# Cyclostationary Crosstalk Cancelation in High-Speed Transmission Lines
## 2 Linear Periodic Time-Varying Filtering
### Sliding Window Filtering  
The sliding window filtering is another efficient realization of the LPTV system for overlapped in the time domain desired signal and interferences [12]. The serial to parallel (s/p) transformation [10] of the time series can be used for SW filtering. The benefit of s/p representation is provided by the developed matrix mathematics. The discrete-time LPTV sliding window filtering with length \(N\) can be expressed using the simultaneous time shift of both the output and input signals by \(kN\) and the property of the periodic two-dimensional IR:  
\[y(n+kN)=\sum\limits_{m=0}^{N-1}h(n+kN,m+kN)x(m+kN)=\sum\limits_{m=0}^{N-1}h( n,m)x(m+kN) \tag{14}\]  
The compact matrix form of the expression (14) can be obtained using s/p transformation of signal \(y(n)\) to the vector signal \(\mathbf{y}[k]\):  
where  
\[\mathbf{H}=\sum_{n=0}^{N-1}\sum_{m=0}^{N-1}\mathbf{u}_{n}h(n,m)\mathbf{u}_{m}^{T} \tag{16}\]  
\[\mathbf{x}[k]=\sum_{m=0}^{N-1}x(m+kN)\mathbf{u}_{m} \tag{17}\]  
\[\mathbf{u}_{m}=\mathbf{P}^{m}\mathbf{u}_{0} \tag{18}\]  
is \(m\)th unit vector obtained by \(m\) permutations of the initial unit vector \(\mathbf{u}_{0}=(1\;0\;\dots\;0)^{T}\);  
\[\mathbf{P}=\left( \begin{array}{cccc}0&0&\dots&0\\ 1&0&\ddots&\vdots\\ \vdots&\ddots&0&0\\ 0&\dots&1&0\end{array} \right) \tag{19}\]  
is a permutation matrix. The scheme of SW filtering is shown in Figure 3. It consists of the s/p transformation of the input signal \(x(n)\), linear transformation (15) of the parallel vector signal \(\mathbf{x}[k]\) and parallel to serial (p/s) transformation of the parallel vector signal \(\mathbf{y}[k]\) for obtaining the output serial signal \(y(n)\). An example of a two-dimensional impulse response \(h(n,m)\) of an SW filter is shown in Figure 4. The chosen number of samples per cyclic period \(N=3\). The relation between the equivalent FRESH and SW type of LPTV filtering can be obtained through s/p transformation for the corresponding HSR components of the input and output signals:  
\[\mathbf{x}[k]=\sum_{n=0}^{N-1}x(n+kN)\mathbf{u}_{n}=\sum_{\mu=0}^{N-1}\mathbf{ F}^{\mu}\mathbf{\widetilde{x}}_{\mu}[k],\;\mathbf{y}[k]=\sum_{n=0}^{N-1}y(n+kN) \mathbf{u}_{n}=\sum_{\
u=0}^{N-1}\mathbf{F}^{\
u}\mathbf{\widetilde{y}}_{\
u }[k] \tag{20}\]  
Figure 4: Two-dimensional impulse response \(h(n,m)\) of the sliding window filter, \(N=3\). Figure 3: Scheme of sliding window filtering. where  
\[\mathbf{\tilde{x}}_{\mu}[k]=\sum_{n=0}^{N-1}\mathbf{\tilde{x}}_{\mu}(n+kN)\mathbf{ u}_{n},\ \mathbf{\tilde{y}}_{v}[k]=\sum_{n=0}^{N-1}\mathbf{\tilde{y}}_{v}(n+kN)\mathbf{u}_{n},\mathbf{F}=\text{diag}(0,F_{N},\ \ldots,\ F_{N}^{N-1}) \tag{21}\]  
The FRESH filtering also has the following compact matrix form:  
\[ \begin{array}{rl}\mathbf{\tilde{y}}(n)&=\sum\limits_{v=0}^{N-1}\mathbf{ \tilde{y}}_{v}(n)\mathbf{u}_{v}=\sum\limits_{v=0}^{N-1}\sum\limits_{\mu=0}^{N -1}\mathbf{u}_{v}G_{v\mu}(n)*\mathbf{\tilde{x}}_{\mu}(n)\\ &=\sum\limits_{v=0}^{N-1}\sum\limits_{\mu=0}^{N-1}\mathbf{u}_{v}G_{v\mu}(n) \mathbf{u}_{\mu}^{T}*\mathbf{\tilde{x}}_{\mu}(n)\mathbf{u}_{\mu}=\mathbf{G}(n )*\mathbf{\tilde{x}}(n)\end{array}  \tag{22}\]  
where the convolution operation is denoted by symbol \(*\),  
\[\mathbf{G}(n)=\sum\limits_{v=0}^{N-1}\sum\limits_{\mu=0}^{N-1}\mathbf{u}_{v}G _{v\mu}(n)\mathbf{u}_{\mu}^{T},\ \mathbf{\tilde{x}}(n)=\sum\limits_{\mu=0}^{N-1}\mathbf{ \tilde{x}}_{\mu}(n)\mathbf{u}_{\mu} \tag{23}\]  
Figure 5 shows the scheme of the frequency shift filtering. The linear transformation has the form of multi-input to multi-output filtering.