Spatial-Spectral Joint Detection for Wideband Spectrum Sensing in Cognitive Radio Networks

SP A T IAL-SPECTRAL JOINT DETE CTION FOR WIDEBAND S PECTR UM SENSING IN COGNITIVE RADIO NETWORKS Zhi Quan † , Shuguang Cui ‡ , Ali H. Sayed † , and H. V incent P oor § † Departmen t of Electrical Enginee ring, University of California, Los Angeles, CA 90095 ‡ Departmen t of Electrical and Compute r Engineering, T exas A&M Uni versity , College Station, TX 77843 § Departmen t of Electrical Enginee ring, Princeton University , Princeton, NJ 08544 Email: { quan , sayed } @ee.u cla.edu, cui@tamu.edu, poor@princeton. edu ABSTRA CT Spectrum sensing is an ess ential functionality that enables cognitive radios to detect s pectral holes and opportu nistically use und er-utilized frequency bands without causing harm ful interferen ce to primary networks. Since indi vidual cognitive radios might not b e able to r eliably detect weak primary sig- nals d ue to channel fading /shadowing, this paper propo ses a cooper ativ e wideb and spectrum sensing scheme, referred to as spatial-spectral joint detection , which is based on a linear combinatio n of the local statistics from spatially distributed multiple co gnitive rad ios. The cooper ativ e sensing prob lem is formulated into an optimization problem, for which subop - timal but efficient solutions can b e o btained thro ugh mathe- matical transforma tion under practical cond itions. Index T erms — Spectrum sensing , distributed d etection, nonlinear optimization, and cognitive radio. 1. INTR ODUCTION As an essential fu nctionality of c ognitive radio (CR) n etworks [1], spectrum sensin g need s to reliably detect weak primar y radio signals of p ossibly-unk nown formats. Genera lly , spec- trum sensing techniqu es can be classified into three categories: energy detection, matched filter cohe rent detectio n [ 2], an d cyclostationary feature detection. Since non -coher ent energy detection is simple and able to gen erate the spectrum-oc cupancy informa tion quickly , we adopt it as the b uilding b lock for con- structing the proposed wideband spectrum sensing schemes. The literature on wideban d spectru m sensing for CR n et- works is limited. An e arlier a pproac h is to use a tuna ble nar- rowband band pass filter at the RF fr ont-end to sense o ne nar- row frequency band at a time, o ver which the existing narrow- band spectrum sensing technique s can be applied. In or der to search over multiple frequency bands at a tim e, the RF front- end needs a wideband architecture, an d spectrum sensing usu- ally oper ates over an e stimate of the power spectral density (PSD) of th e wideban d signal. I n [3] , wa velet transforma tion This research was supported in part by NSF under Grant s ANI-03- 38807, CNS-06- 25637, ECS-06-01266, ECS-07 -25441, CNS-06-2563 7, and by DoD under Grant HO TRN-07-1-0037. was used to estimate th e PSD over a wid e fr equency rang e giv en its multi-resolu tion features. Howe ver, no pr ior work attempts to make decisions o ver multiple f requen cy bands jointly , which is essential for im plementin g efficient CR net- works. In this paper , we consider the situation in which spectrum sensing is compromised by destru ctiv e chann el condition s be- tween the target- under-detection and the detecting cogn iti ve radios, wh ich makes it hard to distinguish b etween a wh ite spectrum and a weak signal. W e propose a coop erative wide- band spectrum sensing scheme that exploits the spatial div er- sity among cognitive radios to im prove the sensing reliabil- ity . The coo peration is based on a linear combination of local statistics fr om spatially distributed cognitive radio s [4] [5], where the signals are assigned different weights accordin g to their positive contributions to joint sen sing. In such a sce- nario, we treat the design of distributed wideban d spectru m sensing as a spa tial-spectral joint detection pr oblem, which is furth er formu lated in to an o ptimization pro blem with th e objective of maximizing the overall opportu nistic thr ough - put under constra ints on the interfer ence to primary users. Throu gh math ematical reform ulation, we derive a suboptimal but efficient solutio n for th e optimization problem, which can considerab ly improve sensing performance. 2. SYSTEM MODEL Consider a primary commu nication system (e.g., multicarrier based) over a wideb and c hannel that is divided into K non- overlapping subch annels. At a par ticular time, some of th e K subchanne ls might not be used by the primary users an d are a vailable for oppo rtunistic spectrum access. Multiuser or- thogon al frequ ency division multiplexing (OFDM) schem es are suitable candidate s for such a scenar io since they ma ke it conv enient to nullify or activ ate some portion of m ultiple nar- row bands. W e model th e d etection prob lem over the subban d k as one to choose betwee n hyp othesis H 0 ,k (“0”), which rep- resents the ab sence of p rimary signals, and hypo thesis H 1 ,k (“1”), which re presents the presence of primar y signals. An illustration whe re only some of th e K ban ds are oc cupied by primary users is illu strated in Fig. 1. The crucial task of spec- Subbands occupied by primary users Spectrum holes 1 0 1 1 1 1 0 0 0 0 Fig. 1 . Ill ustration of the occupanc y of a m ultiband channel. trum sensing is to sense the K freque ncy bands and identif y spectral holes for op portun istic use. For simplicity , we as- sume that the high -layer protoco ls guarantee that all CRs keep quiet durin g the detectio n such that the main spectr al power under detection is emitted by the primary users. Consider a multi-path fading en v ironmen t, wher e h ( l ) , l = 0 , 1 , . . . , L − 1 , de notes the discrete-time c hannel impu lse response between the primar y tran smitter and a CR re ceiv er with L equal to the num ber of resolvable paths. Th e receiv ed baseband signal at the CR front-en d can be expressed as r ( n ) = L − 1 X l =0 h ( l ) s ( n − l ) + v ( n ) , n = 0 , 1 , . . . , N 0 − 1 (1 ) where s ( n ) repr esents the primary tr ansmitted s ignal with cyclic p refix at time n an d v ( n ) is additive co mplex white Gaussian noise with ze ro m ean and variance σ 2 v , i.e., v ( n ) ∼ C N  0 , σ 2 v  . In a m ulti-path fadin g environment, the wid e- band chan nel exhibits freq uency-selective features an d its dis- crete frequen cy resp onses are giv en by H k = 1 √ N 0 L − 1 X n =0 h ( n ) e − j 2 π nk/ N 0 , k = 0 , 1 , . . . , K − 1 (2) where L ≤ N 0 . W e assume that the channel is slowly varying and the chan nel fre quency respon ses { H k } K − 1 k =0 do not vary much d uring a detection inter val. In th e fr equency dom ain, the r eceiv ed sign al at each subcha nnel can be estimated by computin g its discrete Fourier transform (DFT): R k = 1 √ N 0 N 0 − 1 X n =0 r ( n ) e − j 2 π nk/ N 0 = H k S k + V k (3) where S k is the primary signal at subchannel k an d V k = 1 √ N 0 L − 1 X n =0 v ( n ) e − j 2 π nk/ N 0 (4) is the recei ved noise in the frequency domain . Note t hat V k ∼ C N  0 , σ 2 v  since v ( n ) ∼ C N  0 , σ 2 v  and th e DFT is a uni- tary linea r o peration . Wit hout lo ss of genera lity , we assume that th e tran smitted signal S k , c hannel g ain H k , a nd ad ditive noise V k are independ ent of each other . T o decide whether the k -th subchannel is occupied or not, we test the following binary hypo theses: H 0 ,k : R k = V k , k = 0 , 1 , . . . , K − 1 H 1 ,k : R k = H k S k + V k , k = 0 , 1 , . . . , K − 1 (5) For each subchan nel k , we compu te the test statistic as th e sum of received energy over an interval of M samples, i.e., Y k = M − 1 X m =0 | R k ( m ) | 2 k = 0 , 1 , . . . , K − 1 (6) and the decision rule is giv en by Y k H 1 ,k R H 0 ,k γ k (7) where γ k is th e corre sponding decision thr eshold. For sim- plicity , we assume tha t the transmitted signal at ea ch subch an- nel has unit power, i.e. , E  | S k | 2  = 1 ; th is assumption holds when the primar y radio s adopt u niform power tr ansmission strategies given no chann el knowledge at the transmitter s ide. According to th e cen tral limit theo rem for large M , Y k is asymptotically normally distributed with mean E ( Y k ) =  M σ 2 v H 0 ,k M  σ 2 v + | H k | 2  H 1 ,k (8) and v ariance V ar ( Y k ) =  2 M σ 4 v H 0 ,k 2 M  σ 2 v + 2 | H k | 2  σ 2 v H 1 ,k (9) for k = 0 , 1 , . . . , K − 1 . Thu s, a ssuming large M , we have Y k ∼ N ( E ( Y k ) , V ar ( Y k )) . Using the decision ru le in (7) , the pr obabilities of false alarm an d detection at subch annel k can be respectively cal- culated approxim ately as P ( k ) f ( γ k ) = P r ( Y k > γ k |H 0 ) = Q  γ k − M σ 2 v σ 2 v √ 2 M  (10) and P ( k ) d ( γ k ) = Q γ k − M  σ 2 v + | H k | 2  σ v p 2 M ( σ 2 v + 2 | H k | 2 ) ! (11) where Q denotes the tail p robab ility o f th e stand ard n ormal distribution. The choice o f thr eshold γ k leads to a tradeoff between the prob abilities of false alarm and miss P m = 1 − P d . Specifically , a higher threshold will result in a smaller probab ility of false alarm b u t a larger probab ility of miss, and vice versa. 3. SP A TIAL-SPECTRAL JOINT DETECTION Suppose that N spatially distrib uted cognitive radios collabo- rativ ely sen se a wide fr equency band. By combining the local statistics from individual cognitive radios at the fusion cen- ter , which can be o ne of the CRs, the n etwork can m ake a better decision o n the p resence or absen ce of p rimary signals on each of the K subchannels. The coope ration assum es a separate con trol channel, throu gh which th e statistics of in- dividual CRs are transmitted to the fusion center . Let Y k ( n ) denote the received e nergy in the k - th subchann el at cogn itiv e radio n . For each su bchann el, these statistics can be written in a vector as Y k = [ Y k (0) , Y k (1) , . . . , Y k ( N − 1)] T . T o exploit the spatial diversity , we linearly combine the summary statistics f rom spatially d istributed cognitive radios at each subchannel k to o btain a final test statistic: z k = N − 1 X n =0 w k ( n ) Y k ( n ) = w T k Y k (12) where w k = [ w k (0) , w k (1) , . . . , w k ( N − 1)] T are the com - bining coefficients for subchan nel k , which can be compactly written in a matrix as W = [ w 0 w 1 . . . w K − 1 ] . Note that w k ( n ) ≥ 0 , for ev ery k and n . Since the entrie s in Y k are normally distributed, th e test statistics { z k } K − 1 k =0 are also normally distributed with means E ( z k ) = ( M σ 2 v w T k 1 H 0 ,k M w T k  σ 2 v 1 + G k  H 1 ,k (13) where 1 is an all-one vector , and v ariances V ar ( z k ) = ( 2 M σ 4 v w T k w k H 0 ,k 2 M σ 2 v w T k  σ 2 v I + 2 diag( G k )  w k H 1 ,k (14) where G k =  | H k (0) | 2 , | H k (1) | 2 , . . . , | H k ( N − 1) | 2  T are the squared magnitud es of the chan nel gains between the pri- mary transmitter and the N CR rece i vers for subchannel k . In o rder to d ecide the presence o r ab sence of th e pr imary signal in subchanne l k , we use the following binary test z k H 1 ,k R H 0 ,k γ k , k = 0 , 1 , . . . , K − 1 . (15) According ly , the detection perfo rmance in terms of the p rob- abilities of false alarm and detection are gi ven by P ( k ) f ( w k , γ k ) = Q   γ k − M σ 2 v w T k 1 σ 2 v q 2 M w T k w k   (16) and P ( k ) d ( w k , γ k ) = Q   γ k − M w T k  σ 2 v 1 + G k  σ v q 2 M w T k [ σ 2 v I + 2 diag( G k )] w k   (17) For com pactness of notation, we collect the prob abilities of false alar m and detection over the K subchan nels into vectors P f ( W , γ ) and P d ( W , γ ) . Thu s, the pr obabilities of miss can be represented as P m ( W , γ ) = 1 − P d ( W , γ ) . Our go al is to maximize the oppo rtunistic r ate while m eet- ing som e constraints on th e in terferen ce to th e p rimary co m- munication system. Let r k denote the throug hput achiev- able ov er the k -th subchannel if used by cognitive radios, and r = [ r 0 , r 1 , . . . , r K − 1 ] T . Sin ce 1 − P ( k ) f measures the oppor- tunistic spectrum utilization of subchann el k , we define th e aggregate oppo rtunistic throughput capacity as R ( W , γ ) = r T [ 1 − P f ( W , γ )] . (18) For a wid band p rimary system, the impact of interfer- ence ind uced by cog nitiv e devices can be char acterized by a relative prior ity vector over the K subchannels, i.e., c = [ c 0 , c 1 , . . . , c K − 1 ] T , where c k indicates the co st incurre d if the p rimary u ser at sub channel k is interfer ed with. As such, we define the aggregate in terferen ce to the primar y u ser as c T P m ( W , γ ) . Con sequently , the spatial-spectral joint de- tection proble m is formu lated as max W , γ R ( W , γ ) (P1) s . t . c T P m ( W , γ ) ≤ ε (19) P m ( W , γ )  α (20) P f ( W , γ )  β (21) where α = [ α 0 , . . . , α K − 1 ] T and β = [ β 0 , . . . , β K − 1 ] T . Finding the exact solution for th e above pro blem is diffi- cult since fo r any k , P ( k ) f ( w k , γ k ) and P ( k ) d ( w k , γ k ) are ne i- ther con vex nor conca ve fun ctions according to (1 6) and (17). T o jointly optimize W and γ , we can show that ( P1 ) can be reform ulated into an equiv alent form with conv ex constrain ts and an objective function lower b ound ed by a con cav e fu nc- tion under the following conditio ns: 0 < α k ≤ 1 2 and 0 < β k ≤ 1 2 , k = 0 , . . . , K − 1 . (22) Throu gh m aximizing the lower bou nd of the ob jectiv e func- tion, we ar e ab le to ob tain a good app roximatio n to th e opti- mal solution of the original proble m. First, we show how to tran sform the no nconve x con straints in (20 ) and (21) into co n vex co nstraints by exploiting th e monoto nicity of th e Q -func tion. Substituting (16) into the constraint (21), we hav e Q − 1 ( β k ) q 2 M w T k w k ≤ γ k σ 2 v − M w T k 1 (23) where Q − 1 ( β k ) ≥ 0 given β k ≤ 1 / 2 . From (17), the con - straint (20) can be expressed as q 2 M w T k [ σ 2 v I + 2 diag( G k )] w k ≤ γ k − M w T k  σ 2 v 1 + G k  σ v Q − 1 (1 − α k ) (24) giv en α k ≤ 1 / 2 and Q − 1 (1 − α k ) ≤ 0 . Sinc e the left-hand side o n the con straint (2 3) is conve x and the right h and side is linear in ( γ k , w k ), (23) defines a conve x set for ( γ k , w k ) . Similarly , (24) is also a con vex constraint. Then, we reform ulate ( P1 ) by introducing a new v ariable µ k = σ v q 2 M w T k [ σ 2 v I + 2 diag( G k )] w k . (25) By d efining γ ′ k = γ k /µ k and w ′ k = w k /µ k , the constrain ts (23) and (24) can be further written as Q − 1 ( β k ) q 2 M w ′ k T w ′ k ≤ γ ′ k σ 2 v − M 1 T w ′ k (26) and γ ′ k − M  σ 2 v 1 + G k  T w ′ k ≤ σ v Q − 1 (1 − α k ) . (27) Note that (27) is actu ally a lin ear con straint in ( γ ′ k , w ′ k ). Th e constraint (19) now becomes 1 T c − K − 1 X k =0 c k Q  γ ′ k − M  σ 2 v 1 + G k  T w ′ k  ≤ ε, (28) which can be shown to be conv ex b y the following result. Lemma 1 If γ ′ k ≤ M  σ 2 v 1 + G k  T w ′ k , then the functio n Q  γ ′ k − M  σ 2 v 1 + G k  T w ′ k  is conc ave in { γ ′ k , w ′ k } . By chang ing the variables W ′ and γ ′ , P ( k ) f ( w k , γ k ) can be expressed as Q "  γ ′ k σ 2 v − M 1 T w ′ k  s σ 2 v + 2 w ′ k T diag( G k ) w ′ k w ′ k T w ′ k # (29) From the the Rayleigh-Ritz theorem [6], we hav e min n | H k ( n ) | 2 ≤ w ′ k T diag( G k ) w ′ k w ′ k T w ′ k ≤ max n | H k ( n ) | 2 (30) Define a ne w function g k ( γ ′ k , w ′ k ) ∆ = Q  γ ′ k σ 2 v − M 1 T w ′ k  q σ 2 v + 2 min n | H k ( n ) | 2  (31) which can be shown to be con vex by the following result. Lemma 2 If γ ′ k ≥ σ 2 v M 1 T w ′ k , then the function g k ( γ ′ k , w ′ k ) is conve x in { γ ′ k , w ′ k } . Since P ( k ) f ( w k , γ k ) ≤ g k ( γ ′ k , w ′ k ) , the objecti ve functio n in ( P1 ) can be lower bo unded b y P N − 1 k =0 r k [1 − g k ( γ ′ k , w ′ k )] , which is a con cave functio n. Thus, an efficient su boptimal method to solve ( P1 ) is to m aximize the lower b ound o f its objective fun ction, i.e., max W ′ , γ ′ N − 1 X k =0 r k [1 − g k ( γ ′ k , w ′ k )] (P2) st . − K − 1 X k =0 c k Q h γ ′ k − M  σ 2 v 1 + G k  T w ′ k i ≤ ε − 1 T c Q − 1 ( β k ) q 2 M w ′ k T w ′ k ≤ γ ′ k σ 2 v − M 1 T w ′ k γ ′ k − M  σ 2 v 1 + G k  T w ′ k ≤ σ v Q − 1 (1 − α k ) . Implied by th e practical c ondition s in ( 22), this problem is a conv ex optim ization problem and can be solved efficiently . 4. SIMULA T IONS Suppose that two CRs c ooperatively sense a multiband OFDM system with 8 subbands. For each subband, it is expected that the opportun istic spectrum utilization is at least 50% , i.e., β k = 0 . 5 , an d the prob ability that the primary user is interfered is at most α k = 0 . 1 . I t is a ssumed that σ 2 v = 1 and M = 100 . Other paramete rs are g iv en in T able 1. Fig . 2 shows result of solving ( P2 ) , which max imizes the oppo r- tunistic throug hput subject to the con straints o n the interfer- ence. W e obser ve that the joint detection results in much higher op portun istic throu ghput than th e alg orithms withou t cooper ation. Note that the incr ease in the throug hput of the joint o ptimization schem e bec omes r ather slow as we relax the interfe rence constrain t because the in teraction b etween γ and W pushes the system to an op erating p oint a t wh ich the throug hput is more limited by β than by ε . T able 1 . Parameter s used in simulation s G (0) .17 .21 .27 .14 .37 .38 .49 .33 G (1) .21 .17 .21 .21 .17 .43 .15 .35 r 356 327 972 806 755 68 720 15 c .71 5.95 3.91 4.21 .44 2.03 .58 2.85 2 2.5 3 3.5 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 Aggregate Interference Aggregate Opportunistic Throughput (kbps) Joint Opt. CR1 CR2 Fig. 2 . Aggregate opportunistic throughpu t capacity vs. the con- straint on the aggreg ate induced interference. 5. CONCLUSION In this p aper, we ha ve pro posed a sp atial-spectral joint d e- tection fr amework fo r distrib u ted wide band spectrum sens- ing in cog nitiv e rad io networks, within which the coope ration among spatially distrib uted cognitiv e radios is o ptimized over multiple fr equency ban ds. By explo iting the inher ent struc- ture of the form ulation, we have developed sub optimal but efficient solutions for the non- conv ex optimization pr oblem. 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