## 3 Theoretical Analysis
### Prony Algorithm  
The Prony algorithm assumes that data \(x(n)\) to be analyzed is composed of \(p\) exponential functions with arbitrary amplitudes, phases, frequencies and attenuation factors. Its mathematical model is given by  
\[\hat{x}(n)=\sum_{i=1}^{p}b_{i}z_{i}^{n},n=0,1,2,\cdots,N-1, \tag{9}\]  
where \(b_{i}=A_{i}\exp(j\theta_{i})\), \(z_{i}=\exp[(\alpha_{i}+j2\pi f_{i})\Delta t]\), and \(\hat{x}(n)\) is the fitting value of \(x(n)\); \(A_{i},f_{i},\theta_{i}\) and \(\alpha_{i}\) are the amplitude, frequency, initial phase angle and attenuation factor, respectively; \(\Delta t\) is the sampling interval. Equation (9) can be written in a matrix form as follows:  
\[\hat{x}=\mathbf{zb}, \tag{10}\]  
where  
\[\hat{x}=\begin{bmatrix}\hat{x}(0)&\hat{x}(1)&\cdots&\hat{x}(N)\end{bmatrix}^ {T},\mathbf{z}=\begin{bmatrix}\begin{array}{cccc}1&1&&1\\ z_{1}&z_{2}&&z_{p}\\ \vdots&\vdots&&\ddots&\vdots\\ z_{1}^{N-1}&z_{2}^{N-2}&\cdots&z_{p}^{N-1}\end{array}\end{bmatrix},\mathbf{b}= \begin{bmatrix}b_{1}\\ b_{2}\\ \vdots\\ b_{p}\end{bmatrix}. \tag{11}\]  
The following objective function is constructed to minimize the fitting error:  
\[mine=\sum_{n=0}^{N-1}|x(n)-\hat{x}(n)|^{2}. \tag{12}\]It should be noted that Equation (9) is a homogeneous solution of a linear difference equation with constant coefficients. Thus,  
\[\hat{x}(n)=-\sum\limits_{i=1}^{p}a_{i}\hat{x}(n-i),p\leq n\leq N-1. \tag{13}\]  
The error between data \(x(n)\) to be analyzed and fitting data \(\hat{x}(n)\) can be defined as \(e(n)\):  
\[e(n)=x(n)-\hat{x}(n),n=0,1,2,\cdots,N-1. \tag{14}\]  
Hence, Equations (13) and (14) provide  
\[\begin{array}{ll}x(n)&=\hat{x}(n)+e(n)=-\sum\limits_{i=1}^{p}a_{i}\hat{x}(n -i)+e(n)\\ &=-\sum\limits_{i=1}^{p}a_{i}x(n-i)+\sum\limits_{i=0}^{p}a_{i}e(n-i),\end{array} \tag{15}\]  
for which one can define \(u(n)=\sum_{i=0}^{p}a_{i}e(n-i)\). \(x(n)\) can be regarded as the output generated by a \(p\)-order autoregressive model excited by error \(u(n)\), and the parameter \(a_{i}\) can be obtained by solving the canonical equation of the model. Substituting \(a_{i}\) into Equation (16) and computing the root of the polynomial yields parameter \(z_{i}\).  
\[\sum\limits_{i=0}^{p}a_{i}z^{p-i}=0. \tag{16}\]  
From Equations (10) and (11), the least square method is used to solve the matrix, and parameter \(b_{i}\) can be obtained. The least square solution is provided by  
\[\mathbf{b}=\left(\mathbf{z}^{H}\mathbf{z}\right)^{-1}\mathbf{z}^{H}\hat{x}. \tag{17}\]  
Finally, harmonic and interharmonic parameters are obtained through the following equations:  
\[\left\{\begin{array}{ll}A_{i}=\left|b_{i}\right|\\ f_{i}=arctan\left[\frac{Im(z_{i})}{Re(z_{i})}\right]/2\pi\Delta t\\ \theta_{i}=arctan\left[\frac{Im(b_{i})}{Re(b_{i})}\right]/2\pi\Delta t\\ \alpha_{i}=\frac{Im(z_{i})}{\Delta t}\end{array},i=1,2,\cdots,p.\right. \tag{18}\]  
From Equation (9), when performing fitting analysis on data \(x(n)\) using the Prony algorithm, under the assumption that \(x(n)\) is composed of p exponential functions, p sets of data are involved in the results calculated from Equation (18). Each set of data consists of \(A_{i},f_{i},\theta_{i}\) and \(\alpha_{i}\). They contain both non-false components characterizing useful signal parameters and false components characterizing noise signal parameters. Hence, these non-false components need to be accurately screened from these data.