Fast numerical algorithm for the linear canonical transform

IEEE Signal Process. Lett.

2009

100

Abstract-Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. The Fourier and fractional Fourier transforms are special cases of LCTs. We present the exact relation between continuous and discrete LCTs (which generalizes the corresponding relation for Fourier transforms), and also express it in terms of a new definition of the discrete LCT (DLCT), which is independent of the sampling interval. This provides the foundation for approximately computing the samples of the LCT of a continuous signal with the DLCT. The DLCT in this letter is analogous to the DFT and approximates the continuous LCT in the same sense that the DFT approximates the continuous Fourier transform. We also define the bicanonical width product which is a generalization of the time-bandwidth product.Index Terms-Bicanonical width product, fractional Fourier transform, linear canonical series, linear canonical transform.

Section: Discrete Linear Canonical Transformsmentioning

confidence: 99%

mentioning

confidence: 99%

Exact Relation Between Continuous and Discrete Linear Canonical Transforms

Oktem

IEEE Signal Process. Lett.

2009

100

“…This approach should therefore greatly facilitate the solution of both forward and inverse problems in optical diffraction in general and holographic 3DTV in particular. An alternative approach to computing linear canonical transforms which does not involve the FFT has been presented in [11]. A comprehensive discussion of sampling issues with reference to Wigner distributions may be found in [10,30,31].…”

Section: Sampling Issues In Diffractionmentioning

confidence: 99%

Signal processing issues in diffraction and holographic 3DTV

Onural

Signal Processing: Image Communication

2007

Image capture and image display will most likely be decoupled in future 3DTV systems. Due to the need to convert abstract representations of 3D images to display driver signals, and to explicitly consider optical diffraction and propagation effects, it is expected that signal processing issues will be of fundamental importance in 3DTV systems. Since diffraction between two parallel planes can be represented as a 2D linear shift-invariant system, various signal processing techniques naturally play an important role. Diffraction between tilted planes can also be modeled as a relatively simple system, leading to efficient discrete computations. Two fundamental problems are digital computation of the optical field arising from a 3D object, and finding the driver signals for a given optical display device which will then generate a desired optical field in space. The discretization of optical signals leads to several interesting issues; for example, it is possible to violate the Nyquist rate while sampling, but still achieve full reconstruction. The fractional Fourier transform is another signal processing tool which finds applications in optical wave propagation. r

“…We also analyze the evolution of spatial information along the longitudinal direction and show that the spherical reference surfaces should be equally spaced with respect to the fractional Fourier transform order. Our results are relevant to work on both sampling and fast computation of light fields propagating through quadratic-phase systems [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

Optimal representation and processing of optical signals in quadratic-phase systems

Arık

2016

Optics Communications

a b s t r a c tOptical fields propagating through quadratic-phase systems (QPSs) can be modeled as magnified fractional Fourier transforms (FRTs) of the input field, provided we observe them on suitably defined spherical reference surfaces. Non-redundant representation of the fields with the minimum number of samples becomes possible by appropriate choice of sample points on these surfaces. Longitudinally, these surfaces should not be spaced equally with the distance of propagation, but with respect to the FRT order. The non-uniform sampling grid that emerges mirrors the fundamental structure of propagation through QPSs. By providing a means to effectively handle the sampling of chirp functions, it allows for accurate and efficient computation of optical fields propagating in QPSs.