We will use a simple rule to construct our constellation as the Cartesian product of partitioned QAM constellations in which points are alternatively taken and discarded. An example of this kind of constellation is reported in Figure 9.27, where the generic M-QAM constellation is divided into two partitions: gray points that are taken as components in the (e1,e2) plane and black points that are taken as components in the (e3, e4) plane. TABLE 9.5 Minimum Normalized Distance of Constant Power M-4QAM Constellations Obtained from the Vertices of Four Dimensional Regular Polyhedrons Polyhedron Modulation Format Minimum Distance (Normalized to P M-QAM 16 Tetrahedral 8-4QAM V2 = 1.414 12=0.707 8 Hexahedral 16-4QAM 0.471 24 Octahedral 24-4QAM 600 Tetrahedral 120-4QAM 0.618 5 + 1 120 Dodecahedral 600-4QAM V5+ = 0.270 FIGURE 9.27 Partition of a square M-QAM constellation to form by Cartesian product an efficient M-4QAM constellation. Gray and black points individuates the two partitions used to build the final constellation. Telecommunication Networks TABLE 9.6 Minimum Normalized Distance in the Quadrature Space for M-4QAM Constellations Obtained by the Cartesian Product of Regular Partitions of Squared QAM Constellations QAM Sub-Set Minimum Distance Number of Points Modulation Format (Normalized to P M-QAM 64-4QAM 0.202 324-4QAM 0.329 1024-4QAM 0.286 0.046 The minimum distances relative to the constellations that are built with this algorithm are reported in Table 9.6. The results reported in Tables 9.4 through 9.6, besides the outcome of Equation 9.18 for M-DPSK are compared in Figure 9.28. An interesting observation emerges from this figure, which is useful to guide the decision about the number of levels to select when designing a multilevel system based on quadrature modulation. Increasing the number of available degrees of freedom passing from M-QAM to constant power M-4QAM to full M-4QAM is effective in maintaining an high distance among constellation points at high number of levels, while if the number of levels i