Wdeband Spectrum Sensng by Compressed Measurements Davood Mardan Najafabad Department of Electrcal Engneerng Yazd Unversty Emal: d.mardan@stu.yazdun.ac.r Al A. Tadaon Department of Electrcal Engneerng Yazd Unversty Emal: tadaon@yazdun.ac.r Masoud Reza Aghabozorg Sahaf Department of Electrcal Engneerng Yazd Unversty Emal: aghabozorg@yazdun.ac.r Abstract In ths paper we ntroduce a new method for wdeband spectrum sensng whch uses the receved sgnal compressed measurements to determne the occupancy status of each subchannel. In the cogntve rado systems, wdeband spectrum sensng n hgh utlzaton regmes ncreases the speed of the access to the spectrum holes. In order to send data, the prmary users employ dfferent modulaton types n each sub-channel; therefore, the transmtted sgnal s chosen among a sgnal space. Snce just a few numbers of sub-channels are occuped by the prmary user sgnals n each tme nstant, the receved wdeband sgnal has a sparse representaton on the sub-channel sgnal space bases; then, the secondary user can compress the receved sgnal samples. Knowng about the modulaton types and therefore, the sgnal spaces, the nformaton n each subchannel s separated usng a projecton matrx appled to the compressed measurements, drectly. The separated compressed measurements are employed to derve a Generalzed Lkelhood Rato detector and determne the presence or absence of the prmary user sgnal n each sub-channel. We try to determne the sensng matrx so that the flterng and detecton steps and, therefore, wdeband spectrum sensng have low error. Index Terms Cogntve rado; Spectrum sensng; Sparsty; Compressve sensng; I. Introducton Cogntve rado has emerged as a dynamc method to resource allocaton, so that, the unlcensed users, called Secondary Users (SUs), can use the vacant channels belonged to the lcensed user, Prmary Users (PUs). In order to prevent any harmful nterference, the SUs should contnuously sense the spectrum; therefore, the spectrum sensng s one of the most sgnfcant tasks n cogntve rado networks. If we sense one channel of the spectrum at each tme nstant, the whole process takes so much tme n the hgh utlzaton of spectrum. In such regmes, wdeband spectrum sensng methods, whch regard several sub-channels n a wde spectrum smultaneously, can speed the access to the vacant sub-channels. In the cogntve rado networks, the SUs usually have some nformaton about the transmsson parameters n each subchannel used by the PUs. Knowng these nformaton, The SUs can perform spectrum sensng wth lower error rate and lower Computatonal Complexty (CC). Several wdeband spectrum sensng methods have been proposed. Employng the ntroduced method n the works of Tan and Gannaks [1], [2], the SUs use the samples acqured from the receved wdeband sgnal to determne the sub-channel locatons and estmate the sgnal Power Spectral Densty (PSD) for wdeband spectrum sensng. Applyng the Compressve Sensng (CS) concept, we can decrease the complexty of ths method [2], [3]. In these approaches, the authors use a DFT matrx to estmate PSD whch may be replaced by some more accurate methods n practce that leads to the better performance. We also can employ a flter-bank wth narrow band flters to acqure the sgnal samples n each sub-channel. There are several wdeband spectrum sensng methods whch use these separated samples to determne the sub-channel occupancy status. In the [4], [5] SUs employ the output flter-bank samples to realze two Generalzed Lkelhood Rato (GLR) detectors and determne the presence or absence of the PU sgnals. Authors n [6], [7] ntroduced the Mult-band Jont Detecton (MJD) and Mult-band Sensng-tme-adaptve Jont Detecton (MSJD) methods for wdeband spectrum sensng. In these two methods, the SU employs the flter-bank output samples to derve detectors whose thresholds are defned by maxmzng an aggregate opportunstc throughput subject to a lmt an aggregate nterference on the whole wde spectrum. In general, the flter-bank based method, whch employs the narrow band flters to separate nformaton n each sub-channel, mposes a hgh complexty to the SU. Authors n [8] employ Modulated Wdeband Converter (MWC) system to sample the receved wdeband sgnal by sub-nyqust rate. Ths method s an analog-dgtal spectrum sensng whch does not requre any sgnfcant nformaton from the PU sgnals, whle, knowng some nformaton about the PUs (e.g. the PU sgnal modulaton types), we can decrease the Computatonal Complexty (CC) of the wdeband spectrum sensng. In ths paper, we ntroduce a new method whch employs nformaton about the modulaton type and, then, the sgnal space n each sub-channel for wdeband spectrum sensng. Regardng the pont that a large number of sub-channels are vacant, the wdeband receved sgnal has a sparse representaton on a unon of all sub-channel bases. Accordng to the sparsty of the receved sgnal, we compress the samples acqured at or above the Nyqust rate wthout mssng any sgnfcant nformaton. Knowng the sub-channel bases, we can derve several projecton matrces regardng to the sub-channel bases and separate the nformaton n each sub-channel drectly by 978-1-4673-2713-8/12/$31.00 2012 IEEE 000667
the compressed measurements whch decreases the CC. After flterng, the separated compressed measurements s employed to derve detector n each sub-channel. In order to realze detectors, we assume that the nose varance and the PU sgnals are determnstc wth unknown parameters. We also assume that the nose varance s dentcal n all sub-channels. By these assumptons, we try to derve GLR detectors for wdeband spectrum sensng. Choosng an approprate sensng matrx, Monte-Carlo smulatons show that the proposed method outperforms the methods whch use DFT matrx as the flter-bank. The performance of our proposed method s not also far from the flter-bank based method whch uses the deal narrow band flters, whle, the CC of our method s substantally lower. The remanng of ths paper s organzed as follows; n secton II, we show that the wdeband receved sgnal, whch s a sum of all sub-channel sgnals, has a sparse representaton on a unon of sub-channel bases. In secton III, we compress the receved sgnal samples to separate nformaton of each sub-channel and, then, derve the GLR detectors usng these separated compressed measurements. In secton IV, smulaton results evaluate the proposed CS-based method performance and, fnally, we offer a concluson n V. II. Problem Statement In a cogntve rado system, we consder a secondary user whch receves a wdeband sgnal x(t) consstng of K subchannels contamnated by an addtve whte Gaussan nose wth unknown varance σ 2. The N 1 vector x, whch contans the receved sgnal samples acqured at or above the Nyqust rate s represented as, x = x 1 + x 2 +... + x K, (1) where x, {1,..., K} s the sgnal samples of the th subchannel. Regardng the pont that n each sub-channel the prmary user uses a certan modulaton type and then, the transmtted sgnal s chosen among a sgnal space, the samples n the th sub-channel, x, {1,..., K}, belongs to a k dmensonal sgnal space S expanded on k orthogonal bases. Therefore, n accordance wth (1), the receved sgnal samples are represented by a unon of sun-channel bases, as follows, x = Ψ α, (2) where the columns of the N k s ( = k 1 + k 2 +... + k K ) matrx Ψ are the sub-channel bases and α s the representaton of x on the column space of Ψ, R(Ψ). Snce just a few number of subchannels are occuped by the prmary users n each tme nstant [9], α s a sparse vector. Therefore, n accordance wth the compressve sensng theorems, we can compress the receved sgnal samples wth preservng the sgnfcant nformaton of the receved sgnal [10]. Then usng an M N (M < N) sensng matrx Φ, the compressed vector y would be, y = Φ x = Φ (x 1 + x 2 +... + x K ). (3) In the next secton, we employ these compressed measurements to access the nformaton n each sub-channel and, then, use these separated nformaton to derve the GLR detectors and determne the occupancy status of them. III. Proposed CS-Based Method Knowng about the sub-channel modulatons and, then, subchannel bases, the secondary user can separate nformaton of each sub-channel usng compressed measurements, drectly. For ths purpose we can wrte, y = Φx + Φ (x 1 +... + x 1 + x +1 +... + x K ). (4) Let x = x 1 +... + x 1 + x +1 +... + x K, then x s a sgnal n a unon of (K 1) sgnal spaces S j, j {1,..., 1, + 1,..., K}, x S = S 1... S 1 S +1... S K, (5) where s the drect sum of two sgnal spaces [11]. If k k 1 + k 2 +... + k 1 + k +1 +... + k K, then S s a k -dmensonal sgnal space whose bass s a unon of the bases of S j, j {1,..., 1, + 1,..., K}. In the case that dfferent sub-channel bases are not orthogonal, we can represent x as a sum of two components x s and x on S and the orthogonal complement of S,.e. S, respectvely. If P s s the projecton matrx onto orthogonal complement of S then S = P s S. Accordng to ths assumpton and (4), we could wrte, = y = Φ( x s + x + x ). (6) Let x I = x + x ( x I S ), then we have, y = Φ x s + Φ x I. (7) In order to access the nformaton n the th sub-channel, we should fnd a lnear operator P, {1,..., K}, so that by applyng t to the compressed measurements, Φ x I 0. Let Ψ s an N k matrx wth bass of S as ts columns, then the M k matrx Ω s defned as, Ω ΦΨ, {1,..., K}. (8) Accordng to the defnton of Ω, the operator matrx P would be, P = I P Ω, {1,..., K}, (9) where P Ω = Ω Ω [12] and Ω s the Moore-Penrose pseudonverse of Ω. By applyng P to y, we can wrte, y = P y = P Φ x s, {1,..., K}. (10) Therefore, y, {1,..., K} contans the nformaton of the th sub-channel. In ths step we should defne the sensng matrx Φ, so that, we mss the least nformaton of the th sub-channel. For ths purpose, we express a defnton and a theorem on δ-stable characterstc of matrces. Defnton 1 (see [12]): Let U and V are two sub-spaces n R N, the M N matrx A s δ-stable embeddng of (U, V) wth a δ (0, 1) f, (1 δ) u v 2 2 A u A v 2 2 (1 + δ) u v 2 2, (11) for all u U and v V. 978-1-4673-2713-8/12/$31.00 2012 IEEE 000668
Theorem 1 ( see [12]): If the sensng matrx Φ s a δ-stable matrx embeddng of ( S, S ). then and 1 δ 1 δ P Φ x s 2 2 (1 + δ). (12) x s 2 2 P Ω Φ x s 2 2 δ 2 1 + δ x s 2 (1 δ). (13) 2 2 Regardng to ths theorem, f the sensng matrx satsfes the δ-stable embeddng of ( S, S ), we can preserve nformaton of the th sub-channel wth a hgh probablty. Usng separated compressed measurements y, {1,..., K}, we can form the bnary hypothess test and then derve the detectors. Selectng the sensng matrx n accordance wth the condton appled n theorem 1, we can wrte, y = P Φ x s = (I P Ω ) Φ x s = Φ x s P Ω Φ x s Φ x s. (14) Let x s = s + ñ, where s and ñ are components of the PU sgnal, s, and the receved nose, n, n the th sub-channel, projected on S, respectvely. Then we have, y = Φ s + Φ ñ. (15) If we suppose that ñ s a whte Gaussan nose vector wth unknown varance, we can wrte the bnary hypothess test n the th, {1, 2,..., K} sub-channel, as follows, { H 0 : y N(0, σ2 Φ Φ H ), H1 : y N(Φ s, σ 2 Φ Φ H (16) ). Selectng a sensng matrx whch has the orthogonal rows wth the dentcal Eucldean norms, we can rewrte the hypothess test as, { H 0 : y N(0, σ 2 I), H1 : y N(Φ s, σ 2 (17) I), where σ 2 = σ 2 (ΦΦ H ) m,m, m {1,..., M} and (ΦΦ H ) m,m s an element located on the m th row and column of (ΦΦ H ). We assume that s s determnstc and unknown, then, we should obtan the Maxmum Lkelhood (ML) estmaton of the PU sgnals and the nose varance σ 2 under each hypothess. For ths purpose, we acqure the lkelhood functons n each subchannel under H1 and H 0 usng all sub-channel measurements [4], [5], as follows, f (Y; H1 ) = 1 1 M exp{ (2π) KM 2 σ KM 2σ 2 l=1 n D exp{ 1 M (Y 2σ 2,l (Φ s ) l ) 2 } exp{ 1 2σ 2 l=1 M l=1 n {1,2,...,K} (D ) Y 2 n,l } (18) (Y n,l (Φ s n ) l ) 2 }, f (Y; H 0 ) = 1 (2π) KM 2 exp{ 1 2σ 2 1 exp{ σ KM 2σ 2 M M l=1 n {1,2,...,K} (D ) l=1 n (D ) Y 2 n,l } (19) (Y n,l (Φ s n ) l ) 2 }, where Y s a K M matrx whch contans all sub-channel separated compressed measurements, D s a vector wth ndces of m vacant sub-channels and (Φ s n ) l s the l th element of vector Φ s n. Accordng to (18) and (19), the ML estmaton of each unknown parameter under each hypothess would be, { Φ H1 : s k = y k, k ({1,..., K} D). ˆσ 2 = 1 n D y n 2, KM { Φ H0 : s k = y k, k {1,..., K} (D ). ˆσ 2 = 1 n (D ) y n 2, KM (20) (21) If we substtute the estmated parameters n the lkelhood functons, the ML estmaton of vector D, D, s obtaned as a vector whch contans the ndces of m sub-channels wth lowest energes. In order to derve the GLR detectors, we replace the estmated parameters n the lkelhood functons and, then, form the lkelhood rato. Therefore, the rejecton regon n each sub-channel s acqured as, L (Y) = ˆD M. y 2 j ˆD y j 2 λ, (22) where λ s the detector threshold obtaned by a favorte probablty of false alarm whch represents The probablty that a vacant sub-channel s detected occuped by a PU sgnal [13]. In ths step, we should fnd a sensng matrx so that mproves the qualty of detecton. For ths purpose, we try to access a close form equaton for the probablty of detecton. Regardng the pont that detecton statstc n (22) has approxmately a central Ch-square dstrbuton wth M degree of freedom under H0, the probablty of false alarm would be, P fa = P(L > λ ; H 0 ) = 1 F L H 0 (λ ; M) (23) λ t M/2 1 e t/2 = 1 dt, 0 Γ(M/2)2M/2 where F L H0 (. ; M) s the cumulatve dstrbuton functon of L (Y) under H0 and Γ(. ) s the gama functon [14]. Let g(λ ; M/2) = λ t M/2 1 e t/2 dt, then, the detecton threshold 0 Γ(M/2)2 M/2 wth a favorte probablty of false alarm, P fa, s obtaned as, λ = g 1 ( (1 P fa ).Γ(M/2).2 M/2 ; M/2 ), (24) where g 1 (. ; M/2) s the nverse functon of g(. ; M/2). On the other hand under H1, L (Y) has the non-central Ch-square dstrbuton wth M degree of freedom and non-centralty parameter λ defned as [14], M ( ) 2 Φn s λ = xσ, (25) n=1 978-1-4673-2713-8/12/$31.00 2012 IEEE 000669
where Φ n s the n th row of Φ. Therefore, the probablty of detecton would be: M ( ) 2 P d = 1 F L H1 (λ Φn s ; M, λ) = Q M/2 σ, P M. (26) where P M = g ( 1 (1 P f a ).Γ(M/2).2 M/2 ; M/2 ) and F L H1 (. ; M, λ) s the cumulatve dstrbuton functon of L (Y) under H1 and Q M/2(. ) s the Marcum Q-functon [15]. If we create the sensng matrx by choosng randomly M row of an N N DFT matrx and normalze ts columns, we have, n=1 P d = Q M/2 ( Φ s 2 σ N/M, P M ). (27) In the case that we do not compress the measurements and separate the nformaton n each sub-channel drectly wth receved sgnal samples and stated flterng algorthm, the probablty of detecton would be, P dx = Q N/2 ( s 2 σ, P N ), (28) where P N = g 1 ( (1 P fa ).Γ(N/2).2 N/2 ; N/2 ). Comparng (27) and (28) we conclude that f M nclnes to N and the sensng matrx Φ s δ-stable embeddng of ( S, 0), the qualty of detecton wth separated compressed measurements s as good as the qualty of detecton wth uncompressed samples. Therefore n accordance wth the flterng and detecton steps, If the sensng matrx s δ -stable embeddng of (P s S, S ) by a small δ, the proposed wdeband spectrum sensng method has a good performance. IV. smulaton results In order to smulate the proposed wdeband spectrum sensng method, we take N = 160, 320 samples of the receved wdeband sgnal, consstng of K = 20 sub-channels, at Nyqust rate. We assume that the receved wdebend sgnal s contamnated by an addtve whte Gaussan nose and at each tme nstant around 30% of ts sub-channels are occuped by the PU sgnals; therefore, we consder the number of vacant sub-channels m = 10,.e. the SU s sure about the vacancy of 10 sub-channels n each tme nstant. Selectng M rows of a DFT matrx randomly wth a unform dstrbuton and normalzng ts columns, we obtan an M N sensng matrx to compress the receved samples. In order to evaluate the performance of the flterng step, we reconstruct each subchannel samples from the separated compressed measurements usng the reconstructon algorthm Smoothed l 0 -norm (SL0) [16] and, then, compute the Mean Square Error (MSE) n the whole wde spectrum shown n Fg. 1 wth N = 160, 320. As seen n ths fgure when M s at or above 0.3N, MSE s low and approxmately constant. In the case that the number of receved samples s ncreased from N = 160 to N = 320, the rate of sparsty s ncreased too and, then, the flterng performance s mproved. In order to evaluate the performance of the proposed method n terms of the detecton parameters, we use the Monte-Carlo smulaton method and, then, acqure the detecton thresholds so that the probablty of false alarm to be P fa = 0.01. Fg. 2 shows the performance of the proposed method wth dfferent amounts for M n terms of the probablty of detecton versus the Sgnal to Nose Rato (SNR) when the number of the receved samples N = 160. As seen, wth M = 0.4N the proposed method has an acceptable performance. In ths fgure, we also compare the performance of our method wth algorthms whch use a flter-bank to access the nformaton n each sub-channel and realze the detector ntroduced n (22) by the flter-bank output samples. A seen, our proposed method substantally outperforms the method whch apples the DFT matrx as a flter-bank. Let L = 4 as the number of receved samples n each sub-channel, the DFT matrx based method has a rather low computatonal complexty (LN log N 2 +NL) to access the energy n each sub-channel, whle, the detecton threshold n ths method was acqured n the best case where the utlzaton of the wde spectrum s about 0% and, therefore n accordance wth Fg. 3 n hgh utlzaton of spectrum (about 30%), the DFT based method can not satsfy the favorte probablty of false alarm, P fa = 0.01, n hgh SNRs. Our method has also a good performance comparng wth the flterbank based method whch uses deal and mpractcal narrow band flters, whle to access the energy of each sub-channel, the CS-based method mposes just O(MN + KM 2 + KM) as ts computatonal complexty. We can see n Fg. 4 that by ncreasng the number of receved samples, the performance of the wdeband spectrum sensng methods are mproved. In ths fgure we can also observe that the mprovement of the proposed CS based method s more predomnant. The reason of ths preference s that the projecton vector of the receved sample vector on the sub-channel bases s more sparse when N = 320. V. concluson In ths paper we consder the sparsty characterstc of the receved sgnal projecton on a unon of sub-channel sgnal space bases to compress the receved samples. We employed these compressed samples to access the nformaton of each sub-channel and, then, used the separated compressed measurements to form the bnary hypothess test and derve the GLR detectors. Smulaton results showed that the performance of the proposed CS-based method outweghs the method whch apples DFT matrx as a flter-bank. We also saw that, wth a rather low CC, our method has an acceptable performance n compare wth the method whch employs the flter-bank wth the deal and mpractcal narrow band flters. Acknoledgments Ths work was partally supported by Research Insttute for Informaton and Communcaton Technology, ITRC, Iran. 978-1-4673-2713-8/12/$31.00 2012 IEEE 000670
Fg. 1. Mean Square Error (MSE) of the CS-based method versus the dfferent compresson rates for N = 160 and 320, we use the SL0 algorthm to reconstruct the separated compressed measurements. Fg. 3. Probablty of false alarm versus SNR: M = 0.4N, K = 20, N = 160. Fg. 2. Probablty of detecton versus SNR: ths fgure compares the CSbased method performance wth the flter-bank based methods n dfferent amounts for M, K = 20, N = 160, probablty of false alarm P f a = 0.01. References [1] Z. Tan and G. B. Gannaks, A wavelet approach to wdeband spectrum sensng for cogntve rados, cogntve rado orented wreless networks and communcatons, 1st nternatonal conference, June 2006. [2] Z. Tan and G. B. Gannaks, Compressed sensng for wdeband cogntve rados, IEEE Internatonal Conference on Acoustcs, Speech and Sgnal Processng, ICASSP, VOL. 4, Honolulu, HI, pp. 13571360, Apr. 2007. [3] F.Zeng, C. L, and Z. Tan, Dstrbuted Compressve Spectrum Sensng n Cooperatve Multhop Cogntve Networks, IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 5, No. 1, pp. 3748, Feb. 2011. [4] A. Taherpour, S. Gazor and M. Nasr-Kenar, wdeband spectrum sensng n unknown whte Gaussan nose, IET communcatons, VOL. 2, NO. 6, pp. 763-771, July 2008. [5] A. Taherpour, S. Gazor and M. Nasr-Kenar, Invarant wdeband spectrum sensng under unknown varances, IEEE Trans. Wreless Communcatons, VOL. 8, NO. 5, pp. 2182-2186, May 2009. [6] Zh Quan, Shuguang Cu, Al H. Sayed and H. Vncent Poor, Optmal multband jont detecton for spectrum sensng n sogntve sado networks, IEEE Trans. Sgnal Process., VOL. 57, NO. 3, pp. 11281140, Mar. 2009. [7] Pedram Paysarv-Hosen and Norman C. Beauleu, Optmal wdeband Fg. 4. Probablty of detecton versus SNR: n ths fgure we evaluate the performance of proposed method and flter-bank based method wth deal narrow band flters by ncreasng the number of samples from N = 160 to N = 320, K = 20, M = 0.4N and probablty of false alarm P f a = 0.01. spectrum sensng framework for cogntve rado systems, IEEE Trans. Sgnal Process., VOL. 59, NO. 3, pp. 1170-1182, Mar. 2011. [8] Moshe Mshal and Yonna C. Eldar, Wdeband spectrum sensng at sub- Nyqust sates, IEEE Sgnal Processng Magazne, VOL. 28, NO. 4, july 2011. [9] Amr Ghasem and Elvno S. Sousa, Spectrum Sensng n Cogntve Rado Networks: Requrements, Challenges and Desgn Trade-offs, IEEE Communcatons Magazne, VOL. 46, NO. 4, pp. 32-39, Aprl 2008. [10] Davd L. Donoho, Compressed Sensng, IEEE Transactons on Informaton Theory, VOL. 52, NO. 4, APRIL 2006. [11] Stephane Mallat, a wavelet tour of sgnal processng, the sparse way. Elsever Inc, 2009. [12] Mark A. Davenport, Petros T. Boufounos, Mchael B. Wakn and Rchard G. Baranuk, Sgnal Processng Wth Compressve Measurements, IEEE Journal of Selected Topcs n Sgnal Processng, VOL. 4, NO. 2, APRIL 2010. [13] S. M. Kay, Fundamentals of Statstcal Sgnal Processng: Detecton Theory, Upper Saddle Rver, NJ: Prentce-Hall, 1998. [14] Norman Lloyd Johnson, Samuel Kotz and N. Balakrshnan, Contnuous Unvarate Dstrbutons. Volume 2, Wley and Sons, 1995. [15] Albert H.Nuttall, Some ntegrals nvolvng the Q M functon, IEEE Transactons on Informaton Theory, VOL. 21, NO. 1, pp. 95-96, Jan 1975. [16] H. Mohman, M. Babae-Zadeh and C. Jutten, A Fast Approach for Overcomplete Sparse Decomposton Based on Smoothed l 0 Norm, IEEE Trans. Sgnal Process., VOL. 57, NO. 1, Jan. 2009. 978-1-4673-2713-8/12/$31.00 2012 IEEE 000671