## 2 Principles of the Proposed Method  
### The Theory of FrFT  
Fourier transform can well analyze the deterministic signals or stationary random signals. However, for non-stationary signals, the local performance needs to be accurately described by using the two-dimensional joint representation of time domain and frequency domain [22; 23]. FrFT associates relatively independent time-domain and frequency-domain information. It can map a one-dimensional time-domain signal into a two-dimensional time-frequency plane, reflecting the time-frequency distribution characteristics of the signal over time. For signal \(x(t)\), whose domain of definition is \(t\), the expression of FrFT whose transform order is \(p\) can be expressed as:  
\[X_{\alpha}(u)=F^{P}[x(t)]=\int_{-\infty}^{\infty}x(t)K_{\alpha}(t,u)dt \tag{1}\]which also has the inverse transformation:  
\[x(t)=\int_{-\infty}^{\infty}X_{\alpha}(u)K_{\alpha}^{\alpha}(t,u)du \tag{2}\]  
The kernel function of the FrFT is:  
\[K_{\alpha}(t,u)=\left\{ \begin{array}{ll}\sqrt{\frac{1-j\cot\alpha}{2\pi}}\exp( j\frac{t^{2}+\mu^{2}}{2}\cot\alpha-\frac{jtu}{\sin\alpha})&\alpha\
eq n\pi\\ \delta(t-u)&\alpha=2n\pi\\ \delta(t+u)&\alpha=(2n\pm 1)\pi\end{array} \right. \tag{3}\]  
where \(\alpha\) represents the rotation angle introduced by FrFT in the time-frequency domain distribution, and \(p\) is the order of FrFT and is \(p=2\alpha/\pi\). From Equations (1) and (3), it can be seen that FrFT has periodicity and a period of 4. When \(p=4n+1\) (i.e., \(\alpha=2n\pi+\pi/2\)), FrFT becomes a traditional Fourier transform:  
\[X_{2n\pi+\pi/2}(u)=F^{4n+1}[x(t)]=\int_{-\infty}^{\infty}x(t)K_{2n\pi+\pi/2}( t,u)dt=\int_{-\infty}^{\infty}e^{-j2\pi ut}x(t)dt \tag{4}\]  
When \(p=4n-1\) (i.e., \(\alpha=2n\pi-\pi/2\)), FrFT becomes the traditional inverse Fourier transform:  
\[X_{2n\pi-\pi/2}(u)=F^{4n-1}[x(\omega)]=\int_{-\infty}^{\infty}x(\omega)K_{2n \pi-\pi/2}(\omega,u)dt=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{j2\pi u\omega} x(\omega)dt \tag{5}\]  
When \(p=4n\), the FrFT of the signal is itself:  
\[X_{2n\pi}(u)=F^{4n}[x(t)]=\int_{-\infty}^{\infty}x(t)K_{2n\pi}(t,u)dt=x(t) \tag{6}\]  
When \(p=4n+2\), the FrFT of the signal is \(x(-t)\):  
\[X_{2n\pi+\pi}(u)=F^{4n+2}[x(t)]=\int_{-\infty}^{\infty}x(t)K_{2n\pi+\pi}(t,u)dt =x(-t) \tag{7}\]  
Since \(p=2\alpha/\pi\) only appears at the parameter positions of the trigonometric functions and FrFT is periodic in transform order 4, only the interval \(p\in(-2,2]\) is considered. When the order \(p\) changes from 0 to 1, its FrFT smoothly changes from the original function to the traditional Fourier transform, which further proves that the FrFT is correct. When defined by the rotation of the time-frequency plane, the \(p\)-order FrFT can be defined as a linear regular transform, which is described by the transformation matrix:  
\[M=\left[ \begin{array}{cc}A&B\\ C&D\end{array} \right]=\left[ \begin{array}{cc}\cos\alpha&\sin\alpha\\ -\sin\alpha&\cos\alpha\end{array} \right] \tag{8}\]  
where \(\alpha=p\pi/2\), Equation (8) is the two-dimensional rotation matrix on the time-frequency plane, and the transformation of order \(p\) is equivalent to  
to rotate the function by angle \(\alpha_{2}\). If the two operators work at the same time, the original function is continuously rotated by an \(\alpha_{1}+\alpha_{2}\) angle, which is equivalent to \(F^{P_{1}+P_{2}}\) in the original function. From a physical point of view, the traditional Fourier transform is equivalent to a \(90^{\circ}\) rotation in the time-frequency domain, while FrFT can rotate at any angle, and the angle of rotation is related to the order of the FrFT transform. Therefore, it can transform the signal into the fractional domain between time domain and the frequency domain for analysis and processing, which further increases the dimension of the signal. Compared with the definition method of the mathematical model, this definition in the physical sense can better reflect the changes and characteristics brought by FrFT to the time-frequency domain distribution of the signal.