Performance of the Multiscale Sparse Fast Fourier Transform Algorithm

Abstract

How to compute the sparse fast Fourier transform (sFFT) has been a critical topic for a long period of time. sFFT algorithms have faster runtimes and reduced sampling complexities by taking advantage of a signal’s inherent characteristics, namely, that a large number of signals are sparse in the frequency domain (e.g., sensors, video data, audio, medical images.). The first step of sFFT is frequency bucketization through window filters. sFFT algorithms using a flat filter are more convenient and efficient than other algorithms because the filtered signal is concentrated both in the time domain and frequency domain. To date, three sFFT algorithms using the flat filter (the sFFT1.0, sFFT2.0, and sFFT3.0 algorithms) have been proposed by the Massachusetts Institute of Technology (MIT) beginning in 2013. Since then, the sFFT4.0 algorithm using a multiscale approach method has not yet been implemented. This paper will discuss this algorithm comprehensively in theory and implement it in practice. It is proven that the performance of the sFFT4.0 algorithm depends on two parameters. The runtime and sampling complexity are directly correlated to the multiscale parameter and inversely correlated to the extension parameter. The robustness is in directly correlated to the extension parameter and in inverse ratio to the multiscale parameter. Compared with three similar algorithms and four different types of algorithms, the sFFT4.0 algorithm displays an improved runtime and sampling complexity that is ten to one hundred times better than the FFTW algorithm, although the robustness of the algorithm is moderate.