## 4 MMPD Performance Analysis  
### MMPD Phase-Detection Gain Analysis  
The MMPD outputs the phase-detection result in the form of levels, the early output low level is recorded as \"\(-1\)\", and the late output high level is recorded as \"\(+1\)\". Under the DB PAM4 data, the relationship between the sampling phase of the phase detector clock \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Mode** & **erdata\({}_{n-2}\)** & **erdata\({}_{n-1}\)** & **errup\({}_{n-1}\)** & **erlow\({}_{n-1}\)** & **erdata\({}_{n}\)** & **Y\({}_{E}\)** & **Y\({}_{L}\)** & **Phase Info** \\ \hline
000 & x & x & x & x & 0 & 0 & No Info \\
001 & x & 0 & 0 & 0 & x & 0 & 0 & No Info \\
001 & x & 1 & 1 & 0 & x & 0 & 1 & LATE \\
001 & x & 1 & 0 & 1 & x & 1 & 0 & EARLY \\
100 & x & 0 & 0 & 0 & x & 0 & 0 & No Info \\
100 & x & 1 & 1 & 0 & x & 0 & 1 & EARLY \\
100 & x & 1 & 0 & 1 & x & 1 & 0 & LATE \\
011 & x & 0 & 0 & 0 & 0 & 0 & No Info \\
011 & x & 1 & x & x & x & 1 & 0 & EARLY \\ \hline \hline \end{tabular}
\end{table} 
Table 4: MMPD phase-detection truth table for DB PAM-4. Figure 11: Logic gate circuit of Data Selector. and the output voltage is shown in Figure 12. The phase-detection boundary is determined by the voltage thresholds \(+V_{ref}\) and \(-\mathrm{V}_{ref}\) of the error sampler, and the voltage is converted into phase information to obtain the decision phases \(-\varphi_{ref}\) and \(+\varphi_{ref}\). Due to the presence of jitter, the output voltage varies with the phase difference between clock and data. The output voltage can replace by the average value of the output voltage. Such a jitter generally includes Gaussian jitter, uniform jitter, and sinusoidal jitter. When the jitter is Gaussian with mean \(\varphi_{0}\) and variance \(\sigma\), the average output voltage is represented by \(\mu\). The early probability is \(Pr(early|\varphi_{0})\), and the late probability is \(Pr(late|\varphi_{0})\). When the average value of the phase difference is \(\varphi_{0}\), the expression of the average output voltage is \(\mu\)[16]:  
\[\mu=-1\times Pr(early|\varphi_{0})+1\times Pr(late|\varphi_{0}) \tag{5}\]  
According to the Gaussian distribution obtained:  
\[Pr(late|\varphi_{0})=\frac{1}{\sigma\sqrt{2\pi}}\int_{\varphi_{ref}}^{\infty}e ^{\frac{-(x-\varphi_{0})^{2}}{2\sigma^{2}}}dx \tag{6}\]  
Similarly, obtained:  
\[Pr(early|\varphi)\text{=}\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{-\varphi_{ ref}}e^{\frac{-(x-\varphi_{0})^{2}}{2\sigma^{2}}}dx \tag{7}\]  
Substituting (6), (7) into (5) obtained:  
\[\mu=-\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{-\varphi_{ref}}e^{\frac{-(x- \varphi_{0})^{2}}{2\sigma^{2}}}dx+\frac{1}{\sigma\sqrt{2\pi}}\int_{\varphi_{ ref}}^{\infty}e^{\frac{-(x-\varphi_{0})^{2}}{2\sigma^{2}}}dx \tag{8}\]  
Average output curve of variable variance when \(\varphi_{ref}\) = 0.09 UI, as shown in Figure 13. When the phase is in the linear region, the equation can be simplified to obtain the gain in the linear region of the MMPD [8,16]. Figure 12: Relationship between clock sampling phase and PD output voltage. \[\text{let}\ y=\frac{x-q_{0}}{\sqrt{2}\sigma}\text{, then}\] Figure 11:0 Figure 11:1 Figure 11:2  
Using the first-order Taylor formula to approximate, there is some error when calculating the gain in the linear region of the MMPD. Since the approximate area of the first-order Taylor is limited, getting an accurate approximation of the gain, increasing the Taylor order can improve the calculation accuracy and reduce the error. Based on Taylor series, let \(e^{-y^{2}}\approx 1-y^{2}\), \(e^{-y^{2}}\approx 1-y^{2}+\frac{1}{2}y^{4}\), then  
Figure 11:3  
Figure 11:4  
When the error range is less than 0.01, the two data are considered equal, and the algebraic solution of y is approximately 0.381, 0.636. The equivalent integral range corresponding to \(\varphi_{0}\) is \(\varphi_{ref}\pm 0.538\sigma\), \(\varphi_{ref}\pm 0.899\sigma\). Within the same error range, the second-order Taylor expansion has a larger phase margin than the first-order expansion.