Nonlinear Pre-Distortion Based on Indirect Learning Architecture and Cross-Correlation-Enabled Behavioral Modeling for 120-Gbps Multimode Optical Interconnects

In this paper, we present a novel nonlinear pre-distortion scheme enabled by indirect learning architecture and cross-correlation based behavioral modeling. 120-Gbps PAM-4 error free transmission is demonstrated using 30-GHz class VCSEL.

Volterra kernels with first three orders are also utilized. After characterizing and emulating the nonlinear system, the kernels of pre-distortion Volterra series are calculated using ILA. The steps of ILA are expressed as follows [8]. At the first iteration, the input signal passes through the pre-distorter with initial coefficients distribution (the first value of the 1 st order kernel is set to 1. The rest of the coefficients of 1 st order as well as coefficients of the whole 2 nd and 3 rd order kernels of the pre-distortion Volterra series are set to 0). The pre-distorted signal d(n) then passes through the emulated nonlinear system H where the kernels are determined using behavioral modeling. The output y(n) is treated as the input of the pre-distorter and the corresponding output o(n) is subtracted by d(n). The error e(n) is used to train the coefficients of the pre-distorter using Least Mean Square (LMS) algorithm.

Results and Discussions
We first verify the validity of the cross-correlation-based behavioral modeling in both simulation and experiment. The simulation is performed in VPI. The simulation setup is shown in Fig. 1(a) and the parameters used in the simulation are listed in Table. 1. First the Gaussian white noise source is chosen as the input of the VCSEL module in VPI and the response after PD is sent to computer to calculate the cross-correlation with the input noise. The memory length of kernels of the Volterra series that emulate the nonlinear performance of VCSEL and PD are adjusted to ensure the best large signal response fitting. The estimated Volterra kernels are depicted in Fig. 2(a)-(c). From Fig. 3(d), it can be seen that the calculated eye diagram using the estimated Volterra kernels for 50-Gbps PAM-4 signal fits very well with the signal obtained directly from the PD output in simulation.     1(b) shows the experimental setup for the cross-correlation-based behavioral modeling verification. The Gaussian noise signal is generated from a Keysight M8195A arbitrary waveform generator (AWG). The electrical signal is then amplified by a RF amplifier (SHF 807) to ensure that the VCSEL TOSA operates in the nonlinear region. The output electrical signal of photodetector (PD) and the electrical noise signal are both captured by a Keysight Z592A digital storage oscilloscope (DSO) with sampling rate of 160 GSa/s. The cross-correlations between the received noise signal and the noise source are calculated to determine the Volterra kernels which emulate the nonlinearity of the system. In order to make the calculated eye diagrams fit well with the tested ones, the memory length is set to 13, 1 and 0 for the 1 st , 2 nd and 3 rd order kernels, respectively. Fig. 2(c) shows the 1 st order Volterra kernels. The value of the 2 nd order Volterra kernel is -0.0092. Fig. 2(f)-(h) show the eye diagrams for 20-, 40-and 50-Gbps PAM-4 signals. The calculated eye diagrams can reproduce the tested ones satisfactorily. Both the simulation and the experiment results indicate that the Volterra kernels estimated by cross-correlation-based behavioral modeling can emulate the nonlinearity of the transmission system using VCSEL.
After verifying the effectiveness of behavioral modeling and obtaining the Volterra kernels that can emulate the transmission system, ILA is utilized to calculate the Volterra kernels of the pre-distorter for the transmission system including VCSEL and PD in VPI. The calculated pre-distortion Volterra kernels are shown in Fig. 3(a)-(c). The predistorted 120-Gbps PAM-4 signal is imported as the input of the VCSEL module in VPI. The PD output signal is shown in Fig. 3(e). Compared with the signal before pre-distortion as shown in Fig. 3(d), clear eye opening can be observed. Within the data length of the obtained signal, error free is achieved with pre-distortion applied, which indicates the validity of the proposed pre-distortion scheme for 120-Gbps VCSEL based optical interconnect.

Conclusion
In this paper, we present a novel nonlinear pre-distortion scheme based on indirect learning architecture and crosscorrelation-enabled behavioral modeling for multimode optical interconnects. 120-Gbps PAM-4 error free transmission is demonstrated using 30-GHz class VCSEL.

Acknowledgement
This work is supported in part by the National Natural Science Foundation of China under Grant 61605111.