Spectrum analysis for nonuniform sampling of bandlimited and multiband signals in the fractional Fourier domain

This paper explores fractional spectra of signals after two types of nonuniform samplings in the presence of timing offset. Both cases of bandlimited and multiband signals in the fractional Fourier domain are considered, based on which we provide conditions for reconstructing the spectrum and devise relevant methods for compensating the bias introduced by timing offset. To obtain these goals, we divide the nonuniformly sampled signal into uniformly sampled sub-sequences and establish the relationship between the discrete-time fractional Fourier transforms (DTFrFTs) of them and the FrFT of the original signal. It indicates that the obtained DTFrFTs consists of periodic replicas of the FrFT for the original signal with each replica being biased by a phase term introduced by timing offset. Based on the matrix form of the relationship, we implement spectral reconstruction of bandlimited signals under certain conditions by performing the inverse of a matrix determined by the phase terms. Different from bandlimited signals, the DTFrFTs of uniformly sampled sub-sequences for multiband signals are studied within disjoint fractional frequency sub-intervals separated by aliasing boundaries. Moreover, inverse of a matrix with fewer columns and rows is utilized to reconstruct the spectrum. Simulations verify the effectiveness of the proposed methods.