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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.

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.

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.