Compressed Shattering

The central idea of compressed sensing is to exploit the fact that most signals of interest are sparse in some doma in and use this to reduce the number of measurements to encode. However, if the sparsity of the input signal is not preciselyknown, but known to lie within a specified range, compressed sensing as such cannot exploit this fact and would need to use the same number of measurements even for a very sparse signal. In this paper, we propose a novel method called Compressed Shatteri ng to adapt compressed sensing to the specified sparsity range, without changing the sensing matrix by creating shattered s ignals which have fixed sparsity. This is accomplished by first suita bly permuting the input spectrum and then using a filter bank to create fixed sparsity shattered signals. By ensuring that al l the shattered signals are utmost 1-sparse, we make use of a simpl e but efficient deterministic sensing matrix to yield very low number of measurements. For a discrete-time signal of length 1000, wi th a sparsity range of 5−25, traditional compressed sensing requires 175 measurements, whereas Compressed Shattering would only need20− 100 measurements. I. I NTRODUCTION Compressed sensing [1] is a fundamental idea in mathematics, which utilizes thea priori property of signalx(n) of lengthN beingm sparse in some domain, where m << N , along with an appropriately constructed sensing matrix A, to establish a unique solution for an otherwise undetermine d system of linear equations: AM×N · xN×1 = YM×1. (1) The actual solution, is the vector from the solution set, which has the minimuml0 norm. Since this is a NP-hard problem, so we choose the solution which minimizes the l1 norm. It is observed that minimizingl1 norm will give an accurate solution [2] provided the sensing matrix A satisfies the Restricted Isometry Property ( RIP ) property [1]. However, if the sparsity of the input signal is not precisely known, but known to lie within a specified range, traditional compressed sensing as such cannot exploit this fact and would need to use the same number of measurements for all sparsity values in this range. In this case, the compress ed sensing algorithm has to work taking into account the worst case, which corresponds to the signal being least sparse. Fo r example if the input signal is a discrete-time digital signa l of lengthN and can have sparsity anywhere between m1 = 5 to m2 = 25 in the frequency domain, for compressed sensing to work, one has to design the sensing matrix keeping in mind the sparsity valuem2 = 25. For this case, there are 25 frequencies in the signal which will correspond to 48− 50 complex coefficients (depending upon the locations of those 25 frequency coefficients), it was experimentally observed to take about175 (= 7 · 25) measurements for an accurate reconstruction by minimizing thel1 norm. Thus if the input signal had sparsitym = 5, conventional compressed sensing would take 175 measurements (since it has been designed for sparsitym2 = 25) whereas only 40 measurements would have sufficed. Thus, we have unnecessarily used 135 more measurements than needed in this case. In this paper, we propose a novel method called Compressed Shattering to address this particular issue. The central id e of compressed shattering is to adapt compressed sensing to the specified sparsity range by creating shattered signals [3] w hich have fixed sparsity using a filter-bank. Our primary aim is to reduce the number of measurements. II. COMPRESSEDSHATTERING The problem is stated as follows. The input signal is a discrete-time digital signal of lengthN which needs to be sensed. It is sparse within a range [m1,m2] in the frequency domain, where0 ≤ m1 ≤ m2 ≤ ⌊N/2⌋. Here, m-sparse means there are only m non-zero coefficients in the DFT of the input signal, without considering the symmetric comple x conjugate parts. A. Block Diagram: Overview The proposed algorithm is described in Fig. 1. First the input signal is permuted. This results in a permutation of the spectrum ( DFT ) in order to remove any clusters and to spread it out. The permuted spectrum is then passed through a filter-bank, which is a set of T band-pass filters, where 1 ≤ T ≤ ⌊N/2⌋. An inverse permutation operation is done on all the filter outputs to put the spectrum back in its original position. The compressed sensing algorithm is applied on th e output of the filters; using the same sensing matrix on each of them. Depending upon the sparsity level of the original sign al, the filter outputs might be zero or a very sparse signal, and the level of sparsity in the output of each of the filters can be controlled by adjusting the characteristics and number o f filters. In the succeeding subsections, we describe each blo k in detail. Fig. 1. Block diagram which illustrates the compressed shat tering process. σ andσ represent the permutation operation with the respective pa rameters, h1(n), h2(n), . , hT (n) are the impulse responses of T filters of the filter bank,x1(n), x2(n), . , xT (n) are the outputs of theT filters after inverse permutation,y1, y2, . , yT are the measurements corresponding to the T filters, TH represents the threshold operation block, yc1 , yc2 , . , ycr represent the r significant outputs. 0 5 10 15 −10 −5 0 5 10 x(n) 0 5 10 15 0 20 40 60 X(k) 0 5 10 15 −10 −5 0 5 10 x p (n) 0 5 10 15 0 20 40 60 X p (k) n