## 2 Vector Fitting-Cauchy Method  
VCM starts with the frequency response of a system \(Y(s)\) and its transfer function approximation as a ratio of \(u_{k}\) and \(v_{k}\) as in Equation (11), again with \(P=(N-3)/2\) and \(Q=P+1\) as the initial guess for the system order. This selection of the initial resonances is based on [PERSON]'s initial set of poles, before an SVD calculates the approximate system order according to the number of singular values in the linear prediction matrix.  
\[Y(s)=\frac{\sum_{k=0}^{P}u_{k}s_{i}^{k}}{\sum_{k=0}^{Q}v_{k}s_{i}^{k}} \tag{11}\]  
This motivates again a TLS solution to find the transfer function parameters \(a\) and \(b\) from the matrix system in Equation (5). The SVD of this matrix \([C]\) results in Equation (12).  
\[[U][S][V]^{\prime}\begin{bmatrix}u\\ v\end{bmatrix}=0 \tag{12}\]  
The matrix \([S]\) is diagonal with nonnegative elements in decreasing order and, according to the spacing of floating-point numbers \(\delta\), the selection of singular values \(\sigma\) (i.e., the rank \(R\) of the matrix) can be calculated and used in Equation (13), as in the TLS-Cauchy Method, to also set a new value for \(P\) and \(Q\) keeping \(P=Q-1\) always.  
\[P+Q+2=R+1 \tag{13}\]  
Now, this new number of coefficients is used to reconstruct \([C]\) as \([C1]\) and, from there, determine the solution vector from the last column of the matrix \([V1]^{\prime}\) after applying an SVD. These values are \(u_{k}\) and \(v_{k}\), and from there, we have an initial pole and rank guess of the system response. In other words, from Equations (11)-(13), we have only the initial resonances information to start the following new calculation.  
A partial fraction expansion as in Equation (2) gives us the basic poles that are to be included in the TLS problem proposed in Equation (7), producing Equation (14),  
\[Y(s) = \frac{u_{p}s^{P}+u_{P-1}s^{P-1}+\cdots+u_{0}}{v_{Q}s^{Q}+v_{Q-1}s^{Q- 1}+\cdots+v_{0}} = \frac{R_{1}}{s-a_{1}}\,+\,\frac{R_{2}}{s-a_{2}}\,+\,\cdots\,+\, \frac{R_{N}}{s-a_{N}}, \tag{14}\]  
where \(R\) and \(a\) are the residues and their respective poles for a set of damped sinusoidal that could form the original frequency response. If we use Equation (7), we have an overdetermined matrix system to be solved by a least square approach. For a frequency sample, we have  
\[A_{i}x=b_{i},\text{with} \tag{15}\] \[A_{i}=\begin{bmatrix}\frac{1}{st-a_{1}}&\cdots&\frac{1}{st-a_{N}} &1&s_{i}&\frac{-f(s_{i})}{s_{i}a_{1}}&\cdots&\frac{-f(s_{i})}{s_{i}a_{N}} \end{bmatrix},\] (16) \[x=\begin{bmatrix}R_{1}&\cdots&R_{N}&d&h&\bar{R}_{1}&\cdots&\bar{R }_{N}\end{bmatrix}^{T},\quad\text{and}\] (17) \[b_{i}=f(s_{i}) \tag{18}\]  
The vector \([x]\) is the starting point to calculate the zeros in \(\sigma_{fit}(s)\), i.e., the poles of \(f(s)\). After this, the residues associated can be calculated with the rational function approximation  
\[f(s)=\sum_{n=1}^{N}\frac{R_{n}}{s-\bar{R}_{n}}+d+sh, \tag{19}\]  
which is again solved as a least square problem. Reconstructing the signal with residues \(c_{n}\), the poles \(\varepsilon_{N}\), \(d\), and \(sh\), and comparing it to the input signal gives us the validation for the CNRs calculated here. We include an overview of the method in Figure 1.