## 1 Introduction
### Statistical Output Responses (SORs) Derivation Based on Recursive Convolution  
To derive a statistical eye-diagram, including various noises affecting SI/PI, we must define main cursor-PDFs and ISI PDFs. In Figure 6, voltage variation PDFs extracted from the pull-up output response PDF are relocated to the quantized voltage-time planes. In this paper, the main cursor PDF matrixes are defined as Equation (3), and the ISI PDF matrixes are defined as Equation (4). In Figure 6, the main-cursor PDFs and ISI PDFs are also marked. \[f_{\text{m}}^{(1)},\text{ where m}=01,10,00,\&\ 11 \tag{3}\]  
\[f_{\text{m}}^{(\text{k})},\text{ where m}=01,10,00,\&\ 11\text{ and k}=2,3,\ldots,\infty \tag{4}\]  
The responses remaining at the main cursors generating distortions are regarded as an ISI. The ISI can be generated by not only pull-up or pull-down transitions but also steady-state output responses fluctuated by the SSO buffers. Without the SSO buffers, the PDFs of an individual ISI can be represented as the Kronecker delta functions. However, due to voltage uncertainties generated by the SSO buffers, the Kronecker delta functions should be replaced by the ISI-PDFs defined in (4). The total ISI-PDFs, including the various noises, can be calculated by a recursive convolution between ISI-PDFs affecting each other along the voltage axis at every sample time. Results of recursive convolution between \begin{table}
\begin{tabular}{c c c} \hline  & \(\mathbf{P_{01\text{ (or 10)}-\text{DBI}}(N_{01},N_{10},N_{00},N_{11})}\) & \(\mathbf{P_{00\text{ (or 11)}-\text{DBI}}(N_{01},N_{10},N_{00},N_{11})}\) \\ \hline \multirow{2}{*}{Case 1} & If \(N_{01}+N_{10}>N_{T}/2-1\) & If \(N_{01}+N_{10}>N_{T}/2\) \\  & \(P_{01-\text{DBI}}(N_{01},N_{10},N_{10},N_{11})=0\) & \(P_{01-\text{DBI}}(N_{01},N_{10},N_{00},N_{11})=0\) \\  & If \(N_{01}+N_{10}=N_{T}/2-1\) & If \(N_{01}+N_{10}=N_{T}/2\) \\  & \(P_{01-\text{DBI}}(N_{01},N_{10},N_{00},N_{11})\) & \(P_{01-\text{DBI}}(N_{01},N_{10},N_{00},N_{11})=P_{01}(N_{01},N_{10},N_{00},N_{ 11})\) \\  & \(P_{01}(N_{01},N_{10},N_{00},N_{11})\) & \(P_{01}(N_{01},N_{10},N_{00},N_{11})\) \\  & If \(N_{01}+N_{10}<N_{T}/2-1\) & If \(N_{01}+N_{10}<N_{T}/2\) \\  & \(P_{01-\text{DBI}}(N_{01},N_{10},N_{00},N_{11})=P_{01-\text{DBI}}(N_{01},N_{10},N_{00},N_{11})=P_{01}(N_{01},N_{10},N_{00},N_{11})+P_{00}(N_{00},N_{11},N_{01 },N_{11})\) \\ \hline \end{tabular}
\end{table} 
Table 1: Occurrence probability equation summary when DBI coding is adopted. ISI-PDFs at the \(\mathrm{k}^{\text{th}}\) previous unit intervals (UIs) are summarized in Equations (5)-(8). In the (5)-(8), \(\mathrm{ISI_{m}}^{-(\mathrm{k})}(t,v)\), where \(\mathrm{m}=00,\ 11,\ 01,\ \mathrm{and}\ 10\), is ISI-PDF at the \(\mathrm{k}^{\text{th}}\) previous UIs, which can be calculated by taking the convolution between the self-generated ISI, \(f_{\mathrm{m}}^{(\mathrm{k}+1)}\) and ISI-PDFs until the \(\left(\mathrm{k}+1\right)^{\text{th}}\) previous UIs, \(\mathrm{ISI_{m}}^{-(\mathrm{k}+1)}(t,v)\). In this article, ISI-PDFs are defined as (4), where the \(\mathrm{k}\) is counted from 2. Therefore, the self-generated ISIs at the \(\mathrm{k}^{\text{th}}\) previous UIs are defined as \(f_{\mathrm{m}}^{(\mathrm{k}+1)}\), not the \(f_{\mathrm{m}}^{(\mathrm{k})}\). In (5)-(8), new mathematical operators Shift\({}_{\text{0}}\) and Shift\({}_{\text{1}}\) are adopted, which shifts the ISI-PDFs to the logic state \"zero\" or \"one,\" since it is important to redefine the location of the ISI-PDFs to the true voltage level of logic \"one (\(V_{\text{H}}\))\" and \"zero (\(V_{\text{L}}\))\" after the convolution [16]. For example, ISI-PDFs in the logic state \"zero\" should remain in the logic \"zero\" state, which is \(V_{\text{L}}\). Therefore, in (5) and (7), Shift\({}_{\text{0}}\) operator is used to relocate the PDFs to the \(V_{\text{L}}\). The ISI-PDFs in the logic \"one\" should remain in the logic \"one\" state, which is \(V_{\text{H}}\). Therefore, in (6) and (8), Shift\({}_{\text{1}}\) operator is used to relocate the PDFs to the \(V_{\text{H}}\).