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 $L0$, $L1$, and $L2$ 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.

中文翻译：

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混叠滤波器的稀疏快速傅里叶变换算法性能研究

由于稀疏快速傅里叶变换（sFFT）的高效率和广泛的实用性，长期以来已成为一个关键主题。到目前为止，有二十多种不同的sFFT算法通过其独特的方法来计算离散傅里叶变换（DFT）。为了正确使用它们，迫切需要解决的问题是如何在理论和实践上分析和评估这些算法的性能。本文主要讨论使用混叠滤波器的sFFT算法的技术和性能。在第一部分中，本文介绍了三个框架：基于压缩感知（CS）求解器的单发框架，基于二分图的剥离框架和基于二叉树搜索的迭代框架。然后，我们得出了六个相应算法的性能结论：理论上讲是sFFT-DT1.0，sFFT-DT2.0，sFFT-DT3.0，FFAST，R-FFAST和DSFFT算法。在第二部分中，我们进行两类实验，以通过标准测试平台计算不同SNR，不同长度和不同稀疏度的信号，并记录运行时间，采样信号的百分比以及$\mathrm{大号}0$， $\mathrm{大号}\mathrm{1个}$， 和 $\mathrm{大号}\mathrm{2个}$在完全稀疏情况和一般稀疏情况下都会出现错误。这些性能分析的结果是我们优化这些算法并有选择地使用它们的指南。