# Advanced DSP for Coherent Optical Fiber Communication
## 2 Principle of Coherent Detection and Linear Impairments Equalization
A variety of polarization demultiplexing algorithms have been proposed, such as constant modulus algorithm (CMA), least mean square (LMS), recursive least square (RLS), radius directed equalization (RDE) and so on [13; 14]. The details of the CMA algorithm are described below, and the tap weights vector of the equalizer is adapted by the following:  
\[h_{xx} =h_{xx}+\mu\varepsilon_{CMA,X}EX_{out}\cdot EX_{in}^{*}\] \[h_{xy} =h_{xy}+\mu\varepsilon_{CMA,X}EX_{out}\cdot EY_{in}^{*}\] \[h_{yx} =h_{yx}+\mu\varepsilon_{CMA,Y}EY_{out}\cdot EX_{in}^{*} \tag{1}\] \[h_{yy} =h_{yy}+\mu\varepsilon_{CMA,Y}EY_{out}\cdot EY_{in}^{*}\] \[\varepsilon_{CMA,X} =1-|EX_{out}|^{2}\] \[\varepsilon_{CMA,Y} =1-|EY_{out}|^{2} \tag{2}\]  
where \(\mu\) is the iteration factor, \(\varepsilon\) is the result of the error function. The \".\" denotes the vector dot product. \(E^{*}\) denotes for the complex conjugate form. All tap weights are typically set to zero initially, except for the central taps of \(h_{xx}\) and \(h_{yy}\) which are set to unity. The sample rate is twice the symbol rate while the filter tap weights are updated every two samples. The equalization outputs for the two polarizations are:  
\[EX_{out} =h_{xx}\cdot EX_{in}+h_{xy}\cdot EY_{in}\] \[EY_{out} =h_{yx}\cdot EX_{in}+h_{yy}\cdot EY_{in} \tag{3}\]  
Another widely employed algorithm is called LMS. LMS is a type of the stochastic gradient algorithms. Tap weights are determined by the update scheme as follows. Figure 1:0 Figure 1:1 Figure 1:2 Figure 1:3  
Figure 1:4 Figure 1:5  
Figure 1:6 Figure 1:7  
where \(EX_{in}^{*}\), \(EY_{in}^{*}\) represent the complex conjugate of the sampled input signal vectors and \(EX_{out}\), \(EY_{out}\) is the output of the equalizer. Its equalization error is defined by the difference between the reference signal \(R\) and the output signal, which includes both the amplitude and the phase information. \(R\) is the reference signal, \(d_{x}\), \(d_{y}\) are training signals or decision signals, \(\theta_{x}\), \(\theta_{y}\) are the estimated symbol phase after frequency offset compensation and carrier phase estimation (CPE). At the beginning, the equalizer works in the training symbol (TS) mode. Once the equalizer has converged, it moves into a decision-directed (DD) mode. This working scheme is defined as DD-LMS. Before the final stage of restoring the data and applying forward error correction (FEC), specific algorithms are employed to execute the frequency offset compensation and CPE [15; 16; 17; 18; 19; 20; 21]. In theory, coherent detection requires the frequency and the phase of the LO wave to be exactly the same as those of the signal carrier. However, due to the influence of fabrication imperfection of optical devices and environment variation, the frequencies of the transmitter and the LO lasers would not be completely consistent. Besides, the linewidth of the lasers will also introduce the corresponding phase noise, which is always considered as a Wiener process. The large amount of additional phase noise is quite harmful to the phase modulated signal. The purpose of the carrier recovery algorithm in the coherent receiver is to remove the impairments of carrier frequency offset and phase noise by processing a discrete data sample sequence. The principle of the frequency offset estimation is shown in Figure 1:8 In the case of considering the symbol phase only, it is assumed that the sampling value of the \(k_{th}\) symbol received is:  
Figure 1:9  
where \(\theta_{s}(k)\) represents the modulated phase, \(\Delta\omega kT\) is additional phase caused by frequency offset, \(\theta_{L}(k)\) is the phase noise from the laser linewidth, \(\theta_{ASE}(k)\) is related to the amplified spontaneous emission (ASE) noise, \(T\) is the symbol period. In high-speed optical transmission systems, \(\theta_{L}\) varies slowly relative to the symbol rate. By calculating the phase difference between the adjacent symbols, the \(\theta_{L}\) can be removed.