aw detection operated at the electrical level, the decision variable is still a quadratic form of the field quadratures. Next Generation Transmission Systems In these conditions, the saddle point approximation (see Chapter 6) can be used to evalu- ate the error probability if the union bound is used. As a matter of fact, this approximation allows us to decompose the decision process in a set of independent binary decisions. Moreover, among all the addenda entering in the union bound expression of the symbol error probability we can maintain only those relative to first neighbors. The characteristic function of the decision variable relative to the decision on the seg- ment among a couple of first neighbors is given by (9.63) (1-d2s2)2Moe where is the ratio between the optical and the electrical bandwidth SNo is the optical signal to noise ratio d is the distance between first neighbors in the Stokes space A simple closed form expression of the bit error probability it is not easy to achieve, due to the complexity of the calculation involved by the saddle point approximation. Since a simple formula is important for quick evaluation, we have derived an approxi- mated form of the error probability assuming a high signal to noise ratio and the condition SNo >> M2 that is always realized in practice. Moreover, we have decided to put = 2, SO that the electrical and the optical SN coin- cides (the difference is smaller than 2%). Expressing the error probability as a function of SN allows us to consider the depen- dence of the BER on as slow and, for 0.5 < Moe < 4 negligible in a first approximation (see Chapter 6). Under these hypotheses, the bound optimization parameters so can be approximated with a simple expression (9.64) where the numerical value of the approximation parameters are reported in Table 9.7. The ratio between the approximation standard deviation and the average is smaller than 0.6% in all the parameter domain suitable for the applications. TABLE 9.7 Values of the Approximation Parameters That Are Prese