Compressed Sampling in Shift-Invariant Spaces Associated With FrFT

Shift-invariant and sampling spaces play a vital role in the fields of signal processing and image processing. In this paper, we extend the generalized shift-invariant and sampling subspaces from the traditional sampling spaces to the compressed sampling, and develop a compressed sampling method for analog sparse signals based on the shift-invariant spaces associated with fractional Fourier transform (FrFT). First, we show the generalized shift-invariant and sampling subspaces can be used to explain the traditional sampling spaces with single generator or multiple generators in the fractional Fourier domain (FrFD). The non-ideal sampling structures of single channel and multiple channels are special cases of the generalized shift-invariant subspaces. Second, a compressed sampling method for the sparse signals in the FrFD is proposed by reusing the multiple generators of the shift invariant spaces as sparse representation. We combine the sensing matrix of compressed sensing and the framework of sampling scheme in the shift-invariant spaces to construct a compressed sampling method, which perfectly recovered the original signal with a sufficient low sampling rate. By choosing different filters, the proposed framework allows to derive many specific sampling schemes. Finally, a compressed sampling method for multiband signals in the FrFD is proposed based on the forgoing theorems. The numeral simulation validates the theoretical derivations.