One of the most important practical questions which arises when we are designing or using an information transmission or processing system is, "What is the

Consider a signal which is being efficiently communicated (i.e. no redundancy) in the form of a time-dependant analog voltage, . The pattern of voltage variations during a specific time interval,

Using the idea of intersymbol influence we can say that since — there is no redundancy — the values of & will appear to be independent of one another provided that they're far enough apart () to be worth sampling separately. In effect, we can't tell what one of the values is just from knowing the other. Of course, for any

This, of course, is why random noise can produce errors in a received message. The statistical properties of an efficiently signalled message are similar to those of random noise. If the signal and noise were obviously different the receiver could easily separate the noise from the signal and avoid making any errors.

To detect and correct errors we therefore have to make the real signal less ‘noise-like’. This is what we're doing when we use parity bits to add redundancy to a signal. The redundancy produces predictable relationships between different sections of the signal pattern. Although this reduces the system's information carrying efficiency it helps us distinguish signal details from random noise. Here, however, we're interested in discovering the maximum possible information carrying capacity of a system. So we have to avoid any redundancy and allow the signal to have the ‘unpredictable’ qualities which make it statistically similar to random noise.

The amount of noise present in a given system can be represented in terms of its mean noise power

A real signal must have a finite power. Hence for a given set of possible messages there must be some maximum possible power level. This means that the rms signal voltage is limited to some range. It also means that the instantaneous signal voltage must be limited and can't be beyond some specific range, . A similar argument must also be true for noise. Since we are assuming that the signal system is efficient we can expect the signal and noise to have similar statistical properties. This implies that if we watched the signal or noise for a long while we'd find that their level fluctuations had the same peak/rms voltage ratio. We can therefore say that, during a typical message, the noise voltage fluctuations will be confined to some range

The signal and noise are

Consider now dividing this range into bands of equal size. (i.e. each of these bands will cover .) To provide a different label for each band we require symbols or numbers. We can therefore always indicate which band the voltage level occupies at any moment in terms of a

There is no real point in choosing a value for

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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.