Compressed Wideband Sensing in Cooperative Cognitive Radio Networks

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1 Compressed Wideband Sensing in Cooperative Cognitive Radio Networks Zi Tian Department o Electrical and Computer Engineering Micigan Tecnological University Hougton, MI, U.S.A. ztian@mtu.edu Abstract In emerging cognitive radio (CR) networks wit spectrum saring, te irst cognitive task preceding any dynamic spectrum access is te sensing and identiication o spectral oles in wireless environments. Tis paper develops a distributed compressed spectrum sensing approac or (ultra-)wideband CR networks. Compressed sensing is perormed at local CRs to scan te very wide spectrum at practical signal-acquisition complexity. Meanwile, spectral estimates rom multiple local CR detectors are used to collect spatial diversity gain, wic improves te sensing quality especially under ading cannels. New distributed consensus algoritms are developed or collaborative sensing and usion. Using only one-op local communications, tese distributed algoritms converge ast to te globally optimal solutions even or multi-op CR networks, at low communication and computation load scalable to te network size. Keywords: Cognitive Radio, Compressed Spectrum Sensing, Distributed Fusion, Collaborative Sensing, Consensus I. INTRODUCTION In cognitive radio (CR) networks adopting ierarcical spectrum access, unlicensed secondary CR users dynamically seek transmission opportunities over spectral bands tat are temporarily unoccupied by primary communication systems olding te license o te spectrum [1]. Te irst cognitive task is ence to sense and identiy tose unused spectral oles. Te increasingly popular (ultra-)wideband wireless networks not only oer ig trougput and user capacity or primary communication systems, but also purport pronounced dynamic spectrum access opportunities or secondary CR users. On te oter and, spectrum sensing in te wideband regime also aces considerable tecnical callenges. A primary callenge in wideband sensing stems rom te ig RF signal acquisition costs o current analog-to-digital ardware tecnology. Very ig sampling rates are required by conventional spectral estimation metods wic ave to operate at or above te Nyquist rate. Meanwile, te stringent timing requirements or monitoring te dynamically canging spectrum only allow or a limited number o measurements to be collected or sensing, wic makes it callenging to reliably perorm ig-resolution signal reconstruction. Furtermore, wireless ading constitutes a major actor o perormance degradation to traditional spectrum detection tecniques [2]. A CR user may not be able to accurately sense and detect te transmission o a primary system due to several cannel ading eects, including bot large-scale pat loss and small-scale deep ades tat are random and unpredictable. Wen a missed detection arises, te CR may unwittingly transmit over te same cannels used by active primary users, causing detrimental intererence to legacy services. To provide reliable spectrum sensing at aordable complexity, tis paper presents a distributed compressed sensing ramework or wideband communication networks. First, we recognize tat te wireless signals in open-spectrum networks are typically sparse in te requency domain. Tis is due to te low percentage o spectrum occupancy by active radios a act motivating dynamic spectrum access. For sparse signals, recent advances in compressive sampling ave demonstrated te principle o sub-nyquist-rate sampling and reliable signal recovery via computationally easible algoritms [4] [6]. We develop a compressed sensing tecnique or detecting wideband signals at reduced signal sampling and acquisition costs. Next, we deal wit te wireless cannel ading eects by perorming collaborative usion among multiple spatially distributed CRs in te network. Because cannels ade independently among CRs, collaborative usion enables spatial diversity gain and improves te sensing accuracy. Current researc on collaborative sensing ocuses on small-size oneop networks operating in lat ading cannels; e.g., [3]. In contrast, tis paper contributes to develop distributed usion tecniques or multi-op large networks operating in requency selective ading cannels. Consensus-based tecniques [10], [11] are utilized to derive new distributed usion solutions tat converge to te globally optimal solutions, using only oneop local communications among CR neigbors. Te computational complexity o tese distributed usion tecniques is scalable to te network size. Corroborating simulation results are presented to testiy te eectiveness o te proposed compressed sensing and distributed usion tecniques in identiying and locating te spectral ole opportunities in wireless ading environments. II. SIGNAL MODELING AND PROBLEM STATEMENT Consider a (ultra-)wide requency band tat osts bot primary communication systems and secondary CR users. Dierent rom te compressed sensing sceme in [9], we consider a slotted requency segmentation structure in wic te entire wideband cannel is divided into M non-overlapping narrowband subcannels (a.k.a. slots) centered at m } m=0 M /08/$ IEEE. 1 Tis ull text paper was peer reviewed at te direction o IEEE Communications Society subject matter experts or publication in te IEEE "GLOBECOM" 2008 proceedings.

2 Te locations o tese slots are pre-deined and known, as in multi-band radios and OFDM systems, but teir power spectral levels are unknown and dynamically varying, depending on weter tey are occupied by primary users or not in a particular geograpical region and time. Tose temporarily idle subcannels are termed spectral oles and are available or opportunistic spectrum access by secondary users. Tere are J spatially distributed CRs tat collaboratively sense te wide band in order to identiy te spectral oles, giving rise to te spectrum sensing and detection problem. During eac detection period, we assume or simplicity tat iger layer protocols (e.g., te medium access control layer) can guarantee tat all CRs stay silent suc tat only te primary users are emitting spectral power. Suppose tat tere are I active primary users during te detection interval, wose transmitted signals are denoted by s i (t), i =1,...,I. Ater propagating troug a wireless ading cannel, te signal s i (t) reaces te j-t CR receiver in te orm ij (t) s i (t), were denotes convolution and ij (t) is te cannel impulse response tat is typically requency selective over te wide band. We assume tat te cannels are slowly varying and can be treated as time-invariant during te detection interval. Te received signal at CR j is tus given by r j (t) = I i=1 ij(t) s i (t)+w j (t) were te ambient noise w j (t) is wite Gaussian wit zero mean and power spectral density (PSD) σw. 2 To relect te discretized signal response on te M subcannels, we take an M-point discrete Fourier transorm (DFT) on r j (t), were M is larger tan te cannel memory lengt. Collecting te requency-domain samples into an M 1 vector r (j),weave r (j) = I i=1 D(ij) s(i) + w (j) (1) were D (ij) matrix, and (ij) = diag( (ij) ) is an M M diagonal cannel, s (i) and w (j) are te requency-domain discrete versions o ij (t), s i (t) and w j (t) respectively. Tis signal model can be written in a general orm as r (j) = H (j) s(j) + w (j). (2) In te absence o te cannel state inormation (CSI) (ij) } i at eac CR receiver j, it is useul to lump all te transmitted signals as ollows (I M is an identity matrix): H (j) = H := I M ; := I i=1 D(ij) s(i). (3) Wen eac CR knows te cannels (ij), one can adopt [ ] H (j) := D (1j),, D (Ij) ; [ ] = s := (s (1) )T,,(s (I) T )T, j. (4) At eac CR, spectrum sensing boils down to estimating in (2) rom r j (t). In te absence o te CSI, te CR cannot decouple eac product D (ij) s(i) = (ij) s (i) ( denotes element-by-element multiplication); ence, te estimated in (3) entails te unknown cannel gain. Wit CSI, it is possible to estimate individual sources s (i) }I i=1 rom in (4). It is o interest to investigate te impact o te cannel knowledge on spectrum sensing quality, yet te sensing tecniques we will develop are applicable or bot cases, based on te general expression in (2). Depending on te spectrum saring protocol adopted, te CRs migt not be interested in te signal strengt, but simply want to know wic o te M subcannels are unoccupied spectral oles. All CRs avoid transmitting at any occupied subcannels, but dynamically sare te spectral oles among temselves. In tis case, te spectrum sensing task is reduced to spectrum detection. Te goal is to determine te requency occupancy o primary users by detecting a binary state vector d 0, 1} M 1, wose m-t element is deined by d[m] = 1, i m is occupied, i.e., i: s (i) [m] 0 (5) 0, i all primary users are silent on m Spectrum detection is easible even in te absence o te cannel knowledge. In (3), wen [m] 0, at least one o te sources s i () is emitting on m, and te cannel ading ij ( m ) on tis requency is non-zero. In anoter words, occupied subcannels would correspond to non-zero elements in, wereas te rest zero elements in s(j) relect idle requency opportunities or CRs. Exceptions arise wen a deep cannel ade ij ( m ) makes it inaccurate to detect a non-zero s (i) [m] rom s(j) [m], resulting in missed detection. Hence, te spectral detection problem can be tackled by inding te (non-)zero elements in te noise-ree version o te received signal spectrum r (j). Because detection by one CR receiver is subject to missed detection due to cannel ading, it is necessary or a network o spatially diverse CRs to collaborate during te spectral detection pase. III. COLLABORATIVE COMPRESSED SPECTRUM SENSING To acieve ig-perormance spectrum sensing at practical complexity, tis section contributes to develop collaborative sensing scemes tat utilize compressive sampling in te temporal-requency domain and used detection in te spatial domain. Our scemes consist o two steps: i) compressed spectrum sensing at individual CRs to estimate, and make local decisions ˆd (j) } j i needed, at low sampling complexity; and, ii) collaborative, distributed spectral detection/estimation across te network to collect spatial diversity gain. A. Compressed Spectrum Sensing at Individual CRs Let us now turn to eac o te CR receivers j, j =1,...,J. In tis subsection we drop te index j or notational simplicity. Locally, te goal is to estimate s in (2) given H and r(t). To tis end, we develop a compressive sampling tecnique tat reduces te sampling costs in te wideband regime. Our development bears resemblance to te compressed sensing /08/$ IEEE. 2 Tis ull text paper was peer reviewed at te direction o IEEE Communications Society subject matter experts or publication in te IEEE "GLOBECOM" 2008 proceedings.

3 approac in [9], but wit dierent goals: tis work seeks to estimate te spectral sape s given te slotted subcannel structure, wereas [8], [9] aims to ind te unknown requency locations o occupied spectrum segments via edge detection. Te irst step o compressive sampling is to collect timedomain samples. Motivated by te need to reduce te sampling burden in te wideband regime, we adopt a linear random sampler at eac CR to collect a K 1 time-domain sample vector x t rom r(t), K M, as ollows: x t = S T r t (6) were te M 1 vector r t is te discrete-time representation o r(t) and S is an M K projection matrix. Columns s k } K k=1 o S can be viewed as a set o basis unctions or matced ilters used to collect te time-domain samples, wile te measurements x t [k]} K k=1 are in essence te projection o r(t) onto te bases. Te model in (6) subsumes all sampling scemes yielding linear measurements. For example, S = I M represents Nyquist-rate uniorm sampling, wile reduced-rate linear sampling arises wen K < M. We adopt a simple selection matrix S = S c tat randomly retains K(< M) columns o te size-m identity matrix I M. It amounts to collecting samples on te Nyquist sampling grid but skipping randomly (K M) time instants to reduce te average sampling rate. Wit te K measurements x t = S T c r t, we now estimate te requency response s in (2). Noting r t = F 1 M r,weave x t = S T c F 1 M r = S T c F 1 M H s + w (7) were w = S T c F 1 M w is te noise sample vector tat remains to be wite Gaussian. Because te spectrum utilization by te primary network is low a act motivating dynamic spectrum access at te outset, te unknown vector s is sparse wit only a small number o non-zero elements. Te sparsity measure is given by te l-norm s l, l [0, 2), were te l 0 -norm (l =0) indicates exact sparsity [5]. Recent literature as seen te emergence o signal reconstruction tecniques developed under te compressive sampling ramework [6]. For example, te Basis Pursuit (BP) tecnique [7] solves te ollowing linear convex optimization problem: min s s 1, s.t. x t = S T c F 1 M H s. (8) In general, we use s = CS(x; A) to represent a signal recovery algoritm (e.g., BP, OMP, LASSO etc. [6]) or solving te sparse vector s in a linear regression model x = As + w, were w is Gaussian noise. In tis notation, te local spectral estimation solution to (7) can be expressed as ˆ s = CS(x t ; S T c F 1 M H ). (9) B. Distributed Collaborative Spectrum Detection To collect spatial diversity gain, we irst consider CR collaboration or spectrum detection in te absence o cannel knowledge. Te signal model in (3) becomes relevant to reac te detection decision in (5). Locally at eac CR receiver, a decision on te spectrum state vector d can be made by comparing te local spectral estimate ˆ obtained in (9) wit a decision tresold η j : ( ) ˆd (j) = ˆ η j, j =1,...,J. (10) Te tresold η j can be cosen based on a desired level o probability o alse alarms P a, using te well-known Neyman-Pearson binary ypotesis test rule. Globally at te network level, te suicient statistic or optimal decision usion is te average value c = 1 J ˆd J j=1 (j). Tis can be done by a spectrum controller tat collects all local detection outputs and compute c in a centralized manner. However, centralized sensing is sensitive to node ailure and incurs eavy communication overead. Especially or a multiop CR network, extra routing is needed to convey local CR decisions to te spectrum controller. Our goal ere is to design a distributed and decentralized usion rule tat is scalable to te network size J. Local one-op broadcasting is allowed among neigboring CRs, but multi-op routing is to be avoided. To compute c in a distributed manner, we propose to adopt te average-consensus tecnique [10], wic is an algoritm design tecnique or distributed computing over a large network. To represent an average consensus problem, Let G = (N, E) be an undirected connected grap wit node set N = 1,...,J} and edge set E, were eac edge (j, k) E is an unordered pair o distinct nodes witin te one-op communication range. Let c j (0) := ˆd (j) be a real vector associated wit CR node j at time t =0. Te (distributed) average consensus problem is to compute te average (1/J) J j=1 c j(0) at every node, via local communication and computation on te grap. Speciically, eac node j broadcasts c j (t) to its neigbors N j = k (j, k) E}and updates itsel by adding a weigted sum o te local discrepancies, i.e., te dierences between neigboring node values and its own [10]: c j (t +1)=c j (t)+ w jk (c k (t) c j (t)), j, t (11) k N j were w jk is a weigt associated wit te edge (j, k). Tese weigts are algoritm parameters, and some design rules or w jk } are delineated in [10]. Wit properly designed weigts, it can be guaranteed tat lim c j(t) = 1 J c k (0) = 1 J ˆd (k) = c, j =1,...,J. t J J k=1 k=1 (12) Tus, troug local one-op communications, eac CR obtains te averaged statistic c o te entire multi-op network. Subsequently, eac CR can make te usion decision on d straigtorwardly by comparing c j (t) wit a tresold c t at a suiciently large t. Te coice o c t relects ow conservative te network is in taking spectrum opportunities. A conservative CR network decides te presence o a primary user as long as one o te J CRs claims detection, wic corresponds to c t =1/J. A more aggressive network may take a majority vote by setting c t =1/ /08/$ IEEE. 3 Tis ull text paper was peer reviewed at te direction o IEEE Communications Society subject matter experts or publication in te IEEE "GLOBECOM" 2008 proceedings.

4 Putting togeter, te distributed spectrum detection algoritm can be summarized below: s1) eac CR makes its local decision ˆd (j) via compressed sensing in (9) and tresolding in (10), j =1,...,J; s2) eac CR sets c j (0) = ˆd (j), j =1,...,J, and collaboratively iterates troug (11) until convergence c j (t) = c; s3) eac CR makes te usion decision via tresolding ˆd =(c j (t) c t ). C. Distributed Collaborative Spectral Estimation Wen eac CR acquires its own CSI H (j) in (4), it is sensible or te network to use all te local measurements x (j) t } J j=1 in (7) to estimate te common transmit spectrum vector s in (4), giving rise to te collaborative spectral estimation problem. Cannel occupancy decisions made rom s is immune to cannel ading eects, at te expense o CSI estimation eorts at local CRs. Wen a usion center is present, a globally optimal usion estimate o s can be obtained by stacking all measurements into a KJ 1 vector and solving te corresponding linear regression problem as ollows: x (1) ṭ (S (1) M ˆ s = CS. ; H(1). (13) x (J) t (S (J) M H(J) Apparently, tis usion ormula is costly to implement, because te usion center needs to know, in addition to te measurements, te sampling matrices S (j) c } j and cannel matrices H (j) } j rom all CRs, not mention te computational load. To overcome te implementation callenges in centralized usion, we develop a distributed sensing algoritm using consensus tecniques in conjunction wit te alternating direction metod o multipliers. We deine local copies o s and constrain tem to consent wit one-op neigbors, as in [11]. Using te BP algoritm or illustration, eac CR locally carries out te ollowing optimization: (j) min s s.t. 1, x (j) t =(S (j) M H(j) s(j) = s (k), k N j (14) Here CR j ears te broadcasts o s (k) rom all one-op neigbors k N j and uses te constraints (14) to enorce consensus. Tere is no need to excange CSI or sampling matrices among CRs. Wile (14) is conceptually illuminating, we still need to design a distributed algoritm tat is amenable to implementation. Te distributed consensus algoritm in (11) is not directly applicable: it simply perorms data averaging, wereas our spectral estimation problem in (14) requires linear regression on a sparse vector. We combine consensus averaging wit sparsity-constrained linear regression to develop a new iterative procedure or distributed collaborative sensing, as ollows: s1) at t =0, eac CR initializes its local spectral estimate as c j (0) = SC(x (j) t ;(S (j) M H(j) ), j =1,...,J; s2) eac CR locally iterates troug ollowing steps over t: a) compressing sensing on enanced measurements: ( [ ] [ ]) ˆ (j) x (t) =CS t (S (1) c ) ; T F 1 M H(1) c j (t) I M (15) b) consensus averaging: set c j (t) =ˆ (t) and update c j (t+1) = c j (t)+ k N j w jk (c j (t) c k (t)); c) broadcasting o c j (t +1) to one-op neigbors; s3) at t, eac c j (t) converges to te globally optimal ˆ s in (13), j; ten, eac CR detects te requency occupancy state via ( denotes logical OR operation and η t is te detection tresold): ˆd[m] = i ( ˆ s [im + m] η t ), m. (16) In (15), an update on s (j) (t) is sougt not only to satisy te measurement equation in (7), but also to enorce consensus wit te most recent average value troug te constraint c j (t) =I M s (j) (t). In (16), eac CR separates te I transmitted sources s (i) }I i=1 and claims an occupied subcannel as long as tere exists one source i wit a non-zero element s (i) [m] 0. IV. PERFORMANCE SIMULATIONS We consider a wide band o interest tat is partitioned into M = 32 equal-bandwidt subcannels. Primary users randomly occupy some o te subcannels, wit an average spectrum occupancy ratio o 20%. Te wideband cannel experiences requency-selective ading, wic is modeled as a multipat cannel wit N p time-delayed taps and independent Rayleig ading gains on tese taps. Te signal to noise ratio (SNR) is deined to be te signal energy o te wideband signal over te entire spectrum, scaled by te power o te wite noise. Te compression ratio K/M relects te reduced number o samples used wit reerence to te number M needed in ull-rate Nyquist sampling. For te spectral ole detection problem, perormance metrics o interest include te probability o detection P d and te probability o alse alarms P a, wic we average over all subcannels as ollows: P d = E d T (d = ˆd) 1 T d }, P a = E (1 d) T (d ˆd) } M 1 T d were d 0, 1} M 1 is te true requency occupancy state and 1 denotes te all-one vector. CR collaboration enables diversity gain, wic we quantiy via simulations in Fig. 1. Te detection usion sceme is used or illustration, in te absence o any cannel knowledge. It /08/$ IEEE. 4 Tis ull text paper was peer reviewed at te direction o IEEE Communications Society subject matter experts or publication in te IEEE "GLOBECOM" 2008 proceedings.

5 Probability o Detection Probability o Detection #CR = 1, 5, #CR = 1, 3, 5, SNR (db) Fig. 1. Probability o detection or various number o collaborating CRs, or P a = Solid lines: no compression; das lines: compression at K/M = 50%. can be sown tat te detection perormance improves as te number o collaborating CRs J increases. Fig. 1 also illustrates te eect o compressive sampling. Encouragingly, te compressed sensing sceme is able to detect te primary users at strong compression (K/M =0.5), wen te number o samples used (K) is muc less tan tat required by Nyquist rate sampling (M). Wen K/M increases, te robustness to noise improves, so does te detection perormance. It is te inerent spectral occupancy sparsity in te transmitted signals tat enables reduced-rate sampling, wic in turn alleviate te sampling burden and energy consumption o CRs in te wideband regime. On te oter and, Compression incurs perormance degradation, especially in te presence o ambient noise and cannel ading. It is observed tat compression can oset some o te spatial diversity gain enabled by collaboration, but te degradation is relatively small. Tese observations are corroborated by te receiver operating caracteristics (ROC) depicted in Fig. 2. Interestingly, wen te number o collaborating CRs is large, te degradation caused by compression becomes trivial. Te developed distributed algoritms in Section III converge quite ast. Te distributed sensing algoritm in Section III- C converges witin 10 iterations in our simulations. Te distributed detection algoritm in Section III-B converges even aster, typically in 2-3 iterations. V. SUMMARY AND FUTURE WORK Tis paper presents distributed collaborative sensing tecniques in combination wit local compressed sensing to save te overall sensing costs or cognitive radios. Consensus tecniques or bot spectral detection usion and spectral estimation usion are developed, wic converge to teir respective globally optimally solutions using only distributed computation and local communications among one-op neigbors Probability o False Alarm Fig. 2. Receiver Operating Caracteristic (ROC), or SNR = 0 db. Solid lines: no compression; das lines: compression at K/M = 60%. In general, te collected diversity gains vary, depending on te amount o compression and weter te cannel knowledge is available. Wile te CSI knowledge enances sensing perormance, it also requires local cannel estimation wic may be costly or even diicult in multiuser wireless systems. Hence, te cost o cannel estimation needs to be justiied by te sensing perormance gain it oers in collaborative usion. Furtermore, te sensitivity o te usion results to cannel estimation errors need to quantiied in order to properly allocate systems resources to te cannel estimation task. Analysis on te teoretical spatial diversity gain and te value o cannel knowledge will be investigated in uture work. REFERENCES [1] Facilitating Opportunities or Flexible, Eicient, and Reliable Spectrum Use Employing Cognitive Radio Tecnologies, FCC Report and Order, FCC-05-57A1, Marc [2] H. Urkowitz, Energy detection o unknown deterministic signals, Proc. o te IEEE, vol. 55(4), pp , April [3] A. Gasemi, E. Doura, Collaborative spectrum sensing or opportunistic access in ading environments, Proc. IEEE Symp. on DySPAN, Baltimore, MD, Nov [4] E. J. Candes, J. Romberg, T. Tao, Robust Uncertainty Principles: Exact Signal Reconstruction From Higly Incomplete Frequency Inormation, IEEE Trans. Inormation Teory, vol. 52, no. 2, pp , Feb [5] D. L. Donoo, Compressed Sensing, IEEE Trans. on Inormation Teory, vol. 52, pp , April [6] l 1 Magic, a collection o MATLAB routines or solving te convex optimization programs central to compressive sampling. ttp:// [7] S. S. Cen, D. L. Donoo, and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., vol. 20, no. 1, pp , [8] Z. Tian, and G. B. Giannakis, A Wavelet Approac to Wideband Spectrum Sensing or Cognitive Radios, Proc. o Intl. Con. on CROWNCOM, Mykonos, Greece, June [9] Z. Tian, and G. Giannakis, Compressed Sensing or Wideband Cognitive Radios, Proc. o IEEE Intl. Con. on Acoustics, Speec and Signal Processing (ICASSP), Vol. IV, pp. IV , Honolulu, April [10] L. Xiao, S. Boyd, and S.-J. Kim, Distributed Average Consensus wit Least-Mean-Square Deviation, Journal o Parallel and Distributed Computing, 67(1):33-46, [11] I. D. Scizas, A. Ribeiro and G. B. Giannakis, Consensus in Ad Hoc WSNs wit Noisy Links - Part I: Distributed Estimation o Deterministic Signals, IEEE Trans. on Signal Processing, vol. 56, no. 1, pp , January /08/$ IEEE. 5 Tis ull text paper was peer reviewed at te direction o IEEE Communications Society subject matter experts or publication in te IEEE "GLOBECOM" 2008 proceedings.