# Adaptive Pre/Post-Compensation of Cascade Filters in Coherent Optical Transponders
## 5 Simulation Setup and Results
### Results in Fiber Linear Regime  
We start by studying the effect of filtering and PDL on the system performance without any kind of optimization, i.e., the adaptive MIMO filters at the Rx is compensating for all the impairments. In Figure 5, all data are with a frequency offset of the filter models that are set to 0 GHz, therefore, they are modeled as presented in Figure 1b. The ROSNR was measured by sweeping BER versus five OSNR values, then performing an interpolation. We notice that the ROSNR penalty, at 2% (=0.02 or \(2\times 10^{-2}\) BER, versus PDL can be fit to a quadratic function. The penalty of PDL equal to 7.5 dB is aligned with the results reported [33]. As well, as expected, the penalty when aggregating PDL and filtering impairments is bigger than adding the penalty of each impairment tested individually. The reason for the ROSNR penalty is that both impairments are inducing colored noise that affects amplitude. Since the receiver is equalizing the signal so it brings back the two polarizations to be equal and the spectrum to be flat, then the noise will be boosted as well. Therefore, both PDL and filtering can't be corrected, as PMD and chromatic dispersion, but can only be compensated. As well, when the adaptive filter is compensating for more than one impairment, the taps continuously trying to converge to the solution with the minimum mean square error, however, they are prone to misadjustment and hence higher penalty when aggregated. The penalty of 12 xWSSs without PDL is ~1.05 dB, while when added to PDL, it is ~1.2 dB. The higher the noise in the link, i.e., low OSNR, the more advantageous to amplify the signal right at the transmitter since symbols are not corrupted with errors yet. However, to keep the power constant, the peaking of high-frequency components comes at the price of de-emphasizing the low-frequency bins. Hence the trade-off. However, real transmitters have implementation noise due to quantization (limited bits resolution of both filter taps and QAM symbols), their static filters taps are near clipping and effective number of bits (ENOB) at the DAC is a function of frequency; therefore, emphasizing frequency bins at the transmitter might be deteriorating rather than helpful. For \"Tx Equalization\" only, there are two means to implement it. One is zero-forcing which uses the common response of the Rx and apply it with pulse-shaping and required pre-compensation at the static filter. Energy in frequency bins is always normalized to keep output power constant. Figure 5: Required-OSNR penalty as function of PDL and filtering. \[H_{Tx}(f)=H_{RRC}(f)*H_{CMD}^{-1}(f)*\big{|}Hc(f)\big{|}. \tag{17}\]  
The second technique, used to compare our method against, is MMSE-based as it takes SNR into consideration. In Equation (17), \(TF_{\textit{measured}}(f)\) is set equal to inverse of the TF of the total filtering effects. While \(TF_{\textit{target}}(f)\) is simply the flat response aimed at the input of the Rx. \(SNR_{\textit{factor}}\) is the noise adjustment term which can be scanned to get optimal BER. \[Peaking_{Tx}(f)=\frac{TF_{\textit{target}}(f)}{TF_{\textit{measured}}(f)+ SNR_{\textit{factor}}}. \tag{18}\]  
At 2% BER, we notice that when method converges (after around 40 iterations on average with the loop running ones every 10 seconds), most of the compensation (~70%) of the steep roll-off of the cascaded filters (as shown in Figure 1b) occurs at the Tx. While the remaining 30% is applied to the static filter of the receiver. At lower BER, i.e., higher link OSNR, the ratios will change. The reason is that it might not be as beneficial to pre-emphasis data since the noise-induced is small. Such observations are explained by [PERSON]'s optimal solution in presence of pre/post-compensation, Equation (6), where the noise components are showed up in the denominator. As the noise gets larger, the post-compensation gets smaller and vice-versa.