Convolution theorems for the linear canonical transforms

John J. Healy; Xiaolin Li; Min Wan; Liang Zhao

doi:10.1117/12.2520929

23 April 2019 Convolution theorems for the linear canonical transforms

John J. Healy, Xiaolin Li, Min Wan, Liang Zhao

Proceedings Volume 11030, Holography: Advances and Modern Trends VI; 110301A (2019) https://doi.org/10.1117/12.2520929
Event: SPIE Optics + Optoelectronics, 2019, Prague, Czech Republic

Abstract

The linear canonical transforms (LCTs) are a Lie group of transforms including the Fresnel and Fourier transforms that describe scalar wave propagation in quadratic phase systems. As such, they are useful in system analysis and design, and their discretisations are important for opto-numerical systems, e.g. numerical reconstruction algorithms in digital holography. An important topic in the literature is therefore the generalization of Fourier transform properties for the LCTs. A number of authors have proposed convolution theorems for the linear canonical transform, with different goals in mind. In this paper, we compare those methods, with particular attention being paid to the consequences of discretization. In a similar way to how discrete convolution associated with the DFT differs from that associated with the Fourier transform, we must take the chirp-periodic nature of discrete LCTs into account when determining the discrete convolution associated with LCTs. This work is of significance for the simulation of VanderLugt correlators, which have been used for optical implementations of neural networks, and for optical filtering operations and coherent optical signal processing in general.

Citation Download Citation

John J. Healy, Xiaolin Li, Min Wan, and Liang Zhao "Convolution theorems for the linear canonical transforms", Proc. SPIE 11030, Holography: Advances and Modern Trends VI, 110301A (23 April 2019); https://doi.org/10.1117/12.2520929

ACCESS THE FULL ARTICLE

INSTITUTIONAL
Select your institution to access the SPIE Digital Library.

SELECT YOUR INSTITUTION

PERSONAL
Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.

PERSONAL SIGN IN

No SPIE Account? Create one

PURCHASE THIS CONTENT

SUBSCRIBE TO DIGITAL LIBRARY

50 downloads per 1-year subscription

Members: $195

Non-members: $335 ADD TO CART

25 downloads per 1 - year subscription

Members: $145

Non-members: $250 ADD TO CART

PURCHASE SINGLE ARTICLE

Includes PDF, HTML & Video, when available

Members: $17.00

Non-members: $21.00 ADD TO CART

PROCEEDINGS
9 PAGES

DOWNLOAD PAPER SAVE TO MY LIBRARY

GET CITATION

RIGHTS & PERMISSIONS

Get copyright permission Get copyright permission on Copyright Marketplace

KEYWORDS

Convolution

Transform theory

Fourier transforms

Wigner distribution functions

Fractional fourier transform

Analytical research

Hybrid optics

Show All Keywords

Keywords/Phrases

Search In:

Publication Years