Sparse Fast Fourier Transform for Exactly and Generally K-Sparse Signals by Downsampling and Sparse Recovery

Fast Fourier Transform (FFT) is one of the most important tools in digital signal processing. FFT costs O(N \log N) for transforming a signal of length N. Recently, Sparse Fourier Transform (SFT) has emerged as a critical issue addressing how to compute a compressed Fourier transform of a signal with complexity being related to the sparsity of its spectrum. In this paper, a new SFT algorithm is proposed for both exactly K-sparse signals (with K non-zero frequencies) and generally K-sparse signals (with K significant frequencies), with the assumption that the distribution of the non-zero frequencies is uniform. The nuclear idea is to downsample the input signal at the beginning; then, subsequent processing operates under downsampled signals, where signal lengths are proportional to O(K). Downsampling, however, possibly leads to "aliasing." By the shift property of DFT, we recast the aliasing problem as complex Bose-Chaudhuri-Hocquenghem (BCH) codes solved by syndrome decoding. The proposed SFT algorithm for exactly K-sparse signals recovers 1-\tau frequencies with computational complexity O(K \log K) and probability at least 1-O(\frac{c}{\tau})^{\tau K} under K=O(N), where c is a user-controlled parameter. For generally K-sparse signals, due to the fact that BCH codes are sensitive to noise, we combine a part of syndrome decoding with a compressive sensing-based solver for obtaining $K$ significant frequencies. The computational complexity of our algorithm is \max \left( O(K \log K), O(N) \right), where the Big-O constant of O(N) is very small and only a simple operation involves O(N). Our simulations reveal that O(N) does not dominate the computational cost of sFFT-DT.