多尺度稀疏快速傅里叶变换算法性能研究
On Performance of Multiscale Sparse Fast Fourier Transform Algorithm
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。sFFT算法通过利用大量信号在频域中稀疏的信号固有特性（例如，传感器、视频数据、音频、医学图像等）。sFFT的第一阶段是通过这些滤波器之一的频率bucketization：Dirichlet核滤波器，平坦滤波器，混叠滤波器等。与其他sFFT算法相比，使用平坦滤波器的sFFT算法是更方便和有效的，因为滤波后的信号集中在时域和频域。到目前为止，已经有三种sFFT算法sFFT 1.0、sFFT 2.0、sFFT 3.0算法由麻省理工学院（MIT）于2013年提出。尽管如此，使用多尺度方法的sFFT4.0算法尚未实现。本文将从理论上对该算法进行全面的讨论，并在实践中加以实现。证明了sFFT4.0算法的性能取决于两个参数。运行时间和采样复杂度与多尺度参数成正比，与扩展参数成反比。鲁棒性与扩展参数成正比，与多尺度参数成反比。与三种同类算法或其他四类算法相比，sFFT4.0算法具有良好的运行时间和采样复杂度，比fftw算法好十到一百倍，但算法的鲁棒性一般。
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. The sFFT algorithms decrease the runtime and sampling complexity by taking advantage of the signal inherent characteristics that a large number of signals are sparse in the frequency domain(e.g., sensors, video data, audio, medical image, etc.). The first stage of sFFT is frequency bucketization through one of these filters: Dirichlet kernel filter, flat filter, aliasing filter, etc. Compared to other sFFT algorithms, the sFFT algorithms using the flat filter is more convenient and efficient because the filtered signal is concentrated both in the time domain and frequency domain. Up to now, three sFFT algorithms sFFT1.0, sFFT2.0, sFFT3.0 algorithm have been proposed by the Massachusetts Institute of Technology(MIT) in 2013. Still, the sFFT4.0 algorithm using the multiscale approach method has not been implemented yet. This paper will discuss this algorithm comprehensively in theory and implement it in practice. It is proved that the performance of the sFFT4.0 algorithm depends on two parameters. The runtime and sampling complexity are in direct ratio to the multiscale parameter and in inverse ratio to the extension parameter. The robustness is in direct ratio to the extension parameter and in inverse ratio to the multiscale parameter. Compared with three similar algorithms or other four types of algorithms, the sFFT4.0 algorithm has excellent runtime and sampling complexity that ten to one hundred times better than the fftw algorithm, although the robustness of the algorithm is medium.
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