# Adaptive Pre/Post-Compensation of Cascade Filters in Coherent Optical Transponders
## 4 Derivation of Proposed Method  
Without power constraint, pre-shaping (or pre-compensating for frequency loss) data only at the Tx would give the best performance since most of the noise, mainly from ASE in fiber link, has not yet been introduced. However, in practice, transceivers operate with total power constraint over the signal spectrum (\(B\)). \[P_{ch}=\frac{\sum_{f=Freq\ Stqp}^{Freq\ Slqp}10^{\frac{FFT\text{Sim}(f)}{10}}} {B}. \tag{4}\]  
We propose a new method to do the joint compensation (of both vectors \(G\) and \(W\) shown in Figure 2) that takes into consideration the noise and power limitation at the output of the Tx. We propose a gradient-descent approach that compensates for the cascade of ROADM filter effects encountered by an optical signal when routed from the Tx to associated Rx. It is required to transfer the estimated pre-emphasis spectrum to the Rx where all computations can be executed. There are three ways at least to enable the communication: (1) using service channel, usually low data rate, that serves all modems in a node, (2) part of the data overhead in the communicated signal in duplex manner and (3) information of all modems in a software-defined networks ecosystem and a management protocol, such as Network Configuration Protocol (NETCONF), can be used to exchange the coefficients required at the Tx. In this paper, we only look at compensating for the magnitude response. We ignored the phase response because we could show that the adaptive filtering, LMS in our case, is able to correct for phase distortion with no penalty. First, we consider the Wiener solution, [29; 30; 31], in an adaptive equalization context. Once a steady-state is reached, the MIMO adaptive filter at the coherent Rx converges to a near-Wiener solution (as we are dealing with time-varying channel model). Since both polarizations go through the same filtering effect, we can use one of them in our procedure. We are assuming that the received symbols are pre-shaped with \(G\) and impacted by channel response (estimated as \(H_{c}^{-1}\)). In the frequency domain, the input of the adaptive filter of the Rx is:  
\[Y(f)=X(f)G(f)Hc^{-1}(f)+N(f), \tag{5}\]  
where \(X\) is frequency-domain (Fourier transform) representation of the decoded symbols (at the input of the Forward Error Correction (FEC)), representing the estimate of transmitted symbols, and its power spectrum density is defined as \(P_{xx}\). \(Y\) is the Fourier transform of the received symbols at the input of the receiver filters. \(H_{c}\) is the estimate of the static portion of channel frequency response. \(G\) is the frequency domain of pre-compensation coefficients applied at the Tx and \(N\) is the Fourier transform of the total noise, with power spectrum density \(P_{NN}\), estimated from the error used in the adaptive filters' coefficients update. When the cost function is specified the minimum square error, then Wiener equalization/filtering is the optimal solution [30]. Based on the familiar orthogonality principle [31], where the optimal filter is estimated based on the error being orthogonal to the data being used. We define \(S_{XY}\) and \(S_{YY}\) as, respectively, the cross-correlation between the input and the output and the auto-correlation of the input. \[W(f)=s_{XY}/s_{YY} \tag{6}\]The denominator, power spectrum of \(Y(f)\), is derived from Equation (5) as follows:  
\[S_{YY}=E[Y(f)Y(f)^{*}]=P_{XX}(f)\Big{|}G(f)\Big{(}H_{c}^{-1}(f)\Big{)}\Big{|}_{2 }+P_{NN}(f). \tag{7}\]  
The numerator, cross-power spectrum of \(Y(f)\) and \(X(f)\), is derived from Equation (5) as follows:  
\[S_{XY}=E[X(f)Y(f)^{*}]=P_{XX}(f)G^{*}(f)(Hc^{-1}(f))^{*}. \tag{8}\]  
We can show that the Wiener equalizer in the frequency domain, \(W\), is given by the following equation, with * conjugate operators:  
\[W(f)=\frac{P_{XX}(f)G^{*}(f)(Hc^{-1}(f))^{*}}{P_{XX}(f)\Big{|}G(f)\Big{(}H_{c} ^{-1}(f))\Big{|}_{2}+P_{NN}(f)}. \tag{9}\]  
The goal is, through cooperative design, to minimize the mean square error (MSE) between the transmitted symbols and their estimates.