ound in literature, where the bilateral optical bandwidth is used; see Figure 5.9 at page 307). After discrete integration we have (6.58) It is now clear that the sample in front of the threshold comparison circuit is a quadratic form of Gaussian random variable plus a Gaussian independent term. We can assume that the samples that constitute the decision variable Cw are sta- tistically independent: this is the number of samplings at the Nyquist sampling rate of the optical signal that fit into the Nyquist sampling interval of the electrical signal (see Chapter 5). This is rigorously true if the sampling rate corresponds to the Nyquist rate and if the filters (both the optical and the electrical one) are perfect bandpass and low pass filters. Generally these conditions are only approximated by realistic systems; however, even in realistic conditions, assuming that the filters have spectral tails going to zero suf- ficiently fast and that the sampling is not very far from Nyquist condition, the statistical correlation among the samples can be neglected. In this condition the characteristic function of the random variable C (i.e., the bilateral Laplace transform of its probability density) can be evaluated with a known method [26]. Let us individuate a set of bit patterns that allow us to manage the memory due to propa- gation phenomena (in general it will be a set of nine patterns of three bits as done in Section 6.1) and let us call D {b}, the jth pattern. The bit that is under decision is the central bit of the pattern, that will be indicated with bcj The characteristic function F(s) of the decision variable conditioned to the transmission of a certain pattern has the following expression, where s=s,+is; is the Laplace variable: (6.59) where SN is the optical SN ratio that can be written starting from Equation 6.46 as Telecommunication Networks (6.60) with the power of the optical field in the assumed hypothesis indicated with P(bcj) and the sum of the FWM power and the XPM power indicated with Pnli(D) Q = is the