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Computing a k-sparse n-length Discrete Fourier Transform using at most 4k samples and O(k log k) complexity
Given an -length input signal \mbf{x}, it is well known that its
Discrete Fourier Transform (DFT), \mbf{X}, can be computed in
complexity using a Fast Fourier Transform (FFT). If the spectrum \mbf{X} is
exactly -sparse (where ), can we do better? We show that
asymptotically in and , when is sub-linear in (precisely, where ), 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
deterministically chosen samples of the input signal \mbf{x} (where
when ); and (ii) {\bf {computational complexity}}: we can
reliably compute the DFT \mbf{X} using 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 . Our algorithm succeeds with high probability, with the
probability of failure vanishing to zero asymptotically in the number of
samples acquired, .Comment: 36 pages, 15 figures. To be presented at ISIT-2013, Istanbul Turke
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