Digital Computation of Linear Canonical Transforms

Koç, Aykut; Özaktaş, Haldun M.; Candan, Çağatay; Kutay, M. Alper

doi:10.1109/tsp.2007.912890

Cited by 142 publications

(83 citation statements)

References 51 publications

(71 reference statements)

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“…Computation of the output samples from the input samples is fast; it takes ∼N N log time [38]. Moreover, computational accuracy arising from the use of the discrete FRT is not different than that arising from the use of the FFT to compute the common Fourier transform [31]. To summarize, we have shown that we can represent and compute the output field by using an identical number of samples as required for the input.…”

Section: Transverse Sampling Spacingmentioning

confidence: 85%

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

Arık

Özaktaş

2016

Optics Communications

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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.

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Section: Transverse Sampling Spacingmentioning

confidence: 85%

“…One of the possible decompositions involves three stages. The first is a FRT operation, the second is a magnification operation, and the final stage is a chirp multiplication operation [28][29][30][31][32]:…”

Section: Decomposition Of Propagation In Quadratic-phase Systemsmentioning

confidence: 99%

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

Arık

Özaktaş

2016

Optics Communications

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“…All of these produce output vectors which are good approximations to the samples of the continuous transform, limited only by the fundamental fact that a signal cannot have finite extent in more than one domain; since the sampling interval is ensured to satisfy the Nyquist criterion, the output samples can be used to reconstruct good approximations of the continuous output. On the other hand, while the algorithms in [16], [17] also take time, most of them involve more than one FFT and therefore a larger factor in front, in addition to being less transparent. However, this does not automatically mean that these earlier algorithms are slower since the number of samples in these works are not directly comparable to that in this letter, as discussed below.…”

Section: Computation Of Continuous Lctsmentioning

confidence: 99%

“…In contrast, in [16], [17] it is assumed that the signal is confined to a rectangle or ellipse orthogonal to the time-frequency axes in the time-frequency plane, regardless of the parameters of the FRT or LCT to be computed. As noted, it is not possible to directly compare the present algorithm to those in [17] since different families of signals are assumed. Therefore, which algorithm is better will depend strongly on what assumptions are best suited to the family of signals we are dealing with.…”

Section: Computation Of Continuous Lctsmentioning

confidence: 99%

Exact Relation Between Continuous and Discrete Linear Canonical Transforms

Oktem

Özaktaş

2009

IEEE Signal Process. Lett.

100

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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.

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“…1). It is known that the Fresnel integral can be decomposed into an FRT followed by magnification followed by chirp multiplication [2][3][4][5][6]:…”

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confidence: 99%

Fundamental structure of Fresnel diffraction: longitudinal uniformity with respect to fractional Fourier order

2011

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Fresnel integrals corresponding to different distances can be interpreted as scaled fractional Fourier transformations observed on spherical reference surfaces. Transverse samples can be taken on these surfaces with separation that increases with propagation distance. Here, we are concerned with the separation of the spherical reference surfaces along the longitudinal direction. We show that these surfaces should be equally spaced with respect to the fractional Fourier transform order, rather than being equally spaced with respect to the distance of propagation along the optical axis. The spacing should be of the order of the reciprocal of the space-bandwidth product of the signals. The space-dependent longitudinal and transverse spacings define a grid that reflects the structure of Fresnel diffraction.

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Digital Computation of Linear Canonical Transforms

Cited by 142 publications

References 51 publications

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

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

Exact Relation Between Continuous and Discrete Linear Canonical Transforms

Fundamental structure of Fresnel diffraction: longitudinal uniformity with respect to fractional Fourier order

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