# Adaptive Pre/Post-Compensation of Cascade Filters in Coherent Optical Transponders
## 5 Simulation Setup and Results  
### Simulation Environment  
Figure 2 is an overview of the block diagram of the elements used in the simulation to evaluate the methods proposed. The modulation format of choice was 16 QAM with the signal baud-rate equal  
to 35 GBaud and the RRC roll-off factor set to 0.14. The total bandwidth of the signal is 39.9 GHz. The LMS-based adaptive filters each have 17 taps in the time domain. The EO models were ideal, i.e., brick-wall shape with flat passband in spectrum till _Fs_/2. In these simulations, both the frequency and phase of Tx and Rx local oscillators (LO) are equal; therefore, carrier recovery is not active. As well, EO models are set to the ideal flat spectrum. Symbol identification was a simple search for minimum Euclidian distant per polarization for each received time-domain symbol from carrier recovery. Unlike the work done in [11; 12], we are not covering power loss, so we normalized the root-mean square (RMS) of the signal is in simulation. While in the lab experiments, the erbium-doped fiber amplifier (EDFA) was set in power-mode to keep received power the same. The latter was chosen so RMS of the received signal is in the optimum spot for the analog-to-digital converter (ADC). Three set of filters were simulated with each combined with the TF of the common multiplexer/demultiplexer. The 3 dB bandwidth of the final concatenation for 12 and 18 WSSs, Figure 1b, are 30.8 GHz and 32.37 GHz respectively. The model of PDL, used in this correspondence, is the following:  
\[\mathbf{H}_{\mathbf{p}\mathbf{l}\mathbf{l}}=\left[ \begin{array}{cc}\cos(\theta)&\sin(\theta) \\ -\sin(\theta)&\cos(\theta)\end{array} \right]\left[ \begin{array}{cc}1&0\\ 0&k\end{array} \right]\left[ \begin{array}{cc}\cos(\beta)&\sin(\beta)\\ -\sin(\beta)&\cos(\beta)\end{array} \right] \tag{15}\]  
where \(0<k<1\). The attenuation in dB is:  
\[PDI_{dB}=-20*log_{10}(k). \tag{16}\]  
Since \(\theta\) and \(\beta\) are randomly varying, the two polarizations (X and Y) are mixed in a time-varying manner. It is up to the adaptive filter at the Rx to correct and track for the time-varying crosstalk effects. The overall fiber model, Figure 4, used is a concatenation of a channel Mux, filtering effect (N times), noise source to mimic the ASE coming from the Erbium-doped fiber amplifier (EDFA) in each span, bulk PDL element, and CMD Demux. The EDFAs are typically used in every span to compensate for the loss of fiber and insertion loss of optical components. The instantaneous PDL, resulting from contributions of many random birefringences in fibers and optical components along the link, can be chosen from a Maxwell-distribution. Therefore, in deployed fibers, the total PDL observed by the Rx is statistical and varies with time. In both our simulation and experiment, we fixed the value of PDL and applied continuous random rotations before and after, as in Equation (12). The value of \(\theta\) and \(\beta\) was set to rotate at a speed of 800 Hz (maximum rotation measured in buried fibers [32]). The BER was averaged out over the length of the frame required by the FEC. Typically, as used in our simulations, the FEC is preceded with an interleaver to scramble the errors in a burst of symbols (avoiding congested errors that will cause traffic hits). We emulated a turbo FEC of size \(256\times 256\) bits and interleaver of the buffering depth of 24 bits, addition to 4 bits per symbol and 25 pico-seconds per sample, we are simulating 2643 combinations of \(\theta\) and \(\beta\). The clocking of both DAC and ADC is driven by Phase Locked Loop (PLL). At both the Tx and Rx, the dominant sources of jitter in PLLs are the VCO (voltage controller oscillator), reference clocks, frequency dividers and CP (charge pump). With respect to the VCO, it is subject to jitter accumulation  
Figure 4: Simulated channel model with ROADM concatenation, noise and PDL. which manifests itself as high phase noise at low-frequency offsets with respect to the carrier frequency.