# An Insightful Overview of the Wiener Filter for System Identification
## 6 Wiener Filter with the Quadratic Eigenvalue Problem
Relations (45) and (46) can be written in terms of \(\overline{\mathbf{h}}\) as follows:  
\[(\mathbf{R}-\lambda\mathbf{I}_{L})^{2}\overline{\mathbf{h}} =\mathbf{p}, \tag{48}\] \[\overline{\mathbf{h}}^{T}(\mathbf{R}-\lambda\mathbf{I}_{L})^{2} \overline{\mathbf{h}} =\alpha^{2}\] (49) \[=\overline{\mathbf{h}}^{T}\mathbf{p}.\]  
It is easily deduced from this second equation that  
\[\frac{\mathbf{p}^{T}\overline{\mathbf{h}}}{\alpha^{2}}=1. \tag{50}\]  
Developing (48) and using the previous result, it is obtained that  
\[\mathcal{L}\left(\widehat{\mathbf{h}},\lambda\right)=\sigma_{d}^{2}-2\widehat{ \mathbf{h}}^{T}\mathbf{p}+\widehat{\mathbf{h}}^{T}\mathbf{R}\widehat{ \mathbf{h}}-\lambda\left(\widehat{\mathbf{h}}^{T}\widehat{\mathbf{h}}-\alpha ^{2}\right), \tag{44}\]0 \[\mathcal{L}\left(\widehat{\mathbf{h}},\lambda\right)=\sigma_{d}^{2}-2\widehat{ \mathbf{h}}^{T}\mathbf{p}+\widehat{\mathbf{h}}^{T}\mathbf{R}\widehat{ \mathbf{h}}-\lambda\left(\widehat{\mathbf{h}}^{T}\widehat{\mathbf{h}}-\alpha ^{2}\right), \tag{44}\]1  
As a consequence, (51) can be rewritten as  
\[\mathcal{L}\left(\widehat{\mathbf{h}},\lambda\right)=\sigma_{d}^{2}-2\widehat{ \mathbf{h}}^{T}\mathbf{p}+\widehat{\mathbf{h}}^{T}\mathbf{R}\widehat{ \mathbf{h}}-\lambda\left(\widehat{\mathbf{h}}^{T}\widehat{\mathbf{h}}-\alpha ^{2}\right), \tag{44}\]2  
or alternatively,  
\[\mathcal{L}\left(\widehat{\mathbf{h}},\lambda\right)=\sigma_{d}^{2}-2\widehat{ \mathbf{h}}^{T}\mathbf{p}+\widehat{\mathbf{h}}^{T}\mathbf{R}\widehat{ \mathbf{h}}-\lambda\left(\widehat{\mathbf{h}}^{T}\widehat{\mathbf{h}}-\alpha ^{2}\right), \tag{44}\]3  
where  
\[\mathcal{L}\left(\widehat{\mathbf{h}},\lambda\right)=\sigma_{d}^{2}-2\widehat{ \mathbf{h}}^{T}\mathbf{p}+\widehat{\mathbf{h}}^{T}\mathbf{R}\widehat{ \mathbf{h}}-\lambda\left(\widehat{\mathbf{h}}^{T}\widehat{\mathbf{h}}-\alpha ^{2}\right), \tag{44}\]4 \[\mathcal{L}\left(\widehat{\mathbf{h}},\lambda\right)=\sigma_{d}^{2}-2\widehat{ \mathbf{h}}^{T}\mathbf{p}+\widehat{\mathbf{h}}^{T}\mathbf{R}\widehat{ \mathbf{h}}-\lambda\left(\widehat{\mathbf{h}}^{T}\widehat{\mathbf{h}}-\alpha ^{2}\right), \tag{44}\]5 \[\mathcal{L}\left(\widehat{\mathbf{h}},\lambda\right)=\sigma_{d}^{2}-2\widehat{ \mathbf{h}}^{T}\mathbf{p}+\widehat{\mathbf{h}}^{T}\mathbf{R}\widehat{ \mathbf{h}}-\lambda\left(\widehat{\mathbf{h}}^{T}\widehat{\mathbf{h}}-\alpha ^{2}\right), \tag{44}\]6  
and \(\mathbf{Q}(\lambda)\) represents the \(\lambda\)-matrix, which is a degree \(2\)\(L\times L\) matrix polynomial. In (53), the QEP [29] can be recognized, where \(\overline{\mathbf{h}}\) is the right eigenvector corresponding to the eigenvalue \(\lambda\). Consequently, the solution of (43) is \(\widehat{\mathbf{h}}=(\mathbf{R}-\lambda\mathbf{I}_{L})^{-1}\mathbf{p}\), where \(\lambda\) is the smallest eigenvalue of (53). The QEP represents a generalization of the standard eigenvalue problem (SEP), \(\mathbf{Au}=\lambda\mathbf{u}\), and the generalized eigenvalue problem (GEP), \(\mathbf{Au}=\lambda\mathbf{B}\mathbf{u}\). Having a \(\lambda\)-matrix of size \(L\times L\), the QEP has \(2L\) eigenvalues (finite or infinite) containing up to \(2L\) right and \(2L\) left eigenvectors. If there are more than \(L\) eigenvectors, then they cannot, of course, be linearly independent [29]. In the current context, since \(\mathbf{A}_{2}=\mathbf{I}_{L}\) is nonsingular, the QEP has \(2L\) finite eigenvalues. Because \(\mathbf{A}_{0}\), \(\mathbf{A}_{1}\), and \(\mathbf{A}_{2}\) are symmetric, the eigenvalues are real or come in pairs \((\lambda,\lambda^{*})\)[29], where the superscript \({}^{*}\) denotes the complex conjugate operator. An important observation regarding the QEP is that the linearization process is not unique. It is said that a \(2L\times 2L\) matrix \(\mathbf{A}-\lambda\mathbf{B}\) represents a linearization of the \(L\times L\)\(\lambda\)-matrix \(\mathbf{Q}(\lambda)\) if the eigenvalues of these matrices are identical. Due to this process, the QEP can be solved with the GEP. One option is to take  
\[\mathbf{A} =\left[