asing tails, SO that no convergence problem will arise in our calculation. We will also assume in this appendix the following hypotheses: The pulse p(t) at the filter chain input has a zero temporal mean that is [tp(+)dt = 0. For example, it is a Gaussian pulse. The linear filters of the chain have a pulse response a(t) with zero mean, that is, Sta(+)dt =0. This is true, for example, for Gaussian filters, which also have Gaussian pulse response, that is, with a good approximation for a large class of array waveguides (AWGs). Let us imagine having a system constituted by the cascade of two linear filters. Let us assume that two linear operators A and Brepresent the two filters that act on a pulse signal p(t) broadening it SO that (C.1) the pulse width after the operators action is (C.2) and an analogous equation for TB. Changing variables into the integral and remembering the property of the pulse average to be zero, we have (C.3) where TA and TB are the widths of the pulse responses of the filters Appendix C: Inter-Symbol Interference Indexes Summation Rule Let us now apply the two filters together (C.4) the expression of the overall pulse widening is obtained by the following substitution in the integral: 't=xty+z t-v=x (C.5) The variable change has unitary Jacobian, and the integral becomes (C.6) from the integral, since all the pulses have zero mean, the following is derived: (C.7) C.2 Presence of One Nonlinear Operator with Fast Nonlinearity In the cases that interest us, at least one of the involved filters is nonlinear, but with an instantaneous nonlinearity. It is a simplified way to represent the effect of the Kerr nonlin- earity at the system output. The filter B is now composed of two parts: a linear part with memory and an instanta- neous nonlinear part depending on the pulse power, SO that (C.8) As a first step, let us evaluate the pulse widening due to the transit through . This widen- ing is given by (C.9) substituting z=t - I in the integral and remembering that all pulses have zero mean, we obtain