# Adaptive Pre/Post-Compensation of Cascade Filters in Coherent Optical Transponders
## 3 Adaptive Filtering in Coherent Receivers  
In Figure 2, we illustrate the typical blocks in a coherent transponder. In the Tx, data streams are encoded then filtered with a static filter (CD EQ_x/y). The static filter is used for pulse shaping to reduce intersymbol interference and for dispersion pre-compensation. It is shown in [20] that splitting the dispersion compensation between Tx and Rx improves performance in the presence of nonlinearity, such as in long-haul propagation. Tx/Rx Firmware (FW) blocks, shown in Figure 2, represent the software-based routines used to provision/control the digital (such as update filters coefficients and provide telemetry information to the higher application in the networks) and EO portions of a transponder. A dual-polarization optical signal is split and mixed with the output of a local oscillator at the Rx to generate four branches (X/Y polarizations, each has in-phase and quadrature field). The data is digitized using an analog-to-digital converter (ADC) then goes to digital signal processing blocks. At the receiver, static filters (denoted as CD EQ x and CD EQ y in Figure 2) are used as matching the Tx filters to maximize SNR and for dispersion post-compensation. Root raised cosine (RRC) filters are used to form a raised cosine (RC), before adaptive filtering, to minimize inter-symbol interference (ISI). It is typical that the bulk part of dispersion is compensated prior to symbols reaching the butterfly MIMO equalizer. The latter is the core of the Rx and it is adaptive, where Constant Modulus Algorithm (CMA) and Least Mean Square (LMS) are examples of the adaptive method used [21], to decouple the two polarizations and compensate for linear time-variant channel impairments such as Polarization Rotation, Polarization Mode Dispersion (PMD) and Polarization Dependent Loss (PDL). Impairments such as residual dispersion, filters passband ripples, and edges roll-off are quasi-static (i.e., varies very slowly with fiber and components aging/temperature)--they can be estimated using the common factor of the four Multi-Input Multi-Output (MIMO) filters. In [22], the use of a common response was introduced to remove some of the residual dispersion from the adaptive filter and transfer them to the static filter to increase the Rx tolerance of PMD. In our case, instead of using the phase of the common response, we will use the magnitude to help mitigate the filtering effects. Frequency and phase offsets, between Tx and Rx laser oscillators, are tracked by carrier recovery block using methods such as Viterbi-Viterbi [23]. Symbol decoding, or de-mapping, and forward error correction (FEC) work together to extract and correct the sent stream of bits from the received symbols. As described in [24], they both can be based on soft-decision, such likelihood metric, or hard-decision. The \(2\times 2\) MIMO equalizer matrix can be represented as:  
\[H(f)=\left( \begin{array}{cc}H_{xx}(f)&H_{xy}(f)\\ H_{yx}(f)&H_{yy}(f)\end{array} \right), \tag{2}\]  
With the common response that is shown in [22]:  
\[Hc(f)=\sqrt{H_{xx}(f)*H_{yy}(f)-H_{xy}(f)*H_{yx}(f)}. \tag{3}\]  
In practice, the common response is estimated using time averages. The accuracy of the averaged response is improved as we increase the duration of the time average, especially in the presence of strong amplified spontaneous emission (ASE) in the link. With no impairments, such as frequency selective loss and dispersion, the symbol arrives at the Rx within its \"dedicated\" interval. Otherwise, we see the ISI effect where the transition of the symbols scatters to the neighbor intervals. Digital compensation is basically applying different weights and delays, i.e., filtering, to undo the expansion caused by neighboring symbols over the symbol of interest. The size of the filters required depends on the characteristic of the channels. In the frequency domain, the compensation is mainly pre-emphasizing high-frequency components of the data stream.