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ACTA PHOTONICA SINICA

WIGNER DISTRIBUTION FUNCTION WITH GENERALIZED PARAMETERS:A NEW DOUBLE-DOMAIN JOINT REPRESENTATION OF SIGNALS

O Box

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

Based on the fractional Fouriertransform(FRT) ,itis proposed the Wigner distribution function with generalized parameters(WDF-GP) ,which is characterized by three parametersp1,p2 and p3.Then its properties have been briefly discussed.The fact that WDF-GP belongs to the Cohen general bilinearclass was also pointed out.Finally, it is demonstrated that some well known relations,such as the cross-spectral density (CSD) of partial coherent optical field,WDF and the fractional Wigner distribution function (FWDF) can be obtained for special cases of the WDF-GP by virtue of special values of paramters.Itshowed thatthe WDF-GP would possessmore powerful capabil- ity to represent signals than thatof WDF.

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Based on the fractional Fouriertransform(FRT) ,itis proposed the Wigner distribution function with generalized parameters(WDF-GP) ,which is characterized by three parametersp1,p2 and p3.Then its properties have been briefly discussed.The fact that WDF-GP belongs to the Cohen general bilinearclass was also pointed out.Finally, it is demonstrated that some well known relations,such as the cross-spectral density (CSD) of partial coherent optical field,WDF and the fractional Wigner distribution function (FWDF) can be obtained for special cases of the WDF-GP by virtue of special values of paramters.Itshowed thatthe WDF-GP would possessmore powerful capabil- ity to represent signals than thatof WDF.
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The Wigner Distribution Function (WDF), defined for a phase space representation in quantum mechanics, offers an alternate form for representing signals simultaneously in Fourier reciprocal variables such as time, and frequency. A purely nonnegative function that is closely related to the WDF, is the so‐called ’’spectrogram’’. In this paper we describe three types of optical setups for the production of the spectrogram. They use coherent, partially coherent or temporally incoherent light as illumination. As application example speech signals are processed for a speaker verification.
We consider a joint spatial-frequency distribution as a result of superposition of two known Wigner and Ville distributions.The joint distribution is characterized by a certain parameter t that plays a part of measure of superposition: the joint frequency-spatial distribution passes into the Ville one at 0 t = and coincides with the Wigner distribution at 1 t = .For the values 0 1 t < < there exists a row of 'mixed' distributions, which represent superposition of the Wigner and Ville distributions.Explicit expression for the joint distribution of Gaussian signal is obtained by means of analytical calculations.We present graphic illustrations for the shape of joint distribution of Gaussian signal at different values of the joint parameter t .It is demonstrated that the Wigner distribution is formed as a rotation of Ville distribution on the information diagram by the angle proportional to the joint parameter t .
The paper presents a review of some applications of the Wigner distribution function to optics. The Wigner distribution function F(x,u) of a signal describes the signal in space x and spatial frequency u, simultaneously, and can be considered as a local spatial frequency spectrum of the signal. Although derived in terms of Fourier optics, the description of a signal by means of its Wigner distribution function closely resembles the ray concept in geometrical optics. Some examples are given to show this resemblance. Properties of the Wigner distribution function are discussed, showing, for instance, its relation to Heisenberg’s uncertainty principle and to some well‐known radiometric quantities. The concept of the Wigner distribution function is not restricted to deterministic signals; it can easily be extended to stochastic signals. Some examples of stochastic signals are considered. The propagation of signals through linear systems can readily be expressed in terms of Wigner distribution functions. Again, the description of systems by Wigner distribution functions can be interpreted directly in terms of geometrical optics: for Luneburg’s first‐order system, for instance, such a description immediately yields the ray transformation matrix of the system, and the relation between the input and the output Wigner distributioin function reads Fo(x,u) = Fi(Ax+Bu,Cx+Du). Transport equations for the Wigner distribution function in a general linear medium are derived. In the Liouville approximation they take the form of a first‐order partial differential equation, which allows, once more, a geometric‐optical interpretation: along the path of a geometric‐optical light ray the Wigner distribution function has a constant value.