Over the recent years, a new approach for obtaining a succinct approximate representation of ndimensional

We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, and learning theory. We propose a new algorithm for this problem. The algorithm leverages techniques from digital signal processing, notably Gaussian and Dolph-Chebyshev filters. Unlike the typical approach to this problem, our algorithm is not iterative. That is, instead of estimating “large ” coefficients, subtracting them and recursing on the reminder, it identifies and estimates the k largest coefficients in “one shot”, in a manner akin to sketching/streaming algorithms. The resulting algorithm is structurally simpler than its predecessors. As a consequence, we are able to extend considerably the range of sparsity, k, for which the algorithm is faster than FFT, both in theory and practice. 1

We consider the problem of computing the k-sparse approximation to the discrete Fourier transform of an n-dimensional signal. We show: • An O(k log n)-time algorithm for the case where the input signal has at most k non-zero Fourier coefficients, and • An O(k log n log(n/k))-time algorithm for general input signals. Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k = o(n). Further, they are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly k-sparse case is optimal for any k = n Ω(1). We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least Ω(k log(n/k) / log log n) signal samples, even if it is allowed The discrete Fourier transform (DFT) is one of the most important and widely used computational tasks. Its applications are broad and include signal processing, communications, and audio/image/video compression. Hence, fast algorithms for DFT are highly valuable. Currently, the fastest such algorithm is the Fast Fourier