## 3 Adaptive Compensation through Digital Filters  
Simple FIR filters can be placed right after the ADC and before the frequency-domain CD compensation module. The typical usage of those simple FIR filters is to compensate the RF losses of the analog interface, and the tap values of the FIR filters usually remain unchanged during the lifetime of the coherent transponder. However, in this work, the tap coefficients of the FIR filters are dynamically optimized, particularly to compensate the bandwidth narrowing effect. As discussed in Section 2, the value of \(Att_{Edge}\) is directly correlated to the signal's bandwidth. One can use this value to see whether the signal's bandwidth is limited. If so, one can adjust the simple FIR filter to provide more peaking to compensate the limited bandwidth. Two key parameters determine the shape of channel filters--peaking frequency and peaking amount. The peaking amount is the difference between the maximal frequency response and the frequency response close to the direct current (DC). The peaking frequency is where the maximal response locates in the frequency domain. The output of the FIR filter is a convolution between the input signal and the FIR's impulse response, as shown in the equation below. \[y(n)=\sum_{j=1}^{N}h(j)x(n-j) \tag{1}\]  
Figure 4: Attenuation at spectrum edge (\(Att_{Edge}\)) versus the signal’s bandwidth in the receiver. For each setting of signal’s bandwidth, we measure 16 times and take the average value. Figure 3: (**a**) Spectra from the analog-to-digital converter (ADC)’s raw outputs. (**b**) Spectra after applying the Savitzky-Golay filter. (**c**) Spectra of XI tributary under different bandwidths. Results are measured with 14-dB optical signal-to-noise ratio (OSNR). Here, \(x\) is the input signal to the FIR filter, \(h\) is the tap coefficient of the FIR filter, and \(N\) is the total number of taps. In the current DSP ASIC, \(N\) is equal to three. The FIR filter is \(T_{s}\)/2 spaced, where \(T_{s}\) is the symbol period. The tap coefficients of those digital filters are real numbers only. Also, four sets of the FIR filters (for four tributaries) share the same tap coefficients. These simplifications greatly reduce the complexity in the hardware implementation. Next, we search different combinations of the tap coefficients and identify the desired spectral shapes. The desired range for the peaking frequency is from 13 GHz to 19 GHz, and the one for the peaking amount is from 1 dB to 5 dB. To obtain the target spectral shape, we first set the value of main tap to be one. Next, we perform a coarse scan on the values of precursor and postcursor. We adjust those values between \(-0.5\) to \(0.5\) at the step of \(0.1\). By performing a Z-transform, the frequency response of FIR filter is determined. Then, we identify a small range of precursor and postcursor. Within this small range, the peaking amount and the peaking frequency are close to the target values. Furthermore, we optimize the values of precursor and postcursor by performing a fine scan within this small range. With only three taps, we sometimes cannot obtain both the desired peaking amount and the peaking frequency simultaneously. In those situations, we prioritize the peaking amount over the peaking frequency. Finally, we convert the representation of tap coefficients from floating-point number to signed-integer number. The conversion is based on the following rules--the tap coefficients are implemented by 8-bits registers with the most significant bit (MSB) being the sign bit; for the remaining seven bits, two bits are used to represent integer value, and five bits are used to represent fractional value. Figure 5 shows spectral shape dependence of a few FIR filters on the peaking frequency and the peaking amount. One advantage of those simple FIR filters is their approximately linear phase response, which minimizes the ripple in the frequency domain. For each filter shape, there are multiple sets of tap coefficients satisfying the requirement.