and that the ratio between the minimum distance d and the transmitted power P of a M-4QAM constellation is given by (9.32) A few key parameters of some M-4QAM formats of this class are summarized in Table 9.4. For comparison, in the table data relative to M-QAM are also reported. It results that using M-4QAM is always advantageous, but the gain gets smaller and smaller while M increases if squared configurations are used. Just as an example, for M = 16 the gain of squared 16-4QAM versus 16-QAM is 3.2 dB while it is reduced to 0.7 dB if M = 256. If it is important to gain as much as possible from the constellation points distribution a better constellation has to be selected. A great amount of work has been done by mathematicians to study forms and properties of four dimensional constellations [43,44]. TABLE 9.4 Minimum Normalized Distance in the Quadrature Space for Squared M-4QAM Constellations (L,L,L,L4) Modulation Format Minimum Distance (Normalized to M-QAM (2,2,1,1) 4-4QAM V2=1.14 (2,2,2,1) 8-4QAM 2=1.155 2 1 ~ 0.707 =0.707 (2, 2, 2) 16-4QAM V = 0.471 (4,2,2,2) 32-4QAM V13 ~0.277 (4,2,4,2) 64-QAM ~15 =0.2 (4, 2) 128-4QAM 0.127 (4, 4, 4, 4) 256-4QAM 1=0.111 = = 0.094 M-QAM distances are reported for comparison. Next Generation Transmission Systems In the particular case of constant power modulation we can use to define the M-4QAM con- stellations the vertices of convex, regular polyhedrons in four dimensions, as defined in [45]. Table 9.5 reproduces the structure of Table 9.4, but using the new M-4QAM constellations. Even better results can be achieved with a careful optimization of more complex constel- lations [46,47]. It is out of the scope of this book to discuss in detail all the implications of the opti- mum choice of a four dimensional constellation, here we simply limit ourselves to a few examples based on a simple principle: creating a dense lattice of points via the Cartesian product of bidimensional QAM constellations and then select a subset of points for the final four dimensional constellation. 