# An Insightful Overview of the Wiener Filter for System Identification
## 5 Wiener Filter via Kronecker Product Decomposition  
In this section, the investigation on the identification of low-rank systems with the Wiener filter is further discussed. An attempt to minimize the nuclear norm of the system in the regularization part was presented in the previous section, but this approach does not seem very practical, and complexity may be extremely high, especially for large matrices. Here, instead, it is assumed that the rank of the \(L_{1}\times L_{2}\) matrix \(\mathbf{H}\) is known and is much smaller than \(L_{2}\), i.e., \(\text{rank}(\mathbf{H})=P\ll L_{2}\). In this case, \(\mathbf{H}\) can also be expressed as  
\[\mathbf{H}=\sum_{p=1}^{P}\mathbf{h}_{1,p}\mathbf{h}_{2,p}^{T}, \tag{29}\]  
where \(\mathbf{h}_{1,p}\) and \(\mathbf{h}_{2,p}\), of lengths \(L_{1}\) and \(L_{2}\), respectively, are impulse responses. This decomposition of \(\mathbf{H}\) can be explicitly used in the MSE criterion as explained below. As a result  
\[\mathbf{h} =\text{vec}(\mathbf{H})\] \[=\text{vec}\Biggl{(}\sum_{p=1}^{P}\mathbf{h}_{1,p}\mathbf{h}_{2, p}^{T}\Biggr{)}\] \[=\sum_{p=1}^{P}\text{vec}\Bigl{(}\mathbf{h}_{1,p}\mathbf{h}_{2, p}^{T}\Bigr{)}\] \[=\sum_{p=1}^{P}\mathbf{h}_{2,p}\otimes\mathbf{h}_{1,p}, \tag{30}\]  
where \(\otimes\) is the Kronecker product. Furthermore, the filter \(\widehat{\mathbf{h}}\) can also be decomposed as  
\[\widehat{\mathbf{h}}=\sum_{p=1}^{P}\widehat{\mathbf{h}}_{2,p}\otimes\widehat{ \mathbf{h}}_{1,p}, \tag{31}\]  
where \(\widehat{\mathbf{h}}_{1,p}\) and \(\widehat{\mathbf{h}}_{2,p}\) are filters having lengths \(L_{1}\) and \(L_{2}\), respectively. By using the following relationships:  
\[\widehat{\mathbf{h}}_{2,p}\otimes\widehat{\mathbf{h}}_{1,p} =\Bigl{(}\widehat{\mathbf{h}}_{2,p}\otimes\mathbf{I}_{L_{1}} \Bigr{)}\widehat{\mathbf{h}}_{1,p}\] \[=\Bigl{(}\mathbf{I}_{L_{2}}\otimes\widehat{\mathbf{h}}_{1,p} \Bigr{)}\widehat{\mathbf{h}}_{2,p}, \tag{32}\]where by \(\mathbf{I}_{L_{1}}\) and \(\mathbf{I}_{L_{2}}\) are denoted the identity matrices of sizes \(L_{1}\times L_{1}\) and \(L_{2}\times L_{2}\), respectively, into (31), it is obtained that  
\[\widehat{\mathbf{n}} = \sum_{p=1}^{P}\widehat{\mathbf{H}}_{2,p}\widehat{\mathbf{h}}_{1,p} \tag{33}\] \[= \sum_{p=1}^{P}\widehat{\mathbf{H}}_{1,p}\widehat{\mathbf{h}}_{2,p},\]  
where \(\widehat{\mathbf{H}}_{2,p}=\widehat{\mathbf{h}}_{2,p}\otimes\mathbf{I}_{L_{1}}\) and \(\widehat{\mathbf{H}}_{1,p}=\mathbf{I}_{L_{2}}\otimes\widehat{\mathbf{h}}_{1,p}\) are matrices of sizes \(L_{1}L_{2}\times L_{1}\) and \(L_{1}L_{2}\times L_{2}\), respectively. The two expressions from (33) may be rewritten as  
\[\mathbf{h} =\text{vec}(\mathbf{H})\]0 \[\mathbf{h} =\text{vec}(\mathbf{H})\]1  
where  
\[\widehat{\mathbf{H}}_{2} = \left[ \begin{array}{cccc}\widehat{\mathbf{H}}_{2,1}&\widehat{ \mathbf{H}}_{2,2}&\cdots&\widehat{\mathbf{H}}_{2,P}\end{array} \right],\] \[\widehat{\mathbf{H}}_{1} = \left[ \begin{array}{cccc}\widehat{\mathbf{H}}_{1,1}&\widehat{ \mathbf{H}}_{1,2}&\cdots&\widehat{\mathbf{H}}_{1,P}\end{array} \right]\]  
are matrices of sizes \(L_{1}L_{2}\times PL_{1}\) and \(L_{1}L_{2}\times PL_{2}\), respectively, and  
\[\widehat{\mathbf{h}}_{1} = \left[ \begin{array}{cccc}\widehat{\mathbf{h}}_{1,1}^{T}& \widehat{\mathbf{h}}_{1,2}^{T}&\cdots&\widehat{\mathbf{h}}_{1,P}^{T}\end{array}  \right]^{T},\] \[\widehat{\mathbf{h}}_{2} = \left[ \begin{array}{cccc}\widehat{\mathbf{h}}_{2,1}^{T}& \widehat{\mathbf{h}}_{2,2}^{T}&\cdots&\widehat{\mathbf{h}}_{2,P}^{T}\end{array}  \right]^{T}\]  
are vectors of lengths \(PL_{1}\) and \(PL_{2}\), respectively. With this Kronecker product decomposition, it can be seen that \(\widehat{\mathbf{h}}_{1}\) and \(\widehat{\mathbf{h}}_{2}\) need to be identified instead of \(\widehat{\mathbf{h}}\), i.e., \(P(L_{1}+L_{2})\) components instead of \(L=L_{1}L_{2}\). Assume that \(\widehat{\mathbf{h}}_{2}\) (or, equivalently, \(\widehat{\mathbf{H}}_{2}\)) is fixed.