1Computing a k-sparse n-length Discrete Fourier Transform using at most 4k samples and O(k log k) complexity

Given an n-length input signal x, it is well known that its Discrete Fourier Transform (DFT), X, can be computed in O(n log n) complexity using a Fast Fourier Transform (FFT). If the spectrum 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) sample complexity: we need only M = rk deterministically chosen samples of the input signal x (where r < 4 when 0 < δ < 0.99); and (ii) computational complexity: we can reliably compute the DFT X 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. Our approach is based on filterless subsampling of the input signal x using a small set of carefully chosen uniform subsampling patterns guided by the Chinese Remainder Theorem (CRT). The idea is to cleverly exploit, rather than avoid, the resulting aliasing artifacts induced by this subsampling. Specifically, our subsampling operation on x is designed to create aliasing patterns on the spectrum X that “look like ” parity-check constraint