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

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)1. If the DFT ~ X of the signal ~x 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 FFAST algorithm computes an n-length k-sparse DFT ~ X using O(k log n) noise-corrupted time-domain samples, in O(n log n) computations. 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 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. Further, the constants in the big-oh notations are small as is evident from the experimental results provided in Section VII. February 19, 2014 DRAFT