TY - JOUR

T1 - Uncertainty relations and minimum uncertainty states for the discrete Fourier transform and the Fourier series

AU - Forbes, G. W.

AU - Alonso, M. A.

AU - Siegman, A. E.

PY - 2003/6/27

Y1 - 2003/6/27

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0038759688&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/36/25/309

DO - 10.1088/0305-4470/36/25/309

M3 - Article

AN - SCOPUS:0038759688

SN - 0305-4470

VL - 36

SP - 7027

EP - 7047

JO - Journal of Physics A: Mathematical and General

JF - Journal of Physics A: Mathematical and General

IS - 25

ER -