The fractional Fourier transform (FRFT) is one of the most useful tools for the nonstationary signal processing. In this paper, the randomized nonuniform sampling and approximate reconstruction of the nonstationary random signals in the fractional Fourier domain (FRFD) are developed. The nonuniform samples are treated as random perturbations from a uniform grid. The samples used for the sinc interpolation reconstruction are placed on another nonuniform grid which is not necessarily equal to the samples originally acquired. When considering the second-order random statistic characters, the nonuniform sampling is equivalent to the uniform sampling of the signal after a pre-filter in the FRFD, where the frequency response is related to the characteristic function (with its argument scaled by cscα) of the perturbations. The effectiveness of the reconstruction is analyzed and the mean square error (MSE) is computed by utilizing the equivalent filter system. Furthermore, the randomized reconstruction of the chirp period stationary random signal is proposed. At last, the minimum MSE on the special cases of the randomized sampling and reconstruction is discussed. The effectiveness of the proposed reconstruction method is verified by the simulation.
Xu, L., Zhang, F., & Tao, R. (2016). Randomized nonuniform sampling and reconstruction in fractional Fourier domain. Signal Processing, 120, 311–322. https://doi.org/10.1016/j.sigpro.2015.09.016