## 3 Results  
### Simulation System and Result Analysis  
#### 3.1.1 Description of the Simulation Setup  
Figure 2 illustrates the DP-16 QAM simulation setup constructed using VPIphotonics 10.1 software. At the transmitter, the generated pseudo-random binary sequence (PRBS) undergoes the quadrature modulation and then digital-to-analog conversion (DAC) to obtain 69 Gbaud DP-16 QAM electrical signals. The laser at 1550 nm with a 200 kHz line-width is subsequently modulated by an I/Q modulator (one per polarization component), which consists of two nested Mach-Zehnder modulators to accomplish electro-optic conversion. The optical signals are then transmitted over a link comprising 20 spans, each consisting of a 100 km standard single-mode fiber (SSMF) followed by an erbium-doped fiber amplifier (EDFA) for amplification, with noise at 5 dB. Some parameters related to the fiber are as follows: the chromatic dispersion coefficient \(D\) is 16.75 ps/nm/km, the fiber attenuation \(\alpha\) is 0.2 dB/km, and the nonlinearity coefficient \(\gamma\) is 1.31 /W/km. Then, 16,384 symbols of each polarization are transmitted, with each symbol simulated using 8 samples.  
Figure 1: The implementation flow chart of proposed A-DBP.  
After transmission, optical signals traverse an optical band-pass filter (OBPF) and are combined with a local oscillator (LO) at the polarization diversity hybrid to acquire I and Q optical signals of two polarizations. Subsequently, photodetectors (PDs) effectuate photoelectric conversion, and resultant electrical signals are processed in the DSP module implemented using MATLAB 2023b software. The sequence of the subsequent DSP process is shown in Figure 2. Specifically, the algorithm used for carrier frequency recovery is quadratic frequency bias estimation and the algorithm used for carrier phase recovery is a partitioned Viterbi-Viterbi algorithm. It is noted that only two samples per symbol are taken for the DSP module. Due to the different clock sources of AWG and oscilloscope, two samples per symbol are needed for achieving the time synchronization. Meanwhile, the CMA is employed for pre-convergence of the CMMA. During CMA processing in the simulation, we use 11 taps. Additionally, different from the DSP process in [30], we choose to use EVM as CF because there is no predictive term introduced in gradient-dependent algorithms.  
In order to verify the applicability of the proposed algorithm in the links with different types of fibers, two kinds of heterogeneous links are set up for simulation. In Figure 3a, the first hybrid links consist of 10 spans of SSFMs with 100 km per span and 10 spans of SSFMs with 80 km per span. The EDFA gain in the 100 km part and the 80 km part are 20 dB and 16 dB, respectively. In Figure 3b, the second hybrid links are composed of 10 spans of SSFMs and 10 spans of large effective area fibers (LEAFs) with the same length of 100 km per span and the same EDFA gain of 20 dB. The \(\gamma\) of the SSFM and LEAF are 1.31/W/km and 0.75/W/km, respectively.  
Figure 2: The simulation system setup. DAC: digital-to-analog converter. EDFA: erbium-doped fiber amplifier. SMF: single-mode fiber. OBPF: optical band-pass filter. LO: local oscillator. PD: photodetector. ADC: analog-to-digital converter. GSOP: Gram–Schmidt orthogonalization procedure. DC: dispersion compensation. CMA: constant module algorithm. CMMA: cascaded multimodule algorithm. CFR: carrier frequency recovery. CPR: carrier phase recovery. Inverse DC: inverse dispersion compensation. A-DBP: adaptive digital back-propagation.