R-FFAST: A Robust Sub-Linear Time Algorithm for Computing a Sparse DFT

Abstract

The Fast Fourier Transform (FFT) is the most efficiently known way to compute the Discrete Fourier Transform (DFT) of an arbitrary n-length signal, and has a computational complexity of O(n log n). If the DFT X of the signal x has only k non-zero coefficients (where k < n), can we do better? In [1], we addressed this question and presented a novel FFAST (Fast Fourier Aliasing-based Sparse Transform) algorithm that cleverly induces sparse graph alias codes in the DFT domain, via a Chinese-Remainder-Theorem (CRT)-guided sub-sampling operation of the time-domain samples. The resulting sparse graph alias codes are then exploited to devise a fast and iterative onion-peeling style decoder that computes an n length DFT of a signal using only O(k) time-domain samples and O(k log k) computations. TheFFAST algorithm is applicable whenever k is sub-linear in n (i.e. k = o(n)), but is obviously most attractive when k is much smaller than n.In this paper, we adapt the FFAST framework of [1] to the case where the time-domain samples are corrupted by a white Gaussian noise. In particular, we show that the extended noise robust algorithm R-FFAST computes an n-length k-sparse DFT X using O(k log 3 n) 1 noise-corrupted time-domain samples, in O(k log 4 n) computations, i.e., sub-linear time complexity. While our theoretical results are for signals with a uniformly random support of the non-zero DFT coefficients and additive white Gaussian noise, we provide simulation results which demonstrates that the R-FFAST algorithm performs well even for signals like MR images, that have an approximately sparse Fourier spectrum with a non-uniform support for the dominant DFT coefficients.

On Performance of Sparse Fast Fourier Transform Algorithms Using the Flat Window Filter

“…The researcher adopted the FFAST framework to the case that is corrupted by white Gaussian noise. The author showed that the extended noise-robust algorithm R-FFAST [6], [7] computes the DFT using O(K logK ) samples in O(K log 4 N ) runtime. These two algorithms perform well when N is a product of some smaller prime numbers.…”

Peeling Decoding of LDPC Codes with Applications in Compressed Sensing

“…For the complexity of recovery algorithm, the ℓ 1 norm minimization has a computation complexity of O( 3 ). To reduce this complexity, with the sparse measurement matrices, various low computational complexity message-passing algorithms have been introduced for reconstruction of sparse signals in CS [9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In [9], the authors used the random sparse matrix as a new sparse measurement matrix and proposed an iterative algorithm of complexity of O( log( ) log( )), while it only required = O( log( )) measurements.…”

“…To overcome the restriction of the decoding algorithms discussed above, Pawar et al in [19][20][21][22] designed a hybrid mix of the LDPC codes and Discrete Fourier Transform (DFT) framework (LDPC-DFT) and proposed a fast Short-and-Wide Iterative Fast Transform (SWIFT) peeling decoding recovery algorithm. It only needs O( ) measurements and O( ) iterative step to achieve the exact signal recovery in the noiseless case.…”

“…In the presence of noise, both measurements and computational complexity for exact signal recovery are O( log 1.3 ( )). These frameworks also have been used in spectrum sensing in [24]; Hassanieh et al in [24] presented a prime sampling technology that can capture GHz of spectrum in real-time with low speed analog-to-digital converters (ADCs); this prime sampling technology is essentially identical to the framework of [19][20][21][22]25]. These frameworks of [19][20][21][22] also have been used in compressed sensing phase retrieval [26].…”

“…In this study, we are interested in the sparse LDPC-DFT measurement matrices associated with SWIFT recovery algorithm [19][20][21][22], which can reduce both the measurement cost and computational complexity simultaneously. The authors in [19] analyzed the successful recovery probability of SWIFT algorithm via exploiting the connection between CS system and packet-communication system.…”