# Advanced DSP for Coherent Optical Fiber Communication
## 2 Principle of Coherent Detection and Linear Impairments Equalization
\[h_{xx} =h_{xx}+\mu\varepsilon_{CMA,X}EX_{out}\cdot EX_{in}^{*}\]0  
Take the quadrature-phase-shift-keying (QPSK) signal as an example. The QPSK signal has two types of modulated phase which are \((0,\frac{\pi}{2},\pi,\frac{3\pi}{2})\) and \((\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{\pi\pi}{4})\), but the \(\Delta\theta_{s}\) will always be \((0,\frac{\pi}{2},\pi,\frac{3\pi}{2})\). The modulated phase will disappear by a power of four calculation:  
\[h_{xx} =h_{xx}+\mu\varepsilon_{CMA,X}EX_{out}\cdot EX_{in}^{*}\]1  
In high-speed optical transmission systems, the frequency shift is also a slow variation process relative to the symbol rate, so that the frequency offset corresponding to multiple consecutive symbols can be regarded as the same. In this case, a series of consecutive symbols in a block can be processed  
\[h_{xx} =h_{xx}+\mu\varepsilon_{CMA,X}EX_{out}\cdot EX_{in}^{*}\]2 Block diagram of the frequency estimator. \[h_{xx} =h_{xx}+\mu\varepsilon_{CMA,X}EX_{out}\cdot EX_{in}^{*}\]3 Model of the linear fiber transmission link. together to estimate the frequency offset. The effect of \(\theta_{ASE}\) is quite small after the average operation and can be neglected at the optical SNR scenario. Then the frequency offset is obtained through:  
\[h_{xx} =h_{xx}+\mu\varepsilon_{CMA,X}EX_{out}\cdot EX_{in}^{*}\]4  
\[h_{xx} =h_{xx}+\mu\varepsilon_{CMA,X}EX_{out}\cdot EX_{in}^{*}\]5  
After the frequency offset estimation and compensation, carrier phase recovery is used to remove the phase noise component \(\theta_{L}\) and \(\Delta\omega^{\prime}KT\) caused by the linewidth and residual frequency offset of the lasers at the transceiver \(\Delta\omega^{\prime}=\Delta\omega-\Delta\hat{o}\). Viterbi-Viterbi phase estimation algorithm is a commonly used feed-forward digital CPE algorithm [22]. Figure 4 shows the principle diagram of the Viterbi-Viterbi CPE. The processing flow of the algorithm is similar to the aforementioned frequency offset estimation algorithm. The main steps are shown as follows:  
\[ \begin{array}{ll}S^{\prime}(k)&=\exp\{j(\theta_{s}(k)+\theta_{L}(k)+\Delta \omega^{\prime}kT+\theta_{ASE})\}\\ &=\exp\{j(\theta_{s}(k)+\theta_{L^{\prime}}(k)+\theta_{ASE})\}\end{array}  \tag{12}\]  
where \(\theta_{L^{\prime}}(k)\) is the carrier phase noise of the symbol \(k\), \(\theta_{L^{\prime}}(k)=\theta_{L}(k)+\Delta\omega^{\prime}kT\)  
\[\left(S^{\prime}(k)\right)^{4}=\exp\{j(4\theta_{L^{\prime}}(k)+4\theta_{ASE})\} \tag{13}\]  
\(\theta_{L}\) caused by the laser linewidth is the main part of the \(\theta_{L^{\prime}}\). It changes slowly, in contrast to the high-speed symbol stream and can basically be regarded as stable in a continuous symbol block. \(M\) symbols are considered to eliminate the effect of \(\theta_{ASE}\) to increase the accuracy of estimation by means of averaging symbols. \[\sum_{M}\left(S^{\prime}(k)\right)^{4}/M=\exp\{j(4\hat{\theta}_{L^{\prime}})\} \tag{14}\]  
\[\arg(\sum_{M}\left(S^{\prime}(k)\right)^{4}/M)/4=\hat{\theta}_{L}\text{'} \tag{15}\]  
Equation (15) is suitable for the \((0,\frac{\pi}{2},\pi,\frac{3\pi}{2})\) QPSK modulated signal. The residual value of modulated phase \(\pi\) will exist in the symbolic phase after the power of four operations if the \((\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4})\) modulation is applied. Then Equation (15) will be expressed as:  
\[\left(\arg(\sum_{M}\left(S^{\prime}(k)\right)^{4}/M)-\pi\right)/4=\hat{\theta }_{L}\text{'} \tag{16}\]  
The carrier phase error calculated by \(M\) symbols will be shared with \(M\) symbols for phase compensation:  
\[S^{\prime\prime}(k)=S^{\prime}(k)\exp(-j\hat{\theta}_{L}\text{'}) \tag{17}\]  
Some typical achievements have been reported recently. [PERSON] et al. demonstrated a 10 Gbaud PDM-QPSK experiment over 6400 km with DSP equalization [23]. The experimental results of equalized constellation diagrams of the two polarizations are shown in Figure 5. [PERSON] et al.