f log2(1++200) 22 10 (bit/s)/Hz. The SHT, thus, states that given any SNR constraint, there exist some codes (and associated modulation formats) for which error-free transmission is possible within an ISD limit of C(bit/s)/Hz = log2(1 + SNR). As with the (discrete) channel coding theorem (Chapter 13), how to find or to design such codes is, however, not specified by the theory. In the above SHT formulation, it is possible to introduce another important parameter overlooked SO far, which is the noise bandwidth. Assume a communication channel with a capacity per unit time of Cbit/s and a signal bandwidth B. The mean signal power, P, and the mean energy per bit, Es, are related through P = EsCbit/s- The noise power, N, and the mean noise energy per bit, En, are related through N = ENBHz- Thus, the SNR can also be expressed in the form (14.19) which explicitly relates the bit SNR to the bandwidth B. Replacing the above result in Eq. (14.17) yields (14.20) (14.21) 5 For any nonnegative real number qlinear, the decibel scale is defined as qdB = For instance, dB = 100linear, -10 dB = 0.1 inear, and, in particular, 3.0102 dB 2 3 dB = 2linear- 14.1 Gaussian channel ISD ((Bit/s)/Hz) Shannon limit -1.6 dB R > C SNRbit (dB) Figure 14.2 Bandwidth-efficiency diagram plotting information spectral density (ISD) VS. signal-to-noise ratio (bit SNR), according to Eq. (14.22). The two regions R < C and R > C, corresponding to bit rate R and channel capacity Cbit/s, which are separated by the capacity- boundary curve (R = C) are also shown. The Shannon limit corresponds to SNR = 1.6 dB. Base-two exponentiation of both sides of Eq. (14.21), and the introduction of SNRbit = Es/EN as defining the bit SNR, 6 we finally obtain: SNRON=26 (14.22) This formula now provides an explicit and universal relation between the bit SNR (SNRbit) and the channel ISD (C(bit/s)/Hz). It is, thus, possible to plot the function C(bit/s)/Hz = f(SNRbit), as shown in Fig. 14.2, which is referred to as the bandwidth- efficiency diagram. As observed from Fig. 14