Chin. Phys. B, 2010, Vol. 19(1): 014203    DOI: 10.1088/1674-1056/19/1/014203
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# Novel uncertainty relations associated with fractional Fourier transform

Xu Guan-Lei(徐冠雷)a)c), Wang Xiao-Tong(王孝通)a)c), and Xu Xiao-Gang(徐晓刚)b)c)
a Department of Navigation, Dalian Naval Academy, Dalian 116018, China; b Department of Automatization, Dalian Naval Academy, Dalian 116018, Chinac Institute of Photoelectric Technology, Dalian 116018, China
Abstract  In this paper the relations between two spreads, between two group delays, and between one spread and one group delay in fractional Fourier transform (FRFT) domains, are presented and three theorems on the uncertainty principle in FRFT domains are also developed. Theorem 1 gives the bounds of two spreads in two FRFT domains. Theorem 2 shows the uncertainty relation between two group delays in two FRFT domains. Theorem 3 presents the crossed uncertainty relation between one group delay and one spread in two FRFT domains. The novelty of their results lies in connecting the products of different physical measures and giving their physical interpretations. The existing uncertainty principle in the FRFT domain is only a special case of theorem 1, and the conventional uncertainty principle in time-frequency domains is a special case of their results. Therefore, three theorems develop the relations of two spreads in time-frequency domains into the relations between two spreads, between two group delays, and between one spread and one group delay in FRFT domains.
Keywords:  fractional Fourier transform (FRFT)      uncertainty principle      time-frequency spreads      group delay
Received:  21 August 2008      Revised:  23 November 2008      Accepted manuscript online:
 PACS: 03.65.Ta (Foundations of quantum mechanics; measurement theory) 02.30.Nw (Fourier analysis) 02.30.Uu (Integral transforms) 03.65.Db (Functional analytical methods)
Fund: Project supported by the National Natural Science Foundation of China (Grant No. 60473141) and the Natural Science Foundation of Liaoning Province of China (Grant No. 20062191).