# Wigner distribution moments in fractional Fourier transform systems

M.J. Bastiaans, T. Alieva

11 Citations (Scopus)

### Abstract

It is shown how all global Wigner distribution moments of arbitrary order in the output plane of a (generally anamorphic) two-dimensional fractional Fourier transform system can be expressed in terms of the moments in the input plane. Since Wigner distribution moments are identical to derivatives of the ambiguity function at the origin, a similar relation holds for these derivatives. The general input-output relationship is then broken down into a number of rotation-type input-output relationships between certain combinations of moments. It is shown how the Wigner distribution moments (or ambiguity function derivatives) can be measured as intensity moments in the output planes of a set of appropriate fractional Fourier transform systems and thus be derived from the corresponding fractional power spectra. The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived. As an important by-product we get a number of moment combinations that are invariant under (anamorphic) fractional Fourier transformation.
Original language English 1763-1773 11 Journal of the Optical Society of America A, Optics, Image Science and Vision 19 9 https://doi.org/10.1364/JOSAA.19.001763 Published - 2002

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distribution moments
Fourier transforms
Power spectrum
Derivatives
moments
output
ambiguity
power spectra
Byproducts
Fourier transformation

### Cite this

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title = "Wigner distribution moments in fractional Fourier transform systems",
abstract = "It is shown how all global Wigner distribution moments of arbitrary order in the output plane of a (generally anamorphic) two-dimensional fractional Fourier transform system can be expressed in terms of the moments in the input plane. Since Wigner distribution moments are identical to derivatives of the ambiguity function at the origin, a similar relation holds for these derivatives. The general input-output relationship is then broken down into a number of rotation-type input-output relationships between certain combinations of moments. It is shown how the Wigner distribution moments (or ambiguity function derivatives) can be measured as intensity moments in the output planes of a set of appropriate fractional Fourier transform systems and thus be derived from the corresponding fractional power spectra. The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived. As an important by-product we get a number of moment combinations that are invariant under (anamorphic) fractional Fourier transformation.",
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year = "2002",
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language = "English",
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journal = "Journal of the Optical Society of America A, Optics, Image Science and Vision",
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In: Journal of the Optical Society of America A, Optics, Image Science and Vision, Vol. 19, No. 9, 2002, p. 1763-1773.

TY - JOUR

T1 - Wigner distribution moments in fractional Fourier transform systems

AU - Bastiaans, M.J.

AU - Alieva, T.

PY - 2002

Y1 - 2002

N2 - It is shown how all global Wigner distribution moments of arbitrary order in the output plane of a (generally anamorphic) two-dimensional fractional Fourier transform system can be expressed in terms of the moments in the input plane. Since Wigner distribution moments are identical to derivatives of the ambiguity function at the origin, a similar relation holds for these derivatives. The general input-output relationship is then broken down into a number of rotation-type input-output relationships between certain combinations of moments. It is shown how the Wigner distribution moments (or ambiguity function derivatives) can be measured as intensity moments in the output planes of a set of appropriate fractional Fourier transform systems and thus be derived from the corresponding fractional power spectra. The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived. As an important by-product we get a number of moment combinations that are invariant under (anamorphic) fractional Fourier transformation.

AB - It is shown how all global Wigner distribution moments of arbitrary order in the output plane of a (generally anamorphic) two-dimensional fractional Fourier transform system can be expressed in terms of the moments in the input plane. Since Wigner distribution moments are identical to derivatives of the ambiguity function at the origin, a similar relation holds for these derivatives. The general input-output relationship is then broken down into a number of rotation-type input-output relationships between certain combinations of moments. It is shown how the Wigner distribution moments (or ambiguity function derivatives) can be measured as intensity moments in the output planes of a set of appropriate fractional Fourier transform systems and thus be derived from the corresponding fractional power spectra. The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived. As an important by-product we get a number of moment combinations that are invariant under (anamorphic) fractional Fourier transformation.

U2 - 10.1364/JOSAA.19.001763

DO - 10.1364/JOSAA.19.001763

M3 - Article

VL - 19

SP - 1763

EP - 1773

JO - Journal of the Optical Society of America A, Optics, Image Science and Vision

JF - Journal of the Optical Society of America A, Optics, Image Science and Vision

SN - 1084-7529

IS - 9

ER -