## 3 Adaptive Compensation through Digital Filters
It is also important to consider the DC gain of the FIR filter, which is determined by the summation of the absolute value of those tap coefficients. A FIR filter with a high DC gain will lead the signal to being clipped, while a FIR filter with a low DC gain will lead the signal to being submerged under the noise. We optimize the DC gain of the FIR filter to achieve the best performance. We choose three typical filters: 2-dB peaking at 13 GHz, 4-dB peaking at 15 GHz, and 4-dB peaking at 19 GHz, corresponding to the three filters shown in Figure 5. We adjust the DC gain of the FIR filter while maintaining the spectral shape in the frequency domain. We measure the BER under different DC gain and summarize the results in Figure 6. As seen, for each filter's shape, there is an optimal DC gain setting leading to minimum BER. In the following sections, we set the DC gain of the FIR filter to the optimal point. Figure 5: Normalized amplitude and phase responses of simple finite impulse response (FIR) filters with different peaking frequencies and peaking amounts. The results are obtained by applying Z-transform on the tap coefficients of the FIR filter. In general, the spectral response of the FIR filter should be inverse to the overall frequency response of the cascaded ROADMs for equalization. When a signal has a relatively high bandwidth with a small number of cascading ROADMs, the optimal filter has a small peaking amplitude and a small peaking frequency. Otherwise, more noise will be amplified. When a signal has a low bandwidth due to larger number of cascading ROADMs, the optimal filter should have a large peaking amplitude and a large peaking frequency. In this way, any reduction of the signal's bandwidth will be compensated by this adaptive FIR filter. We first study the scenario where no FIR filter is applied. However, we obtain a subpar BER result. The performance degradation is much more severe when various impairments are included. The reason is that the optical and the electrical components in the receiver's data path accumulate the limitation on the received signal's bandwidth. Even when the receiver's signal bandwidth is not limited by the filter-narrowing effect due to the cascading ROADMs, a FIR filter with certain peaking amount is still needed to compensate the bandwidth limitation introduced by various components in the receiver's data path, as discussed in the introduction. We experimentally identify that a mild FIR filter with 2-dB peaking at 13 GHz provides the optimal results across different scenarios. The performance is similar to that reported in [22], where a coherent transponder is built with discrete premium components. This demonstrates that the mild FIR compensates the bandwidth limitation of integrated coherent receiver. To demonstrate the advantage of our proposed method, we also select a strong FIR filter with 4-dB peaking at 19 GHz for further study. It is expected that the mild FIR filter will perform better when the signal's bandwidth is large, while the strong FIR filter will perform better when the signal's bandwidth is small. Through the experimental measurements under different transmission impairments, we validate this prediction. The results are summarized in Figure 7.  
For these two settings, we measure the pre-FEC BER at 16-dB OSNR with a receiver optical power of -18 dBm. This is close to our system's end-of-life specification, which is the summation of the required OSNR for the coherent receiver and the reserved margin for operation. The pre-FEC BER is reported by the DSP ASIC using the output of the FEC decoder. Furthermore, one can use the pre-FEC BER to estimate the SNR. We first convert the measured BER to the Q\({}^{2}\) factor. Next, we choose the baseline as the measured Q\({}^{2}\) factor in the back-to-back scenario with the strong FIR setting. We determine the penalty by calculating the difference between the Q\({}^{2}\) factor under the specific scenario and the Q\({}^{2}\) factor of the baseline scenario.