# Analysis and Modeling of Mueller-Muller Clock and Data Recovery Circuits
## 2 Mueller-Muller Phase Detector
### Linear Gain of MMPD  
Just as it has been done to the BBPD in [7; 8; 14], the binary output characteristic of the MMPD is smoothed out by the jitter inherent in the input data and the recovery clock. First, we assume that the phase error (i.e., the phase difference between the input data and the recovery clock) consists of two terms: one is the deterministic phase error \(u_{\rm e,U}\) (e.g., \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**D (n \(-\) 2)** & **D (n \(-\) 1)** & **D (n)** & **E (n \(-\) 2)** & **E (n \(-\) 1)** & **E (n)** & **Phase Info.** \\ \hline
0 & 0 & 1 & x & 1 & 0 & LATE \\
0 & 0 & 1 & 1 & 0 & x & EARLY \\
0 & 1 & 1 & x & 0 & 1 & LATE \\
0 & 1 & 1 & 0 & 1 & x & EARLY \\
1 & 0 & 0 & x & 0 & 1 & LATE \\
1 & 0 & 0 & 0 & 1 & x & EARLY \\
1 & 1 & 0 & x & 1 & 0 & LATE \\
1 & 1 & 0 & 1 & 0 & 0 & EARLY \\  & & & Other & & & No Info. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Truth table of MMPD. Figure 2: MMPD phase detection waveform. deterministic ISI or sinusoidal jitter) and the other is the random phase error \(u_{\text{e,G}}\) (e.g., Gaussian white noise) [9]. That is:  
\[u=\alpha\times u_{\text{e,G}}+\beta\times u_{\text{e,U}}. \tag{1}\]  
where \(u\) is the average output of the MMPD, and \(\alpha\), \(\beta\) are the weights of the random phase error and the deterministic phase error, respectively. When the phase error between the input data and the recovery clock is dominated by the random error term, we abstract the input jitter using a Gaussian probability density function with a standard deviation of \(\sigma\) and a mean value of 0. Combining the operation of the MMPD in the clock and data recovery circuit, we assign \"1\" to phase \"late\" and \"\(-\)1\" to phase \"early\", and then sum the positive and negative samples with weights given by the probabilities of their occurrence. Therefore, the average output of the MMPD is:  
\[u=(1)P_{\text{r}}(late|\phi)+(-1)P_{\text{r}}(early|\phi). \tag{2}\]  
In Figure 3a, \(\phi\) indicates the average phase difference between the input data and the recovery clock, \(2\phi_{m}\) is the drift bit width, namely the part of the input data whose voltage amplitude is greater than \(+V_{\text{ref}}\) or less than \(-V_{\text{ref}}\). According to the Gaussian distribution probability density function, we obtain  
\[P_{\text{r}}(late|\phi)=\frac{1}{\sigma\sqrt{2\pi}}\int_{\phi_{m}}^{\infty} \exp(\frac{-(x-\phi)^{2}}{2\sigma^{2}})dx. \tag{3}\]  
Let \(y=\frac{x-\phi}{\sqrt{2\sigma}}\) and \(f(y)=e^{-y^{2}}\approx 1-y^{2}\), then  
\[P_{r}(late|\phi)\approx\frac{1}{2}-(\frac{\phi_{\text{m}}-\phi}{\sqrt{2\pi \sigma}}-\frac{(\phi_{\text{m}}-\phi)^{3}}{6\sqrt{2\pi\sigma^{3}}}). \tag{4}\]  
Similarly, we can have  
\[P_{\text{r}}(early|\phi)\approx\frac{1}{2}-(\frac{\phi_{\text{m}}+\phi}{\sqrt{2 \pi\sigma}}-\frac{(\phi_{\text{m}}+\phi)^{3}}{6\sqrt{2\pi\sigma^{3}}}). \tag{5}\]  
Substituting (4) and (5) into (2), we get  
\[u_{\text{e,G}}=(1)P_{\text{r}}(late|\phi)+(-1)P_{\text{r}}(early|\phi)\approx \frac{2\sigma^{2}-\phi_{\text{m}}^{2}}{\sqrt{2\pi\sigma^{3}}}\phi. \tag{6}\]  
When the phase error between the input data and the recovery clock is dominated by the deterministic error term, we abstract the input jitter using a uniform distribution probability density function with a standard deviation of \(\sigma\) and a mean value of 0. From Figure 3b, we get  
\[P_{\text{r}}(late|\phi)=\int_{\phi_{\text{m}}}^{\Delta+\phi}\frac{1}{2\Delta} dx=\frac{\Delta+\phi-\phi_{\text{m}}}{2\Delta}. \tag{7}\]  
Since the variance of the uniform distribution can be expressed as  
\[\sigma^{2}=E[x^{2}]-(E[x])^{2}=\frac{\Delta^{2}}{3}. \tag{8}\]  
Substituting (8) into (7), then  
Figure 2:0Similarly, we can attain  
Figure 2:1  
Substituting (9) and (10) into (2), we get  
Figure 2:2  
Note that the detection density of the input data is not considered in (6) and (11). If the detector density of the input data is \(\lambda\), the gain of the MMPD based on Gaussian jitter or uniform jitter is \(\lambda u\).