Joint compressive spectrum sensing scheme in wideband cognitive radio networks

Transcription

1 J Shanghai Univ (Engl Ed), 2011, 15(6): Digital Object Identifier(DOI): /s Joint compressive spectrum sensing scheme in wideband cognitive radio networks LIANG Jun-hua (ù u), LIU Yang (4 ), ZHANG Wen-jun (Ü ) School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai , P. R. China Shanghai University and Springer-Verlag Berlin Heidelberg 2011 Abstract In this paper, a distributed compressive spectrum sensing scheme in wideband cognitive radio networks is investigated. An analog-to-information converters (AIC) RF front-end sampling structure is proposed which use parallel low rate analog to digital conversions (ADCs) and fewer storage units for wideband spectrum signal sampling. The proposed scheme uses multiple low rate congitive radios (CRs) collecting compressed samples through AICs distritbutedly and recover the signal spectrum jointly. A general joint sparsity model is defined in this scenario, along with a universal recovery algorithm based on simultaneous orthogonal matching pursuit (S-OMP). Numerical simulations show this algorithm outperforms current existing algorithms under this model and works competently under other existing models. Keywords compressive sensing, analog-to-in-formation converter (AIC), wideband congitive radio (CR) network, joint sparsity, spectrum recovery Introduction Cognitive radio (CR) is emerging technology tackling the spectrum scarcity problem caused by current static spectrum allocation schemes in wireless communication networks [1 2]. CRs can sense wireless spectrum and learn from the environment, utilize unoccupied spectrum holes through spectrum sensing, which can dramatically increase the throughput of the whole wireless communication system [3]. Analog to digital conversion (ADC) is the key enabling technology for CR. The Shannon-Nyquist theorem lies at the heart of essentially all ADC devices. It stats that signal bandlimits to B Hz can be perfectly recovered from uniform samples, if the sampling rate is at least 2B samples/s. Today, six decades of after the formulation of this theorem by Shannon, there are various architectures for ADC design: flash, folding, pipelining, time-interleaving to name a few. However, the ultimate goal remains Nyquist sampling, that is conversion at a rate which is twice the highest frequency of the input signal. State-of-the-art Nyquist ADCs can achieve a sampling rate of 550 MHz at 12 bits resolution [4], while typical CR needs operate at as high as tens of GHz. Apparently the improvement trend in state-of-the-art ADCs is not sufficiently fast to fill the requirements of CR technology. The common practice in engineering to overcome the ADC bottleneck is demodulation, namely multiplying the input by the carrier frequency f c of a band of interest, so as to shift the contents of the narrowband transmission from the high frequencies to the origin. Therefore, instead of direct sampling of a wideband input, which may be impossible with existing technology, each band is treated individually. However, demodulation requires knowing the exact carrier frequency. This knowledge presumably requires user assistance, e.g., when tuning a radio to the frequency of a station of interest. Unfortunately, such a user-assisted solution is impractical in typical CR applications, such as military surveillance, radar, electronic warfare and medical imaging. In this paper, the proposed solution as an alternative to demodulation is to sample the entire wideband spectrum directly, which uses compressive sensing theory to significantly decrease the complexity of hardware implementation. Compressive sensing is a novel sampling technique that can sample sparse signals with sub-nyquist rates and performs accurate recovery through l 1 minimization algorithm, which attains a considerable compression rate [5]. Thus, compressive sensing could recover wideband spectrum precisely with much less samples, Received Dec. 21, 2010; Revised Feb. 20, 2011 Project supported by the National Fundamental Research (Grant Nos.2009CB ,2010CB731803), the National Natural Science Foundation of China (Grant Nos , , , ), the Natural High-Technology Research and Development Program of China (Grant Nos.2007AA01Z267, 2009AA01Z248, 2009AA011802) Corresponding author LIANG Jun-hua, Ph D Candidate, liang456@gmail.com

2 J Shanghai Univ (Engl Ed), 2011, 15(6): attaining higher detection probability than traditional energy detection method. In addition, recent emerging analog-to-information converters (AIC) techniques [6 7] guarantee hardware implementation of fast and efficient sampling of real time signals. Noting that the typical wireless wideband signal is sparse enough with a remarkable low occupance rate of the whole spectrum, [8 10] have studied varied compressive spectrum recovery schemes. In [8 10], the spectrum is modeled as sparse signal in wavelet domain. A wavelet approach is proposed to detect the edges and make power spectral density estimations using compressive sensing. In this paper, we prefer a simple but efficient alternative compressive sampling formulation by direct sampling of the wideband spectrum signals. Due to frequency selective fading, random shadowing and probable interference from other wireless facilities, a single CR could not avoid problems such as hidden terminals and results in high miss detection probability. Then cooperative spectrum sensing [11] is proposed to overcome the uncertainty related with single detection. Those types of detection procedure rely on local decisions. In [12 13], joint compressive spectrum recovery is considered, which supposes the signals received at multiple CRs are highly correlated, known as Joint Sparsity [11]. It is important to notice that joint spectrum recovery requires much less samples for each CR comparing with separated recovery scheme, which further reduces power and bandwidth for communication and memory size for storage. However, uncertain channel fading and random shadowing would lead to varying signal strength at different CRs, even placed close to each other, deep shadowing can cause much weaker signal detected at related CRs, as shown in Fig.1. In addition, hidden terminal problems would probably lead to total loss of certain spectrum component at CRs. In this paper, a novel applicable model along with a universal recovery algorithm is proposed covering both existing joint sparse models and simple non-joint models. Fig.1 Illustration of GJSM in CR networks The rest of this paper is organized as follows: In Section 1, compressive sensing backgrounds are introduced. In Section 2, the low rate ADC based sampling architecture is discussed and compressive wideband sampling at single CRs is explored. In Section 3, after defining a new joint sparsity model in CR networks, recovery algorithm and simulation results are discussed. Finally, we conclude our works in Section 4. 1 Compressive sensing backgrounds Suppose an S-sparse signal s R N need to be sampled. Define supp(v) ={i v i 0}, thus supp(s f ) = S. According to compressive sensing theory, the samples have the form y = Φs, wherethem N random measurement matrix Φ projects s onto y R M. Define the restricted isometry constant (RIC) δ s to be the smallest positive constant such that (1 δ s ) x 2 2 Φ T x 2 2 (1 + δ s ) x 2 2, (1) for all subsets T with T S and coefficient sequences {c j },j T. One known result about RIC is that: when δ 2s + δ 3s < 1, the signal s can be exactly recovered by solving ŝ =argmin ŝ 1, s.t. y = Φs. (2) For implementation issues, when the elements of Φ are chosen as i.i.d. random gaussian variables such that Φ i,j N(0, 1 M ), if S CM log N, (3) M then with probability 1 O(e γn ), the signal can be exactly recovered. When the measurements are corrupted with noise, such that y = Φs+w, w 2 σ. The estimated signal ŝ can be recovered by solving ŝ =argmin ŝ 1, s.t. y Φŝ 2 σ. (4) When δ 3s + δ 4s < 2, C s is some constant related to δ 4s, then we have ŝ s 2 C s σ. (5) For the convex optimization problem in (2) and (4), several algorithms like basic pursuit (BP) and orthogonal matching pursuit (OMP) can be applied for signal recovery [14]. WeproposeasimpleOMPbasedjointrecovery algorithm in later sections. 2 Compressive sampling at single CR In CR networks, the application of compressive sampling brings few measurements as well as higher detection rate. In this section, a low rate ADC based CR front-end architecture is firstly proposed, afterwards a compressive sensing formulation is discussed. Consider a slot-segment model of wideband spectrum, where the whole spectrum is divided into K nonoverlapping sub-bands. Suppose each sub-band requires

3 570 J Shanghai Univ (Engl Ed), 2011, 15(6): C samples, totally N = CK points at Nyquist rate should be sampled during each measurement procedure for exact spectrum recovery. In wideband cases, direct sampling requires high speed ADCs and produces huge data for storage and transmission. Recent emerging AIC techniques have made it possible to sample signals at sub-nyquist rate. In this paper, we propose a novel AIC architecture for CR networks based on low rate ADCs. As is shown in Fig.2, suppose during one measurement, a real-time signal r(t) of finiteduration T is sampled. Suppose an N-point sample r t = [r(0),r(t s ),,r(nt s )] is required at Nyquistrate 1 T s, C parallel low speed ADCs operate at a sample 1 rate CT s. During one measurement period, each ADC starts conversion for K times at an interval of CT s continuously. An f = 1 T s system clock controls the synchronization between C ADCs and switching sequences of multiplexers. The received signal r(t) entersc ADCs simultaneously, each ADC data enters M sub-channels, multiplied by random weights ρ m c and accumulated N times. Noting that ρ m c changes to ρ m c [k] atthekth conversion, produced by random number generators as i.i.d. gaussian variables, thus, after each measurement, the mth sub-channel outputs a compressed sample C K N y m = ρ m c [k]r((ck + k)t s)= μ m [n]r t [n], (6) c=1 k=1 n=1 where μ m R N consists of the rearranged weights ρ m c [k]. Thus the output of total M sub-channels will be y = Ur t, (7) where y =[y 1,y 2,,y M ] T is the sampling vector and μ m is the mth row of measurement matrix U. ThisAIC architecture is proposed because only M<<Nmemory units are required and each ADC sample at a low rate 1 of CT s instead of the Nyquist rate 1 T s. Further, more ADCs could be integrated in one AIC to obtain higher spectrum resolution and further reduce sampling rate for individual ADC. Next, considering the signals are corrupted with noise in practise, we have r t = s t + w t, (8) where s t denotes signal before simultaneous ADCs. Suppose w t is independent zero-mean white Gaussian noise with variance σ 2. Thus w t N(0,σ 2 I n ). After one measurement, the measurement vector becomes: y = Ur t = UF N s f + w = Φs f + w, (9) where F N denotes N-point orthogonal DFT matrix which is incoherent with matrix U, and w = Uw t denotes noise at the AIC output. s f is sparse due to low occupance ratio across the whole sensing band, for simplicity, the subscript f is ignored in this paper. When ρ m c is chosen such that ρ m c N(0, 1 M ), it could be proved that Φ=UF N also satisfies (1) with high probability. Further, according to (1), the output noise level can be estimated as Fig.2 AIC structure based on low rate ADCs w 2 2 (1 + δ s ) w t 2 2. (10) These results can be immediately applied in compressive sampling (2), (3) and in joint recovery in next section. 3 Joint recovery at multiple CRs 3.1 Model definition In wideband CR networks, signals received at each CR are highly spatial correlated, while shadowing and hidden terminal problems would cause signal strength detected at CRs varies significantly. This phenomenon indicates different support sets of s j, for all j = 1, 2,,J. Thus a general joint sparsity model (G- JSM) is defined in this scenario. Definition G-JSM s j = Ωθ j + z j, j =1, 2,,J. (11) We suppose each CR signal has the same partial common sparse support Ω and a proprietary term z j. Define Λ c = j supp(s j), for all j =1, 2,,J. Then supp(z j ) = supp(s j )/Λ c. Ω is the sub-matrix of Identity matrix I N, retaining columns indexed by Λ c,and

4 J Shanghai Univ (Engl Ed), 2011, 15(6): θ j denotes varying signal strength on this support set. Clearly, this model fits the scenario where frequency selective channel fading leads to different θ j and hidden terminals contribute to existence of z j. We call this model general joint sparsity model, since when θ j = θ, for all J CRs, this model degenerates to JSM1 [9] : s j = z c + z j, j =1, 2,,J, (12) where z c is the common component between all J signals. A γ weighted l 1 algorithm [9] is proposed for recovery. When z j = 0, this model degenerates to JSM2 [11] : s j = Ωθ j, j =1, 2,,J. (13) Two efficient algorithms OSGA, SOMP [11] are suggested for joint recovery under JSM2. Unfortunately, no existing joint recovery algorithm can be applied directly for our defined general model. We propose a novel joint algorithm which works well universally under different models. 3.2 Recovery algorithm In this subsection, we propose a novel joint recovery algorithm for the G-JSM model. The algorithm consists of two major stages. In the first stage, like SOMP, one column is selected from the measurement matrix jointly at each iteration, however, since not all signals share the same supports, a discarding step follows to make judgements separately. In the second stage, each measurement recovers signal separately using OMP [3],after joint selection failure in the previous stage. We call this method select-discard simultaneous OMP (SD-SOMP) (see Fig.3). For simplicity, we assume all J CRs share the same measurement matrix Φ. According to earlier discussion, the stopping criterion can be made as y j Φŝ j 2 ασ with some constant α or simply set a maximum iteration steps from estimated sparsity level. 3.3 Simulation results Simulation 1 Efficiency of SD-SOMP First, it is easy to see SOMP or OSGA algorithm cannot be applied under our joint sparsity model since these algorithm assume signals all share the same sparse supports. We perform two simulations to show γ weighted algorithm for JSM1 also deteriorates under GJSM. Suppose N=1 024 points signal to be recovered. J =5CR users are present during once measurement procedure. For simplicity, suppose all signals are 12 sparse, with SNR=30 db. First assume all signals share a 11 sparse common component. Clearly, this fits the JSM1 model. The threshold η is set to Figure 4 shows the recovery probability vs. samples per CR. It can be seen γ weighted algorithm works better since it mostly utilize the joint sparsity level. However, when choosing the input: AnM N matrix Φ M 1samplesy j, j =1, 2,,J A discarding threshold η output: N 1 recovered signals s j, j =1, 2,,J initialization: Residuals: ε j y j Support sets: Λ j Reference support set: Λ ref (i) Find the index ω that solve the optimization: JX ω =argmax ε j, φ w ω (14) j=1 φ ω denotes the ωth column of Φ. (ii) if ω Λ ref then else end Λ ref Λ ref {ω} Calculate inner product terms ε j,φ n, suppose that φ ω is related to the kth biggest term. if k>ηn then else end discard ω for the jth measurement. Λ j Λ j {ω} find the index ω separately that matches ω =argmax w ε j, φ ω (15) Λ j Λ j {ω} (iii) Solve the least square problem to get new estimate of ŝ fj ŝ fj =argmin y j Φ Λjŝj 2 (16) where Φ Λj is of size M N by setting φ ω =0,w Λ j, or equivalently, ŝ j = Φ Λ j y j (17) (iv) Update residuals: ε j = y j Φ Λj ŝ j (18) (v) If stoping criterion is matched, iteration ends, else go to step (i). Fig.3 SD-SOMP signals to have 11 common sparsity supports, which fits GJSM. SD-SOMP outperforms this algorithm, as shown in Fig.5. Note all the other parameters remain the same as in Fig.4. In Fig.6, different SNR levels are compared, in high SNR level cases, recovery probability keeps relatively stable resulting from joint support selection. Clearly, more samples or measurements are required in low SNR cases. Simulation 2 Universality of SD-SOMP We provide two simulations to demonstrate the universality of SD-SOMP, resulting from the choice of the discarding threshold η. Figure 7 shows recovery probability vs. discarding threshold under different joint sparsity levels. We set signal length to and each signal

5 572 J Shanghai Univ (Engl Ed), 2011, 15(6): Fig.4 Recovery probability vs. samples per CR under JSM1 supports. In addition, higher joint sparsity level deteriorate slower, further, under highest joint sparsity level, SD-SOMP always outperforms separate recovery. Noting that higher threshold contributes to faster convergence, there is a tradeoff between convergence speed and recovery probability. When joint sparsity level is high, threshold can be set higher, and vice versa. Figure 8 shows the university of SD-SOMP under non-joint sparse cases, and the signals are set to be totally uncorrelated with supp(s i ) supp(s j )=. Thus by decreasing the threshold, SD-SOMP degenerates to traditional separate algorithm. Fig.5 Recovery probability vs. samples per CR under GJSM Fig.8 Recovery probability vs. samples per CR under non JSM Fig.7 Fig.6 Recovery probability vs. different SNRs Recovery probability vs. discarding threshold under GJSM is 10 sparse with SNR=30 db, each CR samples 80 points. 10 CRs are included. First observe that under lower thresholds, SD-SOMP outperforms separate recovery under all joint sparsity levels. As the threshold increases, denoting looser discarding criterion, SD- SOMP will deteriorate caused by falsely selected sparse 4 Conclusions In this paper, we propose a novel AIC structure of CR front-end integrating low rate ADCs and few storage units. Then, we discuss compressive spectrum sampling under this scheme, considering noisy cases. Further, we explore a new joint sparsity model in CR networks and provide a universal SD-SOMP algorithm to perform joint spectrum reconstruction. Simulations show SD- SOMP is actually a robust recovery algorithm under different joint sparsity models. References [1] Wang X, Li Z, Xu P, Xu Y, Gao X, Chen H. Spectrum sharing in cognitive radio networks An auction based approach [J]. IEEE Transactions on System, Man and Cybernetics Part B: Cybernetics, 2010, 40(3): [2] Gao L, Wang X, Xu Y, Zhang Q. Spectrum trading in cognitive radio networks: A contract-theoretic modeling approach [J]. IEEE Journal on Selected Areas in Communications, 2011, 29(4): [3] Huang W T, Wang X B. Throughput and delay scaling of general cognitive networks [C]// The 30th IEEE International Conference on Computer Communications, Shanghai, China. 2011, DOI: /INF- COM [4] Data converters, Texas Instruments Corp. [EB/OL]. analog/docs/dataconvertershome. tsp.

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