# Cyclostationary Crosstalk Cancelation in High-Speed Transmission Lines
###### Abstract  
The theoretical and experimental evaluation of the cyclostationary random data transferring process corrupted by the individually and jointly cyclostationary crosstalk interference is presented. The interference and the message signals were measured by the real time digital oscilloscope. Autocorrelation functions were evaluated by synchronous cyclic averaging procedure. The analyzed periodic two-dimensional impulse response of the time-varying filter allows to obtain the output random process with the same cyclic frequency at the output of the filter by separation of orthogonal stationary waveforms constituting the input cyclostationary random process (CSRP). The filtering of the measured random process was implemented by the cyclic Wiener filter. The evaluation of the two-dimensional autocorrelation function and eye diagrams at the output of the cyclic Wiener filter showed significant reduction of the independent interference components in the estimated message signal. stochastic processes; time-domain analysis; crosstalk interference; electromagnetic compatibility; cyclic Wiener filter  
The bit error rate (BER) can be estimated by statistical processing of the measured or simulated waveforms. Traditionally in communications, performance is characterized by the parameter of the BER defined as the relative frequency of the error:  
\[BER=\lim_{n_{\text{Tot}}\rightarrow\infty}\frac{n_{Err}}{n_{\text{Tot}}} \tag{1}\]  
where \(n_{Err}\) is the number of bit errors in the received data, \(n_{\text{Tot}}\) is the total number of transmitted bits. The BER parameter is essentially the probability of the bit error induced by the noise and interference accompanying the process of data transferring. The BER plot shows the eye diagram opening for different sampling phases during the unit interval [12]. The Figure 1 shows an example of the ordinary statistical processing of waveforms with bit rate 1 Gbit/s captured using the digital oscilloscope [13]. The synchronous ensemble of cyclostationary signals can be represented by eye diagrams. Figure 1a,c present eye diagrams of measured signals without and with crosstalk interference, respectively, for the possible sampling phase during the unit interval with the duration of 1 ns. The crosstalk degrades the signal waveform decreasing the eye-opening width in the middle of the eye diagram. The corresponding BER plots are shown in Figure 1b,d. These plots show the quantity of eye-opening width for the given BER level. The green and yellow contours margin the BER level \(10^{-4}\) for all sampling phases. The log-scaled BER plot can be viewed using a cross-section for the sampling phase corresponding to the middle of the signal bit response (white dash-dot vertical line). Figure 2 shows the comparison of the BER plots for signals without and with crosstalk interference [14]. The eye-opening width for the signal with crosstalk is 0.924 V.  
The characterization based on the BER for the estimated signals using the minimum mean square error criterion is demonstrative for the performance comparison. In practice the reference level for the chosen figure of merit is provided by signal without crosstalk. The framework of the virtual two channel experiment without and with crosstalk is used for the synthesis and analysis of the signal estimation algorithm. Figure 1: (**a**) Eye diagram of the victim signal without crosstalk; (**b**) BER diagram of the victim signal without crosstalk; (**c**) eye diagram of the victim signal with crosstalk; (**d**) BER diagram of the victim signal with crosstalk. Bit rate is 1 Gbit/s. Principal component analysis (PCA) is used to investigate the measured data. The principal components can be computed by performing an eigenvalue decomposition of the data correlation matrix. The crosstalk cancelation can be viewed through the cyclostationary signal minimum MSE estimation. The key factor of the estimation is the evaluated principal components of the victim signal and crosstalk using measured data.