IEEE Proof IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 10, OCTOBER 2005 1 A New Chirp Scaling Algorithm Based on the Fractional Fourier Transform A. S. Amein, Member, IEEE, and J. J. Soraghan, Senior Member, IEEE Abstract—The fractional Fourier transform (FrFT), which is a generalized form of the well-known Fourier transform, has only recently started to appear in the field of signal processing. This has opened up the possibility of a new range of potentially promising and useful applications. In this letter, we develop a new FrFT-based chirp scaling algorithm (CSA) and compare its performance with the classical CSA based on the fast Fourier transform (FFT). Simu- lation results show that the FrFT-based CSA can offer significantly enhanced features compared to the classical FFT-based approach. Index Terms—Chirp, fractional, high resolution, synthetic aper- ture radar (SAR). I. INTRODUCTION T HE fractional Fourier transform (FrFT) was derived by Namias in the 1980s as a new mathematical tool in order to deal with certain problems in quantum mechanics [1]. The first introduction to the application of FrFT in signal processing was published by Almeida [2]. A more recent introduction to the FrFTs and their applications is given in [3], which describes a number of promising research areas for further investigation, including radar applications involving the use and detection of chirp signals, pattern recognition, and synthetic aperture radar (SAR) image processing. The chirp scaling algorithm (CSA) [4] is one of the most well-known and widely used SAR imaging algorithms due to its accurate focusing ability and simplicity of implementation. The algorithm requires only fast Fourier trans- forms (FFTs) and uses complex vector multipliers without any interpolations. This attribute makes hardware and software im- plementations of the CSA simple relative to algorithms that re- quire sophisticated interpolations [19]. Numerous modifications to the CSA operation have been proposed that aim to improve the algorithm’s ability to process spotlight and high-resolution images [5], [6]. In this letter, we introduce a new FrFT-based CSA as an al- ternative to the classical FFT-based algorithm. The proposed al- gorithm, termed the fractional chirp scaling algorithm (FrCSA), avoids the dependence of the conventional CSA on the Taylor series approximations and also removes the need of using the Principle of Stationary Phase (PSP) [7] to approximate some of the integral solutions. In addition, the inherent mathemat- ical characteristics of the FrFT eliminate the requirements for Manuscript received March 15, 2005; revised May 5, 2005. The associate ed- itor coordinating the review of this manuscript and approving it for publication was Dr. Xiang-Gen Xia. The authors are with the Institute of Communication and Signal Processing (ICSP), Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow G67 4BB, U.K. (e-mail: asamein@hotmail.co.uk; j.soraghan@eee.strath.ac.uk). Digital Object Identifier 10.1109/LSP.2005.855547 any additional focusing or side-lobe reduction techniques that are normally deployed in the FFT-based solutions. Preliminary simulation results demonstrate a significant increase in both the signal-to-noise ratio (SNR) and in the sidelobe reduction ratio (SLRR) of the FrCSA compared to the classical FFT-based ver- sion. The rest of this letter is organized as follows. Section II introduces the FrFT and its various mathematical properties. Section III presents the mathematical model for the new FrFT- based CSA. This is followed by some simulation experiments described in Section IV. Finally, some concluding remarks and future work proposals are given in Section V. II. FrFT AND ITS PROPERTIES As the classical Fourier transform (FT) corresponds to a ro- tation in the time-frequency plane over an angle , the FrFT can be considered as a generalized form that corresponds to a rotation over some arbitrary angle with . The continuous one-dimensional (1-D) FrFT is defined by means of the transformation kernel [2], shown in (1) at the bottom of the next page. Given that is the Fourier transform operator and is the fractional Fourier transform operator, then the FrFT possesses the following important properties. 1) Zero rotation: . 2) Consistency with Fourier transforms: . 3) Additivity of rotations: . 4) rotation: . 5) Inverse FrFT: . In addition, the FrFT kernel has the following properties, which will be of interest in this letter. 1) . 2) . 3) . 4) . 5) , where indicates the complex conjugate. Further properties of the FrFT and sample transforms of some common functions can be found in [2] and [3]. Formally, the FrFT of an arbitrary function , with an angle , is defined in (2), shown in the second equation at the bottom of the next page. Equation (2) shows that for angles that are not multiples of , the computation of the FrFT corresponds to the following steps. Step 1) A product by a chirp. Step 2) A Fourier transform [with its argument scaled by ]. Step 3) Another product by a chirp. Step 4) A product by a complex amplitude factor. 1070-9908/$20.00 © 2005 IEEE