## Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space–bandwidth product

JOSA A, Vol. 27, Issue 8, pp. 1885-1895 (2010)

http://dx.doi.org/10.1364/JOSAA.27.001885

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### Abstract

Linear canonical transforms (LCTs) form a three-parameter family of integral transforms with wide application in optics. We show that LCT domains correspond to scaled fractional Fourier domains and thus to scaled oblique axes in the space–frequency plane. This allows LCT domains to be labeled and ordered by the corresponding fractional order parameter and provides insight into the evolution of light through an optical system modeled by LCTs. If a set of signals is highly confined to finite intervals in two arbitrary LCT domains, the space–frequency (phase space) support is a parallelogram. The number of degrees of freedom of this set of signals is given by the area of this parallelogram, which is equal to the bicanonical width product but usually smaller than the conventional space–bandwidth product. The bicanonical width product, which is a generalization of the space–bandwidth product, can provide a tighter measure of the actual number of degrees of freedom, and allows us to represent and process signals with fewer samples.

© 2010 Optical Society of America

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(070.2590) Fourier optics and signal processing : ABCD transforms

(080.2730) Geometric optics : Matrix methods in paraxial optics

(070.2025) Fourier optics and signal processing : Discrete optical signal processing

(070.2575) Fourier optics and signal processing : Fractional Fourier transforms

(050.5082) Diffraction and gratings : Phase space in wave options

**ToC Category:**

Fourier Optics and Signal Processing

**History**

Original Manuscript: March 29, 2010

Manuscript Accepted: May 25, 2010

Published: July 30, 2010

**Citation**

Figen S. Oktem and Haldun M. Ozaktas, "Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space–bandwidth product," J. Opt. Soc. Am. A **27**, 1885-1895 (2010)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-27-8-1885

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