Generalized Sampling Expansions with Multiple Sampling Rates for Lowpass and Bandpass Signals in the Fractional Fourier Transform Domain

Abstract

The objective of generalized sampling expansion (GSE) is the reconstruction of an unknown, continuously defined function $f\left(t\right)$ from samples of the responses from $M$ linear time-invariant (LTI) systems that are each sampled using the $1/M$ th Nyquist rate. In this paper, we investigate the GSE for lowpass and bandpass signals with multiple sampling rates in the fractional Fourier transform (FRFT) domain. First, we propose an improvement of Papoulis’ GSE, which has multiple sampling rates in the FRFT domain. Based on the proposed GSE, we derive the periodic nonuniform sampling scheme and the derivative interpolation method by designing different fractional filters and selecting specific sampling rates. In addition, the Papoulis GSE and the previous GSE associated with FRFT are shown to be special instances of our results. Second, we address the problem of the GSE of fractional bandpass signals. A new GSE for fractional bandpass signals with equal sampling rates is derived. We show that the restriction of an even number of channels in the GSE for fractional bandpass signals is unnecessary, and perfect signal reconstruction is possible for any arbitrary number of channels. Further, we develop the GSE for a fractional bandpass signal with multiple sampling rates. Lastly, we discuss the application of the proposed method in the context of single-image super-resolution reconstruction based on GSE. Illustrations and simulations are presented to verify the validity and effectiveness of the proposed results.