The purpose of studying the sparse representation of signals is to find the optimal approximation of the atoms of the signal under an over-complete dictionary, in which the characteristics of the atoms that match the signal are the characteristics of the signal. At present, the commonly used dictionaries in research include Fourier dictionary, cosine packet dictionary, wavelet packet dictionary, Chirplet dictionary, etc. However, there is little research on Fourier dictionaries from the point of Fourier transform theory. This paper studies three typical algorithms of sparse representation under Fourier dictionary (matching pursuit algorithm, basis pursuit algorithm and the method of frames), and the simulation results show that: 1) the matching pursuit algorithm can adaptively determine the time-frequency parameters of the signal. 2) The basis pursuit algorithm can enhance the ability to identify coupled harmonics, and the precision of reconstructed harmonics can reach 97%. 3) Implementation effect of the method of frames is consistent with fast discrete Fourier transform.
