Cyclostationarity-Based Low Complexity Wideband Spectrum Sensing using Compressive Sampling

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1 Cyclostationarity-Based Low Complexity Wideband Spectrum Sensing using Compressive Sampling Eric Rebeiz, Varun Jain, Danijela Cabric University of California Los Angeles, CA, USA {rebeiz, vjain, Abstract Detecting the presence of licensed users and avoiding interference to them is vital to the proper operation of a Cognitive Radio (CR) network. Operating in a wideband channel requires high Nyquist sampling rates, which is limited by the state-ofthe-art A/D converters. Compressive sampling is a promising solution to reduce sampling rates required in modern wideband communication systems. Among various signal detectors, feature detectors which exploit a signal cyclostationarity are robust against noise uncertainties. In this paper, we exploit the sparsity of the two-dimensional spectral correlation function (SCF), and propose a reduced complexity reconstruction method of the Nyquist SCF from the sub-nyquist samples. The reconstruction optimization is formulated as a regularized least squares problem, and its closed form solution is derived. We show that for a given spectrum sparsity, there exists a lower bound on sampling rates that allows reliable SCF reconstruction. I. INTRODUCTION Motivated by the spectrum scarcity problem, Cognitive Radio (CR) has been proposed as a way to opportunistically allocate unused spectrum licensed to Primary Users (PUs). In this context, secondary users (SUs) sense the spectrum to detect the presence or absence of PUs, and use the unoccupied bands while maintaining a predefined probability of misdetection. Sensing over a wide range of frequencies is highly desirable since sensing multiple channels simultaneously increases the probability of finding unused spectrum. Therefore, a wideband receiver front-end is usually adopted for spectrum sensing in CRs. Unfortunately, radios that operate over a bandwidth on the order of 500 MHz require high sampling rate A/D converters with large dynamic range due to inband PU signals, which are challenging to design. In order to reduce sampling rate requirements, compressive sampling is an appealing option that allows sampling below the Nyquist rate provided signal s sparsity in a given transform domain. Among the spectrum sensing algorithms proposed in the literature [1] [3], we consider in this paper feature based detectors. Feature detectors exploit signal s cyclostationarity by detecting spectral correlation peaks in the SCF, which is sparse in both cyclic and angular frequency domains. They are also highly robust to noise uncertainty which makes other detectors such as energy detectors fail. In addition, feature detectors are able to differentiate among signals of interest This work is part of a collaborative project with Prof. Yonina C. Eldar and Deborah Cohen from Technion - Israel Institute of Technology, Haifa, Israel. (SOIs), interfering signals, and noise by using the SOI s cyclostationary spectral correlation features. In order to detect spectral correlation peaks of a PU signal, feature detectors first estimate the SCF [4] which is a twodimensional spectral map showing the spectral correlation peaks. One approach to perform sub-nyquist cyclostationary feature detection is to first recover the Nyquist samples, then estimate the SCF, and perform feature detection. Such an approach is presented in [5] using the Modulated Wideband Converter (MWC) as a front-end [6]. In the very first paper that considered cyclostationary detection using sub-nyquist samples [], the SCF reconstruction is performed blindly with no a priori knowledge of the carriers and bandwidths of the signals to be detected. Here, we exploit the fact that in a typical CR setting, the sensing radios have some information about the signals to be detected. Then, the goal of CRs is to detect the presence or absence of one or many PUs simultaneously in a wideband channel, with a priori knowledge of their carrier frequencies, symbol rates, and modulation schemes. In this paper, we propose a reduced complexity approach for reconstructing the Nyquist SCF from sub-nyquist samples by formulating an optimization problem which exploits the SCF sparsity, and give a closed form solution to the optimization problem. The rest of the paper is organized as follows. In Section II, we present our system model, give a brief overview on cyclostationarity and the problem statement. In Section III, we relate the SCF of the sub-nyquist samples to the one using Nyquist samples, and give a closed-form solution to the reduced complexity SCF reconstruction problem. In Section IV, we describe the feature-based signal detector. Section V presents the numerical results, and finally Section VI concludes the paper. II. SYSTEM MODEL AND PROBLEM FORMULATION A. System Model In the context of CR spectrum sensing, we focus on the processing of a received wideband signal in a channel of bandwidth B centered at any carrier frequency. The approach and analysis considered here applies to any channel bandwidth, not necessarily supported by state-of-the-art A/D converters. We assume that the wideband signal could be occupied by K PU signals s k (t) k [1,...,K] with known carrier frequencies f ck, symbol periods T k and modulation type that 1644

2 we want to detect the presence of. According to the Nyquist sampling theorem, the sampling rate of the wideband signal should be at least B. The received signal is given by K x(t)= s k (t), where k=1 w k (t) underh k,0 s k (t)= R{a k (nt k )p k (t nt k )e jπfc k t }+w k (t), underh k,1, n= where a k (nt) and p k (t) are the transmitted information symbols and the pulse shaping filter of thek th PU respectively, and w k (t) is the AWGN in the band occupied by the k th transmitter. We assume transmitted information symbols with average power σ a k, a pulse shape filter p k (t) of unit energy, and we define the Signal to Noise Ratio by SNR k = σ a k /σ w k where σ w k is the noise variance in the channel occupied by s k (t). Further, received powers from all the transmitters are assumed to be equal, yielding the same in-band SNR for all present signals. Therefore, the detection performance across the bands of interest will be the same for all K signals. Let H 0 and H 1 correspond to the hypotheses that the desired signal s k (t) is absent or present respectively for any index k [1,...,K]. B. Spectral Correlation Function Overview A random process s(t) is said to be cyclostationary if its second order moments satisfy two conditions: its mean and autocorrelation are both periodic with the same period T. In the context of digital communications, T coincides with the symbol period of the modulation scheme. If we let E[s(t)] = µ s (t) and E[s(t)s(t+τ)] = R s (t,τ), then E[s(t+T)] = µ s (t) and R s (t + T,τ) = R s (t,τ) iff s(t) is cyclostationary. Given a cyclostationary random process, its asymptotic autocorrelation R s (t,τ) can be expanded using the Fourier series as follows R s (t,τ) = Rs(τ)e α jπαt, (1) α where α = l/t for l Z. In (1), Rs(τ) α is the projection of R s (t,τ) onto the exponential basis function with frequency α, given by R α s(τ) = 1 T R s (t,τ)e jπαt dt. () Since R s (t,τ) is periodic in t, the Fourier Series of Rs(τ) α have spectral peaks at α = l/t, l Z. The asymptotic SCF is obtained by taking the Fourier Transform of () with respect to τ, yielding S s (α,f), namely S s (α,f) = R α s(τ)e jπfτ dτ, (3) where α is refered to as the cyclic frequency, and f is the angular frequency. The (α, f) locations of the spectral peaks of the SCF for common digital modulation schemes are summarized in Table I. Note that AWGN is a widesense stationary process and exhibits no cyclic correlation. Therefore, the asymptotic SCF of noise has no spectral features at α 0. TABLE I CYCLIC FEATURES FOR SOME MODULATION CLASSES. Modulation Peaks at (α,f) BPSK ( 1 T,fc), (fc,0), (fc ± 1 T,0) MSK ( 1 1,fc), (fc ± T T,0) QAM ( 1 T,fc) C. Sampling and Computing the Discrete-Time Non- Asymptotic SCF Given the time and energy constraints on the sensing stage, we consider a limited time window of length T sense = N T T s during which the sensing radio acquires a total number of N T incog samples, where T s is the sampling period. Let t [0,T sense ] denote the time variable. Since the computation of the SCF includes computing the auto-correlation, we consider a frame-based model, where each frame is of length N samples, and where the remaining L = N T /N frames (assumed to be integer) are used for statistical averaging. Let x R N denote a frame of samples obtained from sampling x(t) at the Nyquist rate of f s = B. Similarly, let z R M denote a frame of samples resulting from sampling x(t) at a sub-nyquist rate f s = BM N, with a compression ratio M N 1. We further assume that the sub-nyquist samples vector z R M can be obtained from the Nyquist samples x via z = Ax, where A R MxN is the sampling matrix. In practice, the vector z can be obtained directly in the analog front-end using Analog to Information Converter (AIC) [7], or a MWC [6]. All random samplers used in traditional CS literature would work with high probability [8]. In this work, A is chosen to be a random Gaussian matrix in R MxN. In the context of time constrained spectrum sensing, due to the limited number of samples acquired, estimation of the asymptotic SCF is not possible. Thus, the spectrum sensing processor estimates the discrete non-asympotic SCF based on N T samples as follows, S x = D ( 1 L ) x l x T l D, (4) where L is the number of spectral averages, and D is a NxN DFT matrix. The resulting SCF matrix in (4) is a discrete representation of the asymptotic SCF defined in (3). The resolution in both cyclic and angular frequencies of S s is equal to f s /N. As N, the resolution in both α and f decreases and the estimated discrete SCF approaches the asymptotic continuous SCF. The estimated SCF of z can be obtained using (4) by replacing x l by z l. Note that the spectral correlation peaks ofxmight not be present in the SCF ofzdue to the sampling operation, therefore S x has to be reconstructed from the sub-nyquist samples z. III. NYQUIST-SCF RECONSTRUCTION In this section, we find a closed-form expression which reconstructs the Nyquist SCF from sub-nyquist samples. In fact, given that the modulation scheme, carrier frequency and symbol rate of the transmitted signal s k (t) are known, reconstructing the entire SCF from the sub-nyquist samples is not necessary, and only a few (α,f) points in S x are required to perform the signal detection. 1645

3 A. Reconstructing the Nyquist SCF We start by defining the covariance matrices R x R NxN and R z R MxM. The non-asymptotic covariance matrices can be estimated using a limited number of samples by timeaveraging the cross-correlations over L frames, namely R x = 1 L x l x T l and R z = 1 L z l z T l, (5) where the x l and z l denotes the l th frame of Nyquist and sub- Nyquist samples. Given that covariance matrices are symmetric positive semi-definite, only M(M+1) and N(N+1) elements of R z and R x respectively are used for processing. We define the vector r x = [ r x (0,0),r x (1,0),...,r x (N 1,0),r x (1,1),...,r x (N,1),...,r x (0,N 1)] T, (6) where r x R 0.5N(N+1) and ν in r x (n,ν) defines the lag with respect to the variable n. We define the vector r z R 0.5M(M+1) in a similar manner. We relate the vectorized covariance matrices r x and r z defined in (6) to R x and R z respectively through mapping matrices P N {0,1} N x N(N+1), P M {0,1} M x M(M+1), Q N {0,1} N(N+1) xn and Q M {0,1} M(M+1) xm giving the following relationships [], r x = Q N vec(r x ), r z = Q M vec(r z ) vec(r x ) = P N r x, vec(r z ) = P M r z. (7) The elements of r x and r z are rearranged into two matrices R x, R z as follows [] r x (0,0) r x (0,1)... r x (0,N 1) R x r x (1,0) r x (1,1) (8) r x (N 1,0) The SCF matrix S x can be obtained from the covariance matrix R x using the following operation S x = N 1 ν=0 G Nν R x D Nν F N, (9) where F N = [e j π N ν(b N 1 ) ], G Nν = 1 [ e j π N a(n+ν ) ] are NxN matrices, 0.5N(N+1) a,b = [0,..,N 1] correspond to the (α,f) indices respectively, and D Nν is an NxN matrix with its (ν,ν) element being 1 and all the remaining elements being 0. Note that S x is a permutation of S x given in (4) where the PSD elements are now along the first row of S x. The vectorized SCF given by s x = vec(s x ) is related to the vectorized autocorrelation vector r x by [] N 1 s x = (F T N I N ) (D T Nν G Nν )B T Nr x T x r x, (10) ν=0 where is the Kronecker product, and B N {0,1} N(N+1) xn is a mapping matrix relating r x to vec( R x ). Given that z = Ax, it follows that the autocorrelation matrix of x and z are related via R z = AR x A H, where where (.) H denotes the Hermitian conjugate operation. Using relation (7), we can write r z = Q M (A A)P N r x Φr x, (11) where (.) denotes the element-wise complex conjugate operation. We now relate s x to s z. Similarly to (10), we define T z C M x M(M+1). Therefore, s x = T x r x and s z = T z r z. (1) Using the fact that T x is a full column-rank matrix, hence T xt x = IN(N+1) where denotes the pseudo-inverse x N(N+1) operation, we can relate s x to s z as follows: T z r z =T z Φr x (13) s z =T z Φr x = T z ΦT xt x r x Φ s x, (14) where Φ = T z ΦT x C M xn. Using the relation obtained above, we solve for s x by formulating the regularized least squares problem as follows s x s z Φ s x. (15) Given that the SCF is sparse in both α and f, we further introduce an l 1 imization term in the unconstrained optimization problem given by for some λ > 0 R. s x s x 1 +λ s z Φ s x (16) B. Closed Form Solution for SCF Reconstruction As was shown in Section II-B, the spectral correlation peaks are discrete in the cyclic frequency domain α. The resolution in both f and α is solely detered by the FFT size N. Therefore, in order to detect signals in a wideband channel of large bandwidth, the resolution required for the spectral correlation peaks to be proent should be high, and therefore N would be chosen high as well. As a result, the optimization problem proposed in (16) would be computationally expensive as s x is in C N. Since the SCF is sparse, we propose to reconstruct only a few (K f ) points of the SCF where the spectral correlation peaks would be present. Let M f be a matrix R K f xn with elements in each row equal 1 at the indices corresponding to possible spectral peaks in s x. Therefore, we define the following variable ŝ x = M f s x, (17) where ŝ x C K f x1, and where K f is the number of reconstructed cyclic frequencies where signal features exist. We also define the matrix ˆΦ = Φ {j=1:k f }, which selects the K f columns of Φ corresponding to the indices of the spectral correlation peaks. Using this approach, we pose the optimization as follows ŝ x s z ˆΦŝ x (18) 1646

4 which is an optimization in C K f x1, where K f << N. This formulation renders the optimization problem more computationally efficient. Note that since we are only reconstructing a few points of the SCF, the sparsity factor given in the l 1 norm term in (16) has been dropped, reducing it to an unconstrained least squares problem. The least-squares problem given in (18) has a unique solution if the system of equations is overdetered, i.e, if the rank of matrix ˆΦ equals K f. Since the matrix ˆΦ is a rank-deficient matrix, one solution to the optimization problem is obtained by adding a regularization term which will make ˆΦ HˆΦ invertible. Thus, we obtain the closed form expression of the regularized least-squares problem whose solution always exists and given by ŝ x = [λi+ ˆΦ HˆΦ] 1ˆΦ H s z (19) where λ > 0. The presence of the positive semi-definite matrix λi gaurantees an invertible coefficient matrix regardless of whether ˆΦ is rank-deficient or not. It is worth mentioning that the number of frames L used for averaging of the autocorrelation plays a crucial role in the resulting sparsity of the reconstructed SCF. In fact, if L is not large enough, the reconstructed SCF will not be sparse due to the noise fluctuations, and therefore reconstruction of only K f points of the SCF will result in an erroneous SCF. IV. SIGNAL DETECTION ALGORITHM Once the Nyquist-based SCF has been reconstructed by solving (19), we proceed with the detection process in the form of a hypothesis test, where the detection is performed per signal s k (t). The number of reconstructed points K f is equal to the total possible number of present spectral correlation peaks of the received wideband channel, where each present signal could have spectral correlation peaks given in Table I. In this work, we consider single-cycle detectors which base their decisions on collected energies of the spectral peaks in the SCF at a single cyclic frequency α. The number of bins that one could consider at a given cyclic frequency α is a function of the width of the spectral correlation peak in the angular frequency domain f, which is itself dependent on the symbol rate 1/T k and the pulse shape filter p k (t) [9]. Let the test statistic be V which integrates the energy at a given cyclic frequency α i. For modulation classes with more than one cyclic frequency, we choose the one with the largest spectral correlation peak. Therefore, the resulting test statistic is as follows V = ŝ x (α i,f j ), (0) j wheref j refers to the the angular frequency indices over which the spectral correlation peak of interest is spanned. Given that most of noise is generated at the receiver front-end, the threshold to which the test statistic (0) is compared against is computed by turning off the receiving antenna and computing the test statistic under H 0. The estimated distribution of the test statistic under H 0 is used to detere a threshold T to which V is compared against, yielding a constant false alarm ( + ), $ % &' * ( + ), $ % &' *! " # (a) Sub-Nyquist SCF obtained from the autocorrelation vector r z! " - # (b) Reconstructed Nyquist SCF using s z and Eq. (19). Fig. 1. Comparison of sub-nyquist and reconstructed SCFs under SNR = 10 db, M/N=0.75, and L = 00 averages. rate (CFAR). The probability of detection and false alarm are defined as P D = Pr(V > T H 1 ) and P FA = Pr(V > T H 0 ). V. NUMERICAL RESULTS AND DISCUSSION We consider a wideband channel of bandwidth 300 MHz, occupied with up to K = 5 BPSK signals with effective signal bandwidth of 15 MHz, with equal in band SNR for all signals. The elements of the MxN sampling matrix are i.i.d. real Gaussian random variables distributed according to N(0,1/M), and a frame length is set to N = 36 samples. In the first result, we show how the Nyquist SCF is reconstructed from the sub-nyquist samples for a BPSK signal. The compression ratio is set to 0.75, at an SNR = 10 db, L = 00 averages, and K = 1. Fig. 1(a) shows the SCF obtained from processing the sub-nyquist samples z with K f = 9N (reconstructing 9 complete α slices of the Nyquist SCF). Note that although some cyclic frequencies where noise only lies have been reconstructed, that is for illustration purposes and those cyclic frequencies would not be used for detection. Because of the random matrix A with i.i.d. elements which multiplies the correlated elements of x, the resulting vector z loses its spectral correlation, and therefore its SCF looks like one of a stationary process, making the signal detection unfeasible. In contrast, we show that Eq. (19) reconstructs all the spectral correlation peaks of the BPSK signal as is shown in Fig. 1(b). Note that only a few cyclic frequencies were reconstructed here, proving the feasibility of the reduced complexity reconstruction. 1647

5 ^ 4. 9 I J 4 K A L B D M I J N K A L B D O P Q R S T U U V P W X Y Z Z U H G F / 0. / 1. /. / 3 4 : ; < A ; B C D E A ; / [. / 1. / N. /. / \. / 3. / ] 4 : ; < A ; B C D E A ; Fig.. Mean Squared Error (MSE) for different compression ratios for different spectrum sparsities under noiseless conditions. Fig. 3. Required number of frames L for averaging versus compression needed to reach (P FA,P D ) = (0.1,0.9) for K f = 18 (out of N ), with K = 5 signals at SNR = 6 db. In the next experiment, we analyze the the effect of decreasing the compression ratio on the asymptotic test statistic. Under noiseless conditions, we select L to be in the order of 1000 frames to ensure that the reconstructed SCF is K f sparse, i.e. that are at most K f non-zero elements in the SCF. We define the mean squared error as MSE = E [ V V nc / V nc ], (1) where V nc is the resulting test statistic under no compression. Fig. shows the MSE for different channel sparsities, K = 1 and K = 5 signals corresponding to a spectral sparsity of 5% and 5% respectively. The MSE starts increasing below a threshold compression ratio M/N, the point at which the system of equations s z = ˆΦŝ x becomes under-detered. We call the imum compression ratio at which this behavior occurs the Compression Wall. This wall is detered by the sparsity of the SCF, which itself is a function of the present signals in the wideband channel. Therefore, even with an infinite number of samples, there always exists a compression point below which the SCF reconstruction and therefore signal detection starts to degrade. Next, we quantify the number of additional samples required to reach a given (P FA,P D ) = (0.1,0.9) at various compression ratios using our reduced-complexity detector. Fig. 3 shows the trend of L versus the compression ratio under a fixed sparsity of K = 5 signals. As expected, the imum lossless compression ratio (M/N) for the given sparsity is reached at about a compression ratio of 0.4 as shown in the MSE plot in Fig.. Under SNR = 6 db, the required number of averages L increases slightly compared to no compression when the compression ratio is above the compression wall. However, operating at a compression ratio below the compression wall requires an exponentially increasing number of averages L in order to operate at the desired ROC point, which makes the detection infeasible within a constrained sensing time. VI. CONCLUSION In this paper, we presented a reduced-complexity wideband spectrum sensing algorithm which reconstructs portions of the Nyquist SCF from compressed samples via a closedform solution. The existence of such closed-form solution is most relevant when it comes to implementing compressive sensing algorithms on a hardware platform where a convex optimization problem is not trivial to solve. Finally, we analyzed the trade-off between compression ratio and sensing time, and showed that the SCF reconstruction, and therefore signal detection, is feasible as long as the compression ratio is above the compression wall. REFERENCES [1] Y. L. Polo, Y. Wang, A. Pandharipande, and G. Leus, Compressive wideband spectrum sensing, in Proc. IEEE ICASSP, 009. [] Z. Tian, Cyclic feature based wideband spectrum sensing using compressive sampling, in Proc. IEEE ICC, 011. [3] D. Wieruch and V. Pohl, A cognitive radio architecture based on subnyquist sampling, in Proc. IEEE DySPAN, 011. [4] E. Like, V. Chakravarthy, and Z. Wu, Reliable modulation classification at low SNR using spectral correlation, in Proc. IEEE CCNC 007. [5] D. Cohen, E. Rebeiz, V. Jain, Y. C. Eldar, and D. Cabric, Cyclostationary feature detection from sub-nyquist samples, in Proc. IEEE 4th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 011. [6] M. Mishali and Y. Eldar, From theory to practice: Sub-nyquist sampling of sparse wideband analog signals, IEEE J. Sel. Topics Signal Process., vol. 4, no., pp , Apr [7] J. Laska, S. Kirolos, and M. Duarte, Theory and implementation of an analog-to-information converter using random demodulation, in Proc. IEEE ISCAS, 007. [8] M. Davenportand, P. Boufounosand, M. Wakin, and R. Baraniuk, Signal processing with compressive measurements, IEEE J. Sel. Topics Signal Process., vol. 4, no., pp , Apr [9] W. Gardner, W. Brown, and C.-K. Chen, Spectral correlation of modulated signals: Digital modulation, IEEE Trans. Commun., vol. 35, no. 6, pp , Jun