贝叶斯压缩感知算法

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一篇关于贝叶斯压缩感知算法的原理以及应用的外文文献,比较详细
This full text paper was peer reviewed at the direction uf IEEE Communications Sociely subject matler experts for publication in the IEEE DySPAN 2010 proceeding The received signal of k-th secondary user is represented by As illustrated in Figure 2, we define a general correlation factor T, which represents either the temporal or the spatial ()=∑hn()8sm(1)+n(D), correlation level, which is given by I Dn-D, 1 (6) where sm(t)and han (t)represent the primary user's trans M mitting signal and the channel impulse response between In section v. we will demonstrate that different correlation the primary user and the k-th secondary user on the m-th levels provide different performance gains subchannel, respectively. denotes the convolution operator The noise n(t)is assumed to be white Gaussian noise with zero mean and power spectral density(PSD)ok for the k-th IIL COMPRESSED SPECTRUM SENSING AND BAYESIAN secondary user. Similar to the notes in [29], we operate an M LEARNING point discrete Fourier transform for the received signal y(t). In this section, we briefly introduce the compressed sensing We can use an M x 1 vector Y to represent the received and Bayesian learning, thus providing the background for our signal on frequency-domain, which is given by proposed ST-BCSs algorithm M Yk=∑ (3)A Compressed Sensing =1 Briefly speaking, compressed sensing solves an ill-posed where o denotes elementwise multiplication. h, 5m,and inverse problem. Given an N x l observation vector g and an n"are the discrete Fourier transforms of hm(t)sm(t)and N×M(N<M) matrixΦ, MxM matrix yy, the task is respectively. We can construct a diagonal matrix H= find an M X I solution vector f to satisfy an equation, which diag(him )and rewrite(3)in a matrix form, which is given by is given by g=Φ业f Sm+ (4) where 4p is the compressed matrix and y is the projection matrix. The vector f has a sparse representation projected by y. When matrix p, v and vector f satisfy the Restricted Then the detection problem for each secondary user is to do Isometry Property(RIP)[6] it can be solved by following the following binary hypothesis test on 11 subchannels optimization problem frequency domain, i.e f= arg min f‖1 .t. g Φyf (8) Hn sm+n Compressed sensing based Analog-to-Information Conver where m, =1.2..M and k=1.2.k denote the m-th sion (AIC) was proposed in [14][21]. Eldar proposed a subchannel and the k-th secondary user, respectively blind wideband analog reconstruction method [17. based on the autocorrelation reconstruction, [191[30] [31] presented In a wideband wireless application, between two adjacent a scheme of analog signal acquisition, which endows us spectrum sensing periods, it has a low probability that all an implementation structure to acquire the wideband signal subchannels change their occupancy status, since spectrum within an affordable hardware cost. In this paper, based on sensing is implemented in a relative short period (e.g. in 802.22 these implementation structures, we represent the analog signal WRANS dratt, it is supposed to carry out spectrum sensing acquisition in a projection matrix, for simplicity we construct every 24.2 ms [22]). Then the spectrum occupancies between an N x M linear random sampling matrix A, to attain N two adjacent sensing periods are correlated, which offers us tine domain samples from discrete received signal y", which the temporal redundancy information for compressed signal is given b reconstruction. On the other hand, the spectrum observations g=Ay+n of difference cooperative secondary users also have spatial where A can be a Gaussian or Bernoulli random matrix and correlation which provides us spatial redundancy the noise n remains to be white gaussian noise. In wideband spectrum sensing, the vector y can be represented sparsely in the Fourier trans formation domain, which is given by frequency (10) Dn1=[00100 100] where s the inverse fourier transform matrix and wh is the sparse representation. Substituting(10)into(9)we obtain frequency AF + Dn气[101001000 Fig 2: An illustration of the spectrum temporal correlation where O=AF-1 This full text paper was peer reviewed at the direction uf IEEE Communications Sociely subject matler experts for publication in the IEEE DySPAN 2010 proceeding B. Bayesian Learning In an iterative way, sn, and qrm can be update by Am, which To apply the Bayesian learning to recover the compressed Is given by signal [13] [32], the key idea is to establish a hierarchical m km prior. We assume that the noise upon the observations among S cooperative secondary users are independent and identically 入 rn l n distributed (i.i.d.), which is formulated as the following equa q tions(for notational simplicity we drop the secondary users label here) Sm and gm=Qm. Moreover, Sm and Qm can he easily obtained from P(w入)= (0.入2) (12) 2=⊥ 8mbom-0mPtP 8 (24) First, we set up a zero-mean Gaussian prior on each element Bg g (25) W'i, which is satisfied a Normal distribution with zero mean where we define and variance A;. And we define the a posteriori probability for g, which is given by (26) lg-Owl PTp B白∑白B p(gw, a)=(270 )2 exp (13) P is a unitary matrix(PP=I and T is a diagonal matrix And the posterior for w, given g, A, and o2 is the convolution with real-valued elements. Therefore, we can set up an initial of Gaussian random variables [251 A and then attain S and q. A can also be update iteratively When it converges, umP can be attained from(17), as well p(gw,a2)p(w|入 LMP P(g,o )-221-2 exp-(W)2-l(W),(14) IV SPACE-TIME BAYESIAN COMPRESSED SENSING In this section, we propose the novel ST-BCSS algorithm where() represents the vector or matrix conjugate trans- Note that the hierarchical prior in the original BCS is pose, and non-informative. Since we can attain either temporal or W-A (15) spatial redundancy from previous detection results or from (a-2e6+A)-1 other cooperative secondary users, we ∑ (16) information through the prior paramete e can convey these this section we o-2∑eg (17) address how to represent, convey, and multiple a priori With A-diag(A1, A2: .,AM), the goal of BCS is to attain information sources. We assume that the compressed matrices are identical for all cooperative secondary users, which can the a posteriori mean ump. An analytically tractable way te be implemented by using the same random seed to generate solve this problem is to maximization the logarithm of f(x), the projection matrix which is given by C(入) ngp(g,a") log/ p(glw, o2)p(wlA)dw (18)A. Representing the Prior Information In ST-BCSS, prior information can be exchanged between PINlog 2T +logIC +gg,(19) multi-prior entities. Firstly, we present a two-prior-entity sce- nario and then extend to multi-prior scenarios. We use a with gI+0A0H directed graph to illustrate the probabilistic model for each (20) secondary user, as shown in Figure 3 where I is the unit matrix. An efficient algorithm to find a Suppose that we have recovered two spectrum information A that maximizes(18)is the Fast Marginal Likelihood Max entities, wk1 and wk2, from the compressed observation g1 imization(FMLM)[91 [26]. The original FMLM deals with and B4 respectively. From the temporal aspect, the prior real-valued signals and measurement matrices. We derive the information is conveyed from K1 to K2 as represented by the dash arrow. it is single direction information transportation complex-valued FMLM in Appendix. [26] defined sparsity factor, si and the 'quality factor'qi for each column A: in From the spatial aspect, the prior information is exchanged between K1 and K2, represented by the dash and solid arrows matrix 0(the detailed definitions are omitted here). And Xi Moreover, when a secondary user has only one collaborator, can dated using we can exchange the a priori information iteratively which is similar to Turbo decoding. This is especially useful when the capacity of control channel is limited and the secondary (21) user is currently suffering a deep fading detection environ if|g:2≤|s ment. Our numerical simulation demonstrates that after a few This full text paper was peer reviewed at the direction uf IEEE Communications Sociely subject matler experts for publication in the IEEE DySPAN 2010 proceeding Prior Entity KI Prior Entity Kz current compressed observations pk2 compressed matrix白 and previous recovered signal vector w1. Since we need to n=[“,n,…小 a= arg max p(w la, b) extract the a priori information from wi to construct the prior b A-Gamnna(ab) parameter aI from(12) and(28), we establish the probability relation among w, A, and a, by applying the Bayesian Rule, which is given by ~N(0,4 [w p(wa.b)=/m(w(a,bd入 8-N(Bw, o)V 8~N(加,G! where b remains as a constant to represent a uniform belief g level. Substituting (12)and(28)into (29), we apply the Ockham’ s Razor127」 to integrate out the hyperparameter入 Fi a graphic model of two prior entities scenario Then. we obtain iterations of information exchange between two cooperative secondary users, their detection performance will be signif- icantly improved. Here we only address how to convey the a priori information from node K1 to node K2, since the r(a2+ procedure from K2 to K1 is symmetric. Supposed that we (2丌) bi(b+ have recovered a spectrum information w k We extract the a T(ai (30) 2=1 priori information from w to construct the model parameter ak1 In this paper, we consider a Gamma prior for A Since w Thus, ai can be obtained by solving the following equation follows a gaussian distribution while both gamma distribution ai= arg max p(wi lai) and i= 1, 2.,M.(3 and Gaussian distribution belong to the exponential family [4] Thus it provides a conjugate prior distribution between w and Since the prior information is conveyed by a common con A, which makes the inference of the conditional probability of trol channel, which has limited capacity in CR networks, we do w given A tractable. The Gamma prior over A is given by not convey all the a prior information a;(i-1,2,.,M),but convey only those containing important information Figure 5 p(la, b)=r(a)bae demonstrates the relationship between the probability pwi) and the model parameters u;(i=1, 2..M) 选=1 04 b=0.01 eW1-1 0.35 v=1.5 012 0.05 0.40.60.8 800010000 Fig. 5: An illustration of probability p(wi ) over a; Fig. 4: An illustration of Gamma distribution Since i 1, 2M) satisfy the Gaussian di Figure 4 illustrates a Gamma distribution. The parameter b with zero mean, the Bayesian compressed sensing generates a is the scale factor which can be considered as a parameter to sparse solution of w, which means most of w'i(i-1, 2.M1) control the variance of the distribution, while a is the shape equal zero. We define the amount of information provided by factor which can be considered as a parameter to control the data ai as peak of the distribution. The intuitive meanings of the scale D(a3) factor and the sha nape factor are the belief level and the most I(ailP(ui))-p(wilai)log p(i) (-1.2,…M).(32) probable region of the value of A, respectively. In this paper, we assign the same belief level to all prior entities. Therefore, An intuitive explanation of (32) is that those wis with large for the spectrum occupancy reconstruction at K2, we have value have more information, since their probabilities are This full text paper was peer reviewed at the direction uf IEEE Communications Sociely subject matler experts for publication in the IEEE DySPAN 2010 proceeding A good measure for the value of ai [4 is given 6vOrmation Procedure 1 ST-BCSS compressed spectrum recover algo thm with priori information (ai )=logp(aa)log plu 1: Set A; with a initial value and =0 Sequentially compute the initial values: Sn, 4m, 2, u for Therefore, according to the common control channel all m column vector e condition, we apply (33) to select a certain number of ai(iEl), where the set l contains T elements which are if j+ Last Iteration then Select a candidate column vector 0, from e and com the indices of the chosen a Ite the indicator 5:ifn>0&入a<∞t With the a priori information a, the procedure of re- 6 Re- estimate入 constructing the spectrum information is to find aA which 7: else if ni >0&A maximizes the revised logarithm posterior likelihood. The 8: Compute A, by solving (21) and add 0i into the revised logarithm posterior likelihood is given by selected set 9: else if ni≤0then C(A)= logp(g A, 0+)+logp(Ala Delete 0, from selected set and set A,=O p(glw, o2)p(w A)dw +logp(Ala) d if 12: Compute the log likelihood L() 13:if△C(入)< Threshold then 2. v0g 27+ log Igg. 14: i= LastileratioT l +∑(a;lgA-b (34)16:j=j+1 17. end if 18 end if The derivatives of (34)with respect to log i are given by 19: Compute the prior a by (31) 2 20: Apply (32), compute the information metrics for each 2A;(A2+8)2(A-87)24-b 1.2.M 21: Select the ak, k=1,2, ., T with T largest information f(入2,S;,qi2a21b) metrics 入2(A+s) return return w and ak where i∈l. To attain the optimalλ;, we set"(λ)=0.This is equivalent to solving the equation given by f(A2,8a,q,a2b)=0 Prior Entity K. Prior Entity K2 If current i e l, we apply the revised Ai update rule (the 1-(iname( h details of the updating procedure is given in Appendix ) which A-(imma(a h) h Is gIven by -N(0.A w-A(DA I 入=arg{f(入,6;,q2,an,b)=0} (37) For those Ai where i fu, we apply (21) for updating the g parameters rior entity Ks Prior Entity K4 The procedure of spectrum occupancy reconstruction is summarized in procedure l, as show below 4-Gawwnc(a,b) B. Multi-prior Information 1-M,) ,会…,M Consequently, we extend the two-prior-entity processing to w2=[w3,,…,w a multi-prior-cntity one, which is illustrated in Figure 6 e. These priors are from both other cooperative secondary g g rs and the previous recovered spectrum information. For the k-th secondary user with Kn prior entities, the reconstruction lulti-prior information exchange among cooperative secondary users procedure is to find aA which maximizes the log likelihood L. Therefore, we rewrite (34)as This full text paper was peer reviewed at the direction uf IEEE Communications Sociely subject matler experts for publication in the IEEE DySPAN 2010 proceeding for the correlation factors and the test for the st-Bcss algorithm will be our future work c"()= log p(glA, a2)+>logp(Ala p(g w, a )plw A)dw A. Comparison between BP, BCS and ST- BCCS In the first simulation, we compare the receiver Operation +∑logp(Aa). (38)Characteristic (ROC) curves, where Py is the false alarm k=1 probability and Pa is the detection probability, of different Notice that the first term of (38) remains the same as that compressed spectrum algorithms, BP. BCS and ST-BCSS th different amounts of prior information. when there of (34). We define the second term of (38)as is one prior for ST-BCss reconstruction, this prior a is extracted from previous BCs reconstruction. In the two-prior logp(入a) or three-prior situation, one of these priors is extracted from the previous bcs reconstruction, the other is attained Since p(xla)=ILP(, ui), we can extend each A; by from the first round of reconstruction of another cooperative the following expression secondary users. We set an identical correlation factor of T=0.96 for all priors, where T is defined in(6). Here, we =∑logp(A|n4 define the compressed sampling rate as p= M, where M is the length of Nyquist-rate discrete signal before compressing, n is the number of observations after compressing. Then log ii p(ailai) sample rate p=0.5 means that our equivalent sampling rate is only a half of the nyquist sampling rate. The compressed log入 21(8-1)-Knb spectrum sensing are carried out under signal-LO-noise ratio (SNR) equaling -3dB. Notice that most compressed sensing c1 ai log入-b (40) reconstruction algorithms suffer from severe performance degradation due to the strong noise. For instance OMP can where Ci Is a constant and not guarantee any detection probability under this low SNR a4=∑(a4-1) (41) Figure 7 shows the simulation results. In this simulation h=1 each cooperative secondary user conveys 20 prior parameters b Kb (42) ai, i=1, 2...., 20(i. e. T=20 in Procedure 1)to the CR user Therefore, when a secondary user receives prior information carrying out ST-BCSS spectrum recover procedure. We obtain each Roc curve using 1 500 realizations of the spectrum from Kn entities, denoted by a,k= 1, 2... Kn, we replace occupancy and noise. from the simulation results, we observe the parameter @i and b in the spectrum reconstruction with i that compressed spectrum sensin sitive to the and b. The updating procedure remains the same. This prior information fusion mechanisIn provides the robustness against noise level. For a negative SNR, compressed measurements are the variation of the cognitive radio network scale. numerical contaminated by strong noise. Consequently, without any prior information, the compressed spectrum sensing algorithms simulations will demonstrate the performance gain by applying the multi- prior information e.g. BP or BCs, have poor detection performance. On the contrary, due to the temporal or spatial redundancy, the a prior information in the ST-BCSs algorithm significantly improves V. NUMERICAL SIMULATIONS the performance of spectrum reconstruction. Moreover, the In this section, we use numerical simulations to demonstrate more a priori information is provided, the better the detection the performance of the ST-BCCs algorithm proposed in this performance can be attained paper. We consider a wide frequency band of interest segmented into M= 50 subchannels with equal bandwidth To model the sparsity of the spectrum, primary user B. Performance Gain s versus A priori Information randomly choose subchannels and averagely occupy 20% In the second simulation, we demonstrate the performance of the spectrum. In order to apply the ST-BCss algorithm, gains achieved from different amount of a priori information we obtain a priori information from the previous spectrum As defined in Section II, the correlation factor T represents sensing period, whose spectrum occupancy status is slightly the quality of the a priori information. We compare the roc different from current one, as well as two spatial prior entities curves between ST-BCss with a priori information with attained from other cooperative secondary users. Note that, traditional BCS without u priori information. We set three for simplicity, we use the same value for both temporal and different correlation factors for ST-BCSS. The compressed spatial correlation factors. In practice, these two correlation sample rate p is set to be 0.6 and the snr is set to be-3dB factors may be different. However, the true values should be Figure 8 shows the simulation result obtained from field experiment. The measurement experiment This full text paper was peer reviewed at the direction uf IEEE Communications Sociely subject matler experts for publication in the IEEE DySPAN 2010 proceeding this simulation, we set T-1 to exclude the impact of the correlation factor First, we compare the spectrum sensing performance under the same noise level (SNR= 0db) while using various sampling rates. The result Is shown in Fig9. The simulation result verifies that the a priori information provides a BCS without prior significant detection performance gain. When the sample ST-BCSS with 1 prior rate is low (i.e. Rs<0.2), we observe that the Bayesian ST-BCSS with 2 prior ST-BCSS with 3 prior risk is reduced when the number of a priori information is increased. It implies that a priori information can compensate the reduction of measurements. when the sampling rate is 0D.10.20.30.40.50.60.70.80.91 sufficient high, the gain of more a priori information is negligible Fig. 7: The comparison of Roc with different spectrum reconstruction algorithms under SNr=-3dB -e-ST-BCSS with 1 prio "〓ST-Bcss1 with 2 prior " ST-BCSS with 3 priorit ∴,,, 0.5 ST-BCSS with 1 prior. =1 〓ST- Bcss with1 prior:÷0.96 T-BCSS with 1 prior, =0 BCS without prior P 02 0.6 0102030.40.50.60.70.809 Fig. 9: Bayesian risk versus various sampling rates when Fig,8: The comparison of roc versus different correlation SNR=OdB factors with sample rate 0.6 under SNR=-3dB Second, we compare the compressed spectrum reconstruction algorithms carried out when SNR We observe that the performance gain increases as the and SNR 3dB. Figure 10 shows the Bayesian risk correlation factor t increase. This is intuitive. since the higher versus various sampling rates. We notice that the perfornance similarity of the spectrum is, the more information the sec degrades when the noise level increases, which implies ondary user can obtain from the prior source that it needs more measurements to recover the spectrum information. However, with the a priori information, ST- BCss can improve the detection performance. which is C. Bayesian Risk versus Sampling Rate epresented by the dash line with diamond marker In the following simulations, we demonstrate the spectrum sensing performance of BCS and ST-BCCS using various Figure 1l shows the ROC curves under different sampling sampling rates. It is assumed that each subchannel has the rates. We set SNR--3dB. From the simulation results, same probability of being busy or idle. We define the average we observe that the sampling rate and the number of a Bayesian risk rB in the following way: priori information sources have significant impacts on the performance. Again, the ST-BCss considerably outperforms rB=71(1-c1)+丌0C10F+r1co1p (43)the bcs without any a priori information where To and ti denote the probabilities of subchannel being idle or busy, respectively. We assume that To =0.9 and 1=0.1, which implies that the spectrum occupancy D. Turbo Information Exchange is sparse. Cii is the cost when the secondary user claims The informative hierarchical prior infrastructure provide a while the true hypothesis is i. py and pn are the probabilities flexible cooperation schene. In this simulation, we show that, of false alarm and miss detection, respectively. Since it is when the CR user has only one collaborator, the two secondary important to protect the primary user from being interfered users exchange their information iteratively in a way similar by the secondary users, we set asymmetric costs: C10= 1 to Turbo decoding. In our simulation, the SNRs of the two and Co1 =0. 2. Since we focus on the effect of sample rate in secondary users are set to -3dB and-2dB, respectively. We This full text paper was peer reviewed at the direction uf IEEE Communications Sociely subject matler experts for publication in the IEEE DySPAN 2010 proceeding - BCS without prior, SNR=OdB ST-BCSS with 1 prior, SNR=DdB M-SNR=-3dB ST-BCSS with 2 prior, SNr=OdB =PSNR=-2dB ST-BCSS with 2 prior, SNR=-3dB 后 04 02 0.05 Q:999;3:: Sample rate Iteration Fig. 10: Bayesian risk versus various sampling rates and Fig 12: The performance gain attained by turbo prior exchang different snr Ing possible that the reconstruction results are unreliable. Then, the results should be discarded and the secondary users should refrain from transmitting We first apply the Turbo prior exchange method to re a0.5 construct the spectrum. The snrs of the two cooperative dary users are 10dB and 6dB, respectively. Note that I-I ST-BCSS with 2 prior, Rs=0.4 the snr defined here is an equivalent snr, i.e. the two 02 ST-BCSS with 1 prior, Rs=0.5 ST-BCSS with 1 prior, Rs=0.6 cooperative secondary users have the same noise level. We l BCS without prior, Rs=0.6 use only one round of prior exchange. The correlation of the reconstructed spectrum vectors is defined as 040.50.60.70.80.9 a,Wb COTr (44) Fig. Il: ROC. curve versus different sampling rates under b SnR=-3dB where wa and wb are the two reconstrued spectrum vector of two cooperative secondary users, respectively. wa Wb stands for the vector inner product, and. denotes the 2- set the sample rate to Rs =0.6 and set the number of priors norm of vectors for exchanging to 23. The average Bayesian Risk defined in 43 is apply as the performance metric. The numerical simulation result in Fig. 12 shows that, even when the snrs of these EFTwo independent BCS secondary users are relatively low the mechanism of turbo Two temporal adjacent of one BCS Two ST-BCSS information exchange provides a significant performance gain within a few iterations It is observed that, by taking one iteration, the average Bayesian Risk is significantly reduced. Moreover, the simu lation result shows that the turbo prior exchange converges 06 E. Detection of Reconstruction Failures One of the difficulties of applying compressed sensing in spectrum sensing is the uncertainty of the spectrum sparsity When the spectrum of interest is not sparse. the compressed sensing based reconstruction algorithm fails with higl 0.2 0.5 0.7 08 probability. The unreliable reconstruction results may incur Active channels percentage significant risk of violating primary users if they are directly important to detect the failure of the reconstruction. Again, occupancy percentages tion levels against various subchannel used for the decision of spectrum sensing. Therefore, it is Fig. 13: Different correlat we utilize the spatial correlation for the failure detection The principle is that, if two reconstruction results of two Figure 13 shows the correlation levels between the nearby secondary users are significantly different, it is highly reconstructed spectrum occupancy vectors between two This full text paper was peer reviewed at the direction uf IEEE Communications Sociely subject matler experts for publication in the IEEE DySPAN 2010 proceeding secondary users versus different spectrum sparsities when over 90%. The ST-BCSS is also shown to be sensitive to the different types of reconstruction algorithms are used. We number of the amount of a prior information. Moreover, the observe that the correlation decreases as percentage of active ST-BCSs scheme is demonstrated to significantly outperform spectrum increases traditional compressed sensing algorithms, e.g. BP and bcs, for the spectrum sensing in wideband cognitive radio systems Based on the observation in Fig. 13, we can set a threshold We have also proposed a correlation based algorithm for the or the correlation value to detect whether the current spectrum detection of reconstruction failure due to non-sparse spectrum is sparse. Given a certain correlation threshold, if the current for enhancing the robustness of the spectrum sensing correlation value of two reconstructed spectrum vectors is larger than this threshold, we accept the reconstruction resu VIL. ACKNOWLEDGEMENTS Otherwise, the reconstruction results are rejected This work is supported by China scholarship Council(Csc), 111"Project (B08038), National Natural Science Foundation of China(Nos. 60572147), State Key Laboratory of Integrated Service networks Project Grant (ISNO2080002, ISNO90307) National Science Foundation under grants CCF-0830451 and ECCs-0901425 and Xi'an Industrial Science and Technology Project (YF07015) APPENDIX 0.5 Here, we only provide the key derivation steps different from that of the traditional real valued fmlm. the main idea of FMlM is to find a a that maximizes(18), in which the C 0.2 ∑A20n9m+A2:9(45) 2 0.6 C-;+A2202 where,for each updating iteration, it considers only the Fig. 14: ROC curve of the detection of reconstruction failure contribution of a single column Oi in the projection matrix @. Therefore, we can rewrite L(A)as For the detection of reconstruction failure. we first define the success of the spectrum reconstruction as the event that the ()=-oIN log()+log C-a+gc-ig reconstructed occupancies of more than 90 of the spectrum are correct. Then, we define the probability of false alarm, Pf log Ai t log(;+0C- 0) (C-g)2 as the probability of a successful reconstruction result being 入+2C-6; (47) rejected. Similarly, the probability of miss detection, Pm, is defined as the probability of an unsuccessful recon struction Since X is a real number, we consider only the terms with result is accepted. Based on these definitions, we obtain an C-i which is a complex-valued matrix. From(45)we have ROC curve in Fig. 14 which demonstrates the validity of the proposed correlation based detection of reconstruction failure. +∑Amn10m0m (48) We observe that the correlation across different secondarv users can effectively detect the failures of reconstruction due Since the first term of (48) is a real valued diagon matr to the non-spare spectrum, thus significantly improving the and the second term is a complex valued and conjugate robustness of spectrum sensing. symmetric square matrix with diagonal real valued elements, the inverse of ci is still a conjugate symmetric square matrix VI CONCLUSION with real valued diagonal elements. Then we can apply the In this paper, we have proposed a St-BCss scheme for the chur Decomposition to c-1 and obtain spectrum sensing of wideband cognitive radio, which applies a C H (49) probabilistic model to tackle the the noise when reconstruct- ing the spectrum using a limited number of measurements. Where v is an unitary matrix and Q is a diagonal matrix with Based on the hierarchical prior model, the ST-BCSS scheme real valued elements. We rewrite the complex valued terms in exploits the temporal and spatial redundancies to improve the (47)as perfornance of spectrum reconstruction. We have addressed gCig=guSh g (real-valued) the challenge of how to present, convey and fuse multi-prior HC-10,;=0H19nH information The numerical simulation verified that when (real-valued applying the st-bcss scheme, spectrum sensing can be im g)=(02C-ig(0iC-ig plemented under SNR=-3dB with probability of detection (real-valued)

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