Fractional Spectrum Analysis for Nonuniform Sampling in the Presence of Clock Jitter and Timing Offset

This paper focuses on analyzing the fractional spectrum of nonuniform sampling which is affected by clock jitter and timing offset. Both cases of single-channel and recurrent samplings with limited bandwidth in the fractional Fourier domain are considered, for which we study two types of discrete-time fractional Fourier transforms (DTFrFTs). For the first-type DTFrFTs, we perform their statistical analyses, and meanwhile establish relationships between the fractional spectra of the input and output signals of non-ideal single/multi-channel analog-to-digital converter. Such relationships indicate that the statistical mean of each fractional spectrum contains periodic replicas of the spectrum for the input signal, and these replicas are particularly modulated by the characteristic function of the perturbations introduced by the clock jitter. Moreover, we interpret the first-type DTFrFTs from the perspective of linear frequency modulation signal decomposition for nonuniform sampling. The second-type DTFrFTs serve as approximate DTFrFTs, and they especially deal with the general situation when the nonuniform sampling instants are unknown. To this end, we study modified forms of the first-type DTFrFTs with known inputs in kernels, based on which the statistical mean of the approximate fractional spectra for both cases are derived. Then we compensate the fractional spectrum bias with high accuracy by developing optimal filters that minimize the mean square error between the original and the compensated fractional spectra for both cases. Different from single-channel sampling, the compensation for recurrent sampling involves an additional step before optimal filtering, which gives better accuracy. Simulation results show that the proposed methods outperform existing methods.