 the pro- 28.1.3 Noise and the decision process cessed data is compatible with that of the channel, thus line coding for low-pass channels and modulation (carrier keying) The effect of noise on the decision process is treated in many for band-pass channels, are also tasks for the channel coder. textbooks (Carlson, 1986; Sklar, 1988; Lee, 1988). For zero-mean Gaussian noise, the probability of making an erroneous decision is In practice it is not always possible to make a clear distinction given by Equation 28.1, where a is the voltage difference between between source and channel coding functions. Indeed, improved the received signal and its associated decision threshold at the efficiency can sometimes be obtained by combining some tasks decision time and o is the r.m.s. noise voltage. (Hamming, 1980). We now concentrate on the fundamental aspects of digital trans- (28.1) mission with emphasis on transmission over low-pass channels. P e = These same principles apply to band-pass channels which are treated elsewhere. This is known as the Gaussian tail area formula and is often shortened to Equation 28.2. 28.1.2 The baseband waveform P e = r(:) T (28.2) Figure 28.2(a) shows a binary data signal, assumed to be in a suitable form for transmission over a low-pass channel, this is often referred to as the baseband waveform. After transmission over the Unfortunately, there is no analytic solution to this equation and so restricted bandwidth of a channel (and if necessary, some equalisa- tables (Carlson, 1986; Sklar, 1988), bounds or approximation for- 28/4 Digital transmission Decision Output Channel Amplifier circuit stage Equaliser Clock extractor Figure 28.3 A baseband regenerator mulae (Beaulieu, 1989) have to be used. The solid line of Figure ance is usually in favour of binary or 3-level transmission at the 28.4 shows the error probability (P ) plotted as a function of the most. signal-to-noise ratio 20 at the input to the decision circuit. 28.1.4 Waveform shaping and bandwidth This line is equivalent to the case 