The Uncertainty Principle

Consider a pure sine wave tone. The sound of a flute in its upper register is a pretty good approximation to one. Imagine you have a long recording of it and you cut off a section for the purposes of making measurements on it. If you keep cutting bits off so that you have a shorter and shorter record, you eventually end up with a click. Somewhere in between was a point where you stopped being able to recognise it as a tone, an example of the sorites paradox.

Joseph Fourier (1786-1830) developed his series as a tool for his investigations into heat flow. He showed that any shape of waveform as a graph in time can be created by adding together a series of sine waves. Thus any signal can be fully described (mapped) either by its graph against time or against frequency. A formula known as the Fourier Transform enables us to get one from the other.

In our chopping off process we proceeded gradually from a single frequency sine wave to a click, which has a wide range of frequencies. The sine wave was one precise point in the frequency graph, whereas the click occurs at a precise point in time. In between, we observe that the more we locate a signal in the time domain, the less we can locate it in the frequency domain and vice versa. This is the Uncertainty Principle.

In signal processing we are faced with the problem of taking a finite block of signal and using it to represent the whole signal. This is known as the window problem. The sharp corners of the block, in particular, produce lobes of spurious frequencies that severely contaminate the signal. For this reason we use what are called window functions to smooth out the edges. Thus, in estimating the spectrum of a signal, two workers might get different results depending on their choice of window function. This is also a manifestation of the Uncertainty Principle and we cannot accurately estimate the spectrum of a signal from only part of its time record, though the accuracy improves as the length of our record increases.

Of course, the Uncertainty Principle is most closely associated with Heisenberg in the theory of quantum mechanics. It arises through our difficulties in describing the motion of sub-atomic particles, such as electrons. Our senses can only understand two sorts of motion, that of a particle in which a piece of matter moves, or that of a wave in which the matter stays still on average but the energy moves. The motion of an electron is neither, but if we perform an experiment to look for a particle we see a particle and if we look for a wave we see a wave. The Fourier transform comes into operation because of the wavelike properties and the Uncertainty Principle then becomes The more we know about the position of a particle the less we know about its momentum and vice versa.

The principle is more important in our everyday life, however, in the form associating frequency with time. Consider the problem of fitting a linear trend to a sequence of data. This can be regarded as a signal extraction problem, in which we are trying to extract a ramp or sawtooth component of the signal from the unwanted part, which we call noise. The principle applies. The sawtooth has a precise signature in the frequency domain, but it is contaminated by both the noise and the chopping off distortion.

The dilemma of the global warmers, for example, is that they need to prove that a linear trend is located within the last few decades. If they took only one year’s result it would be like our click, exactly located in time but with infinite errors spread across the frequency domain. As they consider more and more years the errors in the frequency domain decrease, but the event is less and less well located in time. The more accurately they measure the trend, the less accurately they know where it is. This is reflected in the simulations performed in our discussion of fitting linear trends.

Naturally, the above verbal discussion is somewhat imprecise and a full explanation would require the use of mathematical language.

John Brignell

March 2003

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