利用混叠滤波器的稀疏快速傅里叶变换算法性能研究
On Performance of Sparse Fast Fourier Transform Algorithms Using the Aliasing Filter
摘要：
计算大小为N的K-稀疏信号的稀疏快速傅里叶变换（sFFT）一直是一个重要的课题。在sFFT中主要有两个阶段：频率桶化和频谱重构。频率桶化相当于通过以下滤波器之一将频率系数散列到B个桶中：Dirichlet内核滤波器，平坦滤波器，混叠滤波器等。频谱重建相当于识别在其桶中隔离的频率。迄今为止，已有四十多种不同的sFFT算法以其独特的方法计算离散傅里叶变换（DFT）。如何从理论和实践上对这些算法的性能进行分析和评价，以便更好地使用它们，是一个迫切需要关注的课题。本文主要讨论利用混叠滤波器的sFFT算法。第一部分介绍了基于压缩感知（CS）求解器的单次框架、基于二分图的剥离框架和基于二叉树搜索的迭代框架。然后从理论上分析了sFFT-DT 1.0、sFFT-DT 2.0、sFFT-DT 3.0、FFAST、R-FFAST和DSFFT算法的性能。在第二部分中，我们利用标准的测试平台对不同信噪比、不同N、不同K的信号进行了两类实验，分别记录了在完全稀疏和一般稀疏情况下的运行时间、采样率和L0、L1、L2误差。实验结果满足理论推导。
英文摘要：
Computing the Sparse Fast Fourier Transform(sFFT) of a K-sparse signal of size N has emerged as a critical topic for a long time. There are mainly two stages in the sFFT: frequency bucketization and spectrum reconstruction. Frequency bucketization is equivalent to hashing the frequency coefficients into B buckets through one of these filters: Dirichlet kernel filter, flat filter, aliasing filter, etc. The spectrum reconstruction is equivalent to identifying frequencies that are isolated in their buckets. More than forty 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. The paper mainly discusses the sFFT Algorithms using the aliasing filter. In the first part, the paper introduces the technique of 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 get the conclusion of the performance of six corresponding algorithms: sFFT-DT1.0, sFFT-DT2.0, sFFT-DT3.0, FFAST, R-FFAST and DSFFT algorithm in theory. In the second part, we make two categories of experiments for computing the signals of different SNR, different N, different K by a standard testing platform and record the run time, percentage of the signal sampled and L0, L1, L2 error both in the exactly sparse case and general sparse case. The result of experiments satisfies the inferences obtained in theory.
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