# Cyclostationary Crosstalk Cancelation in High-Speed Transmission Lines
## 3 Synthesis and Analysis of Cyclic Wiener Filter
The error \(\mathbf{e}[k]\) and the measurement \(\mathbf{y}[k]\) are the uncorrelated random signals so that:  
\[E\Big{\{}\mathbf{e}[k]\mathbf{y}^{T}[k]\Big{\}}=E\Big{\{}\mathbf{y}[k]\mathbf{ e}^{T}[k]\Big{\}}=0 \tag{29}\]  
The correlation matrix of composite vector \(\mathbf{s}[k]\) is analyzed in the same manner as the correlation matrix \(\mathbf{R}_{zz}\) and has a block diagonal form due to the orthogonality of the error and measured signal:  
\[\mathbf{R}_{ss}=E\{\mathbf{s}[k]\mathbf{s}^{T}[k]\}=\left( \begin{array}{cc} \mathbf{Q}&0\\ 0&\mathbf{R}_{yy}\end{array} \right) \tag{30}\]  
Figure 8: (**a**) Synthesis scheme; (**b**) analysis scheme. where  
\[\mathbf{Q}=E\Big{\{}\mathbf{e}[k]\mathbf{e}^{T}[k]\Big{\}} \tag{31}\]  
is an error covariance matrix. The linear transformation of the random composite vectors \(\mathbf{s}[k]\) and \(\mathbf{z}[k]\) for synthesis and analysis schemes can be viewed through the transformation of corresponding correlation matrices using matrix \(\mathbf{B}\):  
\[\mathbf{R}_{ss}=\mathbf{R}\mathbf{R}_{zz}\mathbf{B}=\left( \begin{array}{cc }-\mathbf{I}&\mathbf{W}\\ 0&\mathbf{I}\end{array} \right)\left( \begin{array}{cc}\mathbf{R}_{xx}&\mathbf{ R}_{xy}\\ \mathbf{R}_{yx}&\mathbf{R}_{yy}\end{array} \right)\left( \begin{array}{cc}- \mathbf{I}&0\\ \mathbf{W}^{T}&\mathbf{I}\end{array} \right) \tag{32}\] \[\mathbf{R}_{zz}=\mathbf{R}\mathbf{R}_{ss}\mathbf{B}=\left( \begin{array} []{cc}-\mathbf{I}&\mathbf{W}\\ 0&\mathbf{I}\end{array} \right)\left( \begin{array}{cc}\mathbf{Q}&0\\ 0&\mathbf{R}_{yy}\end{array} \right)\left( \begin{array}{cc}-\mathbf{I}&0\\ \mathbf{W}^{T}&\mathbf{I}\end{array} \right)\]  
The upper triangular matrix \(\mathbf{U}\) and block diagonal matrix \(\mathbf{D}\) corresponding to the Cholesky decomposition of partitioned correlation matrix \(\mathbf{R}_{zz}\) can be used for matrix factorization:  
\[\mathbf{R}_{zz}=\mathbf{UDU}^{T}=\left( \begin{array}{cc}\mathbf{I}&\mathbf{ R}_{xy}\mathbf{R}_{yy}^{-1}\\ 0&\mathbf{I}\end{array} \right)\left( \begin{array}{cc}\mathbf{R}_{zz}/ \mathbf{R}_{yy}&0\\ 0&\mathbf{R}_{yy}\end{array} \right)\left( \begin{array}{cc}\mathbf{I}&0\\ \mathbf{R}_{yy}^{-1}\mathbf{R}_{yx}&\mathbf{I}\end{array} \right) \tag{33}\]  
where  
\[\mathbf{R}_{zz}/\mathbf{R}_{yy}=\mathbf{R}_{xx}-\mathbf{R}_{xy}\mathbf{R}_{ yy}^{-1}\mathbf{R}_{yx} \tag{34}\]  
is the Schur complement of \(\mathbf{R}_{yy}\) in \(\mathbf{R}_{zz}\). The comparison the blocks \(\mathbf{U}\) in (33) and \(\mathbf{B}\) in (32) gives the equation for the cyclic Wiener filter:  
\[\mathbf{W}=\mathbf{R}_{xy}\mathbf{R}_{yy}^{-1} \tag{35}\]  
The error covariance matrix \(\mathbf{Q}\) can be found by comparing correlation matrix \(\mathbf{R}_{ss}\) in (32) and block diagonal matrix \(\mathbf{D}\) in (33):  
\[\mathbf{Q}=\mathbf{R}_{zz}/\mathbf{R}_{yy}=\mathbf{R}_{xx}-\mathbf{R}_{xy} \mathbf{R}_{yy}^{-1}\mathbf{R}_{yx} \tag{36}\]  
In summary of this section, the virtual two channel experiments can be discussed. The analysis problem corresponds to the linear transformation of composite vector \(\mathbf{z}[k]\). This model produces an error that is orthogonal to the measurement provided by the appropriate Wiener filter. The synthesis problem can be modeled using the same linear transformation of the composite vector \(\mathbf{s}[k]\) with a given block diagonal correlation matrix that returns the vector \(\mathbf{z}[k]\). The decomposition of the correlation matrix of vector \(\mathbf{z}[k]\) provides the block Cholesky factors corresponding to the transformed correlation matrix of vector \(\mathbf{s}[k]\). The crosstalk interference canceler is a specific cyclic Wiener filter. The main contribution to the disturbance of the signal is generated by cyclostationary crosstalk in the presence of low stationary noise compared with the desired signal. The characteristics of the synthesized and analyzed cyclic Wiener filters for crosstalk cancelation is presented in Section 4.