*3.4. Other Types of FRFTs*

There are three types of discrete definitions of the FRFT. In Section 3.1, the linear WFRFT is used. The fractional-order matrix is used in Section 3.2. The discrete FRFT, which is called the eigendecomposition type, is used in Section 3.3. Then, there is a sampling-type FRFT.

In [30], a sampling-type FRFT is proposed, and its process can be written as follows:

(a) Chirp multiplication

$$\log(\mathbf{x}\_0) = \exp\left[-ip\mathbf{x}\_0^2 \tan(f/2)\right] f(\mathbf{x}\_0);\tag{68}$$

(b) Chirp convolution

$$\mathbf{g}'(\mathbf{x}) = A\_{\Phi} \int\_{-\infty}^{\infty} \exp[i\pi \csc(\phi)(\mathbf{x} - \mathbf{x}\_{0})^{2}] \mathbf{g}(\mathbf{x}\_{0}) d\mathbf{x}\_{0};\tag{69}$$

(c) Chirp multiplication

$$f\_{\mathfrak{a}}(\mathbf{x}) = \exp\left[-i\pi\mathbf{x}^{2}\tan(\phi/2)\right] \mathbf{g}'(\mathbf{x}).\tag{70}$$

The definition of the sampling type is the numerical simulation of a continuous FRFT. The discretization of the FRFT has been extensively studied [12], and the three main types of DFRFTs are compared, as shown in Table 2. We noticed that the sampling-type FRFT did not satisfy additivity and unitarity.



**Remark 5.** *The M-WFRFT is an extended definition, and its basis function can be expressed as shown in Figure 1. The sampling type FRFT does not satisfy the additivity and unitarity, so it cannot be used as a basis function.*

**Figure 1.** Time-frequency denotation of the M-WFRFT operator.
