance matrix R. Following [9], the ML estimates of the eigenvalues and eigenvectors are λ k,i d j, k,...,r i, û k,i v k, k,...,n Generalized Likelihood-Ratio Test Inserting the ML estimates 6 into the likelihood function 3 yields 7. Now, consider the likelihood functions of the two hyotheses. Then, inserting 7 for both hyotheses into 4 yields Y H, Q Y, Q 0 r0 k r j We state the result in a theorem. q k,0 q j, qk,0 l S k,0 d l qj, i S j, d i K/2 Theorem The GLRT of, where Q i has distinct eigenvalues λ,i > λ 2,i >... > λ ri,i, with known multilicities q,i,...,q ri,i resectively, is where qk,0 r0 k λk,0 r j 6 λj, qj, H η, 8 λ k,i d j, the setss k,i are given by 5, andd j are the eigenvalues of the samle covariance matrix given by 2 sorted in descending order. 4. SPECTRUM SENSING APPLICATIONS In sectrum sensing for cognitive radio, the roblem is to discriminate between noise only and a signal embedded in noise. In the following, we will show a number of sectrum sensing alications, that are secial cases of our general result in Theorem. A standard assumtion is that the noise is zero-mean white, so that y k N0,σ 2 I. That is, under there is only one eigenvalue with multilicity N. This assumtions yields that the numerator in 8 is N N d i N i N N tr R. We will use this assumtion in the sequel of this section. 4.. Multile Receive Antennas SIMO The first secial case we consider is when the detector have multile antennas, which was analyzed in [2, 3, 4, 5]. Assume that there are n r N > receive antennas at the detector. Then, under H, the received signal can be written y k hx k +w k, k,...,k, 9 where h is the channel vector, x k is the transmitted signal samle, and w k is the noise vector. The signal is assumed to be zero-mean Gaussian, i.e. x k N0,γ 2. The noise is the same as under, i.e. w k N0,σ 2 I. Then, the covariance matrix under H is Q γ 2 hh T +σ 2 I. Now we have the following eigenvalues with their corresonding multilicities under the two hyotheses : λ,0 σ 2, q,0 N, { λ, γ 2 h 2 +σ 2, q,, H : λ 2, σ 2, q 2, N. Inserting these into 8 yields the GLRT N tr R N N i2 di N H η. 0 This test is equivalent to [4, eq. 35], showing that the GLRT of [2, 3, 4, 5] is a secial case of Theorem Orthogonal Sace-Time Block Codes Now consider a slightly more general case, when the transmitted signal is encoded with an orthogonal sace-time block code OSTBC. This roblem was considered in [6], and we use the same model here. Assume that there are n r receive antennas and n t transmit antennas. The OSTBC code matrix X C nt T is a linear function ofn s symbolss,...,s ns and their comlex conjugates. The coded symbols columns of X are transmitted over T time intervals. Let Y C nr T be the received matrix that consists of the sace-time coded signal lus noise, i.e. Y HX+W,

4 ex K tr Q 2 i R Y Q i K/2 2π NK/2 det Qi ex K ri 2 k 2π NK/2 ri k m S d m k,i l S d k,i l ex K tr ri 2 k m S k,i v λ mv T N k,i m j djvjvt j K/2 Qi d j qk,i K/2 ri 2π NK/2 det k where H C nr nt is the channel matrix, and W C nr T is a matrix of noise. Following [6], we have assumed erfect time and frequency synchronization. Then, following [6], we can for an OS- TBC equivalently write the model as d j qk,i K/2 ex KN 2 2π NK/2 ex K ri d m 2 k m S k,i 2π NK/2 ri k λ k,i λ k,i K/2 true ML-estimate. Moreover, the ad-hoc detector roosed in [6] is identical to 3. Thus, we have shown that the ad-hoc detector of [6] is also equivalent to the GLRT. This exlains why the numerical erformances of these detectors were identical in [6]. 7 y Gx+w, where G is a real-valued 2n rt 2n s-matrix n s < n rt with the roertyg T G H 2 I, and x [ Res,...,Res ns, Ims,...,Ims ns ]T R 2ns. Now consider K sace-time blocks Y k, or equivalently K vectorsy k, received in a sequence. Moreover, we assume that the channel is slow fading, such that the generator matrix G remains the same during the whole time of recetion. Then, under H, we have the model y k Gx k +w k, k...,k. 2 We assume that the elements of x k are i.i.d. N0,γ 2. In this case, the covariance matrix underh isq γ 2 GG T +σ 2 I. Therefore, we have the following eigenvalue roerties under H { λ, γ 2 H 2 +σ 2, q, 2n s, λ 2, σ 2, q 2, 2n rt 2n s, and under the same as in 0. Using these articular eigenvalue multilicities in Theorem, we obtain the GLR 2nrT tr R 2n rt 2ns 2nrT 2ns λ, λ2, 2n rt 2nrT 2n rt 2nrT 2n rt 2nrT 2n s λ, +2n rt 2n s λ 2, 2nrT λ2, λ, λ2, λ, λ, 2ns λ2, 2nrT 2n s 2ns 2n s λ, +2n rt 2n s λ 2, 2ns λ 2, 2n s λ, λ 2, +2n rt 2n s 2nrT 2nrT By taking the derivative of this GLR, with resect to λ,/ λ 2,, one can show that the GLR is a monotonously increasing function of λ,/ λ 2,. Therefore, the GLRT can be equivalently written λ, λ 2, H η. 3 Now, we have shown that the estimate of the covariance matrix that was roosed in [6], and referred to as near-ml, is actually the 4.3. Signal with Unknown Correlation Structure Now consider the case when the signal correlation is unknown, so that all eigenvalues of the covariance matrix are assumed to have multilicity one. That is, under H q k,, k,...,n. Again, we assume that the noise is white Gaussian, so that there is only one distinct eigenvalue with multilicity N under, as in 0. Using these assumtions in Theorem, we obtain the GLRT N tr R N H N η. j dj This is of course equivalent to the GLRT obtained in [0] for this secial case of the roblem, and also to the shericity test of [] OFDM In this section we consider an OFDM signal with a cyclic refix CP. We will use the vector-matrix model of [7], and show the eigenvalue roerties of the received OFDM signal in an AWGN channel. Again, we assume erfect synchronization. Now, let x k be the N d -vector of data associated with the kth OFDM symbol. This data vector is the outut of the IFFT oeration, used to create the OFDM data. An OFDM symbol is then created by reeating the lastn c elements of x k at the beginning of the symbol. Following [7], the received OFDM symbol can be modelled by 2, where [ ] 0Nc N G d N c I Nc R Nc+N d N d. I Nd Here 0 n m denotes the n m all-zero matrix, and I n denotes the n n identity matrix. Then, G has the roerty G T N G diag,...,,2,...,2 R d N d. 4 }{{}}{{} N d N c Since the matrices G T G and GG T have the same non-zero eigenvalues, this means that the matrix GG T have eigenvalues 2, and 0 with multilicities N c,n d N c an c resectively. Again, the covariance matrix is Q γ 2 GG T + σ 2 I under H, and we get the following eigenvalue roerties λ, 2γ 2 +σ 2, q, N c, λ 2, γ 2 +σ 2, q 2, N d N c, λ 3, σ 2, q 3, N c. N c

5 With these eigenvalue multilicities, the GLR in Theorem becomes N tr R N Nc Nd N c Nc. λ, λ2, λ3, Here, we assumed real-valued OFDM samles, but in reality they are comlex-valued. This is not a restriction. Since the generator matrix G is real valued, we can slit the received vectors into real and imaginary arts and deal with them searately. The only consequence of this is that the dimension of the received vector and the multilicities of the eigenvalues will increase with a factor of two. 5. MONTE-CARLO SIMULATIONS In the following, we will show some numerical results of the roosed GLRT exemlified by a signal encoded with the Alamouti code. We will comare the roosed GLRT, that exloits the known signal structure, with a few eigenvalue-based blind detectors. 5.. Benchmarks There were two blind detectors roosed in [], based on functions of the eigenvalues of the samle covariance matrix. The detectors of [] use the tests A similar test is d H η, d tr R tr R H η. 5 H η. 6 The detector 6 works as a blind detector for any kind of correlated signal, although it was also shown in [2, 3, 4] to be equivalent to the GLRT for the SIMO scenario of Section Numerical Results The Alamouti code is an OSTBC, so the GLRT in this case is given by 3. As a comarison, we show the detection erformance of the detectors resented in Section 5.. Note in assing that none of the detectors requires knowledge of the noise variance. Each detector receivedk 00 code blocks, usingn r 8 receive antennas. The SNR in db is defined as 0log 0 γ 2 /σ 2. Performance is given as the robability of missed detection, P MD, as a function of SNR. The channel coefficients were drawn from a comlex circularly symmetric N0, distribution. The robability of false alarm was chosen top FA The otimal decision thresholds were comuted emirically from a set of noise-only realizations, to achieve the chosen P FA. The results are shown in Figure. It is clear that the GLRT, exloiting the known eigenvalue structure, erforms better than the blind detectors. 6. CONCLUDING REMARKS We generalized and unified numerous recent roblems in sectrum sensing. It should be noted that the general result also includes the roblem of discriminating two signals, of different kind, from one another. This roblem may be of large interest in the context of cognitive radio, when one wishes to distinguish between rimary and secondary user s signals. P MD GLRT 3 d /tr R [2, eq. 6] d / [, Sec. III.A] tr R/ [, Sec. III.B] SNR [db] Fig.. Probability of missed detection versus SNR for detection of an Alamouti coded signal. P FA 0.05, K 00, n r REFERENCES [] Y. Zeng and Y.-C. Liang, Eigenvalue-based sectrum sensing algorithms for cognitive radio, IEEE Trans. on Communications, vol. 57, no. 6, , Jun [2] P. Bianchi, M. Debbah, M. Maida, and J. Najim, Performance of statistical tests for single source detection using random matrix theory, IEEE Trans. on Information Theory, to aear. [3] P. Wang, J. Fang, N. Han and H. Li, Multiantenna-assisted sectrum sensing for cognitive radio, IEEE Trans. on Vehicular Technology, vol. 59, no. 4, , May 200. [4] A. Taherour, M. Nasiri-Kenari and S. Gazor, Multile antenna sectrum sensing in cognitive radios, IEEE Trans. on Wireless Communications, vol. 9, no. 2, , Feb [5] R. Lóez-Valcarce and G. Vazquez-Vilar, Multiantenna sectrum sensing for cognitive radio: overcoming noise uncertainty, in Proc. of the International Worksho on Cognitive Information Processing, , Jun [6] E. Axell and E. G. Larsson, Sectrum sensing of orthogonal sace-time block coded signals with multile receive antennas, in Proc. of IEEE ICASSP 200, , March 200. [7] E. Axell and E. G. Larsson, Otimal and sub-otimal sectrum sensing of OFDM signals in known and unknown noise variance, IEEE Journal on Selected Areas in Communications, Feb. 20. [8] M. Wax, and T. Kailath, Detection of signals by information theoretic criteria, IEEE Trans. on Acoustics, Seech, and Signal Processing, vol. 33, no. 2, , Ar [9] T.W. Anderson, Asymtotic theory for rincial comonent analysis, The Annals of Mathematical Statistics, vol. 34, no., , Mar [0] T. J. Lim, R. Zhang, Y. C. Liang, and Y. Zeng, GLRTbased sectrum sensing for cognitive radio, in Proc. of IEEE GLOBECOM 2008.,. 5, Nov.-Dec [] J.W. Mauchley, Significance test for shericity of a normal n-variate distribution, The Annals of Mathematical Statistics, vol., no. 2, , 940.