nt in the Expression of the Error Probability of Direct Detection M-PolSK -35.241 15.202 -26.489 23.732 -0.1117 -11.320 0.4341 2.809 0.7513 Telecommunication Networks Once so(SN, d) is evaluated, in the approximation of high SN the optimum threshold is always 0.5 SN and the saddle point approximation can be further simplified yielding (9.65) The accuracy of the approximation can be verified comparing the results with the exact saddle point expression evaluated by numerical solution of the bound parameter equation and of the resulting expression of the error probability reported in [37]. The error is never above 0.3 dB as far as the error probability is lower than 10-3 in all the cases reported in [37]. The performances of coherent detection M-PolSK are shown in Figure 9.43 using the con- figurations whose parameters are reported in Table 9.3, the 32 level system constellation is derived by randomly selected 32 points out of the 60 vertices of the snub-dodecahedron and the 64 points constellation by randomly selecting 64 out of the 120 vertices of the Icosidodecahedron. In the figure, the performances of a coherent detection M-PolSK sys- tem using the optimum constellations reported in [35,37] are also plotted in dashed lines. It is evident that the sensitivity gain related to the use of the optimum constellation with respect to a more regular suboptimum one is small and probably in practice it is cancelled by the difficulty to reproduce accurately the optimum configurations due to their more complex structure. 9.3.4.3 M-DPSK with Direct Detection Receiver Last of the modulation formats we have considered is multilevel DPSK with direct detec- tion receiver. Due to its structure based on signal self beating, the receiver belongs to the class of qua- dratic receivers. In this case, the problem of determining the symbol error probability from the characteristic function can be exactly solved as already underlined in Chapter 6 for the binary case. Passing from the symbol error probability to the bit error probability,