*Article* **Multiweighted-Type Fractional Fourier Transform: Unitarity**

**Tieyu Zhao \* and Yingying Chi**

Information Science Teaching and Research Section, School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China; chiyingying@neuq.edu.cn **\*** Correspondence: zhaotieyu@neuq.edu.cn

**Abstract:** The definition of the discrete fractional Fourier transform (DFRFT) varies, and the multiweightedtype 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.

**Keywords:** fractional fourier transform; weighted-type fractional Fourier transform; multiweighted-type fractional fourier transform; unitarity
