Given an n-length input signal \mbf{x}, it is well known that its
Discrete Fourier Transform (DFT), \mbf{X}, can be computed in O(nlogn)
complexity using a Fast Fourier Transform (FFT). If the spectrum \mbf{X} 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 \mbf{x} (where r<4
when 0<δ<0.99); and (ii) {\bf {computational complexity}}: we can
reliably compute the DFT \mbf{X} using O(klogk) 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.Comment: 36 pages, 15 figures. To be presented at ISIT-2013, Istanbul Turke