of binary transmission. For an requirements m-level transmitted signal (requiring m - 1 decision thresholds at the receiver), it can be shown (Bennett and Davey, 1965) that for a In order to use the transmission medium efficiently and to reduce given symbol error probability an improvement in signal-to-noise received noise, the channel bandwidth should be minimised. How- ratio of approximately (m ² - 1)/3 is required relative to the binary ever, this conflicts with the requirement to restrict the spreading of case. This analysis assumes a constant mean signal power and that the pulse waveform at the receiver. For reliable decision making, a all transmit levels are equiprobable. transmitted symbol must be shaped by the channel and its associated equaliser SO that it does not interfere with other received symbols. Figure 28.5 shows a common pulse shaping objective. Symbols are transmitted with time spacing T and shaped before presentation to the decision circuit to yield a peak (at the time a decision is made) m-level with zero amplitude at all other decision times. Such a signal is said to exhibit zero intersymbol interference (i.s.i.) with respect to neigh- bouring received signals. Nyquist (Nyquist, 1928) showed that the minimum transmission 10 log m2- 1 bandwidth (channel plus equaliser) which meets this condition is one which passes all frequencies up to 1/2T and stops all others, as illustrated by the dashed lines of the frequency and time domain pair of Figure 28.6. This result is only of theoretical interest because a transmission characteristic with an infinitely sharp cut-off cannot be Binary realised in practice. Furthermore, small errors in decision timing can lead to large amounts of i.s.i. Both problems can be overcome, but only at the expense of extra bandwidth. A popular solution, which preserves the zero crossing points, is to provide an equaliser Mean signal to noise ratio such that the overall transmission characteristic has a raised cosine shape in the frequency domain. A full (100%) raised cosi