## Simplified fractional Fourier transforms

JOSA A, Vol. 17, Issue 12, pp. 2355-2367 (2000)

http://dx.doi.org/10.1364/JOSAA.17.002355

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### Abstract

The fractional Fourier transform (FRFT) has been used for many years, and it is useful in many applications. Most applications of the FRFT are based on the design of fractional filters (such as removal of chirp noise and the fractional Hilbert transform) or on fractional correlation (such as scaled space-variant pattern recognition). In this study we introduce several types of simplified fractional Fourier transform (SFRFT). Such transforms are all special cases of a linear canonical transform (an affine Fourier transform or an *ABCD* transform). They have the same capabilities as the original FRFT for design of fractional filters or for fractional correlation. But they are simpler than the original FRFT in terms of digital computation, optical implementation, implementation of gradient-index media, and implementation of radar systems. Our goal is to search for the simplest transform that has the same capabilities as the original FRFT. Thus we discuss not only the formulas and properties of the SFRFT’s but also their implementation. Although these SFRFT’s usually have no additivity properties, they are useful for the practical applications. They have great potential for replacing the original FRFT’s in many applications.

© 2000 Optical Society of America

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(070.2590) Fourier optics and signal processing : ABCD transforms

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(070.6110) Fourier optics and signal processing : Spatial filtering

**History**

Original Manuscript: February 25, 2000

Revised Manuscript: May 31, 2000

Manuscript Accepted: August 1, 2000

Published: December 1, 2000

**Citation**

Soo-Chang Pei and Jian-Jiun Ding, "Simplified fractional Fourier transforms," J. Opt. Soc. Am. A **17**, 2355-2367 (2000)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-12-2355

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