Sparse Fast Fourier Transform by downsampling

Abstract

Sparse Fast Fourier Transform (sFFT) [1][2], has been recently proposed to outperform FFT in reducing computational complexity. Assume that an input signal of length N in the frequency domain is K-sparse, whereIn this paper, a new fast sFFT algorithm is proposed and costs O(K log K) averagely without any operations being related to N . The idea is to downsample the original input signal at the beginning. Subsequent processing operates under downsampled signals, which length is proportional to O(K). However, downsampling possibly leads to "aliasing." By shift theorem of DFT, the aliasing problem can be formulated as the "Moment-preserving problem." In addition, a top-down iterative strategy combined with different downsampling factors further saves computational costs. Complexity analysis and experimental results show that our method outperforms FFT and sFFT.

Fast Computing DFT Method for Sparse Signals Based on Downsampling

“…Recently, S. H. Hsieh et al proposed a new concept about sFFT by downsampling in the time domain (sFFT-DT)for the exactly K-sparse signals, assuming that distribution of the non-zero frequency grid is uniform [5]. Their focus was to downsample the original input signal at the beginning; and then, directly conduct operations on downsampled signals, where the length of downsampled signals was kept proportional to ( ) O K .…”

“…Downsampling probably leads to "aliasing", which means that different frequency grids of the original signal map into the same grid of the downsampled signal. In [5], the aliasing problem is reformulated as moment-preserving problem (MPP) and solved via existing approaches.…”

“…Inspired by [5], we proposed a new fast computing DFT method for generally sparse signals. The proposed method consists of three steps.…”

“…As mentioned before, downsampling will lead to "aliasing," where different frequencies become indistinguishable in terms of their positions and values. According to [5], possible positions of nonzero freq. grid can be obtained from the solution of MPP.…”

Time-data trade-off in the sparse Fourier transform

“…The second category is the Aliasing Based Sparse Fourier Transform. This category was invented at least three times independently by [5], [6] and [7] which gives an indication about the importance of such approach.…”