Cooperative Spectrum Sensing Using Random Matrix Theory

Co op erativ e Sp ectrum Sensing Using Random Matrix Theory Leonardo S. Cardoso and Merouane Debbah and Pascal Bianchi Jamal Na jim SUPELEC ENST Gif-sur-Yvette, F rance Paris, F rance { leonardo .cardoso, merouane.debbah and pa scal.bianc hi } @ supelec.fr jamal.na jim@enst.fr No vem b er 20, 2018 Abstract In t h is pap er, using to ols from asymptotic r an d o m matrix theory , a new co op erativ e scheme for fr equency band sensing is int ro duced for b oth A W GN and fading c hann els. Unlike previous w orks in the field, the new scheme do es not require the kn o wledge of the noise statistics or its v ariance and is related to the b eha vior of the largest and smallest eigen v alue of random matrices. Remark ably , sim ulations sho w that the asymp totic claims hold even for a small n umb er of observ ations (whic h mak es it con ve nient for time-v arying top ologies), outp erforming classical energy detection tec hn iques. 1 In tro ducti on It has already b ecome a common understanding that curren t mobile comm unication systems do not ma ke full use o f the av ailable sp ectrum, either due to sparse user a c- cess o r to the system’s inherent deficiencies, a s sho wn by a rep ort from the F ederal Comm unications Commission (FC C) Sp ectrum P olicy T ask F orce [1]. It is en vi- sioned that future systems will b e able to opp ortunistically exploit those sp ectrum ’left-o vers ’, b y means of kno wledge o f the environme nt and cognitio n capability , in order t o a da pt their radio parameters accordingly . Suc h a t echnology has b een pro- p osed b y Joseph Mitola in 2 0 00 and is called cognitiv e radio [2]. D ue to the fact that recen t adv ances on micro-electronics and computer sys tems are p o in ting to a -not so far- era when suc h radios will b e feasible, it is of utmost imp or t ance to dev elop go o d p erforming sensing tec hniques. In its simplest form, sp ectrum sensing means lo oking for a signal in the presence of no ise for a giv en f requency band (it could also encompass being able to classify the signal). This problem has been extensiv ely studied b efore, but it has regained atten tion no w as part of the cognitiv e ra dio researc h efforts. There are sev eral classical tec hniques for this purp ose, suc h as the energy detector (ED) [3–5], the matc hed filter [6 ] and the cyclostationary feature detection [7–9]. These tec hniques ha v e their strengths and weak nesses and are we ll suited for ve ry sp ecific a pplicatio ns. Nev ertheless, the problem of sp ectrum sensing as seen from a cognitiv e radio p ersp ectiv e, has v ery stringen t requiremen ts and limitations, suc h as, • no prior kno wledge o f the signal structure (statistics, noise v ariance v alue, etc...); 1 • the detection of signals in the shortest time p ossible; • ability to detect reliably ev en o ve r heav ily faded environme nts; The works b y Cabric et al. [7 ], Akyildiz et al. [10] and Ha ykin [11] provide a summary of these classical tec hniques from the cognitiv e netw ork p oin t of view. It is clear from these w orks, t ha t none can fully cop e with all t he requiremen ts of the cognitiv e radio net w orks. In simple A W GN (Additiv e White Ga ussian Noise) channe ls, most classical ap- proac hes p erform v ery w ell. Ho w ev er, in the case of fast fading, these tec hniques are no t able to provide satisfactory solutions, in particular to the hidden no de pro b- lem [12 ]. T o this end, sev eral w orks [13–16] hav e lo ok ed in to the case in whic h cognitiv e radio s co op erate for sensing the sp ectrum. These works aim at reducing the probabilit y of false alarm by adding ex tra redundancy to the sensin g pro cess. They also aim at reducin g the num ber of samples collected, and thus , the estima- tion times b y t he use parallel measuring devices. How ev er, ev en though one could exploit t he spatial dime nsion effic iently , these works are based on the same funda- men tal tec hniques, whic h require a priori knowledge of the signal. In this w or k, w e in tro duce an alternativ e metho d for blind (in the sens e that no a priori know ledge is needed) spectrum sensing. This me tho d relies on the use of m ultiple receiv ers to infer on the structure of the receiv ed signals using random ma- trix theory (RMT). W e sho w tha t w e can estimate the sp ectrum o ccupancy reliably with a small amoun t of receiv ed samples. The remainder o f this w ork is divided as follows. In section 2, w e formulate the problem of blind sp ectrum sensing. In section 3 , w e in tro duce the prop osed approac h based on ra ndo m matrix theory . In section 4, w e presen t some practical results whic h confirm that the asymptotic a ssumptions hold ev en for a small a moun t of samples. Then, in section 5, w e show the p erformance results o f the prop osed metho d. Finally , in section 6, w e draw the main conclusions and p oint out further studies. 2 Problem F orm ulation The basic problem concerning sp ectrum sensing is the detection of a signal within a noisy measure. This turns out to b e a difficult task, esp ecially if the receiv ed signal p ow er is v ery low due to pathloss or fading, whic h in the blind spectrum sens ing case is unkno wn. The problem can b e p osed a s a h yp o t hesis test suc h that [3]: y ( k ) =  n ( k ): H 0 h ( k ) s ( k ) + n ( k ): H 1 , (1) where y ( k ) is the receiv ed v ector of samples at instant k , n ( k ) is a noise (not neces - sarily gaussian) of v ariance σ 2 , h ( k ) is the fading comp onent, s ( k ) is the signal which w e wan t to detect, suc h that E [ | s ( k ) | 2 ] 6 = 0, and H 0 and H 1 are the noise-only and signal h yp ot hesis, resp ectiv ely . W e supp ose that the channe l h stays constan t during N blo c ks ( k = 1 ..N ). Classical t ec hniques for sp ectrum sensing based on energy detection compare the signal energy with a know n threshold V T [3–5] deriv ed fro m the statistics of the noise a nd ch annel. The follo wing is considered to b e the decision rule decision =  H 0 , if E [ | y ( k ) | 2 ] < V T H 1 , if E [ | y ( k ) | 2 ] ≥ V T , 2 where E [ | y ( k ) | 2 ] is the energy of the signal and V T is usually tak en as the noise v a r ia nce. One dra wbac k of this approac h is that neither the noise /channel distri- bution nor V T are know n a priori. In real life scenarios V T dep ends on the radio c haracteristics and is hard to b e estimated pro p erly . Moreo v er, in the case of fading and path loss, the energy of the receiv ed signal can b e of the order of the noise, making it difficult to b e detected all the more a s the num ber o f samples N may be v ery limited. Indeed, E [ | y ( k ) | 2 ] is estimated by 1 N N X k =1 | y ( k ) | 2 , whic h is not a go o d estimator for the small sample size case. In t he following, w e pro vide a co op erativ e approach for cognit ive netw orks to detect the signal from a pr imar y system without the need to kno w the noise v ariance using results fro m random matrix theory . 3 Random Matrix Th eory for S p ectrum Sen sing Consider the scenario depicted in Figure 1, in whic h primary users (in white) comm unicate to their dedicated ( primary) base station. Secondary base stations { B S 1 , B S 2 , B S 3 , ..., B S K } are co op erative ly sensing the c hannel in order to identify a white space and exploit the medium. P S f r a g r e p la c e m e n t s p r i m a r y b a s e s t a t i o n B S 1 B S 2 B S 3 B S K Figure 1: Considered scenario for sp ectrum sensing. Before going any further, let us assume the follo wing: • The K base stations in the secondary system share informa t io n b etw een them. This can b e p erformed b y transmission ov er a wired high sp eed bac kb one. • The base statio ns are analyzing the same p ortion of the sp ectrum. Let us consider the following K × N ma t r ix consisting of the samples receiv ed b y all the K secondary base stations ( y i ( k ) is the sample receiv ed b y base station i at instant k ): Y =        y 1 (1) y 1 (2) · · · y 1 ( N ) y 2 (1) y 2 (2) · · · y 2 ( N ) y 3 (1) y 3 (2) · · · y 3 ( N ) . . . . . . . . . y K (1) y K (2) · · · y K ( N )        . The goal of the random matrix theory approa c h is to p erform a test o f indep en- dence of the signals receiv ed b y the v arious base stations. Indeed, in the presence 3 of signal ( H 1 case), all the receiv ed samples are corr elated, whereas when no signal is presen t ( H 0 case), the samples are decorrelated whatev er the fading situation. Hence, in this case, for a fixe K and N → ∞ , the sample co v ariance matrix 1 N YY H con v erge σ 2 I . How ev er, in practice, N can b e of the same o r der of magnitude than K and therefore one can not infer directly 1 N YY H indep endence of the samples. This can b e formalized using to o ls f rom random matrix theory [17]. In the case where the en tries of Y are indep enden t (irresp ectiv ely of the sp ecific probabilit y distribution, which corresponds to the case where no signal is t r ansmitted - H 0 ) results from asymptotic random matrix theory [17] state that: Theorem. Consid er an K × N matrix W whose en tries are indep endent zero- mean complex (or real) random v ariables with v ariance σ 2 N and fourth momen ts of order O ( 1 N 2 ). As K, N → ∞ with K N → α , the empirical distribution of WW H con v erges almost surely to a nonrandom limiting distribution with densit y f ( x ) = (1 − 1 α ) + δ ( x ) + p ( x − a ) + ( b − x ) + 2 π α x where a = σ 2 (1 − √ α ) 2 and b = σ 2 (1 + √ α ) 2 . In terestingly , when there is no signal, the supp ort o f the eigenv alues of the sample co v ariance matrix (in Figure 2, denoted b y ˇ MP) is finite, whatev er the distribution of the noise. The Marc henk o-Pastur law th us serv es as a theoretical prediction under the assumption that matrix is ”all noise”. Deviations from this theoretical limit in the eigen v alue distribution should indicate non-noisy comp onents i.e they should suggest info rmation ab out the ma t r ix. P S f r a g r e p la c e m e n t s ˇ MP a b Figure 2: The Marchenk o - P astur supp ort ( H 0 h yp othesis). In the case in whic h a signal is presen t ( H 1 ), Y can b e rewritten as Y =    h 1 σ 0 . . . . . . h K 0 σ         s (1) · · · s ( N ) z 1 (1) · · · z 1 ( N ) . . . . . . z K (1) · · · z K ( N )      , where s ( i ) a nd z k ( i ) = σ n k ( i ) are r esp ective ly the indep enden t signal and noise with unit v ariance at instan t i a nd base station k . Let us denote by T the matrix: T =    h 1 σ 0 . . . . . . h K 0 σ    . 4 TT H has clearly one eigenv alue λ 1 = P | h i | 2 + σ 2 and all the rest equal to σ 2 . The b eha vior of the eigen v alues of 1 N YY H is r elat ed to the study of the eigen v alue of la rge sample co v a riance matrices of spik ed p opulatio n mo dels [1 8 ]. Let us define the signal to noise ratio (SNR) ρ in t his w ork as ρ = P | h i | 2 σ 2 . Recen t w orks of Baik et al. [18, 19] hav e sho wn that, when K N < 1 and ρ > r K N (2) (whic h are assumptions that are clearly met when the n um b er of samples N are sufficien tly high), the maxim um eigen v alue o f 1 N YY H con v erges a lmost surely to b ′ = ( X | h i | 2 + σ 2 )(1 + α ρ ) , whic h is superior to b = σ 2 (1 + √ α ) 2 seen for the H 0 case. Therefore, whenev er the distribution of the eigen v a lues of the matrix 1 N YY H departs fr o m the Marchenk o -P astur la w (Figure 3) , t he detector know s tha t the signal is presen t. Hence, one can use this in teresting feature to sense the sp ectrum. P S f r a g r e p la c e m e n t s ˇ MP a b b ′ Figure 3: The Marc henk o-P astur supp ort plus a signal comp onent. Let λ i b e the eigenv alues of 1 N YY H and G = [ a, b ], the co o p erativ e sensing algorithm w orks as f ollo ws: 3.1 Noise distribution u nkno wn, v ariance kno wn In this case, the f ollo wing criteria is used: decision =  H 0 : , if λ i ∈ G H 1 : otherw ise (3) Note that refinemen ts of this algorithm (where the pro babilit y of false a larm is tak en into accoun t in the non-asymptotic case) can b e found in [2 0]. The results are based on the computat io n of the asymptotic lar gest eigenv alue distribution in t he H 0 and H 1 case. 5 3.2 Both noise distribution and v ariance unkno wn Note that the ratio of t he maxim um and the minim um eigen v alues in the H 0 h y- p othesis case do es not depend on the noise v ariance. Hence, in order to circum v en t the need for the knowledge of the noise, the following criteria is used: decision = ( H 0 : , if λ max λ min ≤ (1+ √ α ) 2 (1 − √ α ) 2 H 1 : other w ise (4) It should b e noted t hat in this case, one needs to still tak e a sufficien tly high n um b er of samples N suc h that the conditions in Eq. (2) are met. In other w ords, the n um b er of samples scales quadratically with the in v erse of the signal to noise ratio. Note moreo v er that the test H 1 pro vides also a go o d estimator of the SNR ρ . Indeed, the ratio of largest eigen v alue ( b ′ ) and smallest ( a ) of 1 N YY H is related solely to ρ and α i.e b ′ a = ( ρ + 1)(1 + α ρ ) (1 − √ α ) 2 T o our knowledge , this estimator of the SNR has nev er b een put forw ard in the literature b efore. 4 P erformance Analysis The previous theoretical results ha v e sho wn that one is able to distinguish a signal from no ise by the use of only a limiting rat io of the hig hest to the smallest eigenv alue of t he sample co v ariance matrix. F or finite dimensions, the op erating region fo r suc h an alg orithm is still an issue and is related to the asymptotic distribution of a scaling factor o f the ratio [20]. This section pro vides some c hara cterization of this region through the analysis of the ratio b et we en λ max and λ min of 1 N YY H for v a rious matrix sizes. Figures 4 and 5 presen t the λ max /λ min for v arious sizes of Y in the pure noise case, with α = 1 / 2 and α = 1 / 10, res p ectiv ely . F rom the figur es we see that b oth cases pro vide a go o d approx imation of the asymptotic ratio ev en with small matrix sizes. If one ta k es, for example, N = 100 ( K = 50 for α = 1 / 2 and K = 10 for α = 1 / 10) , it can b e seen that the sim ulated cases are resp ectiv ely equal to 81% p ercen t a nd 83% o f the asymptotic limit fo r α = 1 / 2 and α = 1 / 1 0. As exp ected, for a larger Y matrix size, the empirical ratio a ppro ac hes the asymptotic one. Figures 6 and 7 show t he b eha vior of the λ max /λ min for the signal plus noise case for α = 1 / 2 and α = 1 / 10 , resp ective ly . In b oth cases, σ 2 = 1 /ρ (with a ρ of - 5 dB) with P | h i | 2 = 1 (whic h holds under the criteria in Eq. (2)). In this case, λ max λ min = b ′ a , for the pure signal case. In terestingly , for N = 100 ( K = 50 for α = 1 / 2 and K = 10 fo r α = 1 / 10), it can b e seen that the sim ulated case is appro ximately 70% p ercen t and 83 % o f the asymptotic limit fo r α = 1 / 2 and α = 1 / 10, resp ectiv ely . As exp ected, the larger the Y matrix sizes, the closer one gets to the asymptotic ratio. A go o d appro ximation w as obta ined f or v alues of N as lo w as 100 samples. 5 Results Sim ulations w ere carried out t o establish the p erfo r ma nce of the random matrix theory detector sche me in comparison to the co op erative energy detector sc heme 6 0 50 100 150 200 250 300 350 400 0 5 10 15 20 25 30 35 N (number of samples) λ max / λ min H 0 case, fixed α = 1/2 λ max / λ min (Asymptotic) λ max / λ min (Simulated) Figure 4: Beha vior of λ max /λ min for increasing N ( case H 0 , α = 1 / 2 ). 0 200 400 600 800 1000 1200 1400 1600 1800 2000 1 1.5 2 2.5 3 3.5 4 N (number of samples) λ max / λ min H 0 case, fixed α = 1/10 λ max / λ min (Asymptotic) λ max / λ min (Simulated) Figure 5 : Beha vior of λ max /λ min for increasing N (case H 0 , α = 1 / 1 0). based on v oting [15, 16]. The f ramew ork for t he energy detector is exp o sed in section 2, with h ( k ) mo deled a s a rayleigh m ultipath fading of v aria nce 1 /K . The v ariance is normalized to take into accoun t the fact that the energy do es not increase without b ound as the n um b er of base stations increases due to the pa t h loss. A total of 10 secondary base stations w ere sim ulated. F or the voting sc heme, the decision rule is the follo wing: one considers the ov erall spectrum o ccupancy decision to b e the one chose n by most of the secondary base stations. The t hr eshold V T is tak en a s σ 2 (for the kno wn noise v ariance case). F or the random matrix theory based sc heme, a fixed to tal of ( K = 10) base stations w ere adopted. Note that the algorithms can b e optimized for the v ot ing and random matrix theory based rules by adopting decision margins [2 0 ]. Figure 8 depicts the p erformance of t he energy detector sc heme along with the random matr ix theory one fo r N = { 10 , 20 , ..., 60 } samples and a kno wn noise v ari- ance of σ 2 at SNR equal to -5dB. It is imp ortant to stress tha t since K is fixed, α is not constan t a s in the previous section. As clearly s hown, the random matrix theory sche me outp erforms the co op erativ e energy detector case for all n um b er of samples due to its inheren t robustness. Figure 9 plots the p erformance of the random matrix theory sc heme for a n un- kno wn noise v ariance (the v ot ing sc heme can no t b e compared as it relies on the 7 0 50 100 150 200 250 300 350 400 0 5 10 15 20 25 30 35 40 N (number of samples) λ max / λ min H 1 case, SNR = −5 dB, fixed α = 1/2 λ max / λ min (Asymptotic) λ max / λ min (Simulated) Figure 6: Beha vior of λ max /λ min for increasing N ( case H 1 , α = 1 / 2 ). 0 200 400 600 800 1000 1200 1400 1600 1800 2000 1 1.5 2 2.5 3 3.5 4 N (number of samples) λ max / λ min H 1 case, SNR = −5 dB, fixed α = 1/10 λ max / λ min (Asymptotic) λ max / λ min (Simulated) Figure 7 : Beha vior of λ max /λ min for increasing N (case H 1 , α = 1 / 1 0). kno wledge of the noise v ariance). One can see that, indeed, ev en without the know l- edge o f a no ise v ariance, one is still able to ac hiev e a v ery go o d perfo rmance f o r sample sizes greater than 30. 6 Conclus ions In this pap er, we hav e provided a new sp ectrum sensing tec hnique based on random matrix theory and shown its p erformance in comparison to the co op erative energy detector sc heme for b o th a kno wn and unkno wn noise v ariance. Remark ably , the new tec hnique is quite robust and do es not require the kno wledge of the s igna l or noise statistics. Moreo v er, the asymptotic claims turn o ut to b e v alid ev en for a v ery lo w num ber of dimensions. The metho d can b e enhanced (see [20]) b y adj usting the threshold decision, taking in to accoun t the num b er o f samples tho ug h the deriv ation of the proba bilit y o f false ala rm of t he limiting ratio of the larg est to the smallest eigen v alue. 8 10 15 20 25 30 35 40 45 50 55 60 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 N (number of samples) Ratio of correct detections Correct detections at SNR = −5 dB (known noise) ED RMT P S f r a g r e p la c e m e n t s N u m b e r o f s a m p l e s P r o p o r t i o n o f c o r r e c t d e t e c t i o n s Figure 8: Comparison b et w een the ED and random matrix theory approach ( ρ = − 5 dB ). 10 15 20 25 30 35 40 45 50 55 60 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 N (number of samples) Ratio of correct detections Correct detections at SNR = −5 dB (unknown noise) RMT P S f r a g r e p la c e m e n t s N u m b e r o f s a m p l e s P r o p o r t i o n o f c o r r e c t d e t e c t i o n s Figure 9: Random matrix theory appro ac h for an unkno wn noise v ariance. 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