**1. Introduction**

The multiweighted-type fractional Fourier transform (M-WFRFT) is the extended definition of the weighted-type fractional Fourier transform (WFRFT), and its application has been described in detail in our previous research [1]. Here, we focus on summarizing and analyzing the theory of the M-WFRFT. In [2], Zhu et al. proposed the definition of the multifraction Fourier transform, i.e., the M-WFRFT. Researchers have applied this definition to image encryption but have not discussed the properties of the definition itself. Early research work [3–5] laid a solid foundation for the proposal of the M-WFRFT. In 1995, Shih proposed a new type of fractional-order Fourier transform (FRFT), which is called WFRFT because it is a linear summation [3]. This definition has period 4, so it is also called the 4-WFRFT. Subsequently, Liu et al. extended the definition of the WFRFT, and the generalized definition has period M = 4*l*, where *l* = 1,2, ... [4,5]. Zhu's M-WFRFT is proposed on this basis, and its period is any integer M > 4 [2]. However, little is known about the properties of these definitions. Ran et al. sought to present a unified framework with the help of a generalized permutation matrix group and discussed its properties [6]. This research greatly promotes the theoretical development of WFRFTs. Unfortunately, there is no proof of unitarity, and the focus of the previous studies has been the generality of weighted coefficients. Recently, some new definitions based on the M-WFRFT have been proposed [7–11]. For example, Tao et al. proposed multiple-parameter fractional Fourier transforms (MPFRFTs) [8], Ran et al. proposed modified multiple-parameter fractional Fourier transforms (m-MPFRFTs) [9], and Zhao et al. proposed vector power multiple-parameter fractional Fourier transforms (VPMPFRFTs) [10,11]. Unfortunately, the properties of these definitions have not been discussed.

First, Santhanam et al. demonstrated the properties of the WFRFT and proved its unitarity using weighted coefficients [12]. However, this work ignores that the basis function is also a part of the definition. For the M-WFRFT, its basis function is the fractional power of the Fourier transform, so it is not easy to prove its properties. Some recent research results have also failed to prove its properties [13–18]. We proposed a new reformulation of the M-WFRFT to prove its periodicity, additivity and boundary [1]. Unfortunately,

**Citation:** Zhao, T.; Chi, Y. Multiweighted-Type Fractional Fourier Transform: Unitarity. *Fractal Fract.* **2021**, *5*, 205. https://doi.org/ 10.3390/fractalfract5040205

Academic Editors: Thach Ngoc Dinh, Shyam Kamal and Rajesh Kumar Pandey

Received: 3 October 2021 Accepted: 1 November 2021 Published: 8 November 2021

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unitarity is only discussed by means of numerical simulation. This paper is a follow-up of previous research work and mainly seeks to prove and discuss the unitarity of the M-WFRFT. However, the most recent studies have also enlightened our research [19,20].

The remainder of this paper is organized as follows. Section 2 proposes a new reformulation of the M-WFRFT. The unitarity of the M-WFRFT is proven in Section 3. The deviation caused by the numerical simulation is discussed in Section 4. Finally, the conclusions are presented in Section 5.
