Nearly optimal sparse fourier transform

Abstract

We consider the problem of computing the k-sparse approximation to the discrete Fourier transform of an ndimensional signal. We show:• An O(k log n)-time randomized 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 randomized 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).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 to perform adaptive sampling.

An Improved FPGA Implementation of Sparse Fast Fourier Transform

“…In this paper, the hash mapping reconstruction method is used to reconstruct the position of the large frequency coefficients on the N-point sequence. Hash mapping reconstruction method is characterized by simple structure, no iteration, but there are some requirements for the setting of parameters [1,2]. For example, it requires that the B be slightly larger than the square root of the product of N and K, and that N must be an integer power of two, thus zero-padding is needed in some cases.…”

“…However, generally most natural or desired signals can be sparsely represented. Such sparse signature of signals was exploited in many sub-linear algorithms for accelerating the computation of discrete Fourier transform, among which, SFFT [1,2] represents the most efficient alternative. The core idea of the SFFT algorithm is to first convert the FFT operation of a large number of points (N points) into that of much fewer points (B point) by partitioning the frequency coefficients into B buckets, then estimate K large coefficients of the original signal by following certain location-estimation-mapping rules.…”

SFFT based ISAR Imaging

“…Through a simple operation of "frame" dividing that divides a Fourier transform whose length is n into many shorter DFT, this algorithm has reduced computational complexity to ) ) log( (log n nk n O (n stands for the length of signal while k is the degree of the sparsity). Researchers put forward two improved algorithms later: SFFTv2 [1] and SFFTv3 [3], enabling the computational complexity of SFFT decreasing to O (k*logn). At the same time, transformed algorithms used in various fields are developed, such as PS (Phase Shifted) for continuous Fourier transform [4] and 2D-SFFT [5].…”

“…SFFTv3 is another improved version of these two algorithms [3]. All of these three algorithms divide signal into B sections, but differences lies in that: SFFTv1 and SFFTv2 seek for the non-zero points randomly while SFFTv3 determines directly the position of non-zero points in each sections through two permutations of spectra with the same σ and different τ (see theorem 3.2).…”

Fast multi-dimensional NMR acquisition and processing using the sparse FFT

“…SFFT (Ghazi et al 2013;Hassanieh et al 2012a;Hassanieh et al 2012b) is a rapidly developing field of signal processing methods, which is designed for reconstruction of a spectrum from a minimal number of sampled data points and with minimal computations. Similar to CS, the task is to use a small number of measurements in the time domain for reconstructing only the essential, i.e.…”