Spectrum Analysis for Multiband Signals With Nonuniform Sub-Nyquist Sampling in the Fractional Fourier Domain

This paper explores fractional spectrum analysis for periodically and nonuniformly sampled signals with finite disjoint bands in the fractional Fourier domain. Considering that spectrum aliasing occurs in sub-Nyquist sampling in the fractional Fourier domain, we study the conditions for reconstructing the fractional spectrum and devise relevant reconstruction methods. To obtain these goals, we investigate discrete-time fractional Fourier transform of the sampled signal that enables disjoint fractional frequency sub-intervals separated by aliasing boundaries. In order to implement perfect fractional spectrum reconstruction with reduced computations, we minimize the average sampling rate required by nonuniform sampling by means of properly choosing a sampling period. The theoretical lower bound on the optimized average sampling rate is much smaller than the lowest sampling rate required by uniform sampling under the same condition. Moreover, we derive the upper and lower bounds on the mean square error of fractional spectrum reconstruction with incorrect fractional spectrum modeling that assumes the signal as bandlimited to a large fractional frequency interval. Based on the derivations, we then obtain the optimal reconstruction parameters for spectrum reconstruction. Simulation results verify the effectiveness of our proposed methods.