A deterministic sub-linear time sparse fourier algorithm via non-adaptive compressed sensing methods

Abstract

We study the problem of estimating the best B term Fourier representation for a given frequency-sparse signal (i.e., vector) A of length N G B. More precisely, we investigate how to deterministically identify B of the largest magnitude frequencies of Â, and estimate their coefficients, in polynomial (B, log N) time. Randomized sub-linear time algorithms, which have a small (controllable) probability of failure for each processed signal, exist for solving this problem. However, for failure intolerant applications such as those involving mission-critical hardware designed to process many signals over a long lifetime, deterministic algorithms with no probability of failure are highly desirable. In this paper we build on the deterministic Compressed Sensing results of Cormode and Muthukrishnan (CM) [26, 6, 7] in order to develop the first known deterministic sub-linear time sparse Fourier Transform algorithm suitable for failure intolerant applications. Furthermore, in the process of developing our new Fourier algorithm, we present a simplified deterministic Compressed Sensing algorithm which improves on CM's algebraic compressibility results while simultaneously maintaining their results concerning exponential decay.