ecting the points of the constellation. Generally, the noise distribution has an exponential tail, thus the main contribution to the error probability is given by the first neighbors of the considered point. If it is possible to limit the evaluation only to this contribution, Equation 9.54 further simplifies yielding (9.55) where Vk is the number of first neighbors Zk is the projection of the received vector along the direction connecting the considered point with one of the first neighbors Ok is the curve coordinate along this direction Finally, if the constellation is a regular lattice, where all the points have the same num- ber of first neighbors located at a fixed distance and if the noise has the same distribu- tion whatever symbol is transmitted (situation very common for regular lattices), a further simplification of (9.55) is possible since the optimum threshold, for symmetry reasons, is located at half the distance between first neighbors. Next Generation Transmission Systems Thus in this case it is possible to write (9.56) where d is the distance between first neighbors and the other symbol has the same mean- ing as in (9.56) with useless indices eliminated. In the remaining of this paragraph we will use the above equations to analyze the ideal performances of 100 Gbit/s amplified multilevel systems using different types of modula- tion and detection. The general system scheme used for this evaluation is reported in Figure 9.39, which is similar to Figure 6.18, but for the presence of multilevel modulation. We will assume the spans to be of the same length and, due to the presence of the preamplifier, we will neglect shot and thermal noise. As seen in Chapter 6, the ASE noise spectral distribution is given by = + hw (9.57) where Ls is the span length Gil is the in-line amplifier gain Gb is the booster amplifier gain Gp is the preamplifier gain NF is the in-line amplifier noise factor NFb is the booster amplifier noise factor NFp is the preamplifier noise factor Multilevel Multilevel transponder transpo