Multi-channel sampling theorems for band-limited signals with fractional Fourier transform

Abstract

Multi-channel sampling for band-limited signals is fundamental in the theory of multi-channel parallel A/D environment and multiplexing wireless communication environment. As the fractional Fourier transform has been found wide applications in signal processing fields, it is necessary to consider the multi-channel sampling theorem based on the fractional Fourier transform. In this paper, the multi-channel sampling theorem for the fractional band-limited signal is firstly proposed, which is the generalization of the well-known sampling theorem for the fractional Fourier transform. Since the periodic nonuniformly sampled signal in the fractional Fourier domain has valuable applications, the reconstruction expression for the periodic nonuniformly sampled signal has been then obtained by using the derived multi-channel sampling theorem and the specific space-shifting and phase-shifting properties of the fractional Fourier transform. Moreover, by designing different fractional Fourier filters, we can obtain reconstruction methods for other sampling strategies.fractional Fourier transform, fractional band-limited signals, fractional filter, interpolation series, periodic nonuniformly sampled signal

Multi-channel sampling expansion in fractional Fourier domain and its application to superresolution

“…In order to express /(t) involving the output of fractional Fourier filters gk ( nT), we firstly need to derive the reconstruction filters. For this purpose, we form the following systems: Yk (t -nT) (14) …”

Filterbank implementation for multi-channel sampling in fractional Fourier domain

“…In refs. [25,26], we obtained the reconstruction expansions for multi-channel sampling and periodic nonuniform sampling. We apply the identity in this section to obtain efficient reconstruction methods from multi-channel samples and nonuniform samples in the fractional Fourier domain.…”

“…According to the multi-channel sampling theorem in the fractional Fourier domain [25] , a fractional bandlimited signal x(t) can be reconstructed from uniform samples of the outputs g k (t) of M fractional Fourier filters H α,1 (u), H α,2 (u), … , and H α,M (u). The reconstruction formula is s …”

“…Refs. [25,26] presented the spectral properties and the reconstruction formulae for the multi-channel sampling and periodic nonuniform sampling. However, reconstruction of a continuous signal from its nonuniform samples or multi-channel samples using the direct interpolation procedure is complex.…”

Oversampling analysis in fractional Fourier domain

“…[21,22], the spectral relationship between the uniform sampled band-limited signal and the original signal in the fractional Fourier domain is obtained; and in refs. [23,24], the spectral properties for periodic nonuniform sampled signals in the fractional Fourier domain are considered.…”