Multiweighted-Type Fractional Fourier Transform: Unitarity

Abstract

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.

Sliding-Window TD-FrFT Algorithm for High-Precision Ranging of LFM Signals in the Presence of Impulse Noise

“…The echo and transmitted signals are separately transformed using nonlinear FrFTs, and the delay in the LFM signals is calculated based on the peak position offset in the fractional domain. Other researchers [27,28] have introduced a quantum weighted FrFT for application to quantum singular-value decomposition problems and quantum gradient-solving problems. Moreover, a novel random FrFT has been developed [29], inheriting excellent mathematical properties from the FrFT, which can be directly used in optical image encryption and decryption.…”