## 2 Complex Natural Resonances Methods
### Matrix Pencil Method  
This method was presented in 1990 by [6] as an alternative method to [PERSON]'s method for estimating parameters of backscattered signals with noise. It uses the pencil-of-function parameter, resulting in an improvement for noise sensitivity [6], and starts with Equation (4) for time domain samples and a discrete representation of the approximation of the input signal \(y\) into data samples \(y_{k}\). This corresponds to a Hankel matrix \([Y]\) as in (5), with \(L\) as the pencil parameter, \(k=1,2,\ldots,K\) the sample numbers, \(z_{n}=e^{y_{k}T_{s}}\), the sampling period \(T_{s}\), and \(R_{n}\) the complex residues.  
\[y_{k}\approx\sum_{i=1}^{M}R_{n}z_{n}^{k} \tag{4}\]  
\[Y=\begin{bmatrix}y_{o}&y_{1}&\cdots&y_{L}\\ y_{1}&y_{2}&\cdots&y_{L+1}\\ \vdots&\vdots&\ddots&\vdots\\ y_{K-L-1}&y_{K-L}&\cdots&y_{K-1}\end{bmatrix} \tag{5}\]  
To find the critical components of (5), MPM uses \(\lambda\) when Equation (6) is singular. This approach denotes \([Y_{2}]\) as the matrix \([Y]\) with the first column deleted and \([Y_{1}]\) as the matrix \([Y]\) with the last column deleted, and \(H\) as the Hermitian transpose. However, as the solution is not straightforward, several SVDs have to be computed from \([Y]\) to find \(\lambda\) as follows.  
\[[Y_{2}]^{H}-\lambda[Y_{1}]^{H} \tag{6}\]  
Figure 3: Prony’s CNRs calculated.  
Figure 2: Benchmark signal and Prony’s CNR-based reconstruction.  
Applying SVD to \([Y]\) leads to (7) with \([U]\) having the largest \(n\) orthonormalized eigenvectors of \([Y][Y]^{H}\), and \([V]\) the eigenvectors from \([Y]^{H}[Y]\). The singular values composing \([\Sigma]\) are the nonnegative square roots of the eigenvalues of \([Y]^{H}[Y]\). After this matrix has been constructed, and assuming a noisy environment, a new matrix \([Y^{\prime}]\) and the corresponding \([U^{\prime}]\), \([\Sigma^{\prime}]\), and \([V^{\prime}]\) are defined, setting a minimum threshold in order to delete the nonzero and noise-dependent eigenvalues in \([Y]\).  
\[[Y]=[U][\Sigma][V]^{T} \tag{7}\]  
\[\frac{\sigma_{M}}{\sigma_{max}}=10^{-p} \tag{8}\]  
This threshold relies on the selection of a significance parameter, and a way to estimate it is to calculate the ratio between the maximum singular value and all others in the diagonal matrix \([\Sigma]\) to obtain (8), with \(p\) being the number of significant decimal digits in the data [18].  
Now that \([V^{\prime}]\) has \(M\) parameters, is separated into \([V^{\prime}_{1}]\) and \([V^{\prime}_{2}]\), where \([V^{\prime}_{1}]\) is \([V^{\prime}]\) with the last column deleted, and \([V^{\prime}_{2}]\) is \([V^{\prime}]\) with the first column deleted. Equation (6) could now have a singular solution for (9) with a reduced number of samples for the LP matrix.  
\[[V_{2}]^{H}-\lambda[V_{1}]^{H} \tag{9}\]  
The SVD is used to compute the pseudo-inverse of \([V_{1}]^{H}\), which is then multiplied by \([V_{2}]^{H}\) to find the poles \(z\). A least square approach will allow the determination of the residues \(R\) after a Vandermonde matrix is formed, as in Equation (10) [19].  
\[\begin{bmatrix}y_{0}\\ y_{1}\\ \vdots\\ y_{K-1}\end{bmatrix}=\begin{bmatrix}1&1&\cdots&1\\ z_{1}&z_{2}&\cdots&z_{M}\\ \vdots&\vdots&\ddots&\vdots\\ z_{1}^{K-1}&z_{2}^{K-1}&\cdots&z_{M}^{K-1}\end{bmatrix}\begin{bmatrix}R_{1}\\ R_{2}\\ \vdots\\ R_{M}\end{bmatrix} \tag{10}\]  
The implementation of MPM is as shown in the block representation in Figure 4. There is a recursive step variables refinement and setup in which the reconstructed signal is compared with the input signal, and it also works as CNRs validation.  
To test the accuracy of the CNRs extracted, a simple test is then carried out with the benchmark signal \(y(t)=\sin 5t+\sin 10t\). Both the input signal  
Figure 4: General steps for implementing MPM algorithm and comparing input and output signals for CNRs validation.  
struction are compared, and the poles calculated through MPM are shown in Figures 5 and 6, respectively.