s. Electrical signal to noise ratio (dB) Next Generation Transmission Systems Better performances at high number of levels are attained by the constant power M-4QAM, whose constellations are derived from regular convex solids in the four dimen- sional space. In Figure 9.41, the error probability for this kind of modulation is shown, deriving the constellation with 64 points from a random selection of points among the vertices of the 600 Tetrahedral (120 vertices) and the constellation with 512 points from a random selec- tion of 512 among the 600 vertices of the 120 Dodecahedral. No attempt has been made to optimize the choice, although there is a large literature about segmenting of complex constellations and the interested reader is encouraged to start from [62-64] if is interested in methods for an optimum subconstellation selection. Increasing the number of levels, the performance is far better with respect to M-QAM, due to the better exploitation of the available degrees of freedom. Finally, in Figure 9.42, variable power M-4QAM performances are represented in the case of the constellations of Table 9.5. The high number of degrees of freedom renders these modulation formats extremely efficient at very high number of levels, even if at low num- ber of levels the performances are not superior to that of other quadrature based formats. FIGURE 9.41 Plot of the ideal performance of constant power M-4QAM for various values of the Electrical signal to noise ratio (dB) number of levels FIGURE 9.42 Plot of the ideal performance of variable power M-4QAM for various values of the number of levels in the case of the constel- Electrical signal to noise ratio (dB) lations of Table 9.5 Telecommunication Networks 9.3.4.2 M-PolSK with Stokes Parameters Receiver Following the receiver structure to determine the expression of the decision variable and its statistics, it is possible to demonstrate the following properties: The decision variable is a quadratic form of the input field quadratures The decision variable is independ