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

G. W. Forbes*, M. A. Alonso, A. E. Siegman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)7027-7047
Number of pages21
JournalJournal of Physics A: Mathematical and General
Volume36
Issue number25
DOIs
Publication statusPublished - 27 Jun 2003
Externally publishedYes

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