2013/05/03 by Sameer Pawar, Kannan Ramchandran, Pawar, Sameer +1 · 3 citations
Computer Science · Engineering · Mathematics · #Integrated Circuits and Semiconductor Failure Analysis #Sparse and Compressive Sensing Techniques #VLSI and Analog Circuit Testing #cs.DS #cs.IT #cs.MM #math.IT
paper · pdf · doi:10.48550/arxiv.1305.0870
36 pages, 15 figures. To be presented at ISIT-2013, Istanbul Turkey
arxiv created 2015/01/26 · arxiv updated 2015/01/27
Given an n-length input signal \mbfx, it is well known that its Discrete Fourier Transform (DFT), \mbfX, can be computed in O(n log n) complexity using a Fast Fourier Transform (FFT). If the spectrum \mbfX is exactly k-sparse (where k<<n), can we do better? We show that asymptotically in k and n, when k is sub-linear in n (precisely, k ∝ nδ where 0 < δ<1), and the support of the non-zero DFT coefficients is uniformly random, we can exploit this sparsity in two fundamental ways (i) \bf sample complexity: we need only M=rk deterministically chosen samples of the input signal \mbfx (where r < 4 when 0 < δ< 0.99); and (ii) \bf computational complexity: we can reliably compute the DFT \mbfX using O(k log k) operations, where the constants in the big Oh are small and are related to the constants involved in computing a small number of DFTs of length approximately equal to the sparsity parameter k. Our algorithm succeeds with high probability, with the probability of failure vanishing to zero asymptotically in the number of samples acquired, M.