**3. Unitarity**

A complex matrix *U* satisfies

$$
\mathcal{U}\mathcal{U}I^H = \mathcal{U}^H \mathcal{U} = I,\tag{12}
$$

where *H* denotes the conjugate transpose and *I* is the identity matrix. Then, *U* is called a unitary matrix. The greatest difficulty in proving the unitarity of the M-WFRFT is considering the basis function *<sup>F</sup>*4*l*/*M*, *<sup>l</sup>* <sup>=</sup> 0, 1, ··· , *<sup>M</sup>* <sup>−</sup> 1. The basis function is related to the discrete fractional Fourier transform (DFRFT), and the definition of the DFRFT varies. Therefore, we seek to use different types of DFRFT as the basis function to verify the unitarity of M-WFRFT.
