What is the Gabor uncertainty principle?

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This post is adapted from the introduction to my article Hall, M (2006), Resolution and uncertainty in spectral decomposition. First Break 24, December 2006. DOI: 10.3997/1365-2397.2006027. I'm planning to delve into this a bit, partly as a way to get up to speed on signal processing in Python. Stay tuned.

Spectral decomposition is a powerful way to get more from seismic reflection data, unweaving the seismic rainbow.There are lots of ways of doing it — short-time Fourier transform, S transform, wavelet transforms, and so on. If you hang around spectral decomposition bods, you'll hear frequent mention of the ‘resolution’ of the various techniques. Perhaps surprisingly, Heisenberg’s uncertainty principle is sometimes cited as a basis for one technique having better resolution than another. Cool! But... what on earth has quantum theory got to do with it?

A property of nature

Heisenberg’s uncertainty principle is a consequence of the classical Cauchy–Schwartz inequality and is one of the cornerstones of quantum theory. Here’s how he put it:

At the instant of time when the position is determined, that is, at the instant when the photon is scattered by the electron, the electron undergoes a discontinuous change in momen- tum. This change is the greater the smaller the wavelength of the light employed, i.e. the more exact the determination of the position. At the instant at which the position of the electron is known, its momentum therefore can be known only up to magnitudes which correspond to that discontinuous change; thus, the more precisely the position is determined, the less precisely the momentum is known, and conversely. — Heisenberg (1927), p 174-5.

The most important thing about the uncertainty principle is that, while it was originally expressed in terms of observation and measurement, it is not a consequence of any limitations of our measuring equipment or the mathematics we use to describe our results. The uncertainty principle does not limit what we can know, it describes the way things actually are: an electron does not possess arbitrarily precise position and momentum simultaneously. This troubling insight is the heart of the so-called Copenhagen Interpretation of quantum theory, which Einstein was so famously upset by (and wrong about).

Dennis Gabor (1946), inventor of the hologram, was the first to realize that the uncertainty principle applies to signals. Thanks to wave-particle duality, signals turn out to be exactly analogous to quantum systems. As a result, the exact time and frequency of a signal can never be known simultaneously: a signal cannot plot as a point on the time-frequency plane. Crucially, this uncertainty is a property of signals, not a limitation of mathematics.

Getting quantitative

You know we like the numbers. Heisenberg’s uncertainty principle is usually written in terms of the standard deviation of position σx, the standard deviation of momentum σp, and the Planck constant h:

In other words, the product of the uncertainties of position and momentum is small, but not zero. For signals, we don't need Planck’s constant to scale the relationship to quantum dimensions, but the form is the same. If the standard deviations of the time and frequency estimates are σt and σf respectively, then we can write Gabor’s uncertainty principle thus:

So the product of the standard deviations of time, in milliseconds, and frequency, in Hertz, must be at least 80 ms.Hz, or millicycles. (A millicycle is a sort of bicycle, but with 1000 wheels.)

The bottom line

Signals do not have arbitrarily precise time and frequency localization. It doesn’t matter how you compute a spectrum, if you want time information, you must pay for it with frequency information. Specifically, the product of time uncertainty and frequency uncertainty must be at least 1/4π. So how certain is your decomposition?

References

Heisenberg, W (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik 43, 172–198. English translation: Quantum Theory and Measurement, J. Wheeler and H. Zurek (1983). Princeton University Press, Princeton.

Gabor, D (1946). Theory of communication. Journal of the Institute of Electrical Engineering 93, 429–457.

The image of Werner Heisenberg in 1927, at the age of 25, is public domain as far as I can tell. The low res image of First Break is fair use. The bird hologram is form a photograph licensed CC-BY by Flickr user Dominic Alves.