# Identification of Mode Shapes Based on Ambient Signals and the IA-VMD Method
## 4 Prony Algorithm  
The Prony algorithm aims to estimate the parameters for the exponential terms through the following fitting function:  
\[\hat{x}(n)=\sum_{i=1}^{p}b_{i}z_{i}^{n}\;(n=1,2,\cdots,N-1) \tag{13}\]  
where \(x(n)=u_{k}(\Delta t\cdot n)\) is the sampled signal of \(u_{k}(t)\), and \(\Delta t\) is the sample interval. Additionally, \(\hat{x}(n)\) is the Prony fitting function, which is the approximate value of the sampled signal \(x(n)\). \(N\) is the number of sampling points of signals to be analyzed, as acquired by the system. In general, suppose that \(b_{i}\), \(z_{i}\) is a complex number and represents attenuation characteristics and oscillation characteristics of the sampled signal, respectively.  
\[b_{i}=A_{i}e^{i\theta_{i}} \tag{14}\]  
\[z_{i}=e^{(\alpha_{i}+j2\pi f_{i})\Delta t} \tag{15}\]  
where \(A_{i}\), \(\theta_{i}\), \(f_{i}\), and \(\alpha_{i}\) are the amplitude, damping factors, frequency, and phase associated with the \(i\)th mode, respectively.  
(1) The first step of the Prony algorithm is to construct a data matrix:  
\[r(i,j)=\sum_{n=p}^{N-1}x(n-j)x^{*}(n-i)\;(i,j=0,1,\cdots,p_{e}) \tag{16}\]where _r(i,j)_ is the sampled data, and _p\({}_{e}\)_ is the order of the linear prediction model. The Prony algorithm uses the minimum error sum of squares as the estimation principle of model parameters.  
The sample function matrix \(R\) of the expansion order is constructed as follows:  
\[R=\left(\begin{array}{cccc}r(1,0)&r(1,1)&\cdots&r(1,p_{e})\\ r(2,0)&r(2,1)&\cdots&r(2,p_{e})\\ \vdots&\vdots&\ddots&\vdots\\ r(p_{e},0)&r(p_{e},1)&\cdots&r(p_{e},p_{e})\end{array}\right) \tag{17}\]  
where \(p\) is the effective order of matrix \(R\). In general, _p\({}_{e}\)_ is larger than \(p\), but the operation speed will be affected if _p\({}_{e}\)_ is too large. So, _p\({}_{e}\)_ can be taken as an integer less than _N\(/\)_2 in actual calculations.  
(2) Using the singular value decomposition and total least squares (SVD-TLS) method, obtain the effective order \(p\) of the sample function matrix \(R\).  
(3) Establish the linear matrix equation to solve the parameter {_a\({}_{0}\)_,..., _a\({}_{p}\)_}, as follows:  
\[\left(\begin{array}{cccc}r(1,0)&r(1,1)&\cdots&r(1,p)\\ r(2,0)&r(2,1)&\cdots&r(2,p)\\ \vdots&\vdots&\ddots&\vdots\\ r(p,0)&r(p,1)&\cdots&r(p,p)\end{array}\right)\left[\begin{array}{c}a_{0}\\ a_{1}\\ \vdots\\ a_{p}\end{array}\right]=\left[\begin{array}{c}\varepsilon_{p}\\ 0\\ \vdots\\ 0\end{array}\right] \tag{18}\]  
(4) Obtain the characteristic root \(z\) by solving the polynomial equation, as follows:  
\[\varphi(z)=z^{p}+a_{1}z^{p-1}+\cdots+a_{p-1}z+a_{p}=0 \tag{19}\]  
(5) Establish the linear matrix equation to solve the parameter {_b\({}_{0}\)_,..., _b\({}_{p}\)_}, as follows:  
\[\left(\begin{array}{cccc}1&1&\cdots&1\\ Z_{1}&Z_{2}&\cdots&Z_{p}\\ \vdots&\vdots&\ddots&\vdots\\ Z_{1}^{N-1}&Z_{2}^{N-1}&\cdots&Z_{p}^{N-1}\end{array}\right)\left[\begin{array} []{c}b_{1}\\ b_{2}\\ \vdots\\ b_{p}\end{array}\right]=\left[\begin{array}{c}\hat{x}(1)\\ \hat{x}(2)\\ \vdots\\ \hat{x}(p)\end{array}\right] \tag{20}\]  
where \(Z\) = \(\left(\begin{array}{cccc}1&1&\cdots&1\\ Z_{1}&Z_{2}&\cdots&Z_{p}\\ \vdots&\vdots&\ddots&\vdots\\ Z_{1}^{N-1}&Z_{2}^{N-1}&\cdots&Z_{p}^{N-1}\end{array}\right)\), the matrix \(Z\) is a Vandermonde matrix with a dimension of _N\(\times\)P_; it is a full-rank matrix. Additionally, the least-squares solution of Formula (20) can be obtained as follows:  
\[b=\left(Z^{H}Z\right)^{-1}\!Z^{H}\hat{x} \tag{21}\]  
(6) Finally, by Formula (21), the characteristic parameters of the _i_th oscillation mode, including amplitude, phase, frequency, and damping ratio, can be obtained as the following formulas:  
\[\left\{\begin{array}{l}A_{i}=|b_{i}|\\ \theta_{i}=\arctan(\text{Im}(b_{i})/\text{Re}(b_{i}))\\ \alpha_{i}=\text{ln}|Z_{i}|/\Delta t\\ f_{i}=\arctan\left(\text{Im}(Z_{i})\right)/\text{Re}(Z_{i}))/2\pi\Delta t\end{array}\right. \tag{22}\]