Finding
A further illustration of this is that it is theoretically possible to use 2TW (time) derivatives of the signal waveform taken at a single instant, rather than 2TW sample amplitudes spread over the whole of T. It may not be further realised that the formulation of the sampling theorem as 'two samples per cycle of the highest Fourier component frequency' expresses a condition which is sufficient but not always necessary.
Paper
The Sampling Theorem
Abstract
The sampling theorem, as commonly understood, gives a condition which is sufficient but not always necessary. It depends on whether or not the system is designed for signals having a flat spectrum. It is by now a very familiar idea that if a signal is to be digitised, the number of samples should be two per cycle of the highest-frequency Fourier component of the signal. This is usually written as 2TW where T is the duration in time of the signal and W its bandwidth; and in this form, with W as the highest signal (modulation) frequency, after eliminating any carrier, it is applicable to bandpass as well as low-pass systems. There is, of course, the point that in a band-pass system the samples should be arranged in pairs instead of being equally spaced in time, but this merely illustrates the fact that from the point of view of information theory one needs two pieces 0/data per cycle, rather than the amplitudes of an array of samples. A further illustration of this is that it is theoretically possible to use 2TW (time) derivatives of the signal waveform taken at a single instant, rather than 2TW sample amplitudes spread over the whole of T. It may not be further realised that the formulation of the sampling theorem as 'two samples per cycle of the highest Fourier component frequency' expresses a condition which is sufficient but not always necessary. Careful reading of Shannon's paper on 'Communication in the presence of noise" shows that he was considering what sampling was needed to represent any waveform falling within the limits of bandwidth Wand duration T: if one is considering all possible such waveforms, e.g. white noise, then 2TW samples are necessary as well as sufficient. But if the waveforms are limited to a particular statistical class, a smaller number of samples will suffice: this corresponds to the fact that some information about such signals is already available. An example of this is the sampling of noise having a II/spectrum. There has been controversy over a number of years about the variability of II/noise in terms of its 'variance of variance'. Most measurements were digitised, for computer processing, and it seemed natural that samples should be taken at two per cycle of the highest frequency accepted by the apparatus. But Stoisiek and Wolf pointed out that such samples would not be statistically independent if taken from noise having a II/spectrum. The effective number of independent samples would then be less than their actual number and the variability of their mean square value correspondingly greater than expected. From the view-point of information theory it is qualitatively apparent that if something is known about the signal, i.e. that it has a spectrum of II/shape, a smaller amount of specific data will suffice to describe it completely. The quantitative calculation for the II/case is given by Stoisiek and Wolf. A rigorous mathematical analysis has been given by Lee:', including a section on bandlimited stochastic processes; but his mathematical presentation is not directly helpful to engineers. Speech is often said to have the statistical characteristics of noise, but it has a well-defined average spectrum which is far from white. It would be interesting to investigate how far the sampling of speech could be correspondingly reduced and what kind of noise would be introduced as a result of sampling on this minimum basis instead of taking two samples per cycle of the highest frequency.
Authors
D. A. Bell
Journal
International Journal of Electrical Engineering Education