# Wiener Filter Using the Conjugate Gradient Method and a Third-Order Tensor Decomposition
###### Abstract
To this purpose, the initialization relies on (22), so that the settings are  
\[\overline{\underline{\mathbf{g}}}_{11,W(0)}^{(n)} =\overline{\underline{\mathbf{g}}}_{11,W(K_{1})}^{(n-1)}, \tag{65}\] \[\mathbf{z}_{11(0)}^{(n)} =\mathbf{r}_{11}^{(n)}-\mathbf{R}_{11}^{(n)}\overline{\underline{ \mathbf{g}}}_{11,W(0)}^{(n)},\] (66) \[\mathbf{c}_{11(0)}^{(n)} =\mathbf{z}_{11(0)^{\prime}}^{(n)}\] (67) \[\gamma_{11(0)}^{(n)} =\Big{(}\mathbf{z}_{11(0)}^{(n)}\Big{)}^{T}\mathbf{z}_{11(0)}^{(n )}. \tag{68}\]  
Thus, for \(k_{1}=1,2,\ldots,K_{1}\), the CG cycle for the third filter consists of the relations:  
\[\mathbf{q}_{11(k_{1})}^{(n)} =\mathbf{R}_{11}^{(n)}\mathbf{c}_{11(k_{1}-1)^{\prime}}^{(n)} \tag{69}\] \[\mathbf{a}_{11(k_{1})}^{(n)} =\frac{\gamma_{11(k_{1}-1)}^{(n)}}{\Big{(}\mathbf{c}_{11(k_{1}-1 )}^{(n)}\Big{)}^{T}\mathbf{q}_{11(k_{1})}^{(n)}},\] (70) \[\overline{\underline{\mathbf{g}}}_{11,W(k_{1})}^{(n)} =\overline{\underline{\mathbf{g}}}_{11,W(k_{1}-1)}^{(n)}+\mathbf{ a}_{11(k_{1})}^{(n)}\mathbf{c}_{11(k_{1}-1)^{\prime}}^{(n)}\] (71) \[\mathbf{z}_{11(k_{1})}^{(n)} =\mathbf{z}_{11(k_{1}-1)}^{(n)}-\mathbf{a}_{11(k_{1})}^{(n)} \mathbf{q}_{11(k_{1})^{\prime}}^{(n)}\] (72) \[\mathbf{r}_{11}^{(n)} =\left(\overline{\mathbf{G}}_{2,12}^{(n)}\right)^{T}\mathbf{r}_{ \mathbf{x}d}. \tag{64}\]0 (73) \[\mathbf{r}_{11}^{(n)} =\left(\overline{\mathbf{G}}_{2,12}^{(n)}\right)^{T}\mathbf{r}_{ \mathbf{x}d}. \tag{64}\]1 (74) \[\mathbf{r}_{11}^{(n)} =\left(\overline{\mathbf{G}}_{2,12}^{(n)}\right)^{T}\mathbf{r}_{ \mathbf{x}d}. \tag{64}\]2  
The decomposition of the final solution \(\overline{\underline{\mathbf{g}}}_{11,W(K_{1})}^{(n)}\) results in  
\[\overline{\underline{\mathbf{g}}}_{11,W(K_{1})}^{(n)} =\left[ \begin{array}{cc}\left(\overline{\mathbf{g}}_{11,W(K_{1} )}^{1(n)}\right)^{T}&\left(\overline{\mathbf{g}}_{11,W(K_{1})}^{2(n)}\right) ^{T}&\cdots&\left(\overline{\mathbf{g}}_{11,W(K_{1})}^{L_{2}(n)}\right)^{T} \end{array} \right]^{T}, \tag{76}\] \[\overline{\mathbf{g}}_{11,W(K_{1})}^{i(n)} =\left[ \begin{array}{cc}\left(\mathbf{g}_{11,W(K_{1})}^{i(n)} \right)^{T}&\left(\mathbf{g}_{11,W(K_{1})}^{i2(n)}\right)^{T}&\cdots&\left( \mathbf{g}_{11,W(K_{1})}^{iP(n)}\right)^{T}\end{array} \right]^{T},\,i=1,2, \ldots,L_{2}, \tag{77}\]  
and provides the final elements for evaluating the estimated impulse response based on (19). Also, \(\overline{\underline{\mathbf{g}}}_{11,W(K_{1})}^{(n)}\) represents the initialization for the next main iteration of the algorithm, according to (65). Summarizing, using (41), (59), and (77), we obtain  
\[\mathbf{g}_{W}^{(n)}=\sum_{i=1}^{L_{2}}\sum_{j=1}^{P}\mathbf{g}_{2,W(K_{2})}^{ i(n)}\otimes\mathbf{g}_{12,W(K_{1})}^{ij(n)}\otimes\mathbf{g}_{11,W(K_{1})}^{ ij(n)}. \tag{78}\]  
Finally, using the same components, we evaluate the auxiliary matrices (for \(i=1,2,\ldots,L_{2}\) and \(j=1,2,\ldots,P\)):  
\[\mathbf{M}_{12,11}^{ij(n)} =\mathbf{I}_{L_{2}}\otimes\mathbf{g}_{12,W(K_{1})}^{ij(n)}\otimes \mathbf{g}_{11,W(K_{1})}^{ij(n)}, \tag{79}\] \[\mathbf{M}_{11}^{ij(n)} =\mathbf{I}_{L_{2}}\otimes\mathbf{g}_{11,W(K_{1})}^{ij(n)}. \tag{80}\]  
These will be used in the next main iteration of the algorithm, in order to compute the structures from (25) and (42), respectively.