FFAST: An Algorithm for Computing an Exactly $ k$ -Sparse DFT in $O( k\log k)$ Time

Abstract

It is a well-known fact that the Discrete Fourier Transform (DFT) $ \vec {X}$ of an arbitrary $ n$ -length input signal $ \vec {x}$ , can be computed from all the $ n$ time-domain samples in $O( n\log n)$ operations via a Fast Fourier Transform (FFT) algorithm. If the spectrum $ \vec {X}$ is $ k$ -sparse (where $ k\ll n$ ), can we do better? We show that asymptotically in $ k$ and $ n$ , when $ k$ is sub-linear in $ n$ (precisely, $ k= O( n^{ \delta }$ ), where $0 \leq \delta ), and the support of the non-zero DFT coefficients is uniformly random, the fast fourier aliasing-based sparse transform (FFAST) algorithm, proposed in this paper, computes, with asymptotically high probability, the $k$ non-zero DFT coefficients of $\vec {X}$ from $O( k)$ samples of $ \vec {x}$ in $O( k\log k)$ arithmetic operations. Further, the constants in the big Oh notation for both sample and computational cost are small, e.g., when $ \delta , which essentially covers a wide range of sub-linear sparsity cases, the sample cost is less than $4 k$ . Although, in this paper we assume that the samples of the signal $ \vec {x}$ observed by the FFAST are noise-free, a noise-robust extension of the FFAST is provided in a companion (Part II) paper [1] . Our approach is based on filter-less sub-sampling of the input signal using a set of carefully chosen uniform sub-sampling patterns guided by the Chinese Remainder Theorem (CRT) . The idea is to cleverly exploit, rather than avoid, the resulting aliasing artifacts induced by sub-sampling. Specifically, the sub-sampling operation on the time-domain signal $ \vec {x}$ is designed to create aliasing patterns of the non-zero coefficients of the spectrum $ \vec {X}$ to “look like” parity-check constraints of a good erasure-correcting sparse-graph code. Next, we show that computing the sparse DFT $ \vec {X}$ is equivalent to decoding of an appropriate sparse-graph code . The sparse-graph codes further permit a fast peeling-style decoding. Consequently, the resulting DFT computation is low in both the sample and the 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 provide extensive simulation results, that are in tight agreement with our theoretical findings.