 zero duration and infinite amplitude). The transfer 19.2.1 Bandwidth requirements characteristic of the channel should therefore be equalised SO that the output signal has the required spectrum. A commonly used The minimum bandwidth needed to transmit a digital signal at B signal is that having the raised-cosine spectrum given by Equation bauds has been shown by Nyquist (1928) to be Wmin=1/2B Hertz. 19.8 to 19.10. This can be demonstrated as follows. Consider a binary signal consisting of alternate '0's and '1's. This produces a square wave of F(J)=1 for0s/s(1-a) (19.8) frequency 1/2B. Let this be applied to an ideal low pass filter. If the cut off frequency of the filter is as in Equation 19.1 where E is small, the output is a sine wave of frequency 1/2B and the original square wave can be recovered by sampling it at its positive and negative peaks. (19.9) (19.1) Filter cut off frequency (19.10) If the cut off frequency is reduced to the value given in Equation 19.2, the output consists only of the d.c. component and the signal This gain/frequency response rolls off sinusoidally from unity to is lost. zero in the frequency band from for Cut off frequency = 1/8 - (19.2) 100% roll off (i.e. a = 1), the spectrum occupies a bandwidth of 1/T, which is twice the theoretical minimum requirement. Bandwidth is used most efficiently by using as small a roll off a as possible, but Consequently, the minimum bandwidth required is given by Equ- problems of timing and equalisation increase as a is reduced. ation 19.3. (19.3) 19.2.1.1 Equalisation Digital transmission systems can use gain and phase equalisation to In the limit when E tends to zero, Equation 19.4 is obtained. obtain an output signal spectrum corresponding to a pulse waveform 19/4 Digital modulation Input Output Figure 19.1 Block diagram of a transversal equailiser with negligible intersymbol interference, e.g. the raised cosine This can also be done digitally by combining adjacent pulses at the spectrum described above. However, time domain equalisers are sendi