# DLI-Based DP-QPSK Reception Scheme for Short-Range Optical Communication
## 3 The Data Recovery Algorithm  
In this section, the data recovery algorithm for DLI-based DP-QPSK reception is presented. Considering the limitations of the algorithm, assumptions were made first. Signals are usually transmitted in the form of frames in the channel and frame synchronization should be done first at the receiver before subsequent processing. There are two main types of frame synchronization techniques: one is data-aided frame synchronization, in which a predetermined synchronization word is periodically inserted in a frame, and the other is code-property-aided frame synchronization, in which the data is encoded with an error correction encoder. For the former, synchronization is attained through recognizing the synchronization word, while for the latter, synchronization is attained through exploiting the properties of the code. In this study, we only considered the data-aided frame synchronization technique. Hence, we assumed that a predetermined synchronization code was periodically inserted in the raw data before modulation for the receiver to obtain data synchronization. Moreover, the payload in the raw data was random. Based on these two assumptions, the data recovery algorithm was proposed as follows. DP-QPSK modulation lacks the process of differential precoding, while the output of a DLI-based demodulator is the information of phase differences between adjacent symbols. Hence, further data processing is required to recover the raw data. Figure 3 shows the conversion of data flow in a single polarization state in the DLI-based DP-QPSK transmission system. The raw bitstream is modulated to a QPSK signal by the MZM at the transmitting end and the QPSK signal is processed through a DLI and CDR into a demodulated bitstream at the  
receiving end. It is a reverse engineering problem to recover the raw data from the demodulated bitstream and a three-step inverse mapping is required. * Inverse mapping 1 from the demodulated bitstream to the demodulated signal: From Equation (6), we can find the correspondence between them, which is shown in Table 1. Therefore, the inverse mapping can be completed by just referring to Table 1.
* Inverse mapping 2 from the demodulated signal to the QPSK signal: The demodulated signal is the phase difference information and the QPSK signal is the phase information, which means the initial phase \(\varphi_{1}\) is required to complete the inverse mapping. QPSK characterizes different dual-bit symbols with four different phases in two modes, namely, mode A and mode B, corresponding to the parallel modulation and series modulation, respectively. The correspondence is shown in Table 2. Therefore, there are eight possible values for \(\varphi_{1}\) and the eight cases need to be verified one by one in the next step. * Inverse mapping 3 from the QPSK signal to the raw bitstream: The inverse mapping of this step can be completed by referring to Table 2, while for the eight cases caused due to the unknown initial phase, we need to search every bitstream for the frame synchronization code, which only appears in the raw data. Based on the three-step inverse mapping above, the data recovery algorithm can be summarized as follows. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**\(\Delta\varphi_{n}\)** & **0** & \(\pi/2\) & \(\pi\) & \(3\pi/2\) \\ \hline \((i_{u},i_{v})\) & \((\sqrt{2}/2,\sqrt{2}/2)\) & \((-\sqrt{2}/2,\sqrt{2}/2)\) & \((-\sqrt{2}/2,-\sqrt{2}/2)\) & \((\sqrt{2}/2,-\sqrt{2}/2)\) \\ IQ & \(11\) & \(01\) & \(00\) & \(10\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Correspondence between the phase difference and the demodulated symbols. Figure 3: The conversion diagram of the signal flow. The phases of the QPSK signal correspond to the dual-bit symbols of a raw bitstream after modulation. The output of the DLI is the phase difference between adjacent symbols of the QPSK signal. The dual-bit symbols of a demodulated bitstream correspond to the differential phases of a demodulated signal.