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

Abstract

Given an n-length input signal x, it is well known that its Discrete Fourier Transform (DFT), X, can be computed from n samples in O(n log n) operations using a Fast Fourier Transform (FFT) algorithm. If the spectrum X is 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 = O(n δ ) where 0 < δ < 1), and the support of the non-zero DFT coefficients is uniformly random, our proposed FFAST (Fast Fourier Aliasing-based Sparse Transform) algorithm computes the DFT X, from O(k) samples in O(k log k) arithmetic operations, with high probability. Further, the constants in the big Oh notation for both sample and computational cost are small, e.g., when δ < 0.99, which essentially covers almost all practical cases of interest, the sample cost is less than 4k.Our approach is based on filterless subsampling of the input signal x using a 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 subsampling. Specifically, the subsampling operation on x is designed to create aliasing patterns on the spectrum X that "look like" parity-check constraints of a good erasure-correcting sparse-graph code. Next, we show that computing the sparse DFT X is equivalent to decoding of sparse-graph codes. These codes further allow for fast peeling-style decoding. The resulting DFT computation is low in both sample complexity and decoding complexity. We analytically connect our proposed CRT-based aliasing framework to random sparse-graph codes, and analyze the performance of our algorithm using density evolution techniques from coding theory. We also provide simulation results, that are in tight agreement with our theoretical findings.I. INTRODUCTION Spectral analysis using the Discrete Fourier Transform (DFT) has been of universal importance in engineering and scientific applications for a long time. The Fast Fourier Transform (FFT) is the fastest known way to compute the DFT of an arbitrary n-length signal, and has a computational complexity of O(n log n)1 . Many applications of interest involve signals, e.g. audio, image, video data, biomedical signals etc., which have a sparse Fourier spectrum. In such cases, a small subset of the spectral components typically contain most or all of the signal energy, with most spectral components being either zero or negligibly small. If the n-length DFT, X, is k-sparse, where k << n, can we do better in terms of both sample and computational complexity of computing the sparse DFT? We answer this question affirmatively. In particular, we show that asymptotically in k and n, when k is sub-linear in n (precisely, k = O(n δ ) where 0 < δ < 1), and the support of the non-zero DFT coefficients is uniformly random, our proposed FFAST (Fast Fourier Aliasing-based Sparse Transform) algorithm computes the DFT X, from judiciously chosen O(k) samples in O(k log k) arithmetic operations, with high pro...

Time-data trade-off in the sparse Fourier transform

“…However, Pawar and Ramchandran (2013) [5] use coding theoretic tools to identify the minimum required time domain samples with high accuracy. Nevertheless, using the minimum samples leads to high computational complexity.…”

“…SHIFTING AND SUB-SAMPLING IN TIME At a high level the algorithm uses multiple stages, each sub-sampling the original signal in time using a unique subsampling factor. For each stage the DFTs of two sub-sampled in time signals are calculated where one of the signals is shifted in time prior to sub-sampling [5].…”

“…The frequency components that are N/p bins apart will collide into one frequency bin. However, using sub-sampling factors that result in signals with lengths that are either co-prime integers or cyclically shifted sets of co-prime integers will ensure that the components that collide in one of the sub-sampled stages do not collide in the others [5]. Shifting in time will multiply each frequency component by an exponential term:…”

“…1. Removing the recovered components from all sub-sampled signals will iteratively convert more (Multi-tons) to (Single-tons) allowing further recovery [5]. For large N (signal length) and k (sparsity) the number of edges remaining in the graph after each iteration can be estimated using the asymptotic theory Density Evolution which gives an estimation of the density of the remaining edges in the graph after each iteration as follows 2 :…”

“…The second category is the Aliasing Based Sparse Fourier Transform. This category was invented at least three times independently by [5], [6] and [7] which gives an indication about the importance of such approach.…”