# Cyclostationary Crosstalk Cancelation in High-Speed Transmission Lines
## 3 Synthesis and Analysis of Cyclic Wiener Filter  
The signal estimation problem can be viewed through the SW filtering of the measured signals rearranged using serial to parallel transformation. This estimator is referred as a Wiener filter. The equality of the period of the corresponding LPTV filter and cyclic period of the corresponding random cyclostationary processes makes the estimator a cyclic Wiener filter. The matrix notation used for the SW filtering of cyclostationary signals simplifies the description dealing with the stationary vectors. The desired signal is transformed into a real-valued vector \(\mathbf{x}[k]\). The measured signal denoted by the vector \(\mathbf{y}[k]\) with the same length is linear transformed using a real-valued weighting matrix \(\mathbf{W}\). The difference between estimated and desired signals is an estimation error \(\mathbf{e}[k]\). Two equations can be  
Figure 6: Real (\(\mathbf{a}\)–\(\mathbf{c}\)) and imaginary (\(\mathbf{d}\)–\(\mathbf{f}\)) parts of \(1D\) impulse responses \(G_{\
u,\mu}(n)\) of frequency shift filter corresponding to the sliding window filter. Figure 7: Output signals for sliding window filter (blue stems) and equivalent frequency shift filter (red crosses). used for the introduction of the signal estimation problem, avoiding denotation of the filter output signal:  
\[ \begin{array}{l}\mathbf{x}[k]=\mathbf{W}\mathbf{y}[k]-\mathbf{e}[k]\\ \mathbf{e}[k]=\mathbf{W}\mathbf{y}[k]-\mathbf{x}[k]\end{array}  \tag{26}\]  
The obtained dual equations can be mapped into the composite form using the virtual two channel estimation problem. Figure 8 shows the pair of two-channel systems where input and output vectors are composite block vectors. The block vector \(\mathbf{z}[k]\) is obtained as a composition of vectors \(\mathbf{x}[k]\) and \(\mathbf{y}[k]\). The block vector \(\mathbf{s}[k]\) consists of vectors \(\mathbf{e}[k]\) and \(\mathbf{y}[k]\). The scheme in Figure 8 is used for synthesis, and the scheme in Figure 8 is used for an analysis problem. The mentioned schemes are identical if permuting input and output block vectors. The motivation of this step is an application of the partitioned correlation matrices for input and output composite vectors. These matrices contain all the necessary correlations between vectors for solving estimation problems. The input to output relations for the corresponding synthesis and analysis schemes shown in Figure 8 can be written using partitioned matrix \(\mathbf{B}\) for the definition of linear transformation of composite vectors \(\mathbf{z}[k]\) and \(\mathbf{s}[k]\) in accordance with (26) as follows:  
\[ \begin{array}{l}\mathbf{z}[k]=\mathbf{B}\mathbf{s}[k]=\left(\begin{array}[] {c}\mathbf{x}[k]\\ \mathbf{y}[k]\end{array}\right)=\left(\begin{array}{cc}-\mathbf{I}&\mathbf{W }\\ 0&\mathbf{I}\end{array}\right)\left(\begin{array}{c}\mathbf{e}[k]\\ \mathbf{y}[k]\end{array}\right)\\ \mathbf{s}[k]=\mathbf{B}\mathbf{z}[k]=\left(\begin{array}{cc}\mathbf{e}[k]\\ \mathbf{y}[k]\end{array}\right)=\left(\begin{array}{cc}-\mathbf{I}&\mathbf{W }\\ 0&\mathbf{I}\end{array}\right)\left(\begin{array}{c}\mathbf{x}[k]\\ \mathbf{y}[k]\end{array}\right)\end{array}  \tag{27}\]  
where \(\mathbf{I}\) is an identity matrix. The correlation matrix of the block vector \(\mathbf{z}[k]\) is a partitioned block matrix:  
\[\mathbf{R}_{zz}=E\{\mathbf{z}[k]\mathbf{z}^{T}[k]\}=\left( \begin{array}{cc} \mathbf{R}_{xx}&\mathbf{R}_{xy}\\ \mathbf{R}_{yx}&\mathbf{R}_{yy}\end{array} \right) \tag{28}\]  
where \(E\{\bullet\}\) is an expectation operator, \(\mathbf{R}_{xx}=E\{\mathbf{x}[k]\mathbf{x}^{T}[k]\}\) and \(\mathbf{R}_{yy}=E\{\mathbf{y}[k]\mathbf{y}^{T}[k]\}\) are the correlation matrices; \(\mathbf{R}_{xy}=E\{\mathbf{x}[k]\mathbf{y}^{T}[k]\}\) and \(\mathbf{R}_{yx}=E\{\mathbf{y}[k]\mathbf{x}^{T}[k]\}\) are the cross-correlation matrices. The Wiener solution of the estimation problem is based on the principle of orthogonality.