Non-Uniform Sparse Fourier Transform and Its Applications

The fast approximation algorithm of non-uniform discrete Fourier transform (NUDFT) is an important issue in signal processing. In this paper, a novel estimation algorithm is constructed for NUDFT-II, which is the general form of the sparse Fourier transform (SFT). Firstly, we propose the cyclic convolution in the non-uniform frequency domain and derive the product and convolution theorem. Secondly, the relationship is deduced between the inverse NUDFT of a uniformly sampled sequence in the frequency domain and that of a non-uniformly sampled sequence. Then, based on the relationship and cyclic convolution, we establish a random permutation operation, a non-uniform flat window filtering operation, and a frequency subsampling operation for NUDFT-II to compress sparse signal length. Meanwhile, a method to estimate the significant frequencies' locations and values is introduced. Finally, we propose the non-uniform sparse Fourier transform (NUSFT). Simulation results demonstrate that the proposed method has low complexity in sample and runtime, and has high robustness. Furthermore, the NUSFT algorithm has been applied to signal detection and reconstruction, which shows that the required signals are obtained accurately. And the NUSFT can detect frequencies that cannot be detected by the SFT.