## 3 Iterative Wiener Filter  
Let the filter \(\mathbf{g}\) be an estimate of the impulse response \(\mathbf{h}\) and let the symmetric filter:  
\[\mathbf{g}^{+} =\frac{\sum_{p=1}^{p}\big{(}\mathbf{g}_{2,p}\otimes\mathbf{g}_{1,p}+\mathbf{g}_{1,p}\otimes\mathbf{g}_{2,p}\big{)}}{2}\] \[=\mathbf{Q}\mathbf{g} \tag{14}\]  
be an estimate of \(\mathbf{h}^{+}\), with \(\mathbf{g}_{1,p}\) and \(\mathbf{g}_{2,p}\) being estimates of \(\mathbf{h}_{1,p}\) and \(\mathbf{h}_{2,p}\), respectively. We can conveniently rewrite (14) as  
\[\mathbf{g}^{+} =\frac{\sum_{p=1}^{p}\big{(}\mathbf{g}_{2,p}\otimes\mathbf{I}_{ L_{1}}+\mathbf{I}_{L_{1}}\otimes\mathbf{g}_{2,p}\big{)}\mathbf{g}_{1,p}}{2}\] \[=\sum_{p=1}^{p}\mathbf{G}_{\mathbf{g}_{2,p}}\mathbf{g}_{1,p}= \underline{\mathbf{G}}_{2}\mathbf{g}_{1}, \tag{15}\]  
where  
\[\mathbf{G}_{\mathbf{g}_{2,p}}=\frac{\mathbf{g}_{2,p}\otimes\mathbf{I}_{L_{1}} +\mathbf{I}_{L_{1}}\otimes\mathbf{g}_{2,p}}{2}\]  
is a matrix of size \(L\times L_{1}\),  
\[\underline{\mathbf{G}}_{2}=\big{[}\ \ \mathbf{G}_{\mathbf{g}_{2,1}}\ \ \ \mathbf{G}_{\mathbf{g}_{2,2}}\ \ \ \cdots\ \ \ \mathbf{G}_{\mathbf{g}_{2,p}}\ \ \big{]}\]  
is a matrix of size \(L\times PL_{1}\), and  
\[\underline{\mathbf{g}}_{1}=\Big{[}\ \ \mathbf{g}_{1,1}^{T}\ \ \ \mathbf{g}_{1,2}^{T}\ \ \ \cdots\ \ \ \mathbf{g}_{1,p}^{T}\ \Big{]}^{T}\]  
is a vector of length \(PL_{1}\). In the same manner, we can rewrite (14) in another useful form as follows:  
\[\mathbf{g}^{+} =\frac{\sum_{p=1}^{p}\big{(}\mathbf{I}_{L_{2}}\otimes\mathbf{g}_ {1,p}+\mathbf{g}_{1,p}\otimes\mathbf{I}_{L_{2}}\big{)}\mathbf{g}_{2,p}}{2}\] \[=\sum_{p=1}^{p}\mathbf{G}_{\mathbf{g}_{1,p}}\mathbf{g}_{2,p}= \underline{\mathbf{G}}_{1}\underline{\mathbf{g}}_{2}, \tag{16}\]where  
\[\mathbf{G}_{\mathbf{g}_{1,p}}=\frac{\mathbf{I}_{L_{2}}\otimes\mathbf{g}_{1,p}+ \mathbf{g}_{1,p}\otimes\mathbf{I}_{L_{2}}}{2}\]  
is a matrix of size \(L\times L_{2}\),  
\[\underline{\mathbf{G}}_{1}=\left[ \begin{array}{cccc}\mathbf{G}_{\mathbf{g}_{ 1,1}}&\mathbf{G}_{\mathbf{g}_{1,2}}&\cdots&\mathbf{G}_{\mathbf{g}_{1,p}}\end{array} \right]\]  
is a matrix of size \(L\times PL_{2}\), and  
\[\underline{\mathbf{g}}_{2}=\left[ \begin{array}{cccc}\mathbf{g}_{2,1}^{T}& \mathbf{g}_{2,2}^{T}&\cdots&\mathbf{g}_{2,p}^{T}\end{array} \right]^{T}\]  
is a vector of length \(PL_{2}\). Now, we can define the error signal between the desired signal, \(d(t)\), and the estimated signal, \(\tilde{y}(t)=\mathbf{g}^{T}\mathbf{x}(t)\), as follows:  
\[e(t)=d(t)-\tilde{y}(t), \tag{17}\]  
which can be expressed in two different manners:  
\[e(t) =d(t)-\mathbf{g}_{1}^{T}\underline{\mathbf{G}}_{2}^{T}\mathbf{Q}^ {-T}\mathbf{x}(t) \tag{18}\] \[=d(t)-\mathbf{g}_{2}^{T}\underline{\mathbf{G}}_{1}^{T}\mathbf{Q}^ {-T}\mathbf{x}(t). \tag{19}\]  
From (18) and assuming that \(\underline{\mathbf{g}}_{2}\) is fixed, we can write the mean-squared error (MSE) criterion as  
\[\mathcal{J}\left(\underline{\mathbf{g}}_{1}|\underline{\mathbf{ g}}_{2}\right)=E\left[\left|d(t)-\underline{\mathbf{g}}_{1}^{T}\underline{ \mathbf{G}}_{2}^{T}\mathbf{Q}^{-T}\mathbf{x}(t)\right|^{2}\right] \tag{20}\] \[=\sigma_{d}^{2}-2\underline{\mathbf{g}}_{1}^{T}\underline{ \mathbf{G}}_{2}^{T}\mathbf{Q}^{-T}\mathbf{x}_{d}+\underline{\mathbf{g}}_{1}^{ T}\underline{\mathbf{G}}_{2}^{T}\mathbf{Q}^{-T}\mathbf{R}_{\mathbf{x}}\mathbf{Q}^{-1} \underline{\mathbf{G}}_{2}\underline{\mathbf{g}}_{1},\]  
where \(\sigma_{d}^{2}\) is the variance of \(d(t)\) and \(\mathbf{r}_{\text{w}d}=E[\mathbf{x}(t)d(t)]\) is the cross-correlation vector between \(\mathbf{x}(t)\) and \(d(t)\).