# A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model
## Appendix A Identification of Modal Residues and Upper and Lower Residuals with the LSFD Technique  
When the poles \(\lambda_{m}\) are determined from a preliminary LS estimation, the modal residues \(\left[Res\right]_{m}\) and the matrices \(\left[LR\right]\) and \(\left[UR\right]\) of the residual model can be estimated in a linear least squares sense by minimizing the following error equation with respect the remaining unknown invariants in the pole-residue modal model  
\[E_{o}(\Theta,\omega)=\sum_{r=1}^{N_{m}}\frac{\left[Res\right]_{r0}}{j\omega- \lambda_{r}}+\frac{\left[Res^{*}\right]_{r0}}{j\omega-\lambda_{r}^{*}}+a( \omega)\left[LR\right]_{o}+b(\omega)\left[UR\right]_{o}-H_{o}(\omega) \tag{10}\]  
with the subindex \"\(o\)\" designating the \(o\)th row of a matrix, \(E(\Theta,\omega)\) the error matrix and \(\Theta\) representing the row vector containing the unknown parameters given by  
\[\Theta^{T}= \begin{bmatrix}\text{Re}\left(\left[Res\right]\right)_{1o}^{T}& \cdots&\text{Re}\left(\left[Res\right]\right)_{N_{m}o}^{T}&\text{Im}\left( \left[Res\right]\right)_{1o}^{T}&\cdots&\text{Im}\left(\left[Res\right]\right) _{N_{m}o}^{T}&\left[LR\right]_{o}^{T}&\left[UR\right]_{o}^{T}\end{bmatrix} \in \mathbb{R}^{\left(2N_{m}+2\right)\times N_{i}} \tag{11}\]  
Writing down Equation (10) for each frequency line \(f\) (\(f=2\), \(3,\cdots,\)\(N_{f}\)), yields  
\[E_{o}(\Theta)=J\Theta-H_{o}=0 \tag{12}\]  
with \(E_{o}(\Theta)\) and \(H_{o}\) given now by  
\[E_{o}(\Theta)= \begin{bmatrix}E_{o}(\Theta,\omega_{2})\\ \vdots\\ E_{o}(\Theta,\omega_{N_{f}})\end{bmatrix} \in\mathbb{C}^{\left(N_{f}-1\right) \times N_{i}},\quad H_{o}= \begin{bmatrix}H_{o}(\omega_{2})\\ \vdots\\ H_{o}(\omega_{N_{f}})\end{bmatrix} \in\mathbb{C}^{\left(N_{f}-1\right)\times N _{i}} \tag{13}\]  
and the corresponding Jacobian matrix, \(J\in\mathbb{C}^{\left(N_{f}-1\right)\times\left(2N_{m}+2\right)}\), by  
\[J= \begin{bmatrix}\left(\frac{1}{\left[\omega_{2}-\lambda_{1}\right]}+\frac{1} {\left[\omega_{1}-\lambda_{1}\right]}\right)&\cdots&\left(\frac{1}{\left[ \omega_{2}-\lambda_{N_{m}}\right]}+\frac{1}{\left[\omega_{2}-\lambda_{1} \right]}\right)&\left(\frac{j}{\left[\omega_{2}-\lambda_{1}\right]}-\frac{j}{ \left[\omega_{2}-\lambda_{1}\right]}\right)&\cdots&\left(\frac{j}{\left[ \omega_{2}-\lambda_{N_{m}}\right]}-\frac{j}{\left[\omega_{2}-\lambda_{N_{m}} \right]}\right)&a(\omega_{2})&b(\omega_{2})\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots\\ \left(\frac{1}{\left[\omega_{N_{f}}-\lambda_{1}\right]}+\frac{1}{\left[\omega_ {N_{f}}-\lambda_{1}\right]}\right)&\cdots&\left(\frac{1}{\left[\omega_{N_{f }}-\lambda_{N_{m}}\right]}+\frac{1}{\left[\omega_{N_{f}}-\lambda_{1}\right]} \right)&\left(\frac{j}{\left[\omega_{N_{f}}-\lambda_{1}\right]}-\frac{j}{ \left[\omega_{N_{f}}-\lambda_{1}\right]}\right)&\cdots&\left(\frac{j}{\left[ \omega_{N_{f}}-\lambda_{N_{m}}\right]}-\frac{j}{\left[\omega_{N_{f}}-\lambda_ {N_{m}}\right]}\right)&a(\omega_{N_{f}})&b(\omega_{N_{f}})\end{bmatrix}  \tag{14}\]  
It is straightforward to solve Equation (12) for \(\Theta\) in a linear least squares sense. Yet, in order to guarantee the realness of \(\Theta\), \(J\) and \(H_{o}\) are replaced in Equation (12), respectively, by  
\[\left(J\right)_{\text{re}}= \begin{bmatrix}\text{Re}\left(J\right)&\text{Im} \left(J\right)\end{bmatrix} ,\quad\left(H_{o}\right)_{\text{re}}= \begin{bmatrix} \text{Re}\left(H_{o}\right)\\ \text{Im}\left(H_{o}\right)\end{bmatrix}  \tag{15}\]  
and, after some manipulations of the resulting expression, the following equation can be derived to compute the modal residues and the upper and lower matrices of the residual model\[ \begin{bmatrix}\text{Re}\left(\left[\text{\emph{Res}}\right]\right)_{1o}\\ \vdots\\ \text{Re}\left(\left[\text{\emph{Res}}\right]\right)_{N_{\text{no}}o}\\ \text{Im}\left(\left[\text{\emph{Res}}\right]\right)_{1o}\\ \vdots\\ \text{Im}\left(\left[\text{\emph{Res}}\right]\right)_{N_{\text{no}}o}\\ \left[LR\right]_{o}\\ \left[\text{\emph{U}}\right]_{o}\end{bmatrix}