## 2 Review of Phase-Locked Loops
### Computational Delay
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Closed-Loop Transfer Function:** & **F(s) Transform** & **F(s) Transform** & \multicolumn{3}{c}{**N(s) Transform Method**} \\ \cline{3-6}
**H(s)**\(\frac{N(s)(s)}{s_{1}\times N(s)(s)}\) & **Method** & \multicolumn{2}{c}{**Step-invariant**} & **Impulse-invariant** & **Bilinears**\(\rightarrow\frac{1}{2}\frac{1-3}{2\pi\lambda}\) \\  & & \(s\leftarrow\frac{1}{2}\) & & \(s\leftarrow\frac{1}{2}\) & **Bilinears**\(\rightarrow\frac{1}{2}\frac{1-3}{2\pi\lambda}\) \\ \hline First-order loop & & \(\beta_{0}=T_{\rm orb}\) & \(\beta_{1}=T_{\rm orb}\) & \(\beta_{1}=T_{\rm orb}\) & \(\beta_{1}=T_{\rm orb}\) \\ \(H_{1}(s)\) & \(\frac{\alpha_{1}}{s_{1}\times N(s)}\) & - & \(\alpha_{1}=T_{\rm orb}+1\) & \(\alpha_{1}=T_{\rm orb}+2\) \\ \(\alpha_{0}\) & \(\alpha_{1}=48\) & & \(\alpha_{0}=-1\) & \(\alpha_{0}=-1\) & \(\alpha_{0}=T_{\rm orb}-2\) \\ \hline \multirow{6}{*}{Second-order loop} & & \(\beta_{1}=\alpha_{0}T_{\rm orb}\) & \(\beta_{2}=\alpha_{0}T_{\rm orb}\) & \(\beta_{2}=\alpha_{0}T_{\rm orb}\) \\  & & \(\beta_{0}=T_{\rm orb}^{*}-\alpha_{2}T_{\rm orb}\) & \(\beta_{1}=T_{\rm orb}^{*}-\alpha_{2}T_{\rm orb}\) & \(\beta_{1}=T_{\rm orb}^{*}-\alpha_{2}T_{\rm orb}\) \\  & & \(\beta_{1}=1\) & \(\alpha_{1}=T_{\rm orb}^{*}-\alpha_{2}T_{\rm orb}+1\) & \(\beta_{2}=T_{\rm orb}^{*}-\alpha_{2}T_{\rm orb}-2\) & \(\beta_{1}=T_{\rm orb}^{*}-\alpha_{2}T_{\rm orb}+2\) \\  & & \(\alpha_{1}=T_{\rm orb}^{*}-\alpha_{2}T_{\rm orb}+1\) & \(\alpha_{0}=1\) & \(\alpha_{0}=T_{\rm orb}^{*}-\alpha_{2}T_{\rm orb}+2\) \\ \hline \multirow{6}{*}{Second-order loop} & & \(\beta_{1}=T_{\rm orb}^{*}+\alpha_{2}T_{\rm orb}\) & \(\beta_{2}=T_{\rm orb}^{*}+\alpha_{2}T_{\rm orb}\) & \(\beta_{2}=T_{\rm orb}^{*}+\alpha_{2}T_{\rm orb}\) \\  & & \(\beta_{0}=-\alpha_{2}T_{\rm orb}\) & \(\beta_{1}=-\alpha_{2}T_{\rm orb}\) & \(\beta_{1}=T_{\rm orb}^{*}\) \\ \(H_{2}(s)\) & \(\frac{\alpha_{0}}{s_{1}\times N(s)}\) & \(\beta_{1}=1\) & \(\alpha_{1}=T_{\rm orb}^{*}+\alpha_{2}T_{\rm orb}-2\) & \(\beta_{2}=T_{\rm  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Closed-Loop Transfer Function:**} & \multirow{2}{*}{**F(s) Transform**} & \multirow{2}{*}{**Step-Invariant**} & \multicolumn{2}{c}{**N(s) Transform Method**} \\ \cline{3-5}  & & & **Step-Invariant** & **Impulse-Invariant** & **Bilinears\(-\frac{1}{1+1}\) \\ \cline{3-5}  & & & \(s\gets-\frac{1}{1+1}\) & \(s\gets-\frac{1}{1+1}\) & **Bilinears\(-\frac{1}{1+1}\)** \\ \hline \multirow{9}{*}{Third-order loop} & \multirow{9}{*}{Impulse-invariant \(s\gets 1\)} & \(\beta_{1}=T_{0}u_{0}^{3}+a_{3}T_{0}^{4}+b_{3}T_{0}u_{0}^{3}+b_{3}T_{0}u_{0}\) & \(\beta_{1}=T_{0}^{4}a_{3  
The computational delay can be modeled as an additional delay unit for the amount of \(t_{\rm D}\) located before the NCO [14], which results in the modification of the NCO transfer function \(N(z)\) as:  
\[N_{\rm D}(z)=z^{-\frac{t_{\rm D}}{T}}N(z) \tag{6}\]  
where the subscript D means the computational delay included. Utilizing the modified NCO transfer function \(N_{\rm D}(z)\), the closed-loop transfer function \(H(z)\) of (4) is modified to contain the computational delay as follows:  
\[H_{\rm D}(z)=\frac{z^{-\frac{t_{\rm D}}{T}}N(z)F(z)}{1+z^{-\frac{t_{\rm D}}{T} }N(z)F(z)}=\frac{N_{\rm D}(z)F(z)}{1+N_{\rm D}(z)F(z)} \tag{7}\]  
By assuming that the new NCO input is obtained from the integration results at the previous epoch of the loop filter, as a typical example, the \(t_{\rm D}\) becomes \(T\). Then, the \(N_{\rm D}(z)\) contains unit delay as a computational delay as follows:  
\[N_{\rm D}(z)=z^{-1}N(z) \tag{8}\]  
and is used for the stability analysis of the DPLLs in case of computational delay are considered.