# Cyclostationary Crosstalk Cancelation in High-Speed Transmission Lines
## 2 Linear Periodic Time-Varying Filtering
### Frequency Shift Filtering  
The output signal of the discrete-time LPTV system can be obtained using the following general expression  
\[y(n)=\sum_{m=-\infty}^{\infty}h(n,m)x(m) \tag{2}\]  
where \(x(m)\) is an input signal of the LPTV system;  
\[h(n,m)=h(n+kN,m+kN) \tag{3}\]  
is a periodic two-dimensional IR with period \(N\).  
The LPTV filtering procedure can be described particularly as frequency shifting and subsequent linear time invariant filtering [16]. The harmonic series representation (HSR) of the signals is used for such description. When \(N\) is a cyclic period of the random cyclostationary process, the harmonic series decomposition provides wide sense stationary components. The equivalent LTI multi-input to multi-output (MIMO) transformation of the components is used for filtering of the overlapping in the frequency domain desired signal and interferences with the corresponding bandwidths [10].  
The discrete Fourier transform (DFT) of \(L\) samples of the input signal is defined by  
\[X(m)=\frac{1}{L}\sum_{n=0}^{L-1}F_{L}^{-nm}x(m) \tag{4}\]  
where \(F_{L}=e^{j2\pi/L}\).  
The frequency shifting of the input signal by \(\mu\frac{L}{N}\) and the following linear time invariant filtering using rectangular frequency response \(\Pi(m)\) is expressed in the frequency domain as follow:  
\[\widetilde{X}_{\mu}(m)=X\bigg{(}m+\mu\frac{L}{N}\bigg{)}\Pi(m),\ \ \ \mu=0,\ldots,N-1 \tag{5}\]  
where  
\[\Pi(m)=U(m)-U\bigg{(}m-\frac{L}{N}\bigg{)} \tag{6}\]  
\(U(m)\) is a unit step function.  
The time domain response is written using inverse DFT:  
\[\widetilde{x}_{\mu}(n)=\sum_{m=0}^{L-1}F_{L}^{nm}\widetilde{X}_{\mu}(m) \tag{7}\]The initial signal can be written using harmonic series representation that is the summation of the orthogonal components:  
\[x(n)=\sum\limits_{\mu=0}^{N-1}F_{N}^{\mu n}\tilde{x}_{\mu}(n) \tag{8}\]  
where \(F_{N}=e^{\beta 2\pi/N}\).  
The single component of the output signal can be obtained using the equivalent discrete-time multi-input to single output (MISO) LTI filtering:  
\[\tilde{y}_{\
u}(n)=\sum\limits_{\mu=0}^{N-1}\sum\limits_{m=-\infty}^{\infty}G _{\
u\mu}(n-m)\tilde{x}_{\mu}(m)=\sum\limits_{\mu=0}^{N-1}G_{\
u\mu}(n)* \tilde{x}_{\mu}(n) \tag{9}\]  
where the convolution operation is denoted by symbol \(*\). The \(\
u\)th Fourier coefficient of the discrete time two-dimensional function\(g_{\mu}(n^{\prime},n)\) is given by  
\[G_{\
u\mu}(n)=\frac{1}{N}\sum\limits_{n^{\prime}=0}^{N-1}g_{\mu}(n^{\prime},n )F_{N}^{-\
u n^{\prime}} \tag{10}\]  
where \(g_{\mu}(n^{\prime},n)\) is periodic along time \(n^{\prime}\) and non-periodic along time \(n\). The function \(g_{\mu}(n^{\prime},n)\) can be obtained by changing variables \(n=n^{\prime}\) and \(m=n^{\prime}-n\) in the periodic two-dimensional IR \(h_{\mu}(n,m)\) that is the \(\mu\)th version of the IR \(h(n,m)\) frequency shifted by \(\mu\frac{L}{N}\):  
\[g_{\mu}(n^{\prime},n)=g_{\mu}(n^{\prime}+kN,n)=h_{\mu}(n^{\prime},n^{\prime}-n) \tag{11}\]  
where  
\[h_{\mu}(n,m)=h_{\mu}(n+kN,m+kN)=h(n,m)F_{N}^{\mu n} \tag{12}\]  
The resulting output of the LPTV system is written as combination of the orthogonal components:  
\[y(n)=\sum\limits_{\
u=0}^{N-1}F_{N}^{\
u n}\tilde{y}_{\
u}(n) \tag{13}\]