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Journal of Physics A Mathematical and General

Uncertainty relations and minimum uncertainty states for the discrete Fourier transform and the Fourier series

G. W. Forbes, Miguel Angel Alonso, Anthony E.. Siegman

Published Jun 11, 2003• 20 citations

Abstract

The conventional Fourier transform has a well-known uncertainty relation that is defined in terms of the first and second moments of both a function and its Fourier transform. It is also well known that Gaussian functions, when translated to an arbitrary centre and supplemented by a linear phase factor, provide a complete set of minimum uncertainty states (MUSs) that exactly achieves the lower bound set by this uncertainty relation. A similarly general set of MUSs and uncertainty relations are derived here for discrete and/or periodic generalizations of the Fourier transform, namely for the discrete Fourier transform and the Fourier series. These extensions require a modified definition for the width of a periodic distribution, and they lead to more complex uncertainty relations that turn out to depend on the centroid location and mean frequency of the distribution. The derivations lead to novel generalizations of Hermite–Gaussian functions and, like Gaussians, the MUSs can play a special role in a range of Fourier applications.

DOI: 10.1088/0305-4470/36/25/309

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To resolve an issue that was raised in our earlier paper in this journal [69, 340–347 (2001)], a direct analog of the standard uncertainty relation is derived for the discrete Fourier transform (DFT). This inequality gives a simple lower bound for the degree of localization of the DFT of any sequence in terms of just the localization of the original sequence. It is also shown that the earlier uncertainty relations for the Fourier transform and the Fourier series can be derived from this new relation by taking appropriate limits. Because the same cannot be said of the others, this new relation is arguably the most fundamental of the three.

A physicist’s intuition in Fourier theory is generally established from the parallels between Fourier series and transforms. Remarkably, one element of this theory that is especially significant in physics, namely the uncertainty principle, is never treated for Fourier series. We resolve this by first showing that a natural measure of spread for a periodic distribution follows simply upon regarding the distribution as a mass density on a ring. Even though the centroid of this ring is expressed in terms of just a first moment, its distance from the geometric center gives a close analog of variance. We then derive direct analogs of the uncertainty principle for both the Fourier series of a continuous periodic function as well as the fast Fourier transform of discrete data. The results have similar applications to those of the standard uncertainty principle.

We derive an uncertainty relation for two unitary operators which obey a commutation relation of the form UV=e(i phi) VU. Its most important application is to constrain how much a quantum state can be localized simultaneously in two mutually unbiased bases related by a discrete fourier transform. It provides an uncertainty relation which smoothly interpolates between the well-known cases of the Pauli operators in two dimensions and the continuous variables position and momentum. This work also provides an uncertainty relation for modular variables, and could find applications in signal processing. In the finite dimensional case the minimum uncertainty states, discrete analogues of coherent and squeezed states, are minimum energy solutions of Harper's equation, a discrete version of the harmonic oscillator equation.

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on \mathbb{R}^d which may be written as P(x)\exp (-\langle Ax, x\rangle) , with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x) \widehat{f}(y) . We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform.

The uncertainty relation is introduced by using operators which are not necessarily self-adjoint. In general, the uncertainty (lower) limit is different for different functions. Operators are found that define the second moments and the uncertainty limit for their product such that the limit is the same for functions obtained from each other by applying unitary support-size preserving operators, e.g. by shifting in time and frequency. In this case, for all functions of finite energy the limit is related to the minimum support size in the time-frequency domain. Moreover, it is possible to find the discrete-time counterpart of the classical uncertainty relation for continuous-time functions. The operator calculus presents results from signal analysis in a more standard form extensively used in quantum mechanics.