On the linear fractional shift invariant systems associated with fractional Fourier transform

The fractional Fourier transform (FRFT) has been shown to be a powerful analyzing tool in signal processing. Many properties of this transform have been currently derived as counterparts to the corresponding properties of the Fourier transform (FT), including the theory of the linear fractional shift invariant (LFSI) systems. However, the LFSI systems, which are derived as extensions of the linear time invariant (LTI) systems, available in the literature do not generalize very nicely the LTI systems, which state that the output of a LTI system does not depend on the particular time the input is applied, and the FT of the output is the product of the transfer function and the FT of the input in the Fourier domain. In this paper, we propose a new LFSI system structure associated with the FRFT, and the LTI systems are noted as special cases. The eigenfunctions and eigenvalues of the new LFSI systems are presented. Moreover, the conditions for distortionless transmission in the fractional Fourier domain and basic properties of the ideal fractional filter are derived. Some applications of the achieved results are also discussed.