IFOST 2010 Proceedings
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A Novel and Efficient Mixed-Signal Compressed
Sensing for Wide-Band Cognitive Radio
Le Thanh Tan*...
IFOST 2010 Proceedings
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IFOST 2010 Proceedings
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IFOST 2010 Proceedings
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IFOST 2010 Proceedings
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Algorithm 1 The jointly recovery M-SP algorithm
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IFOST 2010 Proceedings
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A novel and efficient mixed-signal compressed sensing for wide-band cognitive radio

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In cognitive radio (CR) networks, unlicensed (cognitive) users can exploit the licensed frequency bands by using spectrum sensing techniques to identify spectrum holes. This paper proposes a distributed compressive spectrum sensing scheme, in which the modulated wide-band converter can apply compressed sensing (CS) directly to analog signals at the sub-Nyquist rate and the central fusion receives signals from multiple CRs and exploits the multiple-measurements-vectors (MMV) subspace pursuit (M-SP) algorithm to jointly reconstruct the spectral support of the wide-band signal. This support is then used to detect whether the licensed bands are occupy or not. Finally, extensive simulation results show the advantages of the proposed scheme. Besides, we also compare the performance of M-SP with M-orthogonal matching pursuit (M-OMP) algorithms.

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A novel and efficient mixed-signal compressed sensing for wide-band cognitive radio

  1. 1. IFOST 2010 Proceedings 1 A Novel and Efficient Mixed-Signal Compressed Sensing for Wide-Band Cognitive Radio Le Thanh Tan*, Hyung Yun Kong* *School of Electrical Engineering University of Ulsan, San 29 of MuGeo Dong, Nam-Gu, Ulsan, Korea 680-749 Email: {tanlh.hkong}@mail.ulsan.ac.kr Abstract-In cognitive radio (CR) networks, unlicensed (cogni­ tive) users can exploit the licensed frequency bands by using spec­ trum sensing techniques to identify spectrum holes. This paper proposes a distributed compressive spectrum sensing scheme, in which the modulated wide-band converter can apply compressed sensing (CS) directly to analog signals at the sub-Nyquist rate and the central fusion receives signals from multiple CRs and exploits the multiple-measurements-vectors (MMV) subspace pursuit (M­ SP) algorithm to jointly reconstruct the spectral support of the wide-band signal. This support is then used to detect whether the licensed bands are occupy or not. Finally, extensive simulation results show the advantages of the proposed scheme. Besides, we also compare the performance of M-SP with M-orthogonal matching pursuit (M-OMP) algorithms. Index Terms-Wide-band spectrum sensing, distributed com­ pressive spectrum sensing, MMV subspace pursuit algorithm, cognitive radio, compressed sensing, modulated wide-band con­ verter. I. INTRODUCTION Cognitive Radio (CR) provides a new paradigm to exploit unoccupied wireless spectrum of the licensed users (LUs) [1], [2]. Moreover, the popular wide-band applications are nowa­ days increasingly interested since they pronounce dynamic spectrum access opportunities for secondary users (SUs). However, spectrum sensing in the wide-band regime also associates with considerable technical challenges. Normally, the RFfront-end can either do narrow-band sensing via a bank of passband filters or use a wide-band RFfront-end followed by DSP blocks to sense the whole bandwidth. Nonetheless, the former has strict constraints of designing filters whereas the latter needs a high-speed analog-to-digital converter (ADC). Recent work in compressed sensing (CS) reveals that a signal having a sparse representation in one basis can be recoveredfrom a small number of projections onto a second basis that is incoherent with the first [3], [4]. Especially, a compressed sampling approach can get the sparse signal at the rates lower than Nyquist sampling; signal reconstruction, which is a solution to a convex optimization problem called min-ll with equality constraints, makes use of the basic pursuit (BP) or some modified methods such as orthogonal matching pursuit (OMP), tree-based OMP (TOMP) [5], [6]. Besides, authors in [5] present a model of a single wide-band CR that uses CS based spectrum sensing schemes. However, this work bases on digital approaches that generally require full-rate sampling before spectrum estimation. Thus, when it comes to practical implementation, there are several issues to be considered. The first problem is that the random projections in CS are done over a discrete-time signal that is obtained by sampling the continuous-time signal at Nyquist rate. So we must avoid to work at Nyquist rate by applying CS directly to the analog signal. Moreover, a model must be appropriate to a practical application by reducing a hardware complexity at a receiver. To cope with the above questions, authors in [7] propose a parallel segmented compressed sensing structure that can directly apply CS to analog signals. Specifically, a received analog signal is time-windowed into segments. Then CS is independently applied to each segment by the bank of mixers and integrators. And OMP is used to jointly recover a signal. Nonetheless, this model developsfrom random demodulator (RD) [8] that has constraints on huge-scale CS matrices, even though a finite-precision of matrix entries also lead to another issue called undesired high correlations between the columns of the CS matrix. Thus, the performances of CS algorithms are dramatically degraded in comparison with an ideal infinite­ precision. Overall, the bank of RD channels [7] duplicates analog issues and a computational complexity is not much improved. In this paper, we design a distributed compressive spectrum sensing model in wide-band environment. Firstly, to acquire analog multiband signals at sub-Nyquist rate, a sampling stage exploits the modulated wide-band converter (MWC), which consists of a bank of modulators and lowpass filters. Then a recovery stage which makes use of the blind-reconstruction approach using the continuous-to-finite block [9], [10] to find the support for signal recovery. It is interesting that this support contains the band edges to determine whether the bands are occupied or not instead of fully reconstructing original signals; thus, we can save a lot of time. To guarantee the exact recovery support S, the reconstruction problem is solved by using the multiple-measurements-vectors (MMV) subspace pursuit (M­ SP) algorithm, which is modifiedfrom the subspace pursuit [11] to apply for the MMV problem [12]. Moreover, based on [13], the novel version of the distributed CS algorithm using M-SP is proposed to jointly and efficiently reconstruct the spectral support at the central fusion. Finally, we compare the performance of our proposed scheme with that of models using the standard MMV orthogonal matching pursuit (standard M­ OMP) and the regularized MMV orthogonal matching pursuit 978-1-4244-9037-0/101$26.00 ©2010 IEEE
  2. 2. IFOST 2010 Proceedings ret) <1>0(1) <1>" JI) Fig. I. Block diagram of compressed sensing of analog signals. T Fig. 2. Demonstration of overlapping windows. (regularized M-OMP) in order to demonstrate the gainsfrom a spatial diversity and make use of the joint sparsity. The rest of this paper is organized as follows. Section II presents compressive spectrum sensing scheme based on RD [7]. Compressive spectrum sensing with modulated wide-band converter is presented in Section III. Section IV proposes the distributed compressive spectrum sensing. Section V demon­ strates simulation results and discussions. Finally, concluding remarks are given in Section VI. II. PRELIMINARIES A. Compressive spectrum sensing scheme based on [7] The compressive spectrum sensing approach presented in [7] bases on the RD [8] which is restricted to discrete multitone signals and has a high computational load. To make this method to work in practice, authors in [7] consider parallel segmented compressed sensing (PSCS) structure as depicted in Fig. 1. Specifically, the received analog signal x(t) is K- sparse over some basis IJf as in (1) for t E [0,T]. P-I X (t)= LanWn (t)= lJfa, (1) s=o where IJf = [wo (t),WI(t),..., Wp-I(t)] consists of P basis elements, a = lao (t),al (t),..., ap-I(t)] has only K « P non-zero elements. We also assume that channel state information (CSI) needs to be made available at the receiver, thus the received signal r(t) can be evaluated by r(t)= x(t)+n(t), (2) where n(t) is an additive white Gaussian noise (AWGN). In this structure, the received signal r(t) is divided into M periods rm(t)= r(t)Wm(t) I��� with a duration time Tc by the window function Wm(t) as shown in Fig. 2. At every duration T, L = MN samples are received by applying a random projection to each segment through N 978-1-4244-9036-3/10/$26.00 ©2010 IEEE ret) p,(t) Fig. 3. Modulated wide-band converter model. parallel channels, i.e., the m-th sample at the n-th channel is calculated by mTm+Tc YmN+n=(rm(t),<I>mN+n(t))=Jr(t) <I>;;'N+n (t)dt (3) mTm where the distribution of <I>;;'N+n (t) can be Gaussian, Bernoulli ensembles and others. It is clearly observed that the PSCS model can reduce the number of parallel ADC channels, i.e., the number of measurements, while the re­ maining incomplete information called salient volume can be jointly reconstructed to the original signal. Because the window function damages heavily at the edges, overlapping is proposed to diminish the loss by averaging the errorfrom reduced data. For signal reconstruction, all received measurements are stacked into a M x N matrix Y and given as [ -T -T -T ]Y= YO,YI" " ,YM-I (4) where Ym = [YmN,YmN+I,...,YmN+N-I]T is the vector of measurements of the m-th segmentfrom N channels. The reconstruction matrix VI= {vf,j}L x P is defined as V�N+n,s= (ws,m(t),<I>mN+n(t))=l;s,m(t) <I>;;'N+n (t)dto (5) where Ws,m(t) = Ws(t)Wm(t).And reconstruction is imple­ mented by using OMP algorithm. It is important to point out that this method does not improve too much to efficiently implement in practical application due to remaining the highly computational burden in digital domain. III. COMPRESSIVE SPECTRUM SENSING WIT H MODULATED WIDE-BAND CONVERTER In this section, we examine the MWC model which applies CS algorithms to the traditionalfrequency-domain [9,10,14]. A. Infinite-measurement-vectors (IMV) modelfor Compressed sampling Fig. 3 shows that the MWC has m channels and in the i-th channel, the input signal r(t) is multiplied by the periodic 2
  3. 3. IFOST 2010 Proceedings 3 Zo (ll) Them*M ret) mixing matrix C ZM I (Il) exp(J27r(M-1)1 IT) Fig. 4. The modified model of modulated wide-band converter. o , lfN� IlvI ' , f� Zo (II) Z'_l (II) Z, (II) Z'+l (II) ZM_l (11) Fig. 5. The relation between the Fourier transform of the input multiband signal r(f) and the lowrate sequences Zt (n). waveform Pi(t), lowpass filtered, and then sampled at the rate liT, where the sampling interval T equals the period of waveforms. For simplicity, we choose the rate liT � B. Fig. 4 depicts the equivalent model to acquire the multi­ band signals. M is chosen as the smallest integer such that M � TfNyq and stands for the compression ratio. In each channel, r(t) is modulated and then filtered to create the sequence Zl(n) as in Fig. 5. For each time-point, the vector Z(n) = [zo(n),. . . , ZM-I(n)]T is compressed into the output vector Y(n)= [YI(n),...,Ym(n)]T as follows: y(n)= Cz (n) (6) The equivalence to Fig. 3 holds due to the periodicity of waveforms Pi(t). Since Pi(t)= Pi(t + T) for all t E R, we have the Fourier expansion +00 ( 2 )pdt)= L Cilexp j;It l=- = (7) where Cil= -f; 1:[Pi(t)exp (-j'2,;It)dt.Choosing the matrix C of Fig. 4 such that its il-th entry is equal to the Fourier coefficient Cil results in the desired equivalence Ydn)= (r(t)pdt)) * h(t)It=nT = Mfil ((r(t)exp (-j�It))* h(t)It=nT) (8) l=O Conceptually, MWC shifts the mixing matrix C = {Cil} into the analog domain, such that each channel realizes a single row of C in analog hardware. B. Signal recovery For signal reconstruction, [9], [10], [14] and references therein suggest an efficient method based on the fact that the bands occupy continuous spectrum intervals, i.e., the vectors z (n) for different time instances share a common spectral Y(II Lowrate reconctruction Frameconstruction Q= Ly(n)yT(n) SolveQ= v"v V Reconstruction joint supportS SolveV=CU s=Usupp(u, ), Fig. 6. The model of signal reconstruction. support called a nonzero location set. Thus, instead of solving separately for every n, we construct a basis Vfrom a set of consecutive sample sets y (n). It is shown in [10] that any suchframe has the same support as the joint sparsity of z(n). The Continuous-to-Finite block (CTF), which is depicted in Fig. 6, implements this principle. The first task in CTF constructs aframe for y (n), which involves computing Q = LY(n)yT(n), n (9) In the presence of noise, (9) is modified as Q' = Q + (T2I and slight rate increase is required. Since Q is positive semi-definite, it can be decomposed as (10) And the joint support is inferredfrom a MMV CS system V= cu, (11) where the sparse matrices have a few nonzero rows and the support S is determined by merging the supports of all columns iTi of the sparsest matrix iT. Once the support Sis found, the pseudo-inverse c1 is computed and then used to recover zs (n).The notation Cs means the column subset of C indicated by S, and similarly zs (n) is the relevant vector subset. The last computation in CTF is solving the MMV system (11) for the sparsest matrix iT which is presented in Section IV. In terms of digital processing, the sequences Zl(n) are first zero padded: { Zl(n) n= nM,n E Z Zl (n) = 0 elsewhere (12) Then, Zl (n) is interpolated to the Nyquist rate, using an ideal (digital) filter. Finally, the interpolated sequences are modulated in time and summed: x (n) = x (nT)= L (Zl(n) * hI(n))exp (21rlfpnT). (13) lES However, in this work, we just consider spectrum sensing so we need to find the support S and Olnit the last step of finding x (n). 978-1-4244-9037-0/101$26.00 ©2010 IEEE
  4. 4. IFOST 2010 Proceedings Yes Quit ilerat'ions No , S goodetlough? Sl ;;;;;;SI-l U {Kil1diceswith Sf = {Kindiceswith the largestpro).coefficients } Fig. 7. Iterations in the proposed M-SP algorithm for K-sparse signals. IV. DISTRIBUTED COMPRESSED SPECTRUM SENSING Let rj(t) be the wide-band analog baseband signal re­ ceived at the j-th CR sensing receiver. Each CR sensing receiver processes the received signal to obtain an Y j = [Yj,l (n),...,Yj,m (n)] of the compressed signal and con­ structs theframe Vj , as in the CS acquisition step described in Section III, theseframes are then sent to the fusion center. The fusion center applies M-SP to jointly reconstruct the spectral support S, which is then used to determine spectrum occupancy. A. Overview of MMV SP (M-SP) approach SP [11] has a provable reconstruction capability comparable to that of linear programming (LP) methods and exhibits the low reconstruction complexity of matching pursuit techniques for sparse signals. We now describe the modified SP algorithm called M-SP to be used for reconstruction of a model with J CRs. We write C in terms of its columns as C= [ C 1 C2 ... CM ] (14) The next step is solving the problem (11) to find the spectral support S by using M-SP. We set the truncation for the subspace Cs of the m x M matrix C as in Definition 1 and the space spanned by the columns of Cs is denoted by span(Cs). D£ifinition 1 (Truncation): Let CE Rmx M and UE RM xr. For an index set S C {I, ...,M}, the matrix Cs consists of the columns of C with indices iE S and the matrix Usdenotes the matrix formed by entries with row indices i E S. Besides, the m x M matrix C also satisfies the RIP as in Definition 2. D£ifinition 2 (RIP): [15], [16] The matrix C E nmx M satisfY the RIP with the parameters (K,8) for K :::; m,0 :::; 8 :::; 1, if for all index sets S C {I"" ,M} and for all qE nisI, (1 - 15K) IIqll� :::; IICsqll� :::; (1+15K) IIqll� (15) and (1 - 15K) :::; Amin (cICs) :::; Amax (cICs) < (1+8K). As in [4], most known families of matrices satisfYing the RIP property are random. If the random matrices whose 978-1-4244-9036-3/101$26.00 ©2010 IEEE entries follow the Gaussian distribution with zero mean and variance 1/M, the RIP for a randomly chosen matrixfrom such ensembles holds with overwhelming probability when­ ever K :::; 1Jog(';}jm) , where 1J is a function of the RIP constant. However, when the reconstruction principle holds for the Fourier ensemble, the random matrices satisfY the RIP with overwhelming probability, given that K :::; 1J[Og�)J6' There also exists a relation between LP reconstruction and the RIP property. The h-LP approach can reconstruct all K­ sparse signals if the RIP must be satisfied with constants 8K, 82K and 83K, which have a condition 15K+82K+83K < 1. However, in [4], the authors improved the above condition to 82K < v'2 - 1. The projection of V onto the subspace span(Cs) is denoted by Vp and can be computed by Vp= proj(V,Cs):= CsC1V. (16) Note that C1= (cICs)-1cI is the pseudo-inverse of the matrix Cs and the subscript T denotes matrix transposition. Corresponding to the projection matrix, the projection residue matrix Vr is defined as Vr= resid(V,Cs):= V -Vp. (17) Fig. 7 illustrates the schematic diagram of iterations in M-SP. Specifically, an estimate is recalculated and updated during each iteration. An index, which is considered reliable in some iteration but shown to be wrong at a later iteration, can be added to or removedfrom the estimated support set at any stage of the recovery process. The expectation is that the recursive refinements of the estimate of the support set will lead to subspaces with strictly decreasing distancefrom the measurement matrix . The following subsection represents the algorithm to solve the above problem. B. The jointly recovery M-SP algorithm The main difference between M-SP and M-OMP is the method to generate S, that is the estimate of the correct support set SaRI. Note that in this algorithm, to update the spectral support, we use the following method: Each j-th CR (j = 1,00', J), at the l-th iteration, we find the nonzero rows in the matrix Aj. Firstly, we use 12-norm to evaluate each row of Aj: i'J-1= IIRowsof [Aj] II= [ t;J.1 tl-1j,2 r (18) and then calculate i'1-1=average{i'J-1}= [ ti-1 tl-12 ... r (19) J where t�-l = J 2: tt/. Finally, the nonzero rows are J=l specified as K indicescoresponding }tothelargestmagnitude entries in thevectori'1-1 (20) 4
  5. 5. IFOST 2010 Proceedings 5 Algorithm 1 The jointly recovery M-SP algorithm Input: A common dictionary m x M matrix C. A m x J data matrix V = [ VI (n) V2 (n) ... VJ (n) 1 re­ ceivedfrom J CR signalsYj = [Yjdn),...,Yjm (n)] , j = 1,. . . , J. Output: The M x J estimated signal Zs lZs,dn) Zs,2(n) ... zs,J (n) ], the estimate of t e correct support set S. Procedure: 1) Initialization: For each j-th CR 0 = 1,..., J), we have the correlation matrix Aj = CTVj then apply (18), (19), (20) to calculate TjO , TO, So, respectively. The residue vector of the projection for the j-th CR is �,j= resid(Vj,Cso) 2) Iteration: At the I - th iteration, we perform the following steps. Step 1: For each j-th CR 0 = 1,..., J), we have the correlation matrix Aj= CTV��:/ then apply (18), (19), (20) to calculate fj-I, f1-1, 51-1, respectively. And then get an updated set 51 = SI-1 U 51-1. Step 2: For each j-th CR, we set the projection coeffi­ cients Up,j= C�,Vj , then apply (18) withAj = Up,j, (19), (20) to calculate TJ, Tl, Sl, respectively. The residue vector of the projection for the j-th CR is V�,j= resid(Vj,CsI). 3) Termination test: The M-SP iteration is terminated when satisfYing min 11;�,jII�_� min 11.v��:/1.12' � = 1, 2,. . . , J. Then let S = S and qUIt the IteratIOn. If the limit is not reached then increased I and return to iteration. 4) Store the results: The estimated signal Zs,j, satisfYing Zs,j Irows{{I,... ,M}-SI} = 0 and Zs,j lSI = C1,yj· The estimate of the correct support set S = Sl. So the algorithm is summarized as Algorithm 1. It is noted that we can improve the performance by a backsolve, i.e., when reaching the termination, we can check the neighbor locations of the entries in the spectral support set S and perform the some iterations to get the exact result. V. SIMULATION RESULTS AND DISCUSSIONS To evaluate the performance of MWC model for compres­ sive wide-band spectrum sensing, we simulate the system on signals contaminated by white Gaussian noise. The signal consists of N = 8 bands, each of width B = 50 MHz, and is given as N X (t) = L VEiBsinc(B(t - 7i))exp (21fJit) (21) i=1 where sinc(x) = sin(x)/x. The energy coefficients are equal to Ei 1 and the time offsets are 7i {0.4,0.7,0.2,0.4,0.2,0.3,0.5, 0.3}f-Lsecs. The carriers fi 0.8 1;' 0.6" 5 � � 04 " -if; 0.2 10 20 30 -T- Standard M-OMP, 1CR ____ Regularized M-OMP, 1CR ,-,-, M-SP 2CRs - - - M-SP 5CRs 40 50 60 70 Sampling channels (m) 80 Fig. 8. Recovery rate with different in at SNR � 10 dB in the case of 1 CR, and varying SNRs from 5dB to l5dB in the case of multiple CRs. 20 15 10 iii' 5 ::s -5 -10 -15 20 30 40 50 60 Sampling chalmels (orn) 70 80 Fig. 9. Regularized M-OMP recovery rate vs both sampling channels and SNRs in the case of 1 CR. 20 15 5 o -5 10 20 Lower bounds of perfect reconstJuction range 30 40 50 60 Sampling channels (m) 70 0.8 0.6 0.4 0.2 80 Fig. 10. M-SP recovery rate vs both sampling channels and SNRs in the case of 1 CR. are chosen randomly in the range [0,fNyq] with fNyq = lOGHz and the sampling stage is fs = fp = fNyq/195 = 51.3MHz. The number of channels is set to m = 80 (m � 2N), where each mixing function Pi(t) alternates sign at most M= Mmin = 195 times. The continuous signals are 978-1-4244-9037-0/101$26.00 ©2010 IEEE
  6. 6. IFOST 2010 Proceedings represented by a grid of 50001 points in the interval 0.5p,secs, the time resolution is T15. Gaussian noise is added and scaled so that the test signal has the desired signal-to-noise ratio (SNR), where the SNR is defined to be 10log (1111:11 11:)• The support of the input signal is reconstructed from in :::; m channels. To reduce IMV system to MMV system, we follow the steps in Fig. 6. Then, theframe V is constructed and the MMV is solved using the M-SP and compared with the standard M-OMP or the regularized M-OMP [12], [17], [18], [19]. It is noted that the reconstruction is successful only if the estimate support S contains the true support SaRI, i.e., the estimate support is allowed to have some additional entries. Fig. 8 illustrates the recovery rate for various numbers in of channels at SNR = 10 dB in the case of 1CR, and at varying SNRsfrom 5dB to 15dB in the case of multiple CRs. The results show that the standard M-OMP has low reconstruction rate especially in low SNR regime (the figure in low SNR regime is omitted due to the limited number of pages). Although we apply some well-known techniques to improve M-OMP, the performance of the regularized M-OMP is lower than that of M-SP. So M-SP is appropriate to apply for jointly reconstructing signal in the case of CR networks. Especially, in the low SNR regime, the correct recovery corresponding to 1 CR case is accomplished when using in ;::: 40 for a stable recovery. However, the performances can be significantly improved by reducing the number of sampling channels when applying M-SP to jointly reconstruct the spectral supportfrom multiple CRs. Furthermore, Fig. 9 and Fig. 10 illustrate the perfect recovery rates vs the number of sampling channels and SNRs. It is easily observed that the perfect-recovery-rate area of M-SP is larger than that of the regularized M-OMP. In comparison with model [7], in this wide-band scenario with N = 8 bands of width B = 50 MHz and fNyq = 10 GHz, we can get K= NB = 400.106tones.In this setting W = 1010, .pI has about [8] R � 1.7Klog(WjK+ 1) = 2.215· 109 rows, resulting in a huge scale CS system. Therefore, the bank of RD channels [7] can't significantly diminish the computational complexity. VI. CONCLUSION In this paper, we presented a distributed compressive spec­ trum sensing scheme using MWC for wide-bandCR networks, in which all operations such as sampling, processing the information contents and recovering the input involve only low-rate computations. Especially, M-SP is exploited to jointly reconstruct the spectral support which is used to detect whether the spectral bands occupy or not. The simulation results prove the prominent advantages of the MWC model using M-SP. ACKNOWLEDGMENT This research was supported by Basic Science Research Program through the National Research Foundation of Ko­ rea(NRF) funded by the Ministry of Education, Science and Technology(No. 2010-0004865). 978-1-4244-9036-3/101$26.00 ©2010 IEEE REFERENCES [I] S. Haykin, "Cognitive radio: Brain-empowered wireless communica­ tions", IEEE Journal on Selected Areas in Communications, voL23, no.2, pp. 201-220, 2005. [2] S. Haykin, D. 1. Thomson and 1. H. Reed, "Spectrwn sensing for cognitive radio", Proceedings of the IEEE, voL97, no.5, pp. 849-877, 2009. [3] D. L. Donoho, "Compressed sensing", IEEE Transactions on Informa­ tion Theory, voL52, noA, pp. 1289-1306, 2006. [4] E. J. Candes and T. Tao, ''Near-optimal signal recovery from random projections: Universal encoding strategies?", IEEE Transactions on Information Theory, voL52, no.12, pp. 5406-5425, 2006. [5] T. Zhi and G. B. Giannakis, "Compressed sensing for wideband cog­ nitive radios", in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. IV-1357-IV-1360, 2007. [6] M. F. Duarte, M. A. Davenport, M. B. Wakin and R. G. Baraniuk, "Sparse signal detection from incoherent projections", in IEEE In­ ternational Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. ill-ill, 2006. [7] Y. Zhuizhuan, S. Hoyos and B. M. Sadler, "Mixed-signal parallel com­ pressed sensing and reception for cognitive radio", in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3861-3864, 2008. [8] J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg and R. G. Baraniuk, "Beyond nyquist: Efficient sampling of sparse bandlimited signals", IEEE Transactions on Information Theory, voL56, no.l, pp. 520-544, 2010. [9] M. Mishali and Y. C. Eldar, ''Blind multiband signal reconstruction: Compressed sensing for analog signals", IEEE Transactions on Signal Processing, voL57, no.3, pp. 993-1009, 2009. [10] M. Mishali and Y. C. Eldar, "Reduce and boost: Recovering arbitrary sets of jointly sparse vectors", IEEE Transactions on Signal Processing, voL56, no.lO, pp. 4692-4702, 2008. [II] D. Wei and O. Milenkovic, "Subspace pursuit for compressive sens­ ing signal reconstruction", IEEE Transactions on Information Theory, voL55, no.5, pp. 2230-2249, 2009. [12] S. F. Cotler, B. D. Rao, E. Kjersti and K. Kreutz-Delgado, "Sparse solutions to linear inverse problems with multiple measurement vectors", IEEE Transactions on Signal Processing, voL53, no.7, pp. 2477-2488, 2005. [13] M. F. Duarte, S. Sarvotham, D. Baron, M. B. Wakin and R. G. Baraniuk, "Distributed compressed sensing of jointly sparse signals", in Conference Record of the Thirty-Ninth Asilomar Conference on Signals, Systems and Computers, pp. 1537-1541, 2005. [14] Y. Bresler, "Spectrwn-blind sampling and compressive sensing for continuous-index signals", in Information Theory and Applications Workshop, pp. 547-554, 2008. [15] E.1.Candes, "Compressive sampling", in Proc. International Congress of Mathematicians, voL3, (Madrid, Spain), pp. 1433-1452 2006. [16] E. 1. Candes and T. Tao, "Decoding by linear programming", IEEE Transactions on Information Theory, voL51, no.12, pp. 4203-4215, 2005. [17] 1. A. Tropp, A. C. Gilbert and M. J. Strauss, "Simultaneous sparse approximation via greedy pursuit", in Proceedings. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. v/72I-v/724 Vol. 725, 2005. [18] 1. A. Tropp and A. C. Gilbert, "Signal recovery from random mea­ surements via orthogonal matching pursuit", IEEE Transactions on Information Theory, voL53, no.12, pp. 4655-4666, 2007. [19] 1. A. Tropp, A. C. Gilbert and M. J. Strauss, "Algorithms for simulta­ neous sparse approximation. Part i: Greedy pursuit", Signal Processing, voL86, no.3, pp. 572-588, 2006. 6

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