## 3 Adaptive Compensation through Digital Filters
For example, in Figure 7d, where the impairment of 25-ps PMD is presented, we notice that the penalty in Q\({}^{2}\) factor for the mild FIR setting is increased from 0.1 dB to 0.5 dB when the signal's bandwidth is reduced from 50 GHz to 20 GHz. This shows that, even though the AEQ for PMD compensation can adjust its spectral response, the impairment due to the signal's bandwidth narrowing effect cannot be fully mitigated by the AEQ alone. By changing the simple Rx FIR filter from the mild setting (2-dB peaking at 13 GHz) to the strong setting (4-dB peaking at 19 GHz), the penalty in Q\({}^{2}\) factor is reduced from 0.5 dB to 0.3 dB with 20 GHz signal's bandwidth, demonstrating the performance improvement from the simple Rx FIR filter on top of the AEQ. Similarly, this performance improvement can also be observed when other types of optical impairments are introduced, as shown in Figure 4:1 Overall, both the AEQ for PMD compensation and the Rx FIR for channel loss compensation can be simultaneously used to improve the tolerance to the bandwidth narrowing effect of the coherent signal passing through multiple cascading ROADMs. A complementary method is to use the PMD equalizer to compensate the dynamic change of the signal's bandwidth and use the simple FIR filter to compensate the slow drift of the signal's bandwidth. In addition to post-compensation at the Rx side, it is beneficial to apply pre-compensation on the Tx side as well. The Tx FIR is located after the FEC layer and before the DAC input. Most applications use those FIR filters to compensate the relatively static loss of the RF channel between the DAC's output and the Mach-Zehnder modulator (MZM) in a set-and-forget approach. Since we can estimate the BW of the signal, we can also adaptively adjust the Tx FIR to pre-compensate the narrowing effect of signal's bandwidth. In addition, the Nyquist filter is widely used to improve the spectral efficiency, which is usually a raised cosine (RC) filter or a root raised cosine (RRC) filter. The roll-off factor for the RC filter or the RRC filter determines the spectral response [23]. Thus, the key parameters of the FIR in the coherent transmitter are the peaking amplitude, the peaking frequency, and the roll-off factor. To determine the tap coefficients, the frequency response of the FIR filter with the desired peaking frequency and the peaking amount is first multiplied with the frequency response of the Nyquist filter. Next, a reverse Z-transform is performed to solve the coefficients of the combined filter. As a demonstration, we apply the peaking filter and the Nyquist filter together to the Tx FIR. The number of taps in the Tx FIR is 17, and the FIR filter is \(T_{\text{s}}\)/2 spaced. We use the RRC filter with the roll-off factor of one as the Nyquist filter. We adjust the peaking amount from 1 dB to 9 \begin{table}
\begin{tabular}{c c c} \hline
**Parameters** & **Rx FIR Filter** & **PMD AEQ (Adaptive Equalizer)** \\ \hline Main function & Static loss compensation for RF channel & SoP tracking and PMD compensation \\ Location & Before CD block & After CD block \\ Tap coefficients & Real value & Complex value \\ Power consumption & Small & Large \\ Complexity & Simple tap-and-delay & Complicate \(2\times 2\) butterfly \\ Feedback mechanism & Spectrum from ADC raw data & Blind equalization or data-aided equalization \\ Loop bandwidth & Low & High \\ Number of taps & Small & Moderate \\ Impact of signal’s bandwidth & Minimal & AEQ may not converge with small bandwidth \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison between Rx FIR filter and PMD adaptive equalizer. dB while keeping the peaking frequency at the Nyquist frequency. We measure the optical spectra using a high-resolution optical spectrum analyzer (Finisar Wave Analyzer 1500) and summarize the results in Figure 8. As seen, the optical spectra clearly show the influence of different peaking values, demonstrating the feasibility of using Tx FIR to pre-compensate the bandwidth narrowing effect.