**2. Reformulation of M-WFRFT**

Shih proposed the WFRFT [3], and its definition can be expressed as

$$F^{\alpha}[f(t)] = \sum\_{l=0}^{3} A\_{l}^{\alpha} f\_{l}(t),\tag{1}$$

with

$$A\_l^a = \cos\left(\frac{(\mathfrak{a} - l)\pi}{4}\right) \cos\left(\frac{2(\mathfrak{a} - l)\pi}{4}\right) \exp\left(\frac{3(\mathfrak{a} - l)i\pi}{4}\right),\tag{2}$$

where *fl*(*t*) = *F<sup>l</sup>* [ *f*(*t*)]; *l* = 0, 1, 2, 3, (*F* denotes Fourier transform). Shih's WFRFT with period 4 is also called the 4-weighted-type fractional Fourier transform (4-WFRFT).

We further improve the weighted coefficient *A<sup>α</sup> <sup>l</sup>* , as shown in Equation (3).

$$\begin{array}{rcl} A\_{I}^{a} & = & \cos\left(\frac{(a-l)\pi}{4}\right) \cos\left(\frac{2(a-l)\pi}{4}\right) \exp\left(\frac{3(a-l)i\pi}{4}\right) \\ & = & \frac{1}{2} \times \left[ \exp\left(\frac{(a-l)\pi i}{4}\right) + \exp\left(\frac{-(a-l)\pi i}{4}\right) \right] \\ & \qquad \times \frac{1}{2} \times \left[ \exp\left(\frac{2(a-l)\pi i}{4}\right) + \exp\left(\frac{-2(a-l)i\pi}{4}\right) \right] \times \exp\left(\frac{3(a-l)i\pi}{4}\right) \\ & = & \frac{1}{4} \left(1 + \exp\left(\frac{2(a-l)\pi i}{4}\right) + \exp\left(\frac{4(a-l)\pi i}{4}\right) + \exp\left(\frac{6(a-l)\pi i}{4}\right)\right) \\ & = & \frac{1}{4} \sum\_{k=0}^{3} \exp\left(\frac{2\pi ik(a-l)}{4}\right). \end{array} \tag{3}$$

Then, we can obtain Equation (4) as

$$
\begin{pmatrix} A\_0^a \\ A\_1^a \\ A\_2^a \\ A\_3^a \end{pmatrix} = \frac{1}{4} \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -i & -1 & i \\ 1 & -1 & 1 & -1 \\ 1 & i & -1 & -i \end{pmatrix} \begin{pmatrix} B\_0^a \\ B\_1^a \\ B\_2^a \\ B\_3^a \end{pmatrix} \tag{4}
$$

where *B<sup>α</sup> <sup>k</sup>* <sup>=</sup> exp2*πik<sup>α</sup>* <sup>4</sup> , *k* = 0, 1, 2, 3. Equation (4) provides ideas for expanding the definition in the future.

Liu et al., generalize Shih's definition, and the generalized definition is shown to have *M*-periodic eigenvalues with respect to the order of Hermite–Gaussian functions (*M* = 4*l*, where *l* = 1,2,3, ...) [4,5].

Subsequently, Zhu et al. proposed a multifractional Fourier transform whose period can be any integer (*M* > 4), so this definition is also called the M-WFRFT [2]. Zhu et al., extended the weighting coefficient *A<sup>α</sup> <sup>l</sup>* , which is more general; the result is shown in Equation (5).

$$\begin{pmatrix} A\_0^a \\ A\_1^a \\ \vdots \\ A\_{M-1}^a \end{pmatrix} = \frac{1}{M} \begin{pmatrix} u^{0 \times 0} & u^{0 \times 1} & \cdots & u^{0 \times (M-1)} \\ u^{1 \times 0} & u^{1 \times 1} & \cdots & u^{1 \times (M-1)} \\ \vdots & \vdots & \ddots & \vdots \\ u^{(M-1) \times 0} & u^{(M-1) \times 1} & \cdots & u^{(M-1) \times (M-1)} \end{pmatrix} \begin{pmatrix} B\_0^a \\ B\_1^a \\ \vdots \\ B\_{M-1}^a \end{pmatrix} \tag{5}$$

where *<sup>u</sup>* <sup>=</sup> exp(−2*πi*/*M*) and *<sup>B</sup><sup>α</sup> <sup>k</sup>* <sup>=</sup> exp2*πik<sup>α</sup> <sup>M</sup>* , *k* = 0, 1, ··· , *M* − 1. Then,

$$\begin{aligned} A\_l^a &= \frac{1}{M} \sum\_{k=0}^{M-1} \exp\left[\frac{2\pi ik(a-l)}{M}\right]; \\ l &= 0, 1, \cdots, M-1. \end{aligned} \tag{6}$$

The M-WFRFT is defined as

$$F\_M^a[f(t)] = \sum\_{l=0}^{M-1} A\_l^a f\_l(t),\tag{7}$$

where the basic functions are *fl*(*t*) <sup>=</sup> *<sup>F</sup>*4*l*/*M*[ *<sup>f</sup>*(*t*)]; *<sup>l</sup>* <sup>=</sup> 0, 1, ··· , *<sup>M</sup>* <sup>−</sup> 1 (*<sup>F</sup>* denotes the Fourier transform).

At present, the M-WFRFT is widely used in image encryption and signal processing [7–11,21–25]. Unfortunately, few researchers have discussed its properties, and the proponents of the definition have not explained the properties. We find that it is not easy to prove the properties of the M-WFRFT (Equation (7)). Some researchers have discussed the properties using the weighted coefficient *A<sup>α</sup> <sup>l</sup>* but ignore that the basis function is also a part of the definition [6,12]. Therefore, we present a new reformulation of the M-WFRFT. As such, Equation (7) can be expressed as

$$\begin{aligned} \left[F\_M^a[f(t)]\right] &= \left.A\_0^a f\_0(t) + A\_1^a f\_1(t) + \dots + A\_{M-1}^a f\_{M-1}(t) \\ &= \left.A\_0^a F\_M^{\frac{M}{M}}[f(t)] + A\_1^a F\_M^{\frac{M}{M}}[f(t)] + \dots + A\_{M-1}^a F\_1^{\frac{4(M-1)}{M}}[f(t)]\right| \\ &= \left(A\_0^a I + A\_1^a F\_M^{\frac{4}{M}} + \dots + A\_{M-1}^a F\_1^{\frac{4(M-1)}{M}}\right) f(t) \\ &= \left(I\_r F\_r^{\frac{4}{M}}, \dots, f^{\frac{4(M-1)}{M}}\right) \begin{pmatrix} A\_0^a \\ A\_1^a \\ \vdots \\ A\_{M-1}^a \end{pmatrix} f(t). \end{aligned} \tag{8}$$

By Equations (5) and (8), Equation (9) is obtained as

$$\mathbf{F}\_{M}^{\mathbf{a}}[f(t)] = \frac{1}{M} \left( I\_{\star} \mathbf{F}\_{\star}^{\mathbf{A}}, \dots, \mathbf{F}\_{\star}^{\frac{M(M-1)}{M}} \right) \begin{pmatrix} \mathbf{u}^{0 \times 0} & \mathbf{u}^{0 \times 1} & \cdots & \mathbf{u}^{0 \times (M-1)} \\ \mathbf{u}^{1 \times 0} & \mathbf{u}^{1 \times 1} & \cdots & \mathbf{u}^{1 \times (M-1)} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{u}^{(M-1) \times 0} & \mathbf{u}^{(M-1) \times 1} & \cdots & \mathbf{u}^{(M-1) \times (M-1)} \end{pmatrix} \begin{pmatrix} \begin{array}{c} \mathbf{B}\_{0}^{\mathbf{a}} \\ \mathbf{B}\_{1}^{\mathbf{a}} \\ \vdots \\ \mathbf{B}\_{M-1}^{\mathbf{a}} \end{array} \end{pmatrix} f(t), \tag{9}$$

where *<sup>u</sup>* <sup>=</sup> exp(−2*πi*/*M*), *<sup>B</sup><sup>α</sup> <sup>k</sup>* <sup>=</sup> exp2*πik<sup>α</sup> <sup>M</sup>* , *k* = 0, 1, ... , *M* − 1 and *F* denotes the Fourier transform. Here, let

$$\begin{cases} \begin{aligned} Y\_0 &= \mathfrak{u}^{0 \times 0}I + \mathfrak{u}^{1 \times 0}F^{\frac{4}{M}} + \dots + \mathfrak{u}^{(M-1) \times 0}F^{\frac{4(M-1)}{M}}, \\ Y\_1 &= \mathfrak{u}^{0 \times 1}I + \mathfrak{u}^{1 \times 1}F^{\frac{4}{M}} + \dots + \mathfrak{u}^{(M-1) \times 1}F^{\frac{4(M-1)}{M}}, \\ Y\_2 &= \mathfrak{u}^{0 \times 2}I + \mathfrak{u}^{1 \times 2}F^{\frac{4}{M}} + \dots + \mathfrak{u}^{(M-1) \times 2}F^{\frac{4(M-1)}{M}}, \\ &\vdots \\ Y\_{M-1} &= \mathfrak{u}^{0 \times (M-1)}I + \mathfrak{u}^{1 \times (M-1)}F^{\frac{4}{M}} + \dots + \mathfrak{u}^{(M-1) \times (M-1)}F^{\frac{4(M-1)}{M}}. \end{aligned} \end{cases} \tag{10}$$

**Definition 1.** A new reformulation of the M-WFRFT as

$$\begin{aligned} \left[T\_{MW}^{\mathbf{a}}[f(t)]\right] &= \frac{1}{M} (Y\_0, Y\_{1\prime}, \dots, Y\_{M-1}) \begin{pmatrix} B\_0^{\mathbf{a}} \\ B\_1^{\mathbf{a}} \\ \vdots \\ B\_{M-1}^{\mathbf{a}} \end{pmatrix} f(t) \\ &= \frac{1}{M} \sum\_{k=0}^{M-1} Y\_k B\_k^{\mathbf{a}} f(t), \end{aligned} \tag{11}$$

where *B<sup>α</sup> <sup>k</sup>* <sup>=</sup> exp2*πik<sup>α</sup> <sup>M</sup>* ; *k* = 0, 1, ··· , *M* − 1.

**Remark 1.** *Our previous work [1] discussed that the new reformulation helps to prove the properties. Unitarity is often used in signal processing. Unfortunately, previous research work only presents simulation verification. This paper will seek to prove and discuss the unitarity.*
