# Cyclostationary Crosstalk Cancelation in High-Speed Transmission Lines
## 4 Experimental Results
### Correlation Analysis of the Measured Signals  
The crosstalk voltage can be assumed as an additive and an independent component of the received signal. The aggressor's signal transmitted in the parallel transmission line has the same cyclic period as the victim's signal [12]. Time-domain measurement of the signals in the transmission lines can be implemented using digital oscilloscopes with sampling rates above decades of gigahertz [18]. The signals measured by real-time oscilloscope were characterized directly in the time-domain. The impulse response of the periodic filter is estimated using time-domain characteristics. The model of the registered signal \(v(n)\) in the assumption of observing the signal of interest \(d(n)\) corrupted by additive interference \(c(n)\) and stationary noise \(w(n)\) is given by:  
\[v(n)=d(n)+c(n)+w(n) \tag{37}\]  
The correlation matrix of the measured signal \(v(n)\) can be expressed as the sum of the correlation matrices \(\mathbf{R}_{dd}\), \(\mathbf{R}_{cc}\) and \(\mathbf{R}_{ww}\) for corresponding jointly uncorrelated components of the signal \(v(n)\):  
\[\mathbf{R}_{vv}=\mathbf{R}_{dd}+\mathbf{R}_{cc}+\mathbf{R}_{ww}=\mathbf{S} \mathbf{\Sigma}\mathbf{S}^{T}+\mathbf{R}_{ww} \tag{38}\]  
where \(\mathbf{S}\mathbf{\Sigma}\mathbf{S}^{T}\) is a decomposition of the correlation matrix for cyclostationary sources using principal components; \(\mathbf{S}=(\mathbf{s}_{d}\,\mathbf{s}_{c})\) is a composite matrix of the eigenvectors \(\mathbf{s}_{d}\) and \(\mathbf{s}_{c}\) of the correlation matrices \(\mathbf{R}_{dd}\) and \(\mathbf{R}_{cc}\), respectively; \(\mathbf{\Sigma}=\text{diag}(\lambda_{d},\lambda_{c})\) is a diagonal matrix with diagonal elements corresponding to the eigenvalues \(\lambda_{d}\) and \(\lambda_{c}\) of the correlation matrices \(\mathbf{R}_{dd}\) and \(\mathbf{R}_{cc}\), respectively; \(\mathbf{R}_{ww}\) is diagonal correlation matrix of wide sense stationary noise. The crosstalk canceler can be expressed as cyclic Wiener filter using equation (35):  
\[\mathbf{C}=\mathbf{R}_{dd}\mathbf{R}_{vv}^{-1} \tag{39}\]  
where  
\[\mathbf{R}_{dd}=\mathbf{R}_{dd}^{T}=\mathbf{R}_{dd} \tag{40}\]  
Figure 9: (**a**) Wiring diagram of the measurement setup; (**b**) photo of the device under test. is a cross correlation matrix of the signal of interest and the received signal;  
\[\mathbf{R}_{vv}^{-1}=\left(\mathbf{S}\Sigma\mathbf{S}^{T}+\mathbf{R}_{ww}\right) ^{-1}=\mathbf{R}_{ww}^{-1}-\mathbf{R}_{ww}^{-1}\mathbf{S}\Big{(}\mathbf{\Sigma} ^{-1}+\mathbf{S}^{T}\mathbf{R}_{ww}^{-1}\mathbf{S}\Big{)}^{-1}\mathbf{S}^{T} \mathbf{R}_{ww}^{-1} \tag{41}\]  
is the inverse of the correlation matrix \(\mathbf{R}_{vv}\) obtained using the matrix inversion lemma or Woodbury identities [19]. The cyclic Wiener filter \(\mathbf{C}\) is the LPTV filter used for the waveform estimation of the signal of interest \(d(n)\). Figures 10 and 11 show the estimated correlation matrices of the cyclostationary sources, the evaluated eigenvalues and scaled eigenvectors corresponding to first eigenvalues of correlation matrices for bit rates 0.4 Gbit/s and 2.5 Gbit/s respectively. Figure 12 shows the cyclic Wiener filter \(\mathbf{C}\) for signals with bit rates 0.4 Gbit/s and 2.5 Gbit/s respectively. The calculated weighting matrix \(\mathbf{C}\) for the data signal estimator has a dumped region (\"canyon\") due to the peak of the crosstalk correlation matrix. Except this region, the weighting matrix repeats the flat form of the data signal correlation matrix. The signal estimation can be physically interpreted as a balanced rejection of fluctuations caused by crosstalk concentrated over time in the middle of the data signal single bit response. As a result, it achieved immunity of the output signal to the crosstalk fluctuations and the reception sufficient part of the data signal to resist the noise. Figure 10: (**a**) The correlation matrix \(\mathbf{R}_{dd}\) of the victim signal without crosstalk; (**b**) the correlation matrix \(\mathbf{R}_{cc}\) of the crosstalk interference signal.