# Comparison of Basic Notch Filters for Semiconductor Optical Amplifier Pattern Effect Mitigation
## 5 ODI vs. MRR Comparison: Results and Discussion  
The ODI and MRR are compared using appropriate performance metrics, which allow one to assess the performance of an SOA subject to a strong pattern effect and subsequently  
Figure 6: Small-signal modulation response of SOA alone (black color) and with the assistance of ODI (red color) or MRR (green color). The dashed line denotes the 3-dB bandwidth level and the arrows the bandwidth extension achieved by the notch filters. the capability of the employed notch filtering schemes to suppress it. These metrics include: (a) The amplitude modulation (\(AM\)), which is defined as \(AM\) (dB) = \(10\log\left(P_{1,high}/P_{1,low}\right)\), where \(P_{1,high}\) (\(P_{1,low}\)) is the peak power of the mark across the amplified data stream, which has the highest (lowest) amplitude level [30]. The \(AM\) provides a measure of the degree of uniformity of the amplified pulses whose pattern-dependent pulse-to-pulse wandering should be reduced by the employed filters below 1 dB [49]. (b) The amplification reduction (\(AR\)), which is defined as \(AR\) (dB) = \(\left|P_{out/ODI,MRR}^{avg}(\text{dBm})-P_{out/SOA}^{avg}(\text{dBm})\right|\), where \(P_{out/ODI,MRR}^{avg}\) and \(P_{out/SOA}^{avg}\) are the total average powers at the OOI or MRR filters and SOA outputs, respectively: The \(AR\) takes into account the fact that, due to the suppression of the red-shifted components by the action of filtering, some useful information contained in them is inevitably lost [21]. This causes the data sequence to receive less amplification than after the SOA alone, which is translated into an amplification penalty [50] that is quantified by the \(AR\). Ideally, the extent of \(AR\) must be such that it can be compensated by the average gain offered by the SOA to the data stream and accordingly allow the net gain of the SOA-notch filter combination to be sufficient for direct signal amplification purposes. (c) The cross-correlation (\(XC\)) coefficient, which is defined as [51], \(XC\) (\(\%\)) = \(\int\limits_{-\infty}^{\infty}\hat{P}_{\text{out/ODI,MRR}}(\text{t})\hat{P}_{ \text{in/SOA}}(\text{t})\,\text{d}t\Bigg{/}\sqrt{\left(\int\limits_{-\infty}^ {\infty}\hat{P}_{\text{out/ODI,MRR}}^{2}(\text{t})\,\text{d}t\right)\left(\int \limits_{-\infty}^{\infty}\hat{P}_{\text{in/SOA}}^{2}(\text{t})\,\text{d}t \right)}\) where the symbol of the caret over the time-dependent powers denotes that the latter are normalized to their maximum value: The \(XC\) is a measure of the degree of similarity of the intensity profile of the amplified signal that is filtered by the OOI or MRR to its profile prior to being inserted in the SOA. The higher the \(XC\), the better is the capability of the given filter to accurately reproduce the original signal. (d) The \(Q^{2}\)-factor represents the OSNR at the input of the receiver decision circuit: in the thermal noise limit [52], where the amplitude fluctuations due to the pattern effect act as noise variance on the marks [19], \(Q=P_{avg}^{1}/\sigma_{avg}^{1}\), where \(P_{avg}^{1}\) is the average and \(\sigma_{avg}^{1}\) is the standard deviation of the peak power of the marks and \(\text{OSNR}~{}=~{}Q^{2}\). Because the OSNR is measured in units of decibels (dB), it is convenient to use the same dimensioning for the \(Q\)-factor. Thus, \(10\log(\text{OSNR})=10\log(Q^{2})=20\log Q\equiv Q^{2}\) (dB). On the other hand, the \(Q\)-factor value in the linear scale must be at least six to ensure that the associated bit error rate is less than \(10^{-9}\)[52] and error-free operation can be achieved. Converting this requirement into logarithmic scale results in \(Q^{2}\) (dB) = \(20\log Q=20\log(6)\simeq 15.6\) dB, which therefore is the limit against which the specific employed metric is addressed in the rest of the paper.