  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.
