On Performance of Sparse Fast Fourier Transform Algorithms Using the Aliasing Filter

• Main Authors: Bin Li, Zhikang Jiang, Jie Chen
• Format: Article
• Language: English
• Published: MDPI AG 2021-05-01
• Series: Electronics
• Subjects: sparse fast Fourier transform (sFFT); aliasing filter; sub-linear algorithms; computational complexity
• Online Access: https://www.mdpi.com/2079-9292/10/9/1117

Computing the sparse fast Fourier transform (sFFT) has emerged as a critical topic for a long time because of its high efficiency and wide practicability. More than twenty different sFFT algorithms compute discrete Fourier transform (DFT) by their unique methods so far. In order to use them properly, the urgent topic of great concern is how to analyze and evaluate the performance of these algorithms in theory and practice. This paper mainly discusses the technology and performance of sFFT algorithms using the aliasing filter. In the first part, the paper introduces the three frameworks: the one-shot framework based on the compressed sensing (CS) solver, the peeling framework based on the bipartite graph and the iterative framework based on the binary tree search. Then, we obtain the conclusion of the performance of six corresponding algorithms: the sFFT-DT1.0, sFFT-DT2.0, sFFT-DT3.0, FFAST, R-FFAST, and DSFFT algorithms in theory. In the second part, we make two categories of experiments for computing the signals of different SNRs, different lengths, and different sparsities by a standard testing platform and record the run time, the percentage of the signal sampled, and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mn>1</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mn>2</mn></mrow></semantics></math></inline-formula> errors both in the exactly sparse case and the general sparse case. The results of these performance analyses are our guide to optimize these algorithms and use them selectively.