# An All-Digital Optical Phase-Locked Loop Suitable for Satellite Downlinks +
## 3 System Implementation and Analysis
### Opl
However, it will increase the value of \(B_{n}T_{s}\), which makes difficult the project of the OPLL since the parameter \(B_{n}T_{s}\) should be reduced. Thus, the OPLL will be analyzed over several \(B_{n}T_{s}\) values to find the phase-error variance and the induced penalty on the BER. The \(B_{n}\) will be changed to show the system behavior as the SR is fixed in two scenarios. The analyses will be conducted for the OPLL using the \(atan2\) discriminator since obtaining the variance and the standard deviation (STD) of the phase error directly from its output is possible. The phase-error variance affects the BER by inducing a power penalty due to an imperfect carrier recovery. The probability of bit error due to the phase error for a QPSK modulation was previously calculated as [31]:  
\[P_{b}(\sigma)=\frac{1}{2}\{\text{erfc}\left[\rho(\cos\sigma+\sin\sigma) \right]+\text{erfc}\left[\rho(\cos\sigma-\sin\sigma)\right]\} \tag{12}\]  
where \(\rho=\sqrt{\frac{RP_{s}}{2R_{s}d}}\), and \(\sigma\) is the STD of the phase error. For example, the standard deviations of \(\sigma=3.1^{\circ}\) and \(\sigma=6^{\circ}\) correspond to power penalties of 0.5 dB and 1 dB at \(1\times 10^{-9}\) BER, respectively. As a result, reducing the phase-error variance lowers the power penalty caused by the carrier synchronization error. In this sense, the analyses will start showing the behavior of the OPLL system under several values of \(B_{n}T_{s}\). Then, the power penalty induced for each condition will be shown for an SNR of 15.5 dB. Figure 6 shows the discriminator output behavior under several values of \(B_{n}T_{s}\) with an MAF of 10 coefficients. These conditions were simulated for an SNR of 15.5 dB at an SR of 10 Gsa/s, where each color represents one iteration, totaling 25 iterations. Figure 6a,b shows the output behavior for a \(B_{n}\) of 0.025 MHz and \(B_{n}\) of 1 MHz where the \(B_{n}T_{s}=2.5\times 10^{-6}\) and \(B_{n}T_{s}=1\times 10^{-4}\), respectively. Reducing the parameter \(B_{n}T_{s}\) lowers the phase-error variance. However, the parameter \(B_{n}T_{s}\) is reduced by lowering the \(B_{n}\) value, which lowers the \(\omega_{n}\) (natural frequency) and slows the OPLL. As a result, it cannot correctly follow the laser variations, as shown in Figure 6a, which shows a fluctuation. This effect is even more pronounced using the SR of 625 MSA/s, which will be discussed next. Figure 7 shows the discriminator output behavior with an MAF of five coefficients for various \(B_{n}T_{s}\) values and an SNR of 15.5 dB. However, the SR is now 625 MSA/s, with each color representing one iteration as before. The output behavior for a \(B_{n}\) of 0.025 MHz, is shown in Figure 7 where \(B_{n}T_{s}=4\times 10^{-5}\). Figure 7 depicts the output behavior for a \(B_{n}\) of 1 MHz, where \(B_{n}T_{s}=1.6\times 10^{-3}\). The number of coefficients of the moving averaging filter also influences the phase variance, which will be discussed next. Figure 6: Discriminator output behavior under various \(B_{n}T_{s}\) values with a sample rate of 10 GSa/s and MAF with 10 coefficients. The laser linewidth was 2 kHz, and the feedback delay was 30 ns. (**a**) Output error for \(B_{n}T_{s}=2.5\times 10^{-6}\), (**b**) Output error for \(B_{n}T_{s}=1\times 10^{-4}\). Figure 7: Discriminator output behavior under various \(B_{n}T_{s}\) values with sample rate of 625 MSA/s and MAF with 5 coefficients. The laser linewidth was 2 kHz, and the feedback delay was 35 ns. (**a**) Output error for \(B_{n}T_{s}=4\times 10^{-5}\), (**b**) Output error for \(B_{n}T_{s}=1.6\times 10^{-3}\). Despite using the same \(B_{n}\) value in both sampling rate scenarios, the OPLL is more impaired at a lower sampling rate. Comparing Figures 6a and 7a, one can see that the OPLL cannot correctly follow the signal variations since it has a fluctuation which in turn induces a fluctuation in the VCO center frequency.