Enhanced Compressive Wideband Frequency Spectrum Sensing for Dynamic Spectrum Access
aa r X i v : . [ c s . I T ] A p r JOURNAL TITLE, VOL. X, NO. X, MONTH YEAR 1
Enhanced Compressive Wideband FrequencySpectrum Sensing for Dynamic Spectrum Access
Yipeng Liu, Qun Wan
Abstract
Wideband spectrum sensing detects the unused spectrum holes for dynamic spectrum access (DSA). Too highsampling rate is the main problem. Compressive sensing (CS) can reconstruct sparse signal with much fewerrandomized samples than Nyquist sampling with high probability. Since survey shows that the monitored signalis sparse in frequency domain, CS can deal with the sampling burden. Random samples can be obtained by theanalog-to-information converter. Signal recovery can be formulated as an L0 norm minimization and a linearmeasurement fitting constraint. In DSA, the static spectrum allocation of primary radios means the bounds betweendi ff erent types of primary radios are known in advance. To incorporate this a priori information, we divide thewhole spectrum into subsections according to the spectrum allocation policy. In the new optimization model, theminimization of the L2 norm of each subsection is used to encourage the cluster distribution locally, while the L0norm of the L2 norms is minimized to give sparse distribution globally. Because the L0 / L2 optimization is notconvex, an iteratively re-weighted L1 / L2 optimization is proposed to approximate it. Simulations demonstrate theproposed method outperforms others in accuracy, denoising ability, etc.
Index Terms
Yipeng Liu is with KU Leuven, Department of Electrical Engineering (ESAT), SCD-SISTA and IBBT Future Health Department,Kasteelpark Arenberg 10, box 2446, 3001 Heverlee, Belgium; e-mail: ([email protected]);.Qun Wan is with the Electronic Engineering Department, University of Electronic Science and Technology of China, Chengdu, 611731,China. e-mail: ([email protected]);.Manuscript revised March 23th, 2012
OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 2 cognitive radio, dynamic spectrum access, wideband spectrum sensing, compressive sensing, sparse signalrecovery.
I. I ntroduction
Cognitive radio (CR) is a very promising technology for wireless communication. Radio spectrum isa precious natural resource. The fixed spectrum allocation is the major way for the spectrum allocationnow. In order to avoid interference, di ff erent wireless services are allocated with di ff erent licensed bands.Currently most of the available spectrum has been allocated. But the increasing wireless services, especiallythe wideband ones, call for much more spectrum access opportunities. The allocated spectrum becomesvery crowded and spectrum scarcity comes. To deal with the spectrum scarcity problem, there are severalways, such as multiple-input and multiple-output (MIMO) communication [1], ultra-wideband (UWB)communication [2], beamforming [3] [4], relay [5], and so on. Investigation demonstrates that most of theallocated bands are in very low utility ratios [6]. CR is proposed to exploit the under-utilization of the radiofrequency (RF) spectrum. It is a paradigm in which the cognitive transmitter changes its parameters toavoid interference with the licensed users. This alteration of parameters is based on the timely monitoringof the factors in the radio environment.Spectrum sensing is one of the main functions of CR. It detects the unused frequency bands, andthen CR users can be allowed to utilize the unused primary frequency bands. Current spectrum sensingis performed in two steps [7]: the first step called coarse spectrum sensing is to e ffi ciently detect thepower spectrum density (PSD) level of primary bands; the second step, called feature detection or multi-dimensional sensing [8], is to estimate other signal space accessible for CR, such as direction of arrival(DOA) estimation, spread spectrum code identification, waveform identification, etc.Coarse spectrum sensing requires fast and accurate power spectrum detection over a wideband andeven ultra-wideband (UWB). One approach utilizes a bank of tunable narrowband bandpass filters. But itrequires an enormous number of RF components and bandpass filters, which leads to high cost. Besides,the number of the bands is fixed and the filter range is always preset. Thus the filter bank way is not OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 3 flexible. The other one is a wideband circuit using a single RF chain followed by high-speed digitalsignal processor (DSP) to flexibly search over multiple frequency bands concurrently [9]. It is flexible todynamic power spectrum density. High sampling rate requirement and the resulting large number of datafor processing are the major problems [10].Too high sampling rate requirement brings challenge to the analog-to-digital converter (ADC). And theresulting large amount of data requires large storage space and heavy computation burden of DSP. Sincesurvey shows sparsity exists in the frequency domain for primary signal, compressive sensing (CS) canbe used to e ff ectively decrease the sampling rate [11] [12] [13]. It assets that a signal can be recoveredwith a much fewer randomized samples than Nyquist sampling with high probability on condition thatthe signal has a sparse representation.In compressive wideband spectrum sensing (CWSS), analog-to-information converter (AIC) can betaken to obtain the random samples from analog signal in hardware as Fig. 1 shows [14] [15]. To get thespectrum estimation, there are mainly two groups of methods [13]. One group is convex relaxation, suchas basis pursuit (BP) [16] [17], Dantzig Selector (DS) [18] , and so on; the other is greedy algorithm,such as matching pursuit (MP) [19], orthogonal matching pursuit (OMP) [20], and so on. Both of theconvex programming and greedy algorithm have advantages and disadvantages when applied to di ff erentscenarios. A short assessment of their di ff erences would be that convex programming algorithm has ahigher reconstruction accuracy while greedy algorithm has less computation complexity. In contrast toBP, basis pursuit denoising (BPDN) has better denoising performance [17] [21].In this paper, the partial Fourier random samples are obtained via AIC with the measurement matrixgenerated by choosing part of separate rows randomly from the Fourier sampling matrix [14]. Based onthe random samples, a generalized sparse constraint in the form of mixed C / C norm is proposed toenhance the recovery performance by exploiting the structure information. It encourages locally clusterdistribution and globally sparse distribution. In the constraint, the estimated spectrum vector is dividedinto sections with di ff erent length according to the a priori information about fixed spectrum allocation. OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 4
The sum of weighted C norms of the sections is minimized. The weighting factor is iteratively updatedas the reciprocal of the energy in the corresponding subband to get more democratical penalty of nonzerocoe ffi cients. Simulation results demonstrate that the proposed generalized sparse constraint based CWSSgets better performance than the traditional methods in spectrum reconstruction accuracy.In the rest of the paper, Section II gives the signal model; Section III states the classical CWSS methods.Section IV provides the generalized sparse constraint based CWSS methods; In section V, the performanceenhancement of the proposed method is demonstrated by numerical experiments; Finally Section VI drawsthe conclusion. II. S ignal M odel According to the FCC report [6], the allocated spectrum is in a very low utilization ratio. It meansthe spectrum is in sparse distribution. Recently a survey of a wide range of spectrum utilization across 6GHz of spectrum in some palaces of New York City demonstrated that the maximum utilization of theallocated spectrum is only 13.1%. It is also the reason that CR can work. Thus it is reasonable that only asmall part of the constituent signals will be simultaneously active at a given location and a certain rangeof frequency band. The sparsity inherently exists in the wideband spectrum [10] [22] [23] [24] [25] [26][27] [28].An N × x can be expanded in an orthogonal complete dictionary Ψ N × N , with therepresentation as x N × = Ψ N × N b N × (1)When most elements of the N × b are zeros, the signal x is sparse. When the number of nonzeroelements of b is S ( S ≪ M < N ), the signal is said to be S -sparse.In traditional Nyquist sampling, the time window for sensing is t ∈ [0 , T ]. N samples are needed torecover the frequency spectrum r without aliasing, where T is the Nyquist sampling duration. A digital OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 5 receiver converts the continuous signal x(t) to a discrete complex sequence y t of length M . For illustrationconvenience, we formulate the sampling model in discrete setting as it does in [10] [22] [23] [24] [25][26] [27] [28]: y t = Ax t (2)where x t represents an N × x t [ n ] = x ( t ) , t = nT , n = , · · · , N , and A is an M × N projection matrix. For example, when A = F N with M = N , model (2) amounts to frequency domainsampling, where F N is the N -point unitary discrete Fourier transform (DFT) matrix. Given the sampleset x t when M < N , compressive spectrum sensing can reconstruct the spectrum of r (t) with the reducedamount of sampling data.To monitor such a broad band, high sampling rate is needed. It is often very expensive. Besides, toomany sampling measurements inevitably ask more storage devices and result in high computation burdenfor digital signal processors (DSP), while spectrum sensing should be fast and accurate. CS provides analternative to the well-known Nyquist-Shannon sampling theory. It is a framework performing non-adaptivemeasurement of the informative part of the signal directly on condition that the signal is sparse [13]. Sinceit is proved that x t has a sparse representation in frequency domain. We can use an M × N random projectionmatrix S c to sample signals, i.e. y t = S c x t , where M < N ; S c is a non-uniform subsampling or randomsubsampling matrix which is generated by choosing M separate rows randomly from the unit matrix I N .The AIC can be used to sample the analog baseband signal x(t) . One possible architecture can bebased on a wideband pseudorandom demodulator and a low rate sampler [14] [15]. First we modulatethe analogue signal by a pseudo-random maximal-length PN sequence. Then a low-pass filter follows.Finally, the signal is sampled at sub-Nyquist rate using a traditional ADC. It can be conceptually modeledas an ADC operating at Nyquist rate, followed by random discrete sampling operation [14]. Then y t isobtained directly from continuous time signal x(t) by AIC. The details about AIC can be found in [14][15]. Here we incorporate the AIC to the spectrum sensing architecture as Fig. 1 shows. OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 6
III. T he C lassical C ompressive W ideband S pectrum S ensing CS theory asserts that, if a signal has a sparse representation in a certain space, one can use therandom sampling to obtain the measurements and successfully reconstruct the signal with overwhelmingprobability by nonlinear algorithms, as stated in section II. The required random samples for recovery arefar fewer than Nyquist sampling.To find the unoccupied spectrum for secondary access, the signal in the monitored band is down-converted to baseband. The analog baseband signal is sampled via the AIC that produces measurementsat a rate below the Nyquist rate.Now we estimate the frequency response of x(t) from the measurement vector y t based on the transfor-mation equality y t = S c F − N r , where r is the N × x(t) ; F N isthe N × N Fourier transform matrix; S c is the M × N matrix which is obtained by randomizing the columnindices and getting the first M columns.Under the sparse spectrum assumption, the FRV can be recovered by solving the combinatorial opti-mization problem ˆr = arg min r k r k s . t . (cid:16) S Tc F − M (cid:17) r = y t (3)Since the optimization problem (3) is nonconvex and generally impossible to solve, for its solution usuallyrequires an intractable combinatorial search. As it does in [10], BP is used to recover the signal: r BP = arg min r k r k s . t . (cid:16) S Tc F − M (cid:17) r = y t (4)This problem is a second order cone program (SOCP) and can therefore be solved e ffi ciently using standardsoftware packages.BP finds the smallest C norm of coe ffi cients among all the decompositions that the signal is decomposedinto a linear combination of dictionary elements (columns, atoms). It is a decomposition principle basedon a true global optimization. OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 7
In practice noise exists in data. Another algorithm called BPDN has superior denoising performancethan BP [21]. It is a shrinkage and selection method for linear regression. It minimizes the sum of theabsolute values of the coe ffi cients, with a bound on the sum of squared errors. To get higher accuracy,we can formulate the BPDN based compressive wideband spectrum sensing (BPDN-CWSS) optimizationmodel as: r BPDN = arg min r k r k s . t . (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) S Tc F − M (cid:17) r − y t (cid:13)(cid:13)(cid:13)(cid:13) ≤ η (5)where η bounds the amount of noise in the data. The computation of the BPDN is a quadratic programmingproblem or more general convex optimization problem, and can be done by classical numerical analysisalgorithms. The solution has been well investigated [21] [29] [30] [31]. A number of convex optimizationsoftware, such as cvx [32], SeDuMi [33] and Yalmip [34], can be used to solve the problem.IV. T he P roposed C ompressive W ideband S pectrum S ensing Among the classical sparse signal recovery algorithms, BPDN achieves the highest recovery accuracy[13]. However, it only takes advantage of sparsity. In wideband CR application, additional a priori information about the spectrum structure can be obtained. The further exploitation of structure informationwould give birth to recovery accuracy enhancement [28] [35] [36] . Besides, It is well-known that theminimization of C norm is the best candidate for sparse constraint. But in order to reach a convexprogramming, the C norm is relaxed to C norm, which leads to the performance degeneration [37].Here a weighting formulation is designed to democratically penalize the elements. It suggests that largeweights could be used to discourage nonzero entries in the recovered FRV, while small weights could beused to encourage nonzero entries. To get the weighted values, a simple iterative algorithm is proposed. A. Wideband spectrum sensing for fixed spectrum allocation
The classical algorithms reconstruct the commonly sparse signal. However, in the coarse widebandspectrum sensing, the boundaries between di ff erent kinds of primary users are fixed due to the static OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 8 frequency allocation of primary radios. For example, the bands 1710 - 1755 MHz and 1805 - 1850 MHzare allocated to GSM1800. Previous CWSS algorithms did not take advantage of the information of fixedfrequency allocation boundaries. Besides, according to the practical measurement, though the spectrumvector is sparse globally, in some certain allocated frequency sections, they are not always sparse. Forexample, in a certain time and area, the frequency sections 1626.5 - 1646.5 MHz and 1525.0 - 1545.0MHz allocated to international maritime satellite are not used, but the frequency sections allocated toGSM1800 are fully occupied. The wideband FRV is not only sparse, but also in sparse cluster distributionwith di ff erent length of clusters. It is the generalization of the so called block-sparsity [35] [36]. Thisfeature is extremely vivid in the situation that most of the monitored primary signals are spread spectrumsignals.Previous classical CWSS does not assume any additional structure on the unknown sparse signal.However in the practical application, the signal may have other structures. Incorporating additionalstructure information would improve the recoverability potentially.Block-sparse signal is the one whose nonzero entries are contained within several clusters. To exploitthe block structure of ideally block-sparse signals, C / C optimization was proposed. The standard blocksparse constraint (SBSC) in the form of C / C optimization can be formulated as [35] [36]:min r K P i = (cid:13)(cid:13)(cid:13) r ( i − d : id (cid:13)(cid:13)(cid:13) ! s . t . (cid:16) S Tc F − M (cid:17) r = y t (6)where K is the number of the divided subbands; d is the length of the divided blocks. Extensiveperformance evaluations and simulations have demonstrated that as d grows the algorithm significantlyoutperforms standard BP algorithm [36].However, in the standard C / C optimization, the estimated sparse signal is divided with the same blocklength, which mismatches the practical situation that the values of the length of the spectrum subbandsallocated to di ff erent radios can not be all the same. Besides, the constraint in (6) does not incorporatethe denoising function. OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 9
To further enhance the performance of CWSS, the fixed spectrum allocation information can be in-corporated in the CWSS algorithm. Based on the a priori information about boundaries, the estimatingPSD vector is divided into sections with their edges in accordance with the boundaries of di ff erent typesof primary users by fixed spectrum allocation. In the BPDN-CWSS, the minimization of the standard C -norm constraint on the whole FRV is replaced by the minimization of the sum of the C norm ofeach divided section of the FRV to encourage the sparse distribution globally while blocked distributionlocally. As it combines C norm and C norm to enforce the sparse blocks with di ff erent block lengths, thenew CWSS model, in the name of variable-length-block-sparse constraint based compressive widebandspectrum sensing (VLBS-CWSS), can be formulated as:min r (cid:0) k r k + k r k + · · · k r K k (cid:1) s . t . (cid:13)(cid:13)(cid:13) y t − S c F − N r (cid:13)(cid:13)(cid:13) ≤ η (7)where r , r , ... , r K are K sub-vectors of r corresponding to d , d , ... , d K − which are the boundariesof the divided sections. η bounds the amount of noise in the data. It can be formulated as: r = r · · · r d | {z } r · · · r d K − + · · · r N | {z } r K T (8)Since the objective function in the VLBS-CWSS (7) is convex and the other constraint is an a ffi ne, itis a convex optimization problem. It can also be solved by a host of numerical methods in polynomialtime. Similar to the solution of the BPDN-CWSS (5), the optimal r of the VLBS-CWSS (7) can also beobtained e ffi ciently using some convex programming software packages. Such as cvx [32], SeDuMi [33],and Yalmip [34], etc.After we get r from (8), power spectrum can be obtained. Several ways can indicate the spectrumholes, such as energy detection [27], edge detection [10], and so on. For example, in energy detection wewill calculate k r k k , k =
1, 2, ... , K . Comparing it with an experimental threshold, the spectrum holes fordynamic access can be clearly given. The energy detection will be used in numerical simulations. OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 10
B. Enhanced variable-length-block-sparse spectrum sensing
In sparse constraint, C norm minimization is relaxed to C norm at the cost of bringing the dependenceon the magnitude of the estimated vector. In the C norm minimization, larger entries are penalized moreheavily than smaller ones, unlike the more democratic penalization of the C norm. Here in the the VLBSconstraint, to encourage sparse distribution of the spectrum in the global perspective, the C norm of aseries of the C norm is minimized. Similarly, the dependence on the power in each subband exits.To deal with this imbalance, the minimization of the weighted sum of the C norm of each blocksis designed to more democratically penalize. The new weighted VLBS constraint based compressivewideband spectrum sensing (WVLBS-CWSS) can be formulated as:min r (cid:0) w k r k + w k r k + · · · + w K k r K k (cid:1) s . t . (cid:13)(cid:13)(cid:13) y t − S c F − N r (cid:13)(cid:13)(cid:13) ≤ η (9)where r , r , ... , r K are defined as (8); η bounds the amount of noise; w = " w w · · · w K T . w i depends on p i ≥ , for i = , · · · , K , where p i corresponds to the power of the primary user exists in the i - th subband.Obviously, the object function of the WVLBS-CWSS (9) is convex. It is a convex optimization problem.In principle this problem is solvable in polynomial time.To realize the WVLBS-CWSS (9), the weighting vector w should be provided. As it is defined before,the computation of the weight w i is in fact the computation of the p i . Here a practical way to iterativelyset the p i is proposed. At each iteration, the p i is the sum of the absolute value of frequency spectrumvector in the corresponding subband. It can be formulated as: p t , i = (cid:13)(cid:13)(cid:13) r t − , i (cid:13)(cid:13)(cid:13) = (cid:12)(cid:12)(cid:12) r t − , d i − + (cid:12)(cid:12)(cid:12) + · · · + (cid:12)(cid:12)(cid:12) r t − , d i (cid:12)(cid:12)(cid:12) (10)where r t − , i is the i - th sub-vector as in (8) at the ( t -1)- th iteration; r t − , d i − + , · · · , r t − , d i are the elementsof the sub-vector r t − , i . After getting the p i , the weighting vector w can be formulated. Here we can get OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 11 it by w i = p i + δ (11)where a small parameter δ > p i does not strictly prohibit a nonzero estimate at the next step.The initial condition of the recursive relation is w i =
1, for all i = , ..., K . That means in the first step,all the blocks are weighted equally. Along with the increase of the iteration times, larger values of p i arepenalized lighter in the WVLBS-CWSS (9) than smaller values of p i . To terminate the iteration at theproper time, the stopping rule can be formulated as k r t − r t − k ≤ ε (12)where r t is the estimated FRV at the t - th iteration; ε bounds the iteration residual.The initial state of the iterative algorithm is the same with the VLBS-CWSS (7). To make a di ff erence,The iterative reweighted algorithm is named as enhanced variable-length-block-sparse constraint basedcompressive wideband spectrum sensing (EVLBS-CWSS).V. S imulation R esults Numerical experiments are presented to illustrate performance improvement of the proposed EVLBS-CWSS for CR. Here we consider a base band signal with its frequency range from 0 Hz to 500 MHz asFig. 2 shows. The primary signals with random phase are contaminated by a zero-mean additive whiteGaussian noise (AWGN) which makes the signal to noise ratio (SNR) be 11.5 dB. Four primary signalsare located at 30 MHz - 60 MHz, 120 MHz - 170 MHz, 300 MHz - 350 MHz, 420 MHz - 450 MHz.Their corresponding frequency spectrum levels fluctuate in the range of 0.0023 - 0.0066, 0.0016 - 0.0063,0.0017 - 0.0063, and 0.0032 - 0.0064, as Fig. 3 shows. Here we take the noisy signal as the received
OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 12 signal x(t) . As CS theory suggests, we sample x(t) randomly at the subsampling ratio 0.40 via AIC asFig. 1. The resulted sub-sample vector is denoted as y t .To make contrast, with the same number of samples, The amplitude of frequency spectrum estimatedby di ff erent methods are given in Fig. 4, Fig.5 and Fig. 6. Fig. 4 shows the result estimated by thestandard BPDN-CWSS (5) where η is chosen to be 0 . k y t k with 1000 tries averaged; Fig. 5 does it bythe VLBS-CWSS (7) where η is chosen to be 0 . k y t k ; Fig. 6 does it by the proposed EVLBS-CWSS(9) where η is chosen to be 0 . k y t k , and δ is chosen to be 0.001.Fig. 6 shows that the proposed EVLBS-CWSS gives the best reconstruction performance. It showsthat there are too many fake spectrum points in the subbands with no active primary signal in Fig 4which is given by the standard BPDN. The noise levels of the spectrum estimated by the B-CWSS andthe VLBS-CWSS are high along the whole monitored band. For the VLBS-CWSS, as in Fig. 5, it hasconsiderable performance improvement, but the noise level in part of the inactive subbands is still high.Some of the estimated spectrum in the inactive subband is a little too high. However, in Fig. 6, the fouroccupied bands clearly show up; the noise levels in the inactive bands are quite low; the variation of thespectrum levels in the boundaries of estimated spectrum are quite abrupt and correctly in accordance withthe generated sparse spectrum in Fig. 2, which would enhance the edge detection performance much.Therefore, the proposed EVLBS-CWSS outperforms the standard BPDN-CWSS and the VLBS-CWSSfor wideband spectrum sensing.Apart from the edge detection, energy detection is the most popular spectrum sensing approach for CR.To test the CWSS performance by energy detection, 1000 Monte Carlo simulations are done with the sameparameters above to give the results of average energy in each section of the divided spectrum vector withthe BPDN-CWSS (5), the VLBS-CWSS (7) and the EVLBS-CWSS (9). The parameter setting is sameas before. The simulated monitored band is divided into 9 sections as Fig 2. The total energy with eachCWSS method is normalized. Table I presents the average energy in each subband with di ff erent recoverymethods, when there are 4 active bands and the sub-sampling ratio is 0.40; Table II does when there are OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 13 k - th subband as: R k ) = k r VLBSk k − k r BPDNk k k r BPDNk k , for active subbands k r BPDNk k − k r VLBSk k k r BPDNk k , for inactive subbands (13) R k ) = k r EVLBSk k − k r BPDNk k k r BPDNk k , for active subbands k r BPDNk k − k r EVLBSk k k r BPDNk k , for inactive subbands (14)where r EVLBSk , r VLBSk and r BPDNk represent values of estimated frequency spectrum vectors in the k - th subband via EVLBS-CWSS, VLBS-CWSS and BPDN-CWSS, respectively. These performance functionscan quantify how much energy increased to enhance the probability of correct energy detection of theactive primary bands and how much denoising performance is enhanced. The values of EDPER in TableI, Table II, Table III and Table IV clearly tell the improvement of the proposed EVLBS-CWSS againstVLBS-CWSS and BPDN-CWSS methods.To further evaluate the performance of EVLBS-CWSS, when the number of active bands is 4 and sub-sampling ratio is 0.40, the residuals k r t − r t − k for 1000 Monte Carlo simulations are measured. Usingthe unnormalized received signal, the measured average power of the random samples y t is 29533. From t = t =
8, the residuals are 361.5066, 261.6972, 55.0035, 17.9325, 15.0799, 13.4075 and 12.6189. Itshows the iteration is almost convergent at t =
5. The iteration would bring the increase of computationcomplexity, but the performance enhancement is obvious and worthwhile.The enhancement of spectrum estimation accuracy qualifies the proposed EVLBS-CWSS as an excellentcandidate for CWSS.
OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 14
VI. C onclusion
In this paper, CS is used to deal with the too high sampling rate requirement problem in the widebandspectrum sensing for CR. The sub-Nyquist random samples is obtained via the AIC with the partialFourier random measurement matrix. Based on the random samples, incorporating the a priori informationof the fixed spectrum allocation, an improved sparse constraint with di ff erent block length is used toenforce locally block distribution and globally sparse distribution of the estimated spectrum. The newconstraint matches the practical spectrum better. Furthermore, the iterative reweighting is used to alleviatethe performance degeneration when the C / C norm minimization is relaxed to the C / C one. Becausethe a priori information about boundaries of di ff erent types of primary users is added and iteration isused to enhance the VLBS constraint performance, the proposed EVLBS-CWSS outperforms previousCWSS methods. Numerical simulations demonstrate that the EVLBS-CWSS has higher spectrum sensingaccuracy, better denoising performance, etc.A cknowledgment This work was supported in part by the National Natural Science Foundation of China under the grant61172140, and ’985’ key projects for excellent teaching team supporting (postgraduate) under the grantA1098522-02. R eferences [1] SM Alamouti,A simple transmit diversity technique for wireless communications. IEEE Journal on Selected Areas in Communications16, 8: 1451 - 1458 (1998)[2] Y Liu, Q Wan, X Chu, Power-e ffi cient ultra-wideband waveform design considering radio channel e ff ects. Radioengineering , 1:179-183 (2011)[3] Y Liu, Q Wan, Robust beamformer based on total variation minimisation and sparse-constraint. Electronics letters , 25: 1697-1699(2010)[4] C Wang, Q Yin, B Shi, H Chen, Q Zou, Distributed transmit beamforming based on frequency scanning. Paper Presented at IEEEInternational Conference on Communications ICC 2011, Kyoto, 5-9 June, 2011 OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 15 [5] C Wang, H Chen, Q Yin, A Feng, AF Molisch, Multi-user two-way relay networks with distributed beamforming. IEEE Transactionson Wireless Communications , 10: 3460-3471 (2011)[6] FCC, Spectrum Policy Task Force Report (2002)ET Docket No. 02-155. http : // transition . fcc . gov / sptf / files / SEWGFinalReport 1 . pdf.accessed 02 Nov 2002[7] A Ghasemi, ES Sousa, Spectrum sensing in cognitive radio networks: requirements, challenges and design trade-o ff s. IEEE Communi-cations Magazine , 4: 32-39 (2008)[8] T Yucek, H Arslan, A survey of spectrum sensing algorithms for cognitive radio applications. IEEE Communications Surveys andTutorials , 1: 116-130 (2009)[9] A Sahai, D Cabric, Spectrum sensing - fundamental limits and practical challenges. A tutorial presented at IEEE DySpan Conference2005, Baltimore, Nov 2005[10] Z Tian, GB Giannakis, Compressed sensing for wideband cognitive radios, Paper presented at International Conference on Acoustics,Speech, and Signal Processing 2007, Honolulu, Hawaii, USA, April 2007[11] D Donoho, Compressed sensing. IEEE Transactions on Information Theory , 4: 289-1306 (2006)[12] EJ Candes, J Romberg, T Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information.IEEE Transactions on Information Theory , 2: 489-509 (2006)[13] EJ Candes, MB Wakin, An introduction to compressive sampling. IEEE Signal Processing Magazine , 2: 21-30 (2008)[14] JN Laska, S Kirolos, Y Massoud, RG Baraniuk, A Gilbert, M Iwen, M Strauss, Random sampling for analog-To-information conversionof wideband signals. Presented at fifth IEEE Dallas Circuits and Systems Workshop, The University of Texas at Dallas, October 29-302006[15] JN Laska, S Kirolos, MF Duarte, TS Ragheb, RG Baraniuk, Y Messoud, Theory and implementation of an analog-To-informationconverter using random demodulation. Paper Presented at 2007 IEEE International Symposium on Circuits and Systems (ISCAS 2007),New Orleans, 27-30 May 2007[16] SS Chen, Basis pursuit. Dissertation, Stanford University, 1995[17] SS Chen, DL Donoho, MA Saunders, Atomic decomposition by basis pursuit. SIAM Journal on Scientific Computing , 1: 33-61(1999)[18] EJ Candes, T Tao, The Dantzig selector: statistical estimation when p is much larger than n. Annals of Statistics , 6: 2313-2351(2007)[19] S Mallat, Z Zhang, Matching pursuit in a time-frequency dictionary. IEEE Transactions on Signal Processing , 12: 3397C3415 (1993)[20] JA Tropp, AC Gilbert, Signal recovery from random measurements via orthogonal matching pursuit. IEEE Transactions on InformationTheory , 12: 4655-4666 (2007)[21] R Tibshirani, Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society B : 267C288 (1996)[22] Z Tian, Compressed wideband sensing in cooperative cognitive radio networks. Paper presented at IEEE Globecom Conference 2008,New Orleans, Dec 2008 OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 16 [23] Z Tian, E Blasch, W Li, G Chen, X Li, Performance evaluation of distributed compressed wideband sensing for cognitive radionetworks. Paper presentted at the ISIF / IEEE International Conference on Information Fusion (FUSION), Cologne, Germany, July, 2008[24] JP Elsner, M Braun, H Jakel, FK Jondral, Compressed spectrum estimation for cognitive radios. Paper presented at 19th Virginia TechSymposium on Wireless Communications, Blacksburg, June 2009[25] Y Wang, A Pandharipande, Y Polo, G Leus, Distributed compressive wide-band spectrum sensing. Paper presented Information Theoryand Applications Workshop, San Diego CA, Feb 2009[26] Y Polo, Y Wang, A Pandharipande, G Leus, Compressive wideband spectrum sensing, Paper presented at International Conference onAcoustics, Speech, and Signal Processing 2009, Taipei, Taiwan, ROC, April 19-24 2009[27] Y Liu, Q Wan, Anti-sampling-distortion compressive wideband spectrum sensing for cognitive radio. International Journal of MobileCommunications , 6: 604-618 (2011)[28] Y Liu, Q Wan, Compressive wideband spectrum sensing for fixed frequency spectrum allocation, (arXiv 2010),http: // arxiv.org / abs / , 2: 407-499 (2004)[30] SJ Kim, K Koh, M Lustig, S Boyd, D Gorinevsky, An interior-point method for large-ccale l1-regularized least squares. IEEE Journalon Selected Topics in Signal Processing , 4: 606-617 (2007)[31] SJ Kim, K Koh, M Lustig, S Boyd, D Gorinevsky, An interior-point method for large-scale l1-regularized least squares. Paper presentedat International Conference on Image Processing, San Antonio, Texas, USA, Sept 16-19, 2007[32] S Boyd, L Vandenberghe, Convex Optimization , (Cambridge University Press, New York NY, 2004)[33] J Sturm, Using sedumi 1.02, A matlab toolbox for optimization over symmetric cones. Optimization Methods and Software , 12:625-653 (1999)[34] J Lofberg, Yalmip: Software for solving convex (and nonconvex) optimization problems. Paper Presented American Control Conference,Minneapolis, MN, USA, June 2006[35] M Stojnic. Strong thresholds for L2 / L1-optimization in block-sparse compressed sensing,” Paper presented at International Conferenceon Acoustics, Signal and Speech Processing 2009, Taipei, Taiwan, ROC, April 2009[36] M Stojnic, F Parvaresh, B Hassibi, On the reconstruction of block-sparse signals with an optimal number of measurements. IEEETransaction on Signal Processing , 8: 3075-3085 (2009)[37] EJ Candes, MB Wakin, S Boyd, Enhancing sparsity by reweighted l1 minimization. The Journal of Fourier Analysis and Applications , 5-6: 877-905 (2008) OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 17
PSD outputsAIC DSPChannelTransmitter Channel 1Transmitter 1 Channel KTransmitter K (cid:35)
Received analogue signal
Fig. 1. The proposed compressive wideband spectrum sensing structure. N o r m a li z ed PS D Fig. 2. The normalized spectrum of noiseless active primary signals in the monitoring band. −3 Frequency (MHz) N o r m a li z ed PS D Fig. 3. The normalized spectrum of noisy active primary signals in the monitoring band.
OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 18 N o r m a li z ed PS D Fig. 4. The compressive wideband spectrum estimation via BPDN-CWSS. N o r m a li z ed PS D Fig. 5. The compressive wideband spectrum estimation via VLBS-CWSS.
OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 19 N o r m a li z ed PS D Fig. 6. The compressive wideband spectrum estimation via EVLBS-CWSS.
OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 20
TABLE IT he total energy in each subband with the three
CWSS methods and the values of
EDPER, when there are active bands and thesub - sampling ratio is OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 21
TABLE IIT he total energy in each subband with the three
CWSS methods and the values of
EDPER, when there are active bands and thesub - sampling ratio is OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 22
TABLE IIIT he total energy in each subband with the three
CWSS methods and the values of
EDPER, when there are active bands and thesub - sampling ratio is OURNAL TITLE, VOL. X, NO. X, MONTH YEAR 23
TABLE IVT he total energy in each subband with the three
CWSS methods and the values of
EDPER, when there are active bands and thesub - sampling ratio is0.30.