Discussion on Weighted Fractional Fourier Transform and Its Extended Definitions

Abstract

The weighted fractional Fourier transform (WFRFT) has always been considered a development of the discrete fractional Fourier transform (FRFT). This paper points out that the WFRFT is a discrete FRFT of eigenvalue decomposition, which will change the consistent understanding of the WFRFT. Extended definitions based on the WFRFT have been proposed and widely used in information processing. This paper proposes a unified framework for extended definitions, and existing extended definitions can serve as special cases of this unified framework. In further analysis, we find that the existing extended definitions are deficient. With the help of a unified framework, we systematically analyze the reasons for the deficiencies. This has great guiding significance for the application of the WFRFT and its extended definitions.

1. Introduction

The fractional Fourier transform (FRFT) can be regarded as a generalization of the Fourier transform (FT). It can not only transform a function from time domain to frequency domain but also to the intermediate domain of any angle. The advantage of doing so is that we can more flexibly choose the angle that is most suitable for observing and analyzing the function. For example, we can use the FRFT to rotate a waveform to the clearest or most ambiguous angle or use the FRFT to rotate a music signal to the easiest or most difficult to identify angle. This characteristic of the FRFT has widened its range of applications. The idea of the FRFT can be traced back to Wiener’s research work in 1929 [1]. It is well known that the characteristic function of the FT is the Hermite–Gauss function, and its eigenvalue is $e^{jn\left(\pi /2\right)}$, $n\in Z$. Wiener used the FT to construct a transform operator, the characteristic function of which is the same as that of the FT, and the corresponding eigenvalue is $e^{jn\alpha }$, where $\alpha$ is the fractional multiple of $\pi /2$. While the transform operator constructed by Wiener is now considered the FRFT, the term FRFT was not used at that time. Subsequent related research is still ongoing [2,3,4]. In 1980, by means of the eigenvalues and eigenfunctions of the FT, Namias first proposed the concept of the FRFT (arbitrary real number) according to the arbitrary power operation of eigenvalues, derived its high-order differential form, and applied it to solve differential equations in quantum mechanics [5]. In this way, the definition of the FRFT is formally determined. With the continuous efforts of researchers, the definition of the FRFT has been further improved, and its physical interpretation and fast discrete algorithm have also been proposed [6,7,8,9]. This leads to strong research interest in this new definition, and the FRFT has become a research hotspot.

In 1995, Shih first proposed the weighted fractional Fourier transform (WFRFT) [10], and its definition is easily extended. Santhanam et al. proposed the discrete FRFT using the rotation angle of the FT [11]. In this paper, our proof shows that it is mathematically equivalent to Shih’s definition. Over the subsequent years, numerous extended definitions have been proposed and extensively used in information security and information processing. Liu et al. proposed the generalized FRFT in 1997, where weighted term parameter k is assigned an integer multiple of 4 and can approach infinity [12,13]. Therefore, the WFRFT can be regarded as a special case of Liu et al.’s definition. Subsequently, Zhu et al. proposed the multifractional Fourier transform (m-FRFT) based on Liu et al.’s definition [14] and applied this algorithm to optical image encryption. This was the first time that the WFRFT had been applied to image encryption after optical encryption with double-random phase encoding [15], which resulted in the development of many image encryption schemes [16,17,18,19,20,21,22,23,24,25]. For example, Tao et al. proposed a multiple parameter fractional Fourier transform (MPFRFT), which is a further extension of Liu et al.’s definition, and this approach was applied to image encryption [16]. The MPFRFT has a larger key space and greatly improves the security of the system. Ran et al. proposed a modified MPFRFT, which has better security [17]. Moreover, similar to the research work [26] of Pei et al., a vector power multiple parameter fractional Fourier transform (VPMPFRFT) was proposed [19] and applied to image encryption to overcome security risks [21]. Recently, we discussed in detail the correlation between the extended definitions of the WFRFT and their security [23].

In signal processing, Ran et al. proposed a new method for analyzing signal sampling and reconstruction using the WFRFT [27,28]. The orthogonal sampling basis of the WFRFT band-limited signal was determined [29,30]. Mei et al. proposed a discrete algorithm that adapts the WFRFT to communication systems and determined modulation and demodulation schemes [31,32]. This further promotes the application of the WFRFT in signal processing. Furthermore, a series of research works were conducted [33,34,35,36,37,38,39,40,41,42,43]; for example, Hui et al. proposed an optimal-order selecting scheme for the 4-WFRFT over doubly selective channels [33]. To ensure the security of the physical layer, the WFRFT system based on a parallel combination spread spectrum was proposed [34,35]. The WFRFT is also used in the design of window filters [36]. Liang et al. proposed a scheme to suppress narrowband interference using the WFRFT domain preprocessing technology [37]. Recently, Li et al. proposed a three-branch transmission scheme using an 8-WFRFT module to improve the bit error rate performance of multiple-input, multiple-output systems with low complexity [40]. The WFRFT is also applied to other research fields, such as image steganography [44], medical image classification [45], breast disease diagnosis [46], and regenerative scanning [47,48].

The wide application of the WFRFT is due to the fact that it is an easy-to-implement discrete FRFT. However, this discrete FRFT is diverse [49,50], and researchers have always classified the WFRFT as an independent discrete algorithm. In this paper, our research shows that the WFRFT is the fractional power of the eigenvalue of the FT. Furthermore, we propose a unified framework; that is, the WFRFT and its extended definitions are special cases of the unified framework. However, we also identify the deficiencies of the extended definitions and analyze the causes of the deficiencies in detail.

The remainder of this paper is organized as follows. The properties and classification of the discrete FRFT are explained in Section 2. The new expressions of the WFRFT and extended definitions are proposed in Section 3. Section 4 proposes a unified framework for the WFRFT. Section 5 discusses the deficiencies of the extended definitions. Finally, the conclusions are presented in Section 6.

2. Discrete Fractional Fourier Transform

The essence of the FT is to represent the signal as the superposition of sinusoidal signals, and the essence of the FRFT is to represent the signal as the superposition of chirp signals. The expression for the FT can be written as follows:

$$g(u)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle {\int }_{-\infty }^{\infty }\exp (-iut)f(t)dt.}$$

(1)

Its integral kernel expression is

$$F^{\pi /2}\left[f(t)\right]=\sqrt{{\displaystyle \frac{1}{2\pi }}}{\displaystyle {\int }_{-\infty }^{\infty }\exp (-iut)f(t)dt.}$$

(2)

Its characteristic function expression is

$$F^{\pi /2}\left[{\varphi }_n(t)\right]=e^{-jn{\displaystyle \frac{\pi }{2}}}{\varphi }_n(u).$$

(3)

The characteristic function ${\varphi }_n(t)=\exp (-t^2/2)H_n(t)$, $H_n(t)={(-1)}^n{e}^{t^2}{\displaystyle \frac{d^n}{d{t}^n}}{e}^{-t^2}$. Equation (3) shows that the FT operator acting on the Hermite characteristic function is equivalent to the eigenvalue multiplied by the corresponding characteristic function. Whether the FT is defined in the form of an integral kernel or in the form of a characteristic function, the corresponding operators $F^{\pi /2}$ are the same, where the eigenvalue is $e^{-in{\displaystyle \frac{\pi }{2}}}$. If the eigenvalues of the FT operator are generalized, that is, the eigenvalue $e^{-in{\displaystyle \frac{\pi }{2}}}$ is changed to $e^{-in\alpha }$, then we can obtain

$$F^{\alpha }\left[{\varphi }_n(t)\right]=e^{-in\alpha }{\varphi }_n(u),$$

(4)

where $\left(\alpha =p\pi /2\right)$, and $p$ is an arbitrary real number. Equation (4) has the same characteristic function as the FT. Furthermore, the integral kernel expression of the FRFT is obtained [5,6].

$$F^{\alpha }\left[f(t)\right]=\sqrt{{\displaystyle \frac{1-i\cot \alpha }{2\pi }}}{\displaystyle {\int }_{-\infty }^{\infty }\exp (\frac{i\cot \alpha }{2}{t}^2+\frac{i\cot \alpha }{2}{u}^2-\frac{itu}{\sin \alpha })}f(t)dt.$$

(5)

The calculation of the discrete FRFT from Equation (5) is much more complicated than that of the discrete FT, which leads to the diversity of the discrete FRFT. The idealized discrete FRFT needs to satisfy the following criteria:

(a)

Approximation: The discrete FRFT and the continuous FRFT are approximate for signal processing;

(b)

Boundary: $F^1=F$, $F$ is the discrete FT operator;

(c)

Unitarity: $F^p\cdot {\left(F^p\right)}^H=I$, $H$ denotes conjugate transpose, $I$ is the identity matrix;

(d)

Additivity: $F^p{F}^q=F^{p+q}$;

(e)

Computation efficiency: The computational complexity of the discrete FRFT should be as low as possible.

In recent years, researchers have proposed a variety of definitions and fast algorithms for the discrete FRFT. Unfortunately, so far, there is no definition that can meet all the above requirements well. At present, the discrete FRFT is divided into three types according to the application requirements [49,50]: (1) sampling type; (2) eigenvalue decomposition type; and (3) weighted type. Our research indicates that the WFRFT is also an eigenvalue decomposition algorithm, which essentially preserves the eigenvector of the FT while the eigenvalues become fractional powers. Next, we will provide a new explanation for the WFRFT and its extended definitions.

3. Weighted Fractional Fourier Transform and Extended Definitions

Researchers have always believed that the WFRFT is an independent discrete FRFT [49,50]. However, the correlation between the WFRFT and the discrete FT has not been clearly explained mathematically. Many extended definitions based on the WFRFT have been proposed, and the inherent connections between these definitions are worth pondering. We will focus on solving these problems in this section.

3.1. Shih’s Weighted Fractional Fourier Transform

In 1995, Shih proposed the WFRFT [10], and in the same year Santhanam et al. also proposed a rotation angle WFRFT [11]. These studies describe different types of FRFT, but our study found that their essence is consistent. Next, we will analyze Shih’s WFRFT in detail. It is defined as follows:

$$F^{\alpha }\left[f(t)\right]={\displaystyle \sum _{l=0}^3{A}_l^{\alpha }{f}_l(t).}$$

(6)

Here, $f_0(t)=f(t)$, $f_1(t)=F\left[f(t)\right]$, $f_2(t)=F^2\left[f(t)\right]$, and $f_3(t)=F^3\left[f(t)\right]$ (F denotes the FT). The weighting coefficient $A_l^{\alpha }$ can be expressed as follows:

$$A_l^{\alpha }=\cos \left({\displaystyle \frac{(\alpha -l)\pi }{4}}\right)\cos \left({\displaystyle \frac{2(\alpha -l)\pi }{4}}\right)\exp \left(-{\displaystyle \frac{3(\alpha -l)i\pi }{4}}\right).$$

(7)

We further organize Equation (7) to obtain

$$\begin{array}{cc}\hfill A_l^{\alpha }& =\cos \left({\displaystyle \frac{(\alpha -l)\pi }{4}}\right)\cos \left({\displaystyle \frac{2(\alpha -l)\pi }{4}}\right)\exp \left(-{\displaystyle \frac{3(\alpha -l)i\pi }{4}}\right)\hfill \\ & ={\displaystyle \frac{1}{2}}\times \left[\exp \left({\displaystyle \frac{(\alpha -l)\pi i}{4}}\right)+\exp \left({\displaystyle \frac{-(\alpha -l)\pi i}{4}}\right)\right]\\ & \times {\displaystyle \frac{1}{2}}\times \left[\exp \left({\displaystyle \frac{2(\alpha -l)\pi i}{4}}\right)+\exp \left({\displaystyle \frac{-2(\alpha -l)\pi i}{4}}\right)\right]\times \exp \left(-{\displaystyle \frac{3(\alpha -l)i\pi }{4}}\right)\\ & ={\displaystyle \frac{1}{4}}\left(1+\exp \left(-{\displaystyle \frac{2(\alpha -l)\pi i}{4}}\right)+\exp \left(-{\displaystyle \frac{4(\alpha -l)\pi i}{4}}\right)+\exp \left(-{\displaystyle \frac{6(\alpha -l)\pi i}{4}}\right)\right)\\ & ={\displaystyle \frac{1}{4}}{\displaystyle \sum _{k=0}^3\exp \left(-\frac{2\pi i}{4}(\alpha -l)k\right)}\end{array}$$

(8)

Equation (8) can also be written as the result of summation:

$$A_l^{\alpha }={\displaystyle \frac{1}{4}}{\displaystyle \frac{1-\exp \left(-2\pi i\left(\alpha -l\right)\right)}{1-\exp \left(-\left(2\pi i\left(\alpha -l\right)\right)/4\right)}}.$$

(9)

We continue to organize Equation (8) to obtain

$$\begin{array}{cc}\hfill A_l^{\alpha }& ={\displaystyle \frac{1}{4}}{\displaystyle \sum _{k=0}^3\exp \left(-\frac{2\pi i}{4}(\alpha -l)k\right)}\hfill \\ & ={\displaystyle \frac{1}{4}}{\displaystyle \sum _{k=0}^3\exp \left(-\frac{2\pi i\alpha k}{4}+\frac{2\pi ilk}{4}\right)}\\ & ={\displaystyle \frac{1}{4}}{\displaystyle \sum _{k=0}^3\exp \left(\frac{2\pi ilk}{4}\right)\exp \left(-\frac{2\pi i\alpha k}{4}\right)}\\ & =IDFT{\left[\exp \left(-{\displaystyle \frac{2\pi i\alpha k}{4}}\right)\right]}_{k=0,1,2,3}\end{array}$$

(10)

where $l=0,1,2,3$, and IDFT denotes the inverse discrete FT. Equation (6) can therefore be expressed as follows:

$$\begin{array}{cc}\hfill F^{\alpha }\left[f(x)\right]& ={\displaystyle \frac{1}{4}}{\displaystyle \sum _{l=0}^3{\displaystyle \sum _{k=0}^3\exp \left[-\frac{2\pi ik(\alpha -l)}{4}\right]f_l(x)}}\hfill \\ & ={\displaystyle \frac{1}{4}}{\displaystyle \sum _{l=0}^3{\displaystyle \sum _{k=0}^3\exp \left(\frac{2\pi ikl}{4}\right)\exp \left(\frac{-2\pi ik\alpha }{4}\right)f_l(x)}}\\ & ={\displaystyle \frac{1}{4}}IDFT\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)\left(\begin{array}{cccc}{F}^0& F^1& F^2& F^3\end{array}\right)f\left(x\right)\\ & ={\displaystyle \frac{1}{4}}\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& i& -1& -i\\ 1& -1& 1& -1\\ 1& -i& -1& i\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)\left(\begin{array}{cccc}{F}^0& F^1& F^2& F^3\end{array}\right)f\left(x\right)\\ & ={\displaystyle \frac{1}{4}}\left(\begin{array}{cccc}{F}^0& F^1& F^2& F^3\end{array}\right)\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& i& -1& -i\\ 1& -1& 1& -1\\ 1& -i& -1& i\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)f\left(x\right)\\ & ={\displaystyle \frac{1}{4}}\left(Y_0,Y_1,Y_2,Y_3\right)\left(\begin{array}{l}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)f\left(t\right)\end{array}$$

(11)

Here, $B_k^{\alpha }=\exp \left(-{\displaystyle \frac{2\pi ik\alpha }{4}}\right)$; $k=0,1,2,3$, and

$$\{\begin{array}{l}{Y}_0=I+F+F^2+F^3\\ Y_1=I+iF-F^2-i{F}^3\\ Y_2=I-F+F^2-F^3\\ Y_3=I-iF-F^2+i{F}^3\end{array}$$

(12)

We know that the discrete FT can be expressed as follows:

$$F=VD{V}^T,$$

(13)

where $D$ is the eigenvalue and $V$ is the eigenvector. The eigenvalues of the FT are 1, i, −1, and −i. Then, Equation (13) can also be written as follows:

$$F={\displaystyle \sum _{k\in E_1}\left(1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}\left(i\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}\left(-1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}\left(-i\right)v_k{v}_k^T}.$$

(14)

Here, $V=\left[v_0,v_1,\cdots ,v_{N-1}\right]$, $v_k$ is the eigenvector and $E_1$, $E_2$, $E_3$, and $E_4$ represent the set of eigenvectors corresponding to the eigenvalues, respectively. Therefore, we can obtain

$$\begin{array}{cc}\hfill F^2& =F\cdot F\hfill \\ & =\left(VD{V}^T\right)\cdot \left(VD{V}^T\right)\\ & =V{D}^2{V}^T\\ & ={\displaystyle \sum _{k\in E_1}{\left(1\right)}^2{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^2{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^2{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^2{v}_k{v}_k^T}\\ & ={\displaystyle \sum _{k\in E_1}\left(1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}\left(-1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}\left(1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}\left(-1\right)v_k{v}_k^T}\end{array}$$

(15)

$$\begin{array}{cc}\hfill F^3& =F\cdot F\cdot F\hfill \\ & ={\displaystyle \sum _{k\in E_1}{\left(1\right)}^3{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^3{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^3{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^3{v}_k{v}_k^T}\\ & ={\displaystyle \sum _{k\in E_1}\left(1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}\left(-i\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}\left(-1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}\left(i\right)v_k{v}_k^T}\end{array}$$

(16)

$$\begin{array}{cc}\hfill F^0& =F^4\hfill \\ & ={\displaystyle \sum _{k\in E_1}{\left(1\right)}^4{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^4{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^4{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^4{v}_k{v}_k^T}\\ & ={\displaystyle \sum _{k\in E_1}\left(1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}\left(1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}\left(1\right)v_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}\left(1\right)v_k{v}_k^T}\end{array}$$

(17)

The results of Equation (12) are obtained

$$\{\begin{array}{l}{Y}_0=F^0+F+F^2+F^3={\displaystyle \sum _{k\in E_1}4\cdot v_k{v}_k^T}\\ Y_1=F^0+iF-F^2-i{F}^3={\displaystyle \sum _{k\in E_4}4\cdot v_k{v}_k^T}\\ Y_2=F^0-F+F^2-F^3={\displaystyle \sum _{k\in E_3}4\cdot v_k{v}_k^T}\\ Y_3=F^0-iF-F^2+i{F}^3={\displaystyle \sum _{k\in E_2}4\cdot v_k{v}_k^T}\end{array}$$

(18)

We know that $B_k^{\alpha }=\exp \left(-{\displaystyle \frac{2\pi ik\alpha }{4}}\right)$; $k=0,1,2,3$,

$$\left(\begin{array}{l}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)=\left(\begin{array}{c}{\left(1\right)}^{\alpha }\\ {\left(-i\right)}^{\alpha }\\ {\left(-1\right)}^{\alpha }\\ {\left(i\right)}^{\alpha }\end{array}\right)$$

(19)

Finally, the WFRFT is represented as follows:

$$\begin{array}{cc}\hfill F^{\alpha }& ={\displaystyle \frac{1}{4}}\left(Y_0,Y_1,Y_2,Y_3\right)\left(\begin{array}{l}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)\hfill \\ & ={\displaystyle \frac{1}{4}}\left(Y_0{B}_0^{\alpha }+Y_1{B}_1^{\alpha }+Y_2{B}_2^{\alpha }+Y_3{B}_3^{\alpha }\right)\\ & ={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{\alpha }{v}_k{v}_k^T}\end{array}$$

(20)

We rephrased the definition of Shih. However, there is another representation of the weighted coefficient of Equation (7), as shown in Equation (21):

$$A_l^{\alpha }=\cos \left({\displaystyle \frac{(\alpha -l)\pi }{4}}\right)\cos \left({\displaystyle \frac{2(\alpha -l)\pi }{4}}\right)\exp \left({\displaystyle \frac{3(\alpha -l)i\pi }{4}}\right).$$

(21)

This expression is equivalent to the above result, and we will prove it here. Furthermore, we can obtain

$$\begin{array}{cc}\hfill A_l^{\alpha }& =\cos \left({\displaystyle \frac{(\alpha -l)\pi }{4}}\right)\cos \left({\displaystyle \frac{2(\alpha -l)\pi }{4}}\right)\exp \left({\displaystyle \frac{3(\alpha -l)i\pi }{4}}\right)\hfill \\ & ={\displaystyle \frac{1}{2}}\times \left[\exp \left({\displaystyle \frac{(\alpha -l)\pi i}{4}}\right)+\exp \left({\displaystyle \frac{-(\alpha -l)\pi i}{4}}\right)\right]\\ & \times {\displaystyle \frac{1}{2}}\times \left[\exp \left({\displaystyle \frac{2(\alpha -l)\pi i}{4}}\right)+\exp \left({\displaystyle \frac{-2(\alpha -l)\pi i}{4}}\right)\right]\times \exp \left({\displaystyle \frac{3(\alpha -l)i\pi }{4}}\right)\\ & ={\displaystyle \frac{1}{4}}\left(1+\exp \left({\displaystyle \frac{2(\alpha -l)\pi i}{4}}\right)+\exp \left({\displaystyle \frac{4(\alpha -l)\pi i}{4}}\right)+\exp \left({\displaystyle \frac{6(\alpha -l)\pi i}{4}}\right)\right)\\ & ={\displaystyle \frac{1}{4}}{\displaystyle \sum _{k=0}^3\exp \left(\frac{2\pi i}{4}(\alpha -l)k\right)}\\ & ={\displaystyle \frac{1}{4}}{\displaystyle \sum _{k=0}^3\exp \left(-\frac{2\pi ilk}{4}+\frac{2\pi i\alpha k}{4}\right)}\\ & ={\displaystyle \frac{1}{4}}{\displaystyle \sum _{k=0}^3\exp \left(-\frac{2\pi ilk}{4}\right)\exp \left(\frac{2\pi i\alpha k}{4}\right)}\\ & =DFT{\left[\exp \left({\displaystyle \frac{2\pi i\alpha k}{4}}\right)\right]}_{k=0,1,2,3}\end{array}$$

(22)

where DFT is a discrete FT, and $l=0,1,2,3$. The WFRFT (Equation (6)) can be expressed as follows:

$$\begin{array}{cc}\hfill F^{\alpha }\left[f(t)\right]& ={\displaystyle \frac{1}{4}}{\displaystyle \sum _{l=0}^3{\displaystyle \sum _{k=0}^3\exp \left[\frac{2\pi ik(\alpha -l)}{4}\right]f_l(t)}}\hfill \\ & ={\displaystyle \frac{1}{4}}{\displaystyle \sum _{l=0}^3{\displaystyle \sum _{k=0}^3\exp \left(-\frac{2\pi ikl}{4}\right)\exp \left(\frac{2\pi ik\alpha }{4}\right)f_l(t)}}\\ & ={\displaystyle \frac{1}{4}}DFT\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)\left(\begin{array}{cccc}{F}^0& F^1& F^2& F^3\end{array}\right)f\left(t\right)\\ & ={\displaystyle \frac{1}{4}}\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& -i& -1& i\\ 1& -1& 1& -1\\ 1& i& -1& -i\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)\left(\begin{array}{cccc}{F}^0& F^1& F^2& F^3\end{array}\right)f\left(t\right)\\ & ={\displaystyle \frac{1}{4}}\left(\begin{array}{cccc}{F}^0& F^1& F^2& F^3\end{array}\right)\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& -i& -1& i\\ 1& -1& 1& -1\\ 1& i& -1& -i\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)f\left(t\right)\\ & ={\displaystyle \frac{1}{4}}\left(Y_0,Y_1,Y_2,Y_3\right)\left(\begin{array}{l}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)f\left(t\right)\end{array}$$

(23)

where $B_k^{\alpha }=\exp \left({\displaystyle \frac{2\pi ik\alpha }{4}}\right)$; $k=0,1,2,3$. Let

$$\{\begin{array}{l}{Y}_0=I+F+F^2+F^3={\displaystyle \sum _{k\in E_1}4\cdot v_k{v}_k^T}\\ Y_1=I-iF-F^2+i{F}^3={\displaystyle \sum _{k\in E_2}4\cdot v_k{v}_k^T}\\ Y_2=I-F+F^2-F^3={\displaystyle \sum _{k\in E_3}4\cdot v_k{v}_k^T}\\ Y_3=I+iF-F^2-i{F}^3={\displaystyle \sum _{k\in E_4}4\cdot v_k{v}_k^T}\end{array}$$

(24)

and

$$\left(\begin{array}{l}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)=\left(\begin{array}{c}{\left(1\right)}^{\alpha }\\ {\left(i\right)}^{\alpha }\\ {\left(-1\right)}^{\alpha }\\ {\left(-i\right)}^{\alpha }\end{array}\right)$$

(25)

Then, Equation (23) can be written as follows:

$$\begin{array}{cc}\hfill F^{\alpha }& ={\displaystyle \frac{1}{4}}\left(Y_0,Y_1,Y_2,Y_3\right)\left(\begin{array}{l}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\end{array}\right)\hfill \\ & ={\displaystyle \frac{1}{4}}\left(Y_0{B}_0^{\alpha }+Y_1{B}_1^{\alpha }+Y_2{B}_2^{\alpha }+Y_3{B}_3^{\alpha }\right)\\ & ={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{\alpha }{v}_k{v}_k^T}\end{array}$$

(26)

Thus, from the two weighted coefficients of Equations (7) and (21), the expression of the WFRFT is consistent.

Through the above theoretical derivation, we obtain the eigenvalues and eigenvectors of the WFRFT as shown in Equations (20) and (26); that is, the eigenvectors of the FT are retained, and the eigenvalues become fractional powers. Equations (20) and (26) can be further expressed as follows:

$$F^{\alpha }=V{D}^{\alpha }{V}^T.$$

(27)

The WFRFT is, in fact, the eigenvalue decomposition type of the discrete FRFT.

3.2. Zhu et al.’s Multifractional Fourier Transform

In 2000, based on existing studies [12,13], Zhu et al. further proposed an extended definition of the WFRFT and applied to image encryption [14]. It is defined as follows:

$$F_M^{\alpha }\left[f(t)\right]={\displaystyle \sum _{l=0}^{M-1}{A}_l^{\alpha }{f}_l(t)}.$$

(28)

Here, $f_l(t)=F^{4l/M}\left[f(t)\right]$, and the weighting coefficient $A_l^{\alpha }$ is

$$A_l^{\alpha }={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}\exp \left[-\frac{2\pi ik(\alpha -l)}{M}\right]}.$$

(29)

This definition extends from four weighted terms to a more general case. The definition of Zhu et al. can be extended to a series of new definitions, which can be realized only by changing Equation (29). For example, in 2008, Tao et al. proposed the MPFRFT [16], which is defined as follows:

$$F_M^{\alpha }(\mathfrak{M},\mathfrak{N})[f(t)]={\displaystyle \sum _{l=0}^{M-1}{A}_l(\alpha ,\mathfrak{M},\mathfrak{N})f_l(t)}.$$

(30)

Here, $f_l(t)=F^{4l/M}\left[f(t)\right]$, and the weighting coefficient $A_l(\alpha ,\mathfrak{M},\mathfrak{N})$ is

$$A_l(\alpha ,\mathfrak{M},\mathfrak{N})={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}\exp \left\{-\frac{2\pi i}{M}\left[\left(m_{k}M+1\right)\alpha (k+n_{k}M)-lk\right]\right\},}$$

(31)

$$\mathfrak{M}=\left(m_0,m_1,\cdots ,m_{M-1}\right)\in {\mathbb{Z}}^M,\mathfrak{N}=\left(n_0,n_1,\cdots ,n_{M-1}\right)\in {\mathbb{Z}}^M.$$

Subsequently, Ran et al. proposed a modified MPFRFT with better security [17], which is defined as follows:

$$F_M^{\alpha }(\Re )[f(t)]={\displaystyle \sum _{l=0}^{M-1}{A}_l\left(\alpha ,\Re \right)f_l(t)}.$$

(32)

Here, $f_l(t)=F^{4l/M}\left[f(t)\right]$, and the weighting coefficient $A_l\left(\alpha ,\Re \right)$ is

$$A_l\left(\alpha ,\Re \right)={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}\exp \left\{\frac{-2\pi i}{M}\left[\alpha \left(k+r_{k}M\right)-lk\right]\right\}},$$

(33)

$$\Re =\left(r_0,r_1,\cdots ,r_{(M-1)}\right)\in {ℝ}^M.$$

The value of the parameter can be a real number, which theoretically improves the security of the system for image encryption.

In 2014, Ran et al. further proposed the VPMPFRFT [19], which can be expressed as follows:

$$F_M^{\overline{\alpha }}(\Re )\left[f(t)\right]={\displaystyle \sum _{l=0}^{M-1}{A}_l\left(\overline{\alpha },\Re \right)f_l(t)}.$$

(34)

Here, $f_l(t)=F^{4l/M}\left[f(t)\right]$, and the weighting coefficient $A_l\left(\overline{\alpha },\Re \right)$ is

$$A_l\left(\overline{\alpha },\Re \right)={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}\exp \left\{\frac{-2\pi i}{M}\left[{\alpha }_k\left(k+r_{k}M\right)-lk\right]\right\}},$$

(35)

$$\overline{\alpha }=\left({\alpha }_0,{\alpha }_1,\cdots ,{\alpha }_{M-1}\right)\in {ℝ}^M,\Re =\left(r_0,r_1,\cdots ,r_{M-1}\right)\in {ℝ}^M.$$

The change in weighting coefficient can bring us a new definition. These definitions are mainly applied in image encryption to improve security. Next, we analyze the correlation between these definitions.

First, Equation (29) can be written as follows:

$$\begin{array}{cc}\hfill A_l^{\alpha }& ={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}\exp \left[-\frac{2\pi ik(\alpha -l)}{M}\right]}\hfill \\ & ={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}\exp \left(\frac{2\pi ikl}{M}\right)}\exp \left({\displaystyle \frac{-2\pi ik\alpha }{M}}\right)\\ & =IDFT{\left[e^{\left(-2\pi i\alpha k/M\right)}\right]}_{k=0,1,2,\cdots \cdots ,M-1}\end{array}$$

(36)

Then, we can obtain

$$\left(\begin{array}{c}{A}_0^{\alpha }\\ A_1^{\alpha }\\ ⋮\\ A_{M-1}^{\alpha }\end{array}\right)={\displaystyle \frac{1}{M}}\left(\begin{array}{cccc}{w}^{0\times 0}& w^{0\times 1}& \cdots & w^{0\times (M-1)}\\ w^{1\times 0}& w^{1\times 1}& \cdots & w^{1\times (M-1)}\\ ⋮& ⋮& \ddots & ⋮\\ w^{(M-1)\times 0}& w^{(M-1)\times 1}& \cdots & w^{(M-1)\times (M-1)}\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ ⋮\\ B_{M-1}^{\alpha }\end{array}\right),$$

(37)

where $w=e^{2\pi i/M}$ and $B_k^{\alpha }=\exp \left(-{\displaystyle \frac{2\pi ik\alpha }{M}}\right),k=0,1,\cdots ,M-1$. Therefore, Equation (28) can be expressed as follows:

$$\begin{array}{cc}\hfill F_M^{\alpha }\left[f\left(t\right)\right]& =A_0^{\alpha }{f}_0\left(t\right)+A_1^{\alpha }{f}_1\left(t\right)+\cdots +A_{M-1}^{\alpha }{f}_{M-1}\left(t\right)\hfill \\ & =A_0^{\alpha }{F}^{0/M}\left[f\left(t\right)\right]+A_1^{\alpha }{F}^{4/M}\left[f\left(t\right)\right]+\cdots +A_{M-1}^{\alpha }{F}^{4\left(M-1\right)/M}\left[f\left(t\right)\right]\\ & =\left(I,F^{4/M},\cdots ,F^{4\left(M-1\right)/M}\right)\left(\begin{array}{c}{A}_0^{\alpha }\\ A_1^{\alpha }\\ ⋮\\ A_{M-1}^{\alpha }\end{array}\right)f\left(t\right)\\ & ={\displaystyle \frac{1}{M}}\left(I,F^{4/M},\cdots ,F^{4\left(M-1\right)/M}\right)\left(\begin{array}{cccc}{w}^{0\times 0}& w^{0\times 1}& \cdots & w^{0\times (M-1)}\\ w^{1\times 0}& w^{1\times 1}& \cdots & w^{1\times (M-1)}\\ ⋮& ⋮& \ddots & ⋮\\ w^{(M-1)\times 0}& w^{(M-1)\times 1}& \cdots & w^{(M-1)\times (M-1)}\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ ⋮\\ B_{M-1}^{\alpha }\end{array}\right)f\left(t\right)\end{array}$$

(38)

In this way, Zhu et al.’ s m-FRFT is expressed as Equation (38). When $B_k^{\alpha }$ takes different results, we can obtain different extended definitions. Thus, Equation (31) can be rewritten as follows:

$$\begin{array}{cc}\hfill A_l(\alpha ,\mathfrak{M},\mathfrak{N})& ={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}\exp \left\{-\frac{2\pi i}{M}\left[\left(m_{k}M+1\right)\alpha (k+n_{k}M)-lk\right]\right\}}\hfill \\ & ={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}\exp \left(\frac{2\pi ilk}{M}\right)\exp \left\{-\frac{2\pi i}{M}\left[\left(m_{k}M+1\right)\alpha (k+n_{k}M)\right]\right\}}\\ & =IDFT{\left[\exp \left\{-{\displaystyle \frac{2\pi i}{M}}\left[\left(m_{k}M+1\right)\alpha (k+n_{k}M)\right]\right\}\right]}_{k=0,1,2,\cdots \cdots ,M-1}\end{array}$$

(39)

Here, if we make

$$B_k^{\alpha }=\exp \left[-{\displaystyle \frac{2\pi i\left(m_{k}M+1\right)\alpha \left(n_{k}M+k\right)}{M}}\right].$$

(40)

Substituting Equation (40) into Equation (38), we can obtain the MPFRFT. Equation (33) can be rewritten as follows:

$$\begin{array}{cc}\hfill A_l\left(\alpha ,\Re \right)& ={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}\exp \left\{-\frac{2\pi i}{M}\left[\alpha \left(k+r_{k}M\right)-lk\right]\right\}}\hfill \\ & ={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}\exp \left(\frac{2\pi ilk}{M}\right)\exp \left\{-\frac{2\pi i}{M}\left[\alpha \left(k+r_{k}M\right)\right]\right\}}\\ & =IDFT{\left[\exp \left\{-{\displaystyle \frac{2\pi i}{M}}\left[\alpha \left(k+r_{k}M\right)\right]\right\}\right]}_{k=0,1,2,\cdots \cdots ,M-1}\end{array}$$

(41)

Here, if we make

$$B_k^{\alpha }=\exp \left\{-{\displaystyle \frac{2\pi i}{M}}\left[\alpha \left(k+r_{k}M\right)\right]\right\}.$$

(42)

Substituting Equation (42) into Equation (38), we can obtain the modified MPFRFT. Equation (35) can be rewritten as follows:

$$\begin{array}{cc}\hfill A_l\left(\overline{\alpha },\Re \right)& ={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}\exp \left\{\frac{-2\pi i}{M}\left[{\alpha }_k\left(k+r_{k}M\right)-lk\right]\right\}}\hfill \\ & ={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}\exp \left(\frac{2\pi ilk}{M}\right)\exp \left\{-\frac{2\pi i}{M}\left[{\alpha }_k\left(k+r_{k}M\right)\right]\right\}}\\ & =IDFT{\left[\exp \left\{-{\displaystyle \frac{2\pi i}{M}}\left[{\alpha }_k\left(k+r_{k}M\right)\right]\right\}\right]}_{k=0,1,2,\cdots \cdots ,M-1}\end{array}$$

(43)

Here, if we make

$$B_k^{\alpha }=\exp \left[-{\displaystyle \frac{2\pi i{\alpha }_k\left(r_{k}M+k\right)}{M}}\right].$$

(44)

Substituting Equation (42) into Equation (38), we can obtain the VPMPFRFT.

In this way, we propose a new expression that can establish relationships between extended definitions.

3.3. Yeh et al.’s Discrete Fractional Fourier Transform

In 2003, Yeh et al. proposed a new computational method for the discrete FRFT [51], which can be defined as follows:

$$F^{\alpha }={\displaystyle \sum _{n=0}^{N-1}{B}_{n,\alpha }{F}^{nb}},$$

(45)

where $b=4/N$. The weighting coefficients $B_{n,\alpha }$ are computed as follows:

$$B_{n,\alpha }=IDFT{\left[e^{-ik\left(\pi /2\right)\alpha }\right]}_k={\displaystyle \frac{1}{N}}{\displaystyle \frac{1-e^{i\cdot N\cdot \left(\pi /2\right)\left(nb-\alpha \right)}}{1-e^{i\cdot \left(\pi /2\right)\left(nb-\alpha \right)}}}.$$

(46)

Comparing Equation (36), it can be clearly seen that Yeh et al.’s definition is very similar to Zhu et al.’s definition, except that the result of $IDFT[\cdot ]$ is different. There are also extended definitions proposed from Yeh et al.’s definition.

In 2010, Tao et al. modified Yeh et al.’s definition and proposed a fractional-order definition for periodic matrices [52]. If a matrix $L$ satisfies $L^P=I$, then we call $L$ a periodic matrix with period $P$. Then, its fractional-order definition can be expressed as follows:

$$L^{\alpha }={\displaystyle \sum _{n=0}^{N-1}{C}_{n,\alpha }{L}^{n\left(P/N\right)}},$$

(47)

where

$$C_{n,\alpha }={\displaystyle \frac{1}{N}}{\displaystyle \frac{1-e^{j2\pi \left(n-\alpha \right)}}{1-e^{j\left(2\pi /N\right)\left(n-\alpha \right)}}}.$$

(48)

Equation (48) can be further expressed as follows:

$$\begin{array}{cc}\hfill C_{n,\alpha }=& {\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=0}^{N-1}\exp \left[j\left(2\pi /N\right)\left(n-\alpha \right)k\right]}\hfill \\ & ={\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=0}^{N-1}\exp \left[j\left(2\pi /N\right)nk\right]}\exp \left[-j\left(2\pi /N\right)\alpha k\right]\\ & =IDFT{\left\{\exp \left[-j\left(2\pi /N\right)\alpha k\right]\right\}}_{n=0,1,\cdots ,N-1}\end{array}$$

(49)

In this way, Equation (49) is very similar to our results above. In 2016, Kang et al. conducted an in-depth analysis of Tao et al.’s definition and proposed a new extended definition [53], which can be expressed as follows:

$$L^{\overline{\alpha }}={\displaystyle \sum _{n=0}^{N-1}{C}_{n,{\alpha }_n}{L}^{n\left(P/N\right)}},$$

(50)

where

$$C_{n,{\alpha }_n}={\displaystyle \frac{1}{N}}{\displaystyle \frac{1-\exp \left[j2\pi \left(n-{\alpha }_n\right)\right]}{1-\exp \left[j\left(2\pi /N\right)\left(n-{\alpha }_n\right)\right]}}.$$

(51)

Equation (51) can be further expressed as follows:

$$\begin{array}{cc}\hfill C_{n,{\alpha }_n}=& {\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=0}^{N-1}\exp \left[j\left(2\pi /N\right)\left(n-{\alpha }_n\right)k\right]}\hfill \\ & ={\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=0}^{N-1}\exp \left[j\left(2\pi /N\right)nk\right]}\exp \left[-j\left(2\pi /N\right){\alpha }_{n}k\right]\\ & =IDFT{\left\{\exp \left[-j\left(2\pi /N\right){\alpha }_{n}k\right]\right\}}_{n=0,1,\cdots ,N-1}\end{array}$$

(52)

Referring to our work in Section 3.1, we can also provide a new reconstruction, which can be represented as follows:

$$\begin{array}{cc}\hfill L^{\overline{\alpha }}=& {\displaystyle \sum _{n=0}^{N-1}{C}_{n,{\alpha }_n}{L}^{n\left(P/N\right)}}\hfill \\ & =\left(L^0,L^{P/N},\cdots ,L^{P\left(N-1\right)/N}\right)\left(\begin{array}{c}{C}_{0,{\alpha }_n}\\ C_{1,{\alpha }_n}\\ ⋮\\ C_{N-1,{\alpha }_n}\end{array}\right)\\ & ={\displaystyle \frac{1}{N}}\left(L^0,L^{P/M},\cdots ,L^{P\left(N-1\right)/N}\right)\left(\begin{array}{cccc}{w}^{0\times 0}& w^{0\times 1}& \cdots & w^{0\times (N-1)}\\ w^{1\times 0}& w^{1\times 1}& \cdots & w^{1\times (N-1)}\\ ⋮& ⋮& \ddots & ⋮\\ w^{(N-1)\times 0}& w^{(N-1)\times 1}& \cdots & w^{(N-1)\times (N-1)}\end{array}\right)\left(\begin{array}{c}{e}^{\left(-2\pi i{\alpha }_{0}0/N\right)}\\ e^{\left(-2\pi i{\alpha }_{1}1/N\right)}\\ ⋮\\ e^{\left(-2\pi i{\alpha }_{N-1}\left(N-1\right)/N\right)}\end{array}\right).\end{array}$$

(53)

where $w=e^{2\pi i/M}$. Let

$$\{\begin{array}{l}{G}_0=w^{0\times 0}{L}^0+w^{1\times 0}{L}^{{\displaystyle \frac{P}{N}}}+\cdots +w^{\left(N-1\right)\times 0}{L}^{{\displaystyle \frac{P\left(N-1\right)}{N}}}\\ G_1=w^{0\times 1}{L}^0+w^{1\times 1}{L}^{{\displaystyle \frac{P}{N}}}+\cdots +w^{\left(N-1\right)\times 1}{L}^{{\displaystyle \frac{P\left(N-1\right)}{N}}}\\ \hfill ⋮\hfill \\ G_{N-1}=w^{0\times \left(N-1\right)}{L}^0+w^{1\times \left(N-1\right)}{L}^{{\displaystyle \frac{P}{M}}}+\cdots +w^{\left(N-1\right)\times \left(N-1\right)}{L}^{{\displaystyle \frac{P\left(N-1\right)}{N}}}\end{array}$$

(54)

Equation (53) can be expressed as follows:

$$L^{\overline{\alpha }}={\displaystyle \sum _{n=0}^{N-1}{G}_n{B}_n^{{\alpha }_n}},$$

(55)

where $B_n^{{\alpha }_n}=\exp \left(-2\pi i{\alpha }_{n}n/N\right)$. Therefore, we provide a new expression for Kang et al.’s definition. In this way, many extended definitions [53] can be reformulated.

For Equation (55), when the periodic matrix is the discrete FT and $B_n^{{\alpha }_n}=\exp \left[-j\left(\pi /2\right){\alpha }_{n}n\right]$, the result is the MPDFRFT.

For Equation (55), when the periodic matrix is the discrete cosine transform, and $B_n^{{\alpha }_n}=\exp \left(-j\pi {\alpha }_{n}n\right)$, the result is the multiple parameter discrete fractional cosine transform (MPDFRCT).

For Equation (55), when the periodic matrix is discrete sine transform, and $B_n^{{\alpha }_n}=\exp \left(-j\pi {\alpha }_{n}n\right)$, the result is the multiple parameter discrete fractional sine transforms (MPDFRST).

For Equation (55), when the periodic matrix is discrete Hartley transform, and $B_n^{{\alpha }_n}=\exp \left(-j\pi {\alpha }_{n}n\right)$, the result is multiple parameter discrete fractional Hartley transforms (MPDFRHT).

For Equation (55), when the periodic matrix is discrete Hadamard transform, and $B_n^{{\alpha }_n}=\exp \left(-j\pi {\alpha }_{n}n\right)$, the result is multiple parameter discrete fractional Hadamard transforms (MPDFRHT).

The definitions of Yeh et al. and Tao et al. can also serve as special cases for the new expression. The new expression proposed in this section is very similar to that in Section 3.2, so we provide a unified framework for the extended definitions.

4. Unified Framework

In this section, we will establish a unified framework with the help of the analysis of various extended definitions in Section 3. Let $T$ denote a periodic matrix (strictly speaking, the matrix should be symmetrical) and $T^P=I$ ( $P$ is the period of the matrix $T$, and $I$ denotes the identity matrix). The unified framework can be expressed as follows:

$$T^{\overline{\alpha }}={\displaystyle \frac{1}{M}}\left(T^0,T^{P/M},\cdots ,T^{P\left(M-1\right)/M}\right)\left(\begin{array}{cccc}{w}^{0\times 0}& w^{0\times 1}& \cdots & w^{0\times (M-1)}\\ w^{1\times 0}& w^{1\times 1}& \cdots & w^{1\times (M-1)}\\ ⋮& ⋮& \ddots & ⋮\\ w^{(M-1)\times 0}& w^{(M-1)\times 1}& \cdots & w^{(M-1)\times (M-1)}\end{array}\right)\left(\begin{array}{c}{B}_0^{{\alpha }_0}\\ B_1^{{\alpha }_1}\\ ⋮\\ B_{M-1}^{{\alpha }_{M-1}}\end{array}\right),$$

(56)

where $w=e^{2\pi i/M}$, and when $B_k^{{\alpha }_k}$ takes different values, we can obtain different definitions.

For example, in the definition of Shih, the periodic matrix is the FT ( $T=F$), and the period of the FT matrix $P=4$. When $M=4$, ${\alpha }_0={\alpha }_1={\alpha }_2={\alpha }_3=\alpha$ and $B_k^{{\alpha }_k}=\exp \left(-{\displaystyle \frac{2\pi i\alpha k}{4}}\right)$; $k=0,1,2,3$, and we obtain the WFRFT of Shih.

Furthermore, if $M$ is an integer greater than 4, ${\alpha }_0={\alpha }_1=\cdots ={\alpha }_{M-1}=\alpha$ and $B_k^{{\alpha }_k}=\exp \left(-{\displaystyle \frac{2\pi i\alpha k}{M}}\right)$; $k=0,1,2,\cdots ,M-1$, and we obtain Zhu et al.‘s m-FRFT [14]. In Zhu et al.’s definition, when $B_k^{{\alpha }_k}=\exp \left[-{\displaystyle \frac{2\pi i\left(m_{k}M+1\right)\alpha \left(n_{k}M+k\right)}{M}}\right]$, we obtain Tao et al.’s MPFRFT [16]. When $B_k^{{\alpha }_k}=\exp \left\{-{\displaystyle \frac{2\pi i}{M}}\left[\alpha \left(r_{k}M+k\right)\right]\right\}$, we obtain Ran et al.’s modified MPFRFT [17]. When $B_k^{{\alpha }_k}=\exp \left[-{\displaystyle \frac{2\pi i{\alpha }_k\left(r_{k}M+k\right)}{M}}\right]$, we obtain Ran et al.’s VPMPFRFT [19].

In Equation (56), when $T$ is the FT matrix, $M$ is an integer greater than 4, ${\alpha }_0={\alpha }_1=\cdots ={\alpha }_{M-1}=\alpha$ and $B_k^{{\alpha }_k}=\exp \left[-jk\left(\pi /2\right)\alpha \right]$. Therefore, we obtain the discrete FRFT of Yeh et al. in Section 3.3. When T is a periodic matrix, ${\alpha }_0={\alpha }_1=\cdots ={\alpha }_{M-1}=\alpha$, and $B_k^{{\alpha }_k}=\exp \left[-j\left(2\pi /M\right)\alpha k\right]$, we can obtain the definition of Tao et al. If $B_k^{{\alpha }_k}=\exp \left[-j\left(2\pi /M\right){\alpha }_{k}k\right]$, then we obtain the definition of Kang et al.

Therefore, the WFRFT and extended definitions in Section 3.1 and Section 3.2 are special cases of the unified framework.

5. Discussion

In this section, we will use the unified framework to provide a detailed proof of the deficiencies of extended definitions as well as discuss and analyze their impact on the algorithm.

5.1. Deficiency of the Extended Definition (I)

Firstly, we determine that the extended definition based on the WFRFT has only four effective weighting terms.

In Section 4, in our proposed unified framework, when $\overline{\alpha }=\alpha$, $M=4$, $T=F$, $B_k^{\alpha }=\exp \left(-{\displaystyle \frac{2\pi i\alpha k}{4}}\right)$, and $k=0,1,2,3$, the transform is considered a WFRFT. When $M$ is an integer greater than 4 and $B_k^{\alpha }=\exp \left(-{\displaystyle \frac{2\pi ik\alpha }{M}}\right),k=0,1,\cdots ,M-1$, the extended definition applies. We take $M=5$ as an example for analysis, and its expression can be written as follows:

$$F_5^{\alpha }\left[f\left(x\right)\right]={\displaystyle \frac{1}{5}}\left(\begin{array}{cccc}{F}^0& F^{{\displaystyle \frac{4}{5}}}& \cdots & F^{{\displaystyle \frac{16}{5}}}\end{array}\right)\left(\begin{array}{cccc}{w}^{0\times 0}& w^{0\times 1}& \cdots & w^{0\times 4}\\ w^{1\times 0}& w^{1\times 1}& \cdots & w^{1\times 4}\\ ⋮& ⋮& \ddots & ⋮\\ w^{4\times 0}& w^{4\times 1}& \cdots & w^{4\times 4}\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ ⋮\\ B_4^{\alpha }\end{array}\right)f\left(x\right),$$

(57)

where $w=e^{2\pi i/5}$ and $B_k^{\alpha }=\exp \left(-{\displaystyle \frac{2\pi ik\alpha }{5}}\right);k=0,1,\cdots ,4$. Equation (57) can also be written as follows:

$$\begin{array}{cc}\hfill F^{\alpha }& ={\displaystyle \frac{1}{5}}\left(\begin{array}{ccccc}{Y}_0& Y_1& Y_2& Y_3& Y_4\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\\ B_4^{\alpha }\end{array}\right)\hfill \\ & ={\displaystyle \frac{1}{5}}\left(Y_0{B}_0^{\alpha }+Y_1{B}_1^{\alpha }+Y_2{B}_2^{\alpha }+Y_3{B}_3^{\alpha }+Y_4{B}_4^{\alpha }\right)\end{array}$$

(58)

Let

$$\{\begin{array}{l}{Y}_0=w^{0\times 0}{F}^0+w^{1\times 0}{F}^{{\displaystyle \frac{4}{5}}}+\cdots +w^{4\times 0}{F}^{{\displaystyle \frac{16}{5}}}\\ Y_1=w^{0\times 1}{F}^0+w^{1\times 1}{F}^{{\displaystyle \frac{4}{5}}}+\cdots +w^{4\times 1}{F}^{{\displaystyle \frac{16}{5}}}\\ Y_2=w^{0\times 2}{F}^0+w^{1\times 2}{F}^{{\displaystyle \frac{4}{5}}}+\cdots +w^{4\times 2}{F}^{{\displaystyle \frac{16}{5}}}\\ Y_3=w^{0\times 3}{F}^0+w^{1\times 3}{F}^{{\displaystyle \frac{4}{5}}}+\cdots +w^{4\times 3}{F}^{{\displaystyle \frac{16}{5}}}\\ Y_4=w^{0\times 4}{F}^0+w^{1\times 4}{F}^{{\displaystyle \frac{4}{5}}}+\cdots +w^{4\times 4}{F}^{{\displaystyle \frac{16}{5}}}\end{array}$$

(59)

From Equation (20), we know that

$$\{\begin{array}{l}{F}^0={\displaystyle \sum _{k\in E_1}{\left(1\right)}^0{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^0{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^0{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^0{v}_k{v}_k^T},\\ F^{4/5}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{4/5}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{4/5}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{4/5}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{4/5}{v}_k{v}_k^T},\\ F^{8/5}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{8/5}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{8/5}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{8/5}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{8/5}{v}_k{v}_k^T},\\ F^{12/5}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{12/5}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{12/5}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{12/5}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{12/5}{v}_k{v}_k^T},\\ F^{16/5}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{16/5}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{16/5}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{16/5}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{16/5}{v}_k{v}_k^T}.\end{array}$$

(60)

Equation (60) is substituted into Equation (59), and we obtain

$$\{\begin{array}{l}{Y}_0={\displaystyle \sum _{k\in E_1}5\cdot v_k{v}_k^T}\\ Y_1=0\\ Y_2={\displaystyle \sum _{k\in E_4}5\cdot v_k{v}_k^T}\\ Y_3={\displaystyle \sum _{k\in E_3}5\cdot v_k{v}_k^T}\\ Y_4={\displaystyle \sum _{k\in E_2}5\cdot v_k{v}_k^T}\end{array}$$

(61)

Thus, Equation (58) is written as follows:

$$\begin{array}{cc}\hfill F^{\alpha }& ={\displaystyle \frac{1}{5}}\left(\begin{array}{ccccc}{Y}_0& Y_1& Y_2& Y_3& Y_4\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\\ B_4^{\alpha }\end{array}\right)\hfill \\ & ={\displaystyle \frac{1}{5}}\left(Y_0{B}_0^{\alpha }+Y_1{B}_1^{\alpha }+Y_2{B}_2^{\alpha }+Y_3{B}_3^{\alpha }+Y_4{B}_4^{\alpha }\right)\\ & ={\displaystyle \sum _{k\in E_1}{B}_0^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{B}_2^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{B}_3^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{B}_4^{\alpha }{v}_k{v}_k^T}\end{array}$$

(62)

In Equation (62), there are only four effective weighting terms. Next, we will verify the extended definition of $M=6$, and based on the previous results, we can directly obtain Equation (63),

$$\{\begin{array}{l}{Y}_0=w^{0\times 0}{F}^0+w^{1\times 0}{F}^{{\displaystyle \frac{4}{6}}}+\cdots +w^{5\times 0}{F}^{{\displaystyle \frac{20}{6}}}\\ Y_1=w^{0\times 1}{F}^0+w^{1\times 1}{F}^{{\displaystyle \frac{4}{6}}}+\cdots +w^{5\times 1}{F}^{{\displaystyle \frac{20}{6}}}\\ Y_2=w^{0\times 2}{F}^0+w^{1\times 2}{F}^{{\displaystyle \frac{4}{6}}}+\cdots +w^{5\times 2}{F}^{{\displaystyle \frac{20}{6}}}\\ Y_3=w^{0\times 3}{F}^0+w^{1\times 3}{F}^{{\displaystyle \frac{4}{6}}}+\cdots +w^{5\times 3}{F}^{{\displaystyle \frac{20}{6}}}\\ Y_4=w^{0\times 4}{F}^0+w^{1\times 4}{F}^{{\displaystyle \frac{4}{6}}}+\cdots +w^{5\times 4}{F}^{{\displaystyle \frac{20}{6}}}\\ Y_5=w^{0\times 5}{F}^0+w^{1\times 5}{F}^{{\displaystyle \frac{4}{6}}}+\cdots +w^{5\times 5}{F}^{{\displaystyle \frac{20}{6}}}\end{array}$$

(63)

From Equation (20), we know that

$$\{\begin{array}{l}{F}^0={\displaystyle \sum _{k\in E_1}{\left(1\right)}^0{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^0{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^0{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^0{v}_k{v}_k^T},\\ F^{4/6}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{4/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{4/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{4/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{4/6}{v}_k{v}_k^T},\\ F^{8/6}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{8/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{8/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{8/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{8/6}{v}_k{v}_k^T},\\ F^{12/6}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{12/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{12/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{12/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{12/6}{v}_k{v}_k^T},\\ F^{16/6}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{16/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{16/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{16/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{16/6}{v}_k{v}_k^T},\\ F^{20/6}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{20/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(-i\right)}^{20/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{20/6}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(i\right)}^{20/6}{v}_k{v}_k^T}.\end{array}$$

(64)

Substituting Equation (64) into Equation (63), we can obtain

$$\{\begin{array}{l}{Y}_0={\displaystyle \sum _{k\in E_1}6\cdot v_k{v}_k^T}\\ Y_1=0\\ Y_2=0\\ Y_3={\displaystyle \sum _{k\in E_4}6\cdot v_k{v}_k^T}\\ Y_4={\displaystyle \sum _{k\in E_3}6\cdot v_k{v}_k^T}\\ Y_5={\displaystyle \sum _{k\in E_2}6\cdot v_k{v}_k^T}\end{array}$$

(65)

In this way, the extension definition is written as follows:

$$\begin{array}{cc}\hfill F^{\alpha }& ={\displaystyle \frac{1}{6}}\left(\begin{array}{cccccc}{Y}_0& Y_1& Y_2& Y_3& Y_4& Y_5\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ B_2^{\alpha }\\ B_3^{\alpha }\\ B_4^{\alpha }\\ B_5^{\alpha }\end{array}\right)\hfill \\ & ={\displaystyle \frac{1}{6}}\left(Y_0{B}_0^{\alpha }+Y_1{B}_1^{\alpha }+Y_2{B}_2^{\alpha }+Y_3{B}_3^{\alpha }+Y_4{B}_4^{\alpha }+Y_5{B}_5^{\alpha }\right)\\ & ={\displaystyle \sum _{k\in E_1}{B}_0^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{B}_3^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{B}_4^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{B}_5^{\alpha }{v}_k{v}_k^T}\end{array}$$

(66)

Equation (66) indicates that there are only four effective weighting terms.

Furthermore, we provide proof of the extended definition in its general form. Its expression can be written as follows:

$$\begin{array}{cc}\hfill F_M^{\alpha }\left[f\left(t\right)\right]& ={\displaystyle \frac{1}{M}}\left(I,F^{{\displaystyle \frac{4}{M}}},\cdots ,F^{{\displaystyle \frac{4\left(M-1\right)}{M}}}\right)\left(\begin{array}{cccc}{w}^{0\times 0}& w^{0\times 1}& \cdots & w^{0\times (M-1)}\\ w^{1\times 0}& w^{1\times 1}& \cdots & w^{1\times (M-1)}\\ ⋮& ⋮& \ddots & ⋮\\ w^{(M-1)\times 0}& w^{(M-1)\times 1}& \cdots & w^{(M-1)\times (M-1)}\end{array}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ ⋮\\ B_{M-1}^{\alpha }\end{array}\right)f\left(t\right).\hfill \\ & ={\displaystyle \frac{1}{M}}\left(Y_0,Y_1,\cdots ,Y_{M-1}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ ⋮\\ B_{M-1}^{\alpha }\end{array}\right)f\left(t\right)\\ & ={\displaystyle \frac{1}{M}}{\displaystyle \sum _{k=0}^{M-1}{Y}_k}{B}_k^{\alpha }f\left(t\right)\end{array}$$

(67)

where $w=e^{2\pi i/M}$ and $B_k^{\alpha }=\exp \left(-{\displaystyle \frac{2\pi ik\alpha }{M}}\right);k=0,1,\cdots ,M-1$.

$$\{\begin{array}{l}{Y}_0=w^{0\times 0}I+w^{1\times 0}{F}^{{\displaystyle \frac{4}{M}}}+\cdots +w^{\left(M-1\right)\times 0}{F}^{{\displaystyle \frac{4\left(M-1\right)}{M}}}\\ Y_1=w^{0\times 1}I+w^{1\times 1}{F}^{{\displaystyle \frac{4}{M}}}+\cdots +w^{\left(M-1\right)\times 1}{F}^{{\displaystyle \frac{4\left(M-1\right)}{M}}}\\ Y_2=w^{0\times 2}I+w^{1\times 2}{F}^{{\displaystyle \frac{4}{M}}}+\cdots +w^{\left(M-1\right)\times 2}{F}^{{\displaystyle \frac{4\left(M-1\right)}{M}}}\\ \hfill ⋮\hfill \\ Y_{M-1}=w^{0\times \left(M-1\right)}I+w^{1\times \left(M-1\right)}{F}^{{\displaystyle \frac{4}{M}}}+\cdots +w^{\left(M-1\right)\times \left(M-1\right)}{F}^{{\displaystyle \frac{4\left(M-1\right)}{M}}}\end{array}$$

(68)

and

$$\{\begin{array}{l}{F}^0={\displaystyle \sum _{k\in E_1}{\left(1\right)}^0{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^0{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^0{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^0{v}_k{v}_k^T},\\ F^{4/M}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{4/M}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{4/M}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{4/M}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{4/M}{v}_k{v}_k^T},\\ F^{8/M}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{8/M}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{8/M}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{8/M}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{8/M}{v}_k{v}_k^T},\\ \hfill ⋮\hfill \\ F^{4\left(M-2\right)/M}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{4\left(M-2\right)/M}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{4\left(M-2\right)/M}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{4\left(M-2\right)/M}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{4\left(M-2\right)/M}{v}_k{v}_k^T},\\ F^{4\left(M-1\right)/M}={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{4\left(M-1\right)/M}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(i\right)}^{4\left(M-1\right)/M}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{\left(-1\right)}^{4\left(M-1\right)/M}{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{\left(-i\right)}^{4\left(M-1\right)/M}{v}_k{v}_k^T}.\end{array}$$

(69)

Equations (68) and (69) are simplified as follows:

$$Y_k=w^{0\times k}{F}^0+w^{1\times k}{F}^{{\displaystyle \frac{4}{M}}}+\cdots +w^{\left(M-1\right)\times k}{F}^{{\displaystyle \frac{4\left(M-1\right)}{M}}}$$

(70)

and

$$F^{4l/M}={\displaystyle \sum _{j\in E_1}{\left(1\right)}^{4l/M}{v}_j{v}_j^T}+{\displaystyle \sum _{j\in E_2}{\left(i\right)}^{4l/M}{v}_j{v}_j^T}+{\displaystyle \sum _{j\in E_3}{\left(-1\right)}^{4l/M}{v}_j{v}_j^T}+{\displaystyle \sum _{j\in E_4}{\left(-i\right)}^{4l/M}{v}_j{v}_j^T}.$$

(71)

Equation (71) can be further written as follows:

$$F^{4l/M}=\left(\begin{array}{cccc}1& 1& 1& 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\left(1\right)}^{4l/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_2}{\left(i\right)}^{4l/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_3}{\left(-1\right)}^{4l/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_4}{\left(-i\right)}^{4l/M}{v}_j{v}_j^T}\end{array}\right).$$

(72)

Substituting Equation (72) into Equation (70), we obtain

$$\begin{array}{cc}\hfill Y_k=& w^{0\times k}{F}^0+w^{1\times k}{F}^{{\displaystyle \frac{4}{M}}}+\cdots +w^{\left(M-1\right)\times k}{F}^{{\displaystyle \frac{4\left(M-1\right)}{M}}}\hfill \\ & =\left(\begin{array}{cccc}1& 1& 1& 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\left(1\right)}^0{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_2}{\left(i\right)}^0{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_3}{\left(-1\right)}^0{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_4}{\left(-i\right)}^0{v}_j{v}_j^T}\end{array}\right)w^{0\times k}+\left(\begin{array}{cccc}1& 1& 1& 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\left(1\right)}^{4/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_2}{\left(i\right)}^{4/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_3}{\left(-1\right)}^{4/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_4}{\left(-i\right)}^{4/M}{v}_j{v}_j^T}\end{array}\right)w^{1\times k}+\cdots +\left(\begin{array}{cccc}1& 1& 1& 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\left(1\right)}^{4\left(M-1\right)/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_2}{\left(i\right)}^{4\left(M-1\right)/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_3}{\left(-1\right)}^{4\left(M-1\right)/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_4}{\left(-i\right)}^{4\left(M-1\right)/M}{v}_j{v}_j^T}\end{array}\right)w^{\left(M-1\right)\times k}\\ & =\left(\begin{array}{cccc}1& 1& 1& 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\displaystyle \sum _{l=0}^{M-1}{\left(1\right)}^{4l/M}{w}^{l\times k}{v}_j{v}_j^T}}\\ {\displaystyle \sum _{j\in E_2}{\displaystyle \sum _{l=0}^{M-1}{\left(i\right)}^{4l/M}{w}^{l\times k}{v}_j{v}_j^T}}\\ {\displaystyle \sum _{j\in E_3}{\displaystyle \sum _{l=0}^{M-1}{\left(-1\right)}^{4l/M}{w}^{l\times k}{v}_j{v}_j^T}}\\ {\displaystyle \sum _{j\in E_4}{\displaystyle \sum _{l=0}^{M-1}{\left(-i\right)}^{4l/M}{w}^{l\times k}{v}_j{v}_j^T}}\end{array}\right)\\ & =\left(\begin{array}{cccc}1& 1& 1& 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{l=0}^{M-1}{\left(1\right)}^{4l/M}{w}^{l\times k}}\cdot {\displaystyle \sum _{j\in E_1}{v}_j{v}_j^T}\\ {\displaystyle \sum _{l=0}^{M-1}{\left(i\right)}^{4l/M}{w}^{l\times k}}\cdot {\displaystyle \sum _{j\in E_2}{v}_j{v}_j^T}\\ {\displaystyle \sum _{l=0}^{M-1}{\left(-1\right)}^{4l/M}{w}^{l\times k}}\cdot {\displaystyle \sum _{j\in E_3}{v}_j{v}_j^T}\\ {\displaystyle \sum _{l=0}^{M-1}{\left(-i\right)}^{4l/M}{w}^{l\times k}}\cdot {\displaystyle \sum _{j\in E_4}{v}_j{v}_j^T}\end{array}\right)\end{array}$$

(73)

The next proof is to represent the eigenvalues $\left(\begin{array}{cccc}1& i& -1& -i\end{array}\right)$ using $e^{h\pi i/2}$, $h=0,1,2,3$. In this way, we can calculate the following equation,

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=0}^{M-1}{\left(e^{h\pi i/2}\right)}^{4l/M}{w}^{l\times k}}& ={\displaystyle \sum _{l=0}^{M-1}{e}^{2\pi ilh/M}{e}^{2\pi ilk/M}}\hfill \\ & ={\displaystyle \sum _{l=0}^{M-1}{e}^{2\pi il\left(k+h\right)/M}}\end{array}$$

(74)

We thus obtain

$${\displaystyle \sum _{l=0}^{M-1}{e}^{2\pi il\left(k+h\right)/M}}=\{\begin{array}{l}M, if h =0,k=0\\ M, if h=1,k=M-1\\ M, if h =2,k=M-2\\ M, if h =3,k=M-3\\ 0, others\end{array}$$

(75)

Equation (68) can be written as follows:

$$\{\begin{array}{l}{Y}_0={\displaystyle \sum _{j\in E_1}M\cdot v_j{v}_j^T},\\ Y_1=0,\\ ⋮\\ Y_{M-4}=0,\\ Y_{M-3}={\displaystyle \sum _{j\in E_4}M\cdot v_j{v}_j^T,}\\ Y_{M-2}={\displaystyle \sum _{j\in E_3}M\cdot v_j{v}_j^T,}\\ Y_{M-1}={\displaystyle \sum _{j\in E_2}M\cdot v_j{v}_j^T.}\end{array}$$

(76)

The expression of Equation (67) is as follows:

$$\begin{array}{cc}\hfill F_M^{\alpha }\left[f\left(t\right)\right]& ={\displaystyle \frac{1}{M}}\left(Y_0,Y_1,\cdots ,Y_{M-1}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ \hfill ⋮\hfill \\ B_{M-1}^{\alpha }\end{array}\right)f\left(t\right)\hfill \\ & ={\displaystyle \sum _{k\in E_1}{B}_0^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_4}{B}_{M-3}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_3}{B}_{M-2}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{B}_{M-1}^{\alpha }{v}_k{v}_k^T}\end{array}$$

(77)

We can easily find that there are only four effective weighting terms in the extended definition.

This indicates that the extended definitions based on the WFRFT are deficient, as there are only four weighting terms in the extended definitions.

5.2. Deficiency of the Extended Definition (II)

In this section, we will analyze the extended definition of the weighted fractional Hartley transform as an example. The discrete Hartley transform (DHT) is

$$H={\displaystyle \frac{1}{\sqrt{N}}}\left[\cos \left({\displaystyle \frac{2\pi mn}{N}}\right)+\sin \left({\displaystyle \frac{2\pi mn}{N}}\right)\right].$$

(78)

Its fractional-order transform can be expressed as follows:

$$H^{\alpha }=V{D}^{\alpha }{V}^H.$$

(79)

Because the eigenvalues of the Hartley transform are 1 and −1, Equation (79) can also be expressed as follows:

$$\begin{array}{cc}\hfill H^{\alpha }& ={\displaystyle \sum _{k\in E_1}{\left(1\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{\left(-1\right)}^{\alpha }{v}_k{v}_k^T}\hfill \\ & =\left(\begin{array}{cc}1& 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{k\in E_1}{\left(1\right)}^{\alpha }{v}_k{v}_k^T}\\ {\displaystyle \sum _{k\in E_2}{\left(-1\right)}^{\alpha }{v}_k{v}_k^T}\end{array}\right)\end{array}$$

(80)

The extended definition [53] based on the weighted fractional Hartley transform can be expressed as follows:

$$T^{\alpha }={\displaystyle \frac{1}{M}}\left(I,H^{{\displaystyle \frac{2}{M}}},\cdots ,H^{{\displaystyle \frac{2\left(M-1\right)}{M}}}\right)\left(\begin{array}{cccc}{w}^{0\times 0}& w^{0\times 1}& \cdots & w^{0\times (M-1)}\\ w^{1\times 0}& w^{1\times 1}& \cdots & w^{1\times (M-1)}\\ ⋮& ⋮& \ddots & ⋮\\ w^{(M-1)\times 0}& w^{(M-1)\times 1}& \cdots & w^{(M-1)\times (M-1)}\end{array}\right)\left(\begin{array}{c}{B}_0^{{\alpha }_0}\\ B_1^{{\alpha }_1}\\ ⋮\\ B_{M-1}^{{\alpha }_{M-1}}\end{array}\right)f\left(t\right),$$

(81)

where $w=e^{2\pi i/M}$ and $B_k^{{\alpha }_k}=\exp \left(-j\pi {\alpha }_{k}k\right)$, $k=0,1,2,\cdots ,M-1$. Let

$$\{\begin{array}{l}{Y}_0=w^{0\times 0}I+w^{1\times 0}{H}^{{\displaystyle \frac{2}{M}}}+\cdots +w^{\left(M-1\right)\times 0}{H}^{{\displaystyle \frac{2\left(M-1\right)}{M}}}\\ Y_1=w^{0\times 1}I+w^{1\times 1}{H}^{{\displaystyle \frac{2}{M}}}+\cdots +w^{\left(M-1\right)\times 1}{H}^{{\displaystyle \frac{2\left(M-1\right)}{M}}}\\ Y_2=w^{0\times 2}I+w^{1\times 2}{H}^{{\displaystyle \frac{2}{M}}}+\cdots +w^{\left(M-1\right)\times 2}{H}^{{\displaystyle \frac{2\left(M-1\right)}{M}}}\\ ⋮\\ Y_{M-1}=w^{0\times \left(M-1\right)}I+w^{1\times \left(M-1\right)}{H}^{{\displaystyle \frac{2}{M}}}+\cdots +w^{\left(M-1\right)\times \left(M-1\right)}{H}^{{\displaystyle \frac{2\left(M-1\right)}{M}}}\end{array}$$

(82)

Equation (82) can be abbreviated as follows:

$$Y_k=w^{0\times k}I+w^{1\times k}{H}^{{\displaystyle \frac{2}{M}}}+\cdots +w^{\left(M-1\right)\times k}{H}^{{\displaystyle \frac{2\left(M-1\right)}{M}}}.$$

(83)

Substituting Equation (80) into Equation (83), we can obtain

$$\begin{array}{cc}\hfill Y_k& =w^{0\times k}I+w^{1\times k}{H}^{{\displaystyle \frac{2}{M}}}+\cdots +w^{\left(M-1\right)\times k}{H}^{{\displaystyle \frac{2\left(M-1\right)}{M}}}\hfill \\ & =\left(\begin{array}{cc}1& 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\left(1\right)}^0{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_2}{\left(-1\right)}^0{v}_j{v}_j^T}\end{array}\right)w^{0\times k}+\left(\begin{array}{cc}1& 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\left(1\right)}^{2/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_2}{\left(-1\right)}^{2/M}{v}_j{v}_j^T}\end{array}\right)w^{1\times k}+\cdots +\left(\begin{array}{cc}1& 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\left(1\right)}^{2\left(M-1\right)/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_2}{\left(-1\right)}^{2\left(M-1\right)/M}{v}_j{v}_j^T}\end{array}\right)w^{\left(M-1\right)\times k}\\ & =\left(\begin{array}{cc}1& 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\displaystyle \sum _{l=0}^{M-1}{\left(1\right)}^{2l/M}{w}^{l\times k}{v}_j{v}_j^T}}\\ {\displaystyle \sum _{j\in E_2}{\displaystyle \sum _{l=0}^{M-1}{\left(-1\right)}^{2l/M}{w}^{l\times k}{v}_j{v}_j^T}}\end{array}\right)\\ & =\left(\begin{array}{cc}1& 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{l=0}^{M-1}{\left(1\right)}^{2l/M}{w}^{l\times k}}\cdot {\displaystyle \sum _{j\in E_1}{v}_j{v}_j^T}\\ {\displaystyle \sum _{l=0}^{M-1}{\left(-1\right)}^{2l/M}{w}^{l\times k}}\cdot {\displaystyle \sum _{j\in E_2}{v}_j{v}_j^T}\end{array}\right)\end{array}$$

(84)

The eigenvalues of the Hartley transform are 1 and −1, which we can represent as $e^{h\pi i}$ with $h=0,1$. Therefore, we can obtain

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=0}^{M-1}{\left(e^{h\pi i}\right)}^{2l/M}{w}^{l\times k}}& ={\displaystyle \sum _{l=0}^{M-1}{e}^{2\pi ilh/M}{e}^{2\pi ilk/M}}\hfill \\ & ={\displaystyle \sum _{l=0}^{M-1}{e}^{2\pi il\left(k+h\right)/M}}\end{array}$$

(85)

then

$${\displaystyle \sum _{l=0}^{M-1}{e}^{2\pi il\left(k+h\right)/M}}=\{\begin{array}{l}M, if h =0,k=0\\ M, if h =1,k=M-1\\ 0, others\end{array}$$

(86)

Thus, Equation (82) is written as follows:

$$\{\begin{array}{l}{Y}_0={\displaystyle \sum _{j\in E_1}M\cdot v_j{v}_j^T},\\ Y_1=0,\\ ⋮\\ Y_{M-2}=0,\\ Y_{M-1}={\displaystyle \sum _{j\in E_2}M\cdot v_j{v}_j^T.}\end{array}$$

(87)

The extended definition based on the weighted fractional Hartley transform can be expressed as follows:

$$\begin{array}{cc}\hfill H_M^{\alpha }\left[f\left(t\right)\right]& ={\displaystyle \frac{1}{M}}\left(Y_0,Y_1,\cdots ,Y_{M-1}\right)\left(\begin{array}{c}{B}_0^{\alpha }\\ B_1^{\alpha }\\ ⋮\\ B_{M-1}^{\alpha }\end{array}\right)f\left(t\right)\hfill \\ & ={\displaystyle \sum _{k\in E_1}{B}_0^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{k\in E_2}{B}_{M-1}^{\alpha }{v}_k{v}_k^T}\end{array}$$

(88)

The extension definition has only two weighting terms.

In this section, the extended definition of the weighted fractional Hartley transform is analyzed, and the results show that there are only two effective weighting terms. In Section 3.3, other extended definitions also have such deficiencies.

5.3. Deficiency of the Extended Definition (III)

The above analysis indicates that the extended definition based on the WFRFT has four weighting terms, while the extended definition based on the weighted fractional Hartley transform has two weighting terms. We speculate that this is related to the period of the matrix, so in this section we analyze the extended definition of general periodic matrices.

The $N\times N$ matrix $T$ is a periodic matrix satisfying $T^P=I$, and its eigen decomposition form is $T=VD{V}^H$. The eigenvalues of the periodic matrix $T$ satisfy ${\lambda }^P=1$, and these $P$ eigenvalues can be expressed as $\lambda =\left\{e^{2\pi i0/P},e^{2\pi i1/P},\cdots ,e^{2\pi i\left(P-1\right)/P}\right\}$. We can obtain

$$T={\displaystyle \sum _{j\in E_1}{e}^{2\pi i0/P}{v}_k{v}_k^T}+{\displaystyle \sum _{j\in E_2}{e}^{2\pi i1/P}{v}_k{v}_k^T}+\cdots +{\displaystyle \sum _{j\in E_P}{e}^{2\pi i\left(P-1\right)/P}{v}_k{v}_k^T}$$

(89)

Its fractional-order transform can be represented as follows:

$$\begin{array}{cc}\hfill T^{\alpha }=& {\displaystyle \sum _{j\in E_1}{\left(e^{2\pi i0/P}\right)}^{\alpha }{v}_k{v}_k^T}+{\displaystyle \sum _{j\in E_2}{\left(e^{2\pi i1/P}\right)}^{\alpha }{v}_k{v}_k^T}+\cdots +{\displaystyle \sum _{j\in E_P}{\left(e^{2\pi i\left(P-1\right)/P}\right)}^{\alpha }{v}_k{v}_k^T}\hfill \\ & =\left(\begin{array}{cccc}1& 1& \cdots & 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\left(e^{2\pi i0/P}\right)}^{\alpha }{v}_k{v}_k^T}\\ {\displaystyle \sum _{j\in E_2}{\left(e^{2\pi i1/P}\right)}^{\alpha }{v}_k{v}_k^T}\\ ⋮\\ {\displaystyle \sum _{j\in E_P}{\left(e^{2\pi i\left(P-1\right)/P}\right)}^{\alpha }{v}_k{v}_k^T}\end{array}\right)\end{array}$$

(90)

Its extended definition can be expressed as follows:

$$T^{\overline{\alpha }}={\displaystyle \frac{1}{M}}\left(T^0,T^{P/M},\cdots ,T^{P\left(M-1\right)/M}\right)\left(\begin{array}{cccc}{w}^{0\times 0}& w^{0\times 1}& \cdots & w^{0\times (M-1)}\\ w^{1\times 0}& w^{1\times 1}& \cdots & w^{1\times (M-1)}\\ ⋮& ⋮& \ddots & ⋮\\ w^{(M-1)\times 0}& w^{(M-1)\times 1}& \cdots & w^{(M-1)\times (M-1)}\end{array}\right)\left(\begin{array}{c}{B}_0^{{\alpha }_0}\\ B_1^{{\alpha }_1}\\ ⋮\\ B_{M-1}^{{\alpha }_{M-1}}\end{array}\right)$$

(91)

From the unified framework, we can know that $B_k^{{\alpha }_k}$ can take different results. Let

$$\{\begin{array}{l}{Y}_0=w^{0\times 0}{T}^0+w^{1\times 0}{T}^{{\displaystyle \frac{P}{M}}}+\cdots +w^{\left(M-1\right)\times 0}{T}^{{\displaystyle \frac{P\left(M-1\right)}{M}}}\\ Y_1=w^{0\times 1}{T}^0+w^{1\times 1}{T}^{{\displaystyle \frac{P}{M}}}+\cdots +w^{\left(M-1\right)\times 1}{T}^{{\displaystyle \frac{P\left(M-1\right)}{M}}}\\ Y_2=w^{0\times 2}{T}^0+w^{1\times 2}{T}^{{\displaystyle \frac{P}{M}}}+\cdots +w^{\left(M-1\right)\times 2}{T}^{{\displaystyle \frac{P\left(M-1\right)}{M}}}\\ ⋮\\ Y_{M-1}=w^{0\times \left(M-1\right)}{T}^0+w^{1\times \left(M-1\right)}{T}^{{\displaystyle \frac{P}{M}}}+\cdots +w^{\left(M-1\right)\times \left(M-1\right)}{T}^{{\displaystyle \frac{P\left(M-1\right)}{M}}}\end{array}$$

(92)

Equation (92) can be simplified as follows:

$$Y_k=w^{0\times k}{T}^0+w^{1\times k}{T}^{{\displaystyle \frac{P}{M}}}+\cdots +w^{\left(M-1\right)\times k}{T}^{{\displaystyle \frac{P\left(M-1\right)}{M}}}.$$

(93)

Substituting Equation (90) into Equation (93), we can obtain

$$\begin{array}{cc}\hfill Y_k& =w^{0\times k}{T}^0+w^{1\times k}{T}^{{\displaystyle \frac{P}{M}}}+\cdots +w^{\left(M-1\right)\times k}{T}^{{\displaystyle \frac{P\left(M-1\right)}{M}}}\hfill \\ & =\left(\begin{array}{cccc}1& 1& \cdots & 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\left(e^{2\pi i0/P}\right)}^0{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_2}{\left(e^{2\pi i1/P}\right)}^0{v}_j{v}_j^T}\\ ⋮\\ {\displaystyle \sum _{j\in E_4}{\left(e^{2\pi i\left(P-1\right)/P}\right)}^0{v}_j{v}_j^T}\end{array}\right)w^{0\times k}+\left(\begin{array}{cccc}1& 1& \cdots & 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\left(e^{2\pi i0/P}\right)}^{P/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_2}{\left(e^{2\pi i1/P}\right)}^{P/M}{v}_j{v}_j^T}\\ ⋮\\ {\displaystyle \sum _{j\in E_4}{\left(e^{2\pi i\left(P-1\right)/P}\right)}^{P/M}{v}_j{v}_j^T}\end{array}\right)w^{1\times k}+\cdots +\left(\begin{array}{cccc}1& 1& \cdots & 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\left(e^{2\pi i0/P}\right)}^{P\left(M-1\right)/M}{v}_j{v}_j^T}\\ {\displaystyle \sum _{j\in E_2}{\left(e^{2\pi i1/P}\right)}^{P\left(M-1\right)/M}{v}_j{v}_j^T}\\ ⋮\\ {\displaystyle \sum _{j\in E_4}{\left(e^{2\pi i\left(P-1\right)/P}\right)}^{P\left(M-1\right)/M}{v}_j{v}_j^T}\end{array}\right)w^{\left(M-1\right)\times k}\\ & =\left(\begin{array}{cccc}1& 1& \cdots & 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{j\in E_1}{\displaystyle \sum _{l=0}^{M-1}{\left(e^{2\pi i0/P}\right)}^{Pl/M}{w}^{l\times k}{v}_j{v}_j^T}}\\ {\displaystyle \sum _{j\in E_2}{\displaystyle \sum _{l=0}^{M-1}{\left(e^{2\pi i1/P}\right)}^{Pl/M}{w}^{l\times k}{v}_j{v}_j^T}}\\ ⋮\\ {\displaystyle \sum _{j\in E_4}{\displaystyle \sum _{l=0}^{M-1}{\left(e^{2\pi i\left(P-1\right)/P}\right)}^{Pl/M}{w}^{l\times k}{v}_j{v}_j^T}}\end{array}\right)\\ & =\left(\begin{array}{cccc}1& 1& \cdots & 1\end{array}\right)\left(\begin{array}{c}{\displaystyle \sum _{l=0}^{M-1}{\left(e^{2\pi i0/P}\right)}^{Pl/M}{w}^{l\times k}}\cdot {\displaystyle \sum _{j\in E_1}{v}_j{v}_j^T}\\ {\displaystyle \sum _{l=0}^{M-1}{\left(e^{2\pi i1/P}\right)}^{Pl/M}{w}^{l\times k}}\cdot {\displaystyle \sum _{j\in E_2}{v}_j{v}_j^T}\\ ⋮\\ {\displaystyle \sum _{l=0}^{M-1}{\left(e^{2\pi i\left(P-1\right)/P}\right)}^{Pl/M}{w}^{l\times k}}\cdot {\displaystyle \sum _{j\in E_4}{v}_j{v}_j^T}\end{array}\right)\end{array}$$

(94)

where $w=e^{2\pi i/M}$, $h=0,1,2,\cdots ,P-1$,

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=0}^{M-1}{\left(e^{2\pi ih/P}\right)}^{Pl/M}{w}^{l\times k}}& ={\displaystyle \sum _{l=0}^{M-1}{e}^{2\pi ilh/M}{e}^{2\pi ilk/M}}\hfill \\ & ={\displaystyle \sum _{l=0}^{M-1}{e}^{2\pi il\left(k+h\right)/M}}\end{array}$$

(95)

Therefore,

$${\displaystyle \sum _{l=0}^{M-1}{e}^{2\pi il\left(k+h\right)/M}}=\{\begin{array}{l}M, if h =0,k=0\\ M, if h =1,k=M-1\\ ⋮\\ M, if h =P,k=M-P\\ 0, others\end{array}$$

(96)

Equation (91) is represented as follows:

$$\begin{array}{cc}\hfill T^{\overline{\alpha }}& ={\displaystyle \frac{1}{M}}\left(Y_0,Y_1,\cdots ,Y_{M-1}\right)\left(\begin{array}{c}{B}_0^{{\alpha }_0}\\ B_1^{{\alpha }_1}\\ ⋮\\ B_{M-1}^{{\alpha }_{M-1}}\end{array}\right)\hfill \\ & ={\displaystyle \frac{1}{M}}\left(Y_0{B}_0^{{\alpha }_0}+0\cdots +0+Y_{M-P}{B}_{M-P}^{{\alpha }_{M-P}}+\cdots +Y_{M-1}{B}_{M-1}^{{\alpha }_{M-1}}\right).\end{array}$$

(97)

From the above proof, we know that the extended definition only has $P$ weighted terms, which are related to the period of the matrix. This deficiency will have a significant impact on the application of the extended definitions and requires sufficient attention from researchers.

6. Conclusions

In this paper, we propose that the WFRFT is the result of the FT changing its eigenvalue into a fractional power while retaining the eigenvector, which gives us a new understanding of the WFRFT. We clarified the existing extended definitions of the WFRFT, thus establishing a unified framework for weighted class transforms wherein the previously proposed definitions can be used as special cases. Within this unified framework, we analyzed an extended definition based on the WFRFT, which has only four effective weighting terms. The extended definition based on the weighted fractional Hartley transform has only two effective weighting terms. Furthermore, we demonstrated that the effective weighting terms of the general extended definition are related to the period of the matrix. In this way, our research indicates that the extended definitions of the weighted fractional-order transform are deficient, and this deficiency has a significant impact on its application. This research achievement has important reference value for the application of the WFRFT and extended definitions.