On scaling properties of fractional Fourier transform and its relation with other transforms K.K. Sharma * , S.D. Joshi Department of Electrical Engineering, Indian Institute of Technology, New Delhi, India Received 22 November 2004; received in revised form 4 July 2005; accepted 5 July 2005 Abstract Several properties of fractional Fourier transform (FRFT) have been studied recently and many are being investi- gated at present. In this article, scaling property of the FRFT is generalized and some of its applications are suggested. Some extensions of the sampling relations in the FRFT domain are also presented. The issues related to connections between the FRFT and other signal transforms such as scale transform, fractional Mellin transform, and chirp z-trans- form, are also investigated. Ó 2005 Elsevier B.V. All rights reserved. PACS: 42.30 k; 32.30; 33.20; 78.30; 78.40 Keywords: Fractional Fourier transform; Fractional Mellin transform; Chirp-z transform; Sampling theorems 1. Introduction Fourier transform is one of the most widely used tools in signal processing and optics. Fractional Fou- rier transform (FRFT) is a generalization of the conventional Fourier transform and has received much attention in recent years [1,3–5,17]. A number of useful properties of the FRFT have been derived in [5,6,10], which are generalizations of the corresponding properties of the conventional Fourier transform. Many other interesting properties of the FRFT, including its product and convolution theorems, are cur- rently well known in signal processing community and can be found in [1,10,13]. The FRFT has also been proved to relate to other signal analysis tools such as Wigner distribution, neural network, wavelet trans- form, short-time Fourier transform and various chirp related operations [5,8]. 0030-4018/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.07.013 * Corresponding author. Tel.: +91 141 2551693; fax: +91 011 26581069. E-mail address: kksharma_mrec@yahoo.com (K.K. Sharma). Optics Communications 257 (2006) 27–38 www.elsevier.com/locate/optcom