## 2 Problem Description, Simulation Model and Methodology
### Numerical Simulation Description and Measurement Methodology  
Figure 3 depicts a diagram of the numerical model developed to study the OSNR penalties caused by the EEPN for model A, corresponding to the system impulse response of Figure 2b, and model B, corresponding to the system impulse response of Figure 2c. After generating the 3 modulated and independent signals, the transmitter adds the same amount of multiplicative phase noise TX\({}_{\text{pn}}\) to each mode. In model B, the MUX was modeled with a rank 6 RCU matrix [42]. Mode- and polarization-independent additive white Gaussian noise (AWGN) was added to each polarization on  
Figure 2: (**a**) Diagram of the simple modeling cases in an FMF transmission system. (**b**) Simplified weakly-coupled FMF impulse response when there is no mode mixing in the fiber and mode-selective DE/MUXes are used (model A). Differential mode delay (DMD). (**c**) Weakly-coupled FMF frequency response with non-mode-selective DE/MUXes (model B). On both examples (**b**,**c**), only the X polarization for LP\({}_{01}\) and LP\({}_{11}^{a}\) modes is plotted for clarity. each mode to set the desired OSNR. The fiber was modeled as a channel with ideal coupling within the mode-groups, null coupling between mode-groups and no CD with a rank 2 RCU matrix for the \(\text{LP}_{\text{01}}\) mode-group and a rank 4 RCU matrix for the \(\text{LP}_{\text{11}}\) mode-group. \(\text{LP}_{\text{11}}^{\text{a}}\) and \(\text{LP}_{\text{11}}^{\text{b}}\) modes were delayed with respect to \(\text{LP}_{\text{01}}\) by \(\tau_{2}\) symbols to account for the link differential mode group delay. In model B, the DEMUX was modeled with a rank 6 RCU matrix. Finally, the receiver added the same multiplicative phase noise \(\text{RX}_{\text{pn}}\) to each mode due to the LO. In model A, the DE/MUXes were assumed to be ideal mode-selective, i.e., no mode coupling at all, and were modeled with rank 6 identity matrices. The simulated baud rate was 25 GBd and a pseudo-random binary sequence (PRBS) of order 15 was used with all the combinations of quadratures, polarizations and modes, decorrelated with each other over at least 1111 symbols, which was longer than the maximum DMD spread under study to avoid equalization artifacts. The receiver DSP chain, as depicted in Figure 1, consisted of signal normalization, clock recovery, time-domain equalization-CPR loop, hard-detection and error counting. Each synthetic data trace, shown in Figure 4a, consisted of 131,072 symbols. Before equalization, 8192 symbols were stripped at the beginning and at the end to discard erroneous data introduced by circular convolutions. The equalizer was a complex 2 taps-per-symbol \(6\times 6\) time-domain equalizer [25] using the BPS CPR algorithm [18], where taps were updated on each symbol which enables high tracking speed of the phase noise. The equalizer taps were initially updated with a training sequence for the first 8192 symbols and then the equalizer was switched to decision-directed mode, where the error signal was computed with the hard-detected symbols. In order to avoid any simulation bias due to equalizer poor convergence or singularities, the equalizer initial taps were initialized with the inverse of the channel response, which was random for each realization. Note that the frequency offset was null. Before hard-detection, 16,384 and 8192 symbols were stripped at the beginning and at the end, respectively, to avoid errors due to equalizer pre-convergence and CPR, respectively. Data traces having carrier phase cycle slips were discarded. The cycle slip detector operated by first dividing the received bit-stream into blocks and counting the number of errors per block. The cycle slip detector was triggered if the number of errors in any block were greater than a threshold. Figure 3: Simplified simulation diagram for model A and model B. In model A, the MUX and DEMUX are modeled with rank 6 identity matrices whereas in model B the DE/MUXes are modeled with rank 6 random complex unitary matrices.