# System-Level Statistical Eye Diagram for Signal Integrity
## 2 Statistical Eye Diagram
Thus, the SBR _V(t)_ is obtained by the following equation:  
\[SBR(t)=IFIT\{FFT\{p(t)\}\times H(f)\} \tag{1}\]  
where,  
\[p(t)=\left\{ \begin{array}{ll}V_{p},&0\leq t\leq 1 UI\\ 0,&t>1 UI\end{array} \right. \tag{2}\]  
Figure 2: **(a)** The worst contour by the PDA method; **(b)** statistical eye diagram by the statistical approach. The PDA provides the inner-most contour of the eye diagram. Thus, limited information is provided. In contrast, the statistical eye diagram provides the probability distribution function (PDF) depending on the sampling time. The color represents the probability depending on the sampling time. The channel response \(H(f)\) is the insertion loss \(S_{21}\) in the frequency domain. Therefore, the ISI noise by a single bit is evaluated with the SBR. The next step is to define the main cursors which are equivalent to the degraded ONE in the current sampling time. Hence, the main cursors have a length of \(1\ U\!I\). Typically, the first and last main cursors have the smallest voltage gap compared to the others to avoid a mismatched voltage in the eye diagram. The timing window for the main cursors is shifted until the first and last cursors have the smallest gap within the given range. The timing window should have a length of \(1\ U\!I\). If not, it violates the definition of an eye diagram. The given range is determined based on the peak value of the SBR. As long as the above timing window includes the peak value, the timing window is shifted left or right. In other words, the main cursors are defined as follows:  
\[X_{main}(t)=SBR(t_{main}\leq t\leq t_{main}+1\ UI) \tag{3}\]  
where \(SBR(t_{main})\approx SBR(t_{main}+1 UI)\). Figure 3 shows the Bit PDF based on the main cursors. After defining the main cursors from the SBR, the rest of the waveforms are defined as the pre- and post-cursors, respectively:  
\[X_{pre}(t)=SBR(t<t_{main}) \tag{4}\]  
\[X_{post}(t)=SBR(t_{main}+1 UI<t) \tag{5}\]  
The defined post-cursors are divided into multiple intervals with a length of \(1\ U\!I\):  
\[X_{post}^{N}=X_{post}(t_{main}+1 UI+(N-1)\times 1 UI<t\leq t_{main}+1 UI+N\times 1 UI) \tag{6}\]  
where \(N\) is a positive integer. Likewise, the divided pre-cursors are also divided as follows:  
\[X_{pre}^{N}=X_{pre}(t_{main}-N\times 1 UI\leq t<t_{main}-(N-1)\times 1 UI) \tag{7}\]  
Figure 3: Bit PDF in the statistical eye diagram. The PDF for the main cursors is from the channel response for bit ONE. The amplitude PDF is defined at the sampling time \(\tau_{sampling}\). The probability density function (PDF) of each cursor is defined as follows:  
\[PDF_{post}^{N}(a|\tau)=\delta(a)+\delta\Big{(}a-X_{post}^{N}(\tau)\Big{)} \tag{8}\]  
where \(0\leq\tau\leq 1 UI\). Consecutive convolutions are applied to the above pre- and post-cursors to calculate the probability distribution function (PDF) of the ISI:  
(9)  
\[PDF_{post}(a|\tau)=PDF_{post}^{1}(a|\tau)\varoccleft PDF_{post}^{2}(a|\tau)\varoccleft PDF_{post}^{ \infty}(a|\tau)\varoccleft PDF_{post}^{\infty}(a|\tau)\varoccleft PDF_{post}  
encoding becomes more beneficial for advanced high-performance systems. Because the intensity of the magnetic and electric fields increases at higher frequencies, the amount of the crosstalk is proportional to the data rate. Thus, crosstalk mitigation becomes essential. The advanced high-performance systems also have more data channels. This means that an analysis of the crosstalk would complicated. In this case, TMDS encoding reduces the crosstalk noise which may simplify the analysis. In order to apply the statistical approach to the encoded channel, the stochastic models for encoders were introduced. The stochastic model represents the behavior of an encoder in terms of probability. The statistical approach commonly uses the SBR which is the simplest channel response. As both encodings focus on the bit transitions, a statistical eye diagram has to be constructed with the bit transitions. Thus, the proposed method was based on the double-edge response (DER) [21].