# Analysis and Modeling of Mueller-Muller Clock and Data Recovery Circuits
## 4 Jitter Analysis of MM-CDR  
As discussed in Section 3.2 and [16], the MM-CDR loop characteristics vary with the error jitter \(\sigma_{\text{ER}}\). To obtain an accurate MM-CDR model, the exact error jitter is required. One possible way is to accurately simulate the entire system at one time, measure the error jitter, and then build a linear model. Nevertheless, this will greatly lengthen the design cycle. In contrast, if we estimate the error jitter in advance based on the circuit characteristics, we can not only obtain accurate CDR performance metrics, but also shorten the design cycle. Here, we focus on the effect of the two main sources of jitter on the error jitter: input data jitter and PI quantization noise [17]. In this work, we assume that there is a clean channel and the noise of the input data is generated by the oscillator at the transmitter. For simplicity, we only consider the main noise of the oscillator (i.e., thermal noise). Its noise power spectral density (PSD) as a function of the offset frequency \(f\) is given by [18],  
\[S_{\text{DAT}}=\frac{n}{f^{2}}, \tag{18}\]  
where \(n\) is the coefficient of the oscillator thermal noise. The period jitter of the input data \(\sigma_{\text{DAT}}\) can be obtained by integrating the phase noise with the transfer function \(\left(1-z^{-1}\right)\)[19]. \[\sigma_{\text{DAT}}{}^{2}=(\frac{T}{2\pi})^{2}\int_{0}^{\infty}\left|1-z^{-1} \right|^{2}\times S_{\text{DAT}}df\approx\frac{nT^{3}}{2}. \tag{19}\]  
Substituting (19) into (18), the noise of input data can be written as  
\[S_{\text{DAT}}=\frac{2\sigma_{\text{DAT}}{}^{2}}{T^{3}f^{2}}. \tag{20}\]  
Figure 8: Jitter transfer of different input jitters. Figure 9: Jitter tolerance of different input jitters. According to [20], in order to reduce the loop bandwidth and jitter peaking during design, we usually provide a larger damping factor \(\zeta\), so the transfer function can be approximated by that of a one-pole system. Using time instead of phase, the error transfer function is  
\[\frac{t_{\text{ER}}}{t_{\text{IN}}}=\frac{s}{s+w_{\text{-3 dB}}}\left(w_{\text{- 3 dB}}=K_{\text{MD}}\alpha,\alpha=\frac{K_{\text{V}}K_{p}T_{\text{PI}}}{2^{N_{ \text{SR}}}N_{\text{PI}}T_{\text{DLF}}}\right)\text{.} \tag{21}\]  
where \(w_{\text{-3 dB}}\) is the bandwidth of MM-CDR; \(T_{\text{PI}}\) is the period of phase interpolator's input clock; \(T_{\text{DLF}}\) is the period of digital filter. Combining (20) and (21), the error jitter contributed by the input data jitter is  
\[\sigma_{\text{ER,in}}^{2}=\frac{T^{2}}{4\pi^{2}}\int_{0}^{\infty}\frac{2\sigma _{\text{DAT}}^{2}}{T^{3}f^{2}}\bigg{|}\frac{s}{s+w_{\text{-3 dB}}}\bigg{|}^{2} df=\frac{\sigma_{\text{DAT}}^{2}}{2 Tw_{\text{-3 dB}}}\text{.} \tag{22}\]  
According to Section 2.2, we indicate with \(\sigma_{\text{ER,in}}\) the rms value of the jitter out of the MMPD, we can write:  
\[K_{\text{MD}}=\lambda\frac{2\sigma_{\text{ER,in}}^{2}-\phi_{\text{m}}^{2}}{ \sqrt{2\pi}\sigma_{\text{ER,in}}^{3}}\text{.} \tag{23}\]  
Note that we consider the error jitter \(\sigma_{\text{ER,in}}\) caused by the input data noise as the total error jitter, and in Section 5 we will give the reasonableness analysis. Combining (21)-(23), we have,  
\[\sigma_{\text{ER,in}}\approx\frac{\sigma_{\text{DAT}}^{2}\sqrt{2\pi}+\sqrt{2 \pi\sigma_{\text{DAT}}^{4}+32\phi_{\text{m}}^{2}T^{2}\alpha^{2}\lambda^{2}}}{ 8T\alpha\lambda}\text{.} \tag{24}\]  
Moreover, the quantization noise of PI is not negligible, and we consider that the PI oscillates between two phases. Assuming that the data phase has a delay of \"\(x\)\" with respect to the recovery clock, the phase of the recovery clock with respect to the previous moment when PI is rotated is \(\Delta=T_{\text{PI}}/N_{\text{PI}}\), where \(\Delta\) is the minimum step of PI.