R-FFAST: A Robust Sub-Linear Time Algorithm for Computing a Sparse DFT
The fast Fourier transform is the most efficiently known way to compute the discrete Fourier transform (DFT) of an arbitrary <inline-formula> <tex-math notation="LaTeX">$ n$ </tex-math></inline-formula>-length signal, and has a computational complexity of <inline-formula> <tex-math notation="LaTeX">$O( n\log n)$ </tex-math></inline-formula>. If the DFT <inline-formula> <tex-math notation="LaTeX">$ \vec {X}$ </tex-math></inline-formula> of the signal <inline-formula> <tex-math notation="LaTeX">$ \vec {x}$ </tex-math></inline-formula> has only <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> non-zero coefficients (where <inline-formula> <tex-math notation="LaTeX">$ k< n$ </tex-math></inline-formula>), 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 <inline-formula> <tex-math notation="LaTeX">$ k$ </tex-math></inline-formula>-sparse DFT of an <inline-formula> <tex-math notation="LaTeX">$ n$ </tex-math></inline-formula>-length signal using only <inline-formula> <tex-math notation="LaTeX">$O( k)$ </tex-math></inline-formula> time-domain samples and <inline-formula> <tex-math notation="LaTeX">$O( k\log k)$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$ k$ </tex-math></inline-formula>-sparse DFT of an <inline-formula> <tex-math notation="LaTeX">$ n$ </tex-math></inline-formula>-length signal using <inline-formula> <tex-math notation="LaTeX">$O( k\log ^{3} n)$ </tex-math></inline-formula> noise-corrupted time-domain samples in <inline-formula> <tex-math notation="LaTeX">$O( k\log ^{4} n)$ </tex-math></inline-formula> complexity, i.e., <italic>sub-linear sample and time complexity</italic>. In <xref rid="sec9" ref-type="sec">Section IX</xref>, 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.