dress two issues: (a) the relation between the waveform modulation format and the ISD, and (b) for a given modulation format, the relation between SNR and BER (or symbol error rate, SER, for multi-level formats). With such knowledge, we will then be able to fill out the diagram with different families of points, i.e., to plot iso-format or iso-SER/BER curves, and observe the corresponding tradeoffs. An overview of the different types of modulation formats is beyond the scope of this book. Here, we shall use a limited selection of modulation formats and their ISD, SNR, or SER properties to provide illustrative cases of bandwidth-efficiency tradeoffs. The following assumes some familiarity with telecommunications signaling (i.e., intensity, frequency, phase, amplitude, and multi-level waveform modulation).7 Nonreturn-to-zero (NRZ) format The nonreturn-to-zero (NRZ) format, also called on-off keying (OOK), corresponds to the most basic and common modulation scheme. Bit encoding consists of turning the signal power on or off during each bit period, to represent 1 or 0, respectively. Its ISD is intrinsically limited to 1 (bit/s)/Hz8 and its BER is given by the generic formula: (14.23) where erf(x) is the complementary error function.9 7 S. Haykin, Digital Communications (New York: J. Wiley & Sons, 1988), J.G. Proakis, Digital Communica- tions, 4th edn. (New York: McGraw Hill, 2001). Actually, the 1 (bit/s)/Hz ISD for NRZ/OOK signals represents the asymptotic limit of channel capacity, which is rapidly reached for SNRbit 5 dB. The demonstration of such a result is left as an exercise. 9 By definition, erfc(x) = 1 - erf(x) with erf(x) = being the error function. The latter can be approximated through the expansion: erf(x) = 1-(alt+art2 a3t3 + a4t4 + ast5) exp(-x2), with t = 1/(1+px), p = 0.3275, a = 0.2548, a2 = -0.2844, az 1.4214, a4 = -1.4531 and a5 = 1.0614. Consistently with the exact definition, in the limit x 0 we have erf(x) = 0. It is customary to define 1/VSNRbit hence from Eq. (14.23), BER = erfc(Q/V2)/2 with 