# Analysis and Modeling of Mueller-Muller Clock and Data Recovery Circuits
## 3 Small Signal Model of MM-CDR  
### Linear Voting Model  
Majority voting is implemented in CDR by adding the N-bit parallel PD output together and then taking the sign of results. Compared to the boxcar filter, voting is able to reduce the loop delay and lower the output noise of the MMPD. Hence, the digital CDR generally adopts voting as the decimation operation.  
The gain of voting relies on the MMPD output and needs to be analyzed jointly with the MMPD. Taking a bank of four Mueller-Muller phase detectors via voting as an example, we give all the possible inputs leading to the output results in Table 2.  
If the input jitter is Gaussian, the probability of the MMPD output \(0\), \(-1\), \(1\) is determined when the phase difference is fixed. According to the principle of probability statistics, the probability when the voting output is \"late\" (i.e., 1) is  
\[P_{\mathit{vote}}(late)=C_{4}^{1}P_{0}^{3}P_{la}+C_{4}^{2}P_{0}^{2}P_{la}^{2}+C_ {4}^{2}C_{2}^{1}P_{la}^{2}P_{0}P_{ea}+C_{4}^{1}P_{la}^{3}P_{ea}+C_{4}^{1}P_{la} ^{3}P_{0}+P_{la}^{4}. \tag{12}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Voting Output** & \multicolumn{8}{c}{**All Circumstances**} \\ \hline
1 & 1000 & 1100 & 110\(-\)1 & 111\(-\)1 & 1110 & 1111 \\ \(-1\) & \(-1000\) & \(-1\)\(-\)100 & \(-1\)\(-\)101 & \(-1\)\(-\)1\(-\)11 & \(-1\)\(-\)1\(-\)10 & \(-1\)\(-\)1\(-\)1\(-\)1 \\
0 & 0000 & 001\(-\)1 & 11\(-\)1\(-\)1 & & & \\ \hline \end{tabular} Note: All inputs in the table for the voting are in no particular order.  
\end{table}
Table 2: Input and output of vote.  
Figure 4: Simulated gain of MMPD. (**a**) Gaussian jitter at small phase difference, (**b**) uniform jitter at small phase difference, (**c**) Gaussian jitter at large phase difference, and (**d**) uniform jitter at large phase difference.  
where \(P_{0}\), \(P_{lu}\), and \(P_{ea}\) are the probabilities when the MMPD output are \"no decision\" (i.e., 0), \"late\" (i.e., 1), and \"early\" (i.e., \(-1\)) respectively.  
\[P_{vote}(\text{early})=C_{4}^{1}P_{0}^{3}P_{ea}+C_{4}^{2}P_{ea}^{2}P_{0}^{2}+C_{ 4}^{2}C_{2}^{1}P_{ea}^{2}P_{0}P_{la}+C_{4}^{1}P_{ea}^{3}P_{la}+C_{4}^{1}P_{ea}^{ 3}P_{0}+P_{ea}^{4}. \tag{13}\]  
Finally, we obtain the united gain of the MMPD and the voting:  
\[u=(1)P_{vote}(late)+(-1)P_{vote}(early). \tag{14}\]  
To compare the impact of the voter with Boxcar on the loop and get the gain of the voter, we simulate the joint gain of MMPD and Vote, as shown by the red line in Figure 5, whose gain can be expressed as \(K_{\text{MD}}\times K_{\text{V}}\). At the same time, we also simulate the joint gain of MMPD and boxcar filter, as depicted by the blue line in Figure 5, whose gain can be expressed as \(K_{\text{MD}}\times 1\). The gain ratio of these two cases can be obtained through simulation as well, and finally the gain of the voter can be derived. Figure 5a,b show that the MMPD-based voting is greatly affected by jitter and drift bit width, with the gain of the voter around 0.54 when the jitter is much larger than the drift bit width, and around 0.78 when the drift bit width is close to the jitter. Therefore, to perform analysis and fully understand the impact of voting, it is important to obtain accurate jitter and drift bit width.