 Appendix C: Inter-Symbol Interference Indexes Summation Rule lb(2p(p(b(p(a (C.10) The nonlinear broadening is complex to evaluate, depending on f(); however, Equation C.10 is sufficient to reach our goal. With the same procedure of the previous section, we can write the expression of the overall pulse widening as TT+b(fp( (C.11) C.3 Comments and Examples As a comment, it is to be observed that the derivation we have done holds exactly only in very specific situations, where strict hypotheses like exponential tails and zero mean holds for all the pulses, both the signal and the pulse response of the filters. Naturally, if these conditions are not fulfilled completely, the property will not hold exactly, but in a regime of small distortion, we can frequently use this fundamental rule. As an example, let us consider a squared pulse 10 ps long with a rise time of 3 ps, as shown in Figure C.1. Let us filter the pulse with a first Gaussian filter of insufficient band- width, equal to 61 GHz instead of the required minimum of 100 GHz. As a consequence of the filtering, the pulse spreads, as shown in Figure C.2, and assumes a standard deviation of 11.6 ps that is exactly the result of Equation C.3. Input pulse Output pulse Filter pulse response Time (A.U.) FIGURE C.1 Pulse at the input of the filter chain and after the first filter. Appendix C: Inter-Symbol Interference Indexes Summation Rule Original pulse Filter output pulse Filter input pulse Time (A.U.) FIGURE C.2 Pulse at the input and at the output of the second filter. Filtering again the pulse with another filter with an even smaller bandwidth (25.88 GHz in this case), the pulse broadens again reaching a width, as shown in Figure C.2. The final width is exactly what we have foreseen in Equation C.7. Appendix D: Fiber Optical Amplifiers: Analytical Modeling D.1 Introduction Optical amplifiers are key devices to determine the performance of optical transmission systems. In order to understand the impact of behavior of different amplifiers on the char- acteristics 