Probability and Information Theory with Applications to by P. M. Woodward

By P. M. Woodward

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If we ignore the quantum effects which occur at very high frequencies, pure white noise with W infinite is an impossible idealisation. The sampling points would be separated by vanishingly small intervals of time and the waveform would be randomly infinite at all points. This is a further justification for the arbitrary bandwidth limitation. W e have now seen that noise energy, if it is like thermal noise, is distributed uniformly over both frequency (up to a very high frequency) and time. In time T and a band W, there are 2WT degrees of freedom and an energy %N0is associated with each.

Having described this, however, we may perhaps forget all about binary digits and proceed by a looser, more intuitive method ( W O O D W A R D and DAVIES, 1 9 5 2 ) , using the ideas of communication rather than storage as a model. ) When a communication is received, the state of knowledge of the recipient or "observer" is changed, and it is with the measurement of such changes that communication theory has to deal. Before reception, each of the possible message states has a certain probability of occurring; afterwards, one particular state X will have been singled out in the mind of the observer, the uncertainty described by its initial probability P{X) will be removed and information gained.

This geometrical concept has been used with great effect by SHANNON in a paper (1949) on communication in the presence of noise. 8 UNIFORM GAUSSIAN NOISE Of all forms of noise, the simplest to discuss is that which is as random as possible within the constraints imposed by the system in which it occurs. Thermal noise is of this nature, and the description which follows is limited to noise having the same statistical structure as classical thermal noise. It is the most fundamental type of noise to understand, and in order to describe it mathematically, we shall find it convenient to impose an arbitrary frequency limitation, excluding all frequencies numerically greater than W.

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