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

Abstract

The fast Fourier transform 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 $ \vec {X}$ of the signal $ \vec {x}$ has only $k$ non-zero coefficients (where $ k ), can we do better? We addressed this question and presented a novel fast Fourier aliasing-based sparse transform (FFAST) algorithm that cleverly induces sparse-graph alias codes in the DFT domain, via a Chinese-remainder-theorem-guided sub-sampling operation in the time-domain. The induced sparse-graph alias codes are then exploited to devise a fast and iterative onion-peeling style decoder that computes $ k$ -sparse DFT of an $ n$ -length signal using only $O( k)$ time-domain samples and $O( k\log k)$ computations. In this paper, we generalize the FFAST framework by Pawar and Ramchandran to the noisy setting where the time-domain samples are corrupted by white Gaussian noise. We show that the noise-robust R-FFAST algorithm computes a $ k$ -sparse DFT of an $ n$ -length signal using $O( k\log ^{3} n)$ noise-corrupted time-domain samples in $O( k\log ^{4} n)$ complexity, i.e., sub-linear sample and time complexity . In Section IX , we provide extensive simulation results validating the empirical performance of the R-FFAST algorithm, e.g., we show that the R-FFAST algorithm computes a 50-sparse DFT of an ≈ 10 million length signal using only 4800 noisy samples with an effective signal-to-noise ratio of 5 dB. We also provide comparison of the run-time performance of several existing sparse Fourier transform implementations with that of the R-FFAST and show that it is almost 20 times faster, for comparable settings, than the state-of-the-art algorithm, while simultaneously providing better support recovery guarantees. 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 demonstrate that the R-FFAST algorithm performs well even for signals like magnetic resonance images, that have an approximately sparse Fourier spectrum with a non-uniform support for the dominant DFT coefficients.