典型稀疏快速傅里叶变换算法的经验评价
Empirical Evaluation of Typical Sparse Fast Fourier Transform Algorithms
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算法利用信号在频域中稀疏的固有特性，降低了运算时间和采样复杂度。目前已提出的sFFT算法有十余种，根据滤波器、结构、定位方法、估计方法等不同，可分为多种类型。本文从理论上对这些算法的实现技术进行了全面的分析。它们的性能在实践中得到了充分的检验和验证。理论分析包括以下内容：信号的五种运算，三种频率桶化方法，五种定位方法，四种估计方法，桶化引起的两个问题，解决这两个问题的三种方法，四种算法框架。对上述技术和方法进行了详细介绍，并举例说明了上述研究。在理论研究的基础上，利用标准的测试平台对不同信噪比、N、K的信号进行了实验，记录了8种不同sFFT算法的运行时间、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. 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. More than ten sFFT algorithms have been proposed, which can be classified into many types according to filter, framework, method of location, method of estimation. In this paper, the technology of these algorithms is completely analyzed in theory. The performance ofthem is thoroughly tested and verified in practice. The theoretical analysis includes thefollowing contents: five operations of signal, three methods of frequency bucketization, five methods of location, four methods of estimation, two problems caused by bucketization, three methods to solve these two problems, four algorithmic frameworks. All the above technologies and methods are introduced in detail and examples are given to illustrate the above research. After theoretical research, we make experiments for computing the signals of different SNR, N , K by a standard testing platform and record the run time, percentage of the signal sampled and L0 , L1 , L2 error with eight different sFFT algorithms. The result of experiments satisfies the inferences obtained in theory.
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