red because the waveform energy depends on the particular symbol being transmitted, unlike the previous modulation formats. Using Eqs. (14.23)-(14.26), and the numerical approximation for erfc(x), one can compute the SER associated with each modulation format for different values of M. Since we are interested in iso-SER plots, we may choose, for instance, SER = 10-3, 10-5, 10-11 for each of these plots and find the corresponding SNRbit values numerically in each case. 11 result shown 14.5. 12 One observes from the figure that there exist qualitative differences between the different formats. Comparing first binary formats (M = 2, SER = BER), it is seen that at equal BERs, NRZ requires substantially higher SNRs than any other format. This feature is explained by the fact that the other formats use coherent receivers, which have higher sensitivity: for equal SNRs, coherent receivers yield lower BERs than the direct-detection receivers used in NRZ. All binary formats have an ISD of 1 (bit/s)/Hz. 13 Therefore, the only two degrees of freedom are the format and the BER. Binary QAM is seen to have the lowest SNR requirement at given BER. The choice of a modulation format being primarily dictated by practical engineering considerations (complexity of transmitter or receiver, feasibility, reliability, cost), the BER is not necessarily a decisive factor. Furthermore, we know from the SHT that for any SNR satisfying SNRbit > 1.6 dB, there exist (error-correction) codes making it possible to reduce the BER to arbitrarily small values. How practical such codes are to implement, and how powerful (coding gain) remain two key questions for system design. Consider next the M-ary format plots shown in Fig. 14.5. An additional degree of freedom is provided by the number of modulation levels, M, which makes it possible to modify the ISD. As previously seen, M-FSK is not bandwidth-efficient 10 The modulated signal has the form f(t) = a(t) cos(wt) + b(t) sin(wt), where a(t), b(t) are independently modulated with M levels, forming a s