On uncertainty principle for signal concentrations with fractional Fourier transform

Authors:
Jun Shi;Xiaoping Liu;Naitong Zhang
Affiliations:
Communication Research Center, Harbin Institute of Technology, Harbin 150001, China and Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA;Communication Research Center, Harbin Institute of Technology, Harbin 150001, China;Communication Research Center, Harbin Institute of Technology, Harbin 150001, China and Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China
Venue:
Signal Processing
Year:
2012

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Abstract

The fractional Fourier transform (FRFT) - a generalized form of the classical Fourier transform - has been shown to be a powerful analyzing tool in signal processing. This paper investigates the uncertainty principle for signal concentrations associated with the FRFT. It is shown that if the fraction of a nonzero signal's energy on a finite interval in one fractional domain with a certain angle @a is specified, then the fraction of its energy on a finite interval in other fractional domain with any angle @b(@b@a) must remain below a certain maximum. This is a generalization of the fact that any nonzero signal cannot have arbitrarily large proportions of energy in both a finite time duration and a finite frequency bandwidth. The signals which are the best in achieving simultaneous concentration in two arbitrary fractional domains are derived. Moreover, some applications of the derived theory are presented.