The multiple-parameter fractional Fourier transform Authors Jun Lang Department of Electronic Engineering Beijing Institute of Technology Ran Tao Department of Electronic Engineering Beijing Institute of Technology QiWen Ran Department of Mathematics Harbin Institute of Technology Yue Wang Department of Electronic Engineering Beijing Institute of Technology Article

First Online: 18 June 2008 Received: 28 November 2007 Accepted: 26 March 2008 DOI :
10.1007/s11432-008-0073-6

Cite this article as: Lang, J., Tao, R., Ran, Q. et al. Sci. China Ser. F-Inf. Sci. (2008) 51: 1010. doi:10.1007/s11432-008-0073-6
Abstract The fractional Fourier transform (FRFT) has multiplicity, which is intrinsic in fractional operator. A new source for the multiplicity of the weight-type fractional Fourier transform (WFRFT) is proposed, which can generalize the weight coefficients of WFRFT to contain two vector parameters \(
\mathfrak{M},\mathfrak{N} \in \mathbb{Z}^M
\) . Therefore a generalized fractional Fourier transform can be defined, which is denoted by the multiple-parameter fractional Fourier transform (MPFRFT). It enlarges the multiplicity of the FRFT, which not only includes the conventional FRFT and general multi-fractional Fourier transform as special cases, but also introduces new fractional Fourier transforms. It provides a unified framework for the FRFT, and the method is also available for fractionalizing other linear operators. In addition, numerical simulations of the MPFRFT on the Hermite-Gaussian and rectangular functions have been performed as a simple application of MPFRFT to signal processing.

Keywords multiple-parameter fractional Fourier transform weight-type fractional Fourier transform multiplicity of the fractional Fourier transform signal processing Supported by the National Natural Science Foundation of China (Grant No. 60572094), the Doctorship Foundation of China Educational Department (Grant No. 1010036620602) and the National Natural Science Foundation of China for Distinguished Young Scholars (Grant No. 60625104)

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