Multiweighted-Type Fractional Fourier Transform: Unitarity

The definition of the discrete fractional Fourier transform (DFRFT) varies, and the multiweighted-type fractional Fourier transform (M-WFRFT) is its extended definition. It is not easy to prove its unitarity. We use the weighted-type fractional Fourier transform, fractional-order matrix and eigendecomposition-type fractional Fourier transform as basic functions to prove and discuss the unitarity. Thanks to the growing body of research, we found that the effective weighting term of the M-WFRFT is only four terms, none of which are extended to M terms, as described in the definition. Furthermore, the program code is analyzed, and the result shows that the previous work (Digit Signal Process 2020: 104: 18) based on MATLAB for unitary verification is inaccurate.

Automated summary

This study investigates the discrete fractional Fourier transform and its extended definition, the multiweighted-type fractional Fourier transform, and proves its unitarity. By using basic functions for discussion, it is found that the effective weighting term of the M-WFRFT is only four terms, different from the M terms described in the definition. The analysis of program code reveals inaccuracies in previous works based on MATLAB for unitary verification.