A robust R-FFAST framework for computing a k-sparse n-length DFT in O(k log n) sample complexity using sparse-graph codes

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 X⃗ of the signal x⃗ has only k non-zero coefficients (where k <; n), can we do better? In [1], we presented a novel FFAST (Fast Fourier Aliasing-based Sparse Transform) algorithm that cleverly induces sparse graph codes in the DFT domain, via a Chinese-Remainder-Theorem (CRT)-guided sub-sampling operation of the time-domain samples. The resulting sparse graph code is then exploited to devise a simple and fast 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, in the absence of any noise. In this paper, we extend the FFAST framework of [1] to the case where the time-domain samples are corrupted by white Gaussian noise. In particular, we show that the extended noise robust algorithm R-FFAST computes an n-length k-sparse DFT X⃗ using O(k log n) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> noise-corrupted time-domain samples, in O(n log n) computations <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . 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.