**5. Conclusions**

In this paper, we present a new reformulation of the M-WFRFT to prove its unitarity. The M-WFRFT uses the DFRFT as the basis function, and the diversity of the DFRFT leads to different definitions of the M-WFRFT. We use the linear weighted-type, fractional-order matrix and eigendecomposition-type FRFT as the basis functions and prove the unitarity of the M-WFRFT. The results show that M-WFRFTs based on these three definitions have unitarity. However, with greater research, the results also show that the effective weighted sum of the M-WFRFT is only four terms. That is to say, as an extended definition of the WFRFT, the M-WFRFT shows no increase in its weighting term. It has great reference value for the application of the M-WFRFT. Furthermore, we note the deviation between the numerical simulation and the theoretical analysis, which reveals that the unitary verification based on MATLAB is inaccurate for the previous work. Finally, we analyze two examples and establish the reasons for the deviation. In other words, the fractional power operation directly based on MATLAB can only obtain one root at a time.

**Author Contributions:** Methodology, T.Z.; software, T.Z.; validation, T.Z. and Y.C.; investigation, Y.C.; writing—original draft preparation, T.Z.; writing—review and editing, T.Z.; supervision, T.Z.; funding acquisition, T.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study was supported by the Fundamental Research Funds for the Central Universities (N2123016); and the Scientific Research Projects of Hebei colleges and universities (QN2020511).

**Acknowledgments:** We would like to express our great appreciation to the editors and reviewers.

**Conflicts of Interest:** The authors declare no conflict of interest.
