n2+n3+N4 (9.11) In many practical cases, for example, M-ary Quadrature Amplitude Modulation (M-QAM) and M-ary Phase Shift Keying (M-PSK), the same pulse is used to modulate all the quadratures, thus we will assume this condition and we will neglect the index for g;(t), calling the modulation pulse simply g(t). In any case, it is straightforward to Next Generation Transmission Systems generalize the derivations that follow to the case of different modulating pulses for the different quadratures. Once a general description of this class of modulation formats is set up, we have first to demonstrate that a practical modulator for such signals can be built and then evaluate the transmitted signal power spectral density to verify the expected shrink with respect to a standard NRZ signal at the same bit rate. A possible scheme of a modulator for a general modulation format of this class is reported in Figure 9.17. This is a cascade of an amplitude, a phase, and a polarization modulator [31]. It is clear that in practice, when particular modulation formats are adopted, this will not be the opti- mum modulator and other schemes will be used. Just to consider only one aspect, the modulators cascade causes a great loss to be introduced on the field to be transmitted. However, the scheme of Figure 9.17 demonstrates that whatever modulation format of this class is considered, a possible modulation scheme exists. As far as the spectrum occupation is considered, we can define the so-called optical field spectrum matrix S in the base (pt, p_) through its elements as (9.12) where 30 indicates the Fourier transform <x> indicates the ensemble average Since the pulse g(t), in order not to provoke ISI, is completely contained in the symbol interval and due to the independence among symbols transmitted in different intervals, (9.12) rewrites (9.13) where G(w, T) is the power spectral density of the transmission pulse (s), is the average of the square of the transmitted symbols 8ih is one if j = h, otherwise it is zero Multilevel encod