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Ukrainian Journal of Physical Optics

Joint spatial-frequency distribution of Gaussian signals and prospects of its applications for description of optical systems

Yu.M. Kozlovskii

Published Jan 1, 2008• 0 citations
Publisher

Abstract

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 .

DOI
10.3116/16091833/9/2/105/2008

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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 .
Mutual space-frequency distribution is proposed and it is shown that Wigner and Weyl distribution functions are only particular cases of these distribution. Mutual distribution for Gaussian signal is analytically obtained. The simple connection between Wigner and Weyl distributions is established. It is shown that Wigner distribution forms as the rotational displacement of Weyl distribution on informational diagram of conjugate coordinates (x;p)(x;p)(x;p) on an angle proportional to the mutual parameter ttt. The results of direct calculations of mutual distribution for Gaussian signal in the mutual domain are presented.
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.
The description of a signal by means of a local frequency spec-trum resembles such things as the score in music, the phase space in mechanics, and the ray concept in geometrical optics. Two types of local frequency spectra are presented: the Wigner distribution function and the sliding-window spectrum, the latter having the form of a cross-ambiguity function. The Wigner distribution function in particular can provide a link between Fourier optics and geometrical optics; many properties of the Wigner distribution function, and the way in which it prop-agates through linear systems, can be interpreted in geometric-optical terms. The Wigner distribution function is linearly related to other signal representations like Woodward's ambiguity function, Rihaczek's complex energy density function, and Mark's physical spectrum. An advantage of the Wigner distribution function and its related signal representations is that they can be applied not only to deterministic signals but to stochastic signals as well, leading to such things as Walther's generalized radiance and Sudarshan's Wolf tensor. On the other hand, the sliding-window spectrum has the advantage that a sampling theorem can be formulated for it: the sliding-window spectrum is completely determined by its values at the points of a certain space-frequency lattice, which is exactly the lattice suggested by Gabor in 1946. The sliding-window spectrum thus leads naturally to Gabor's expansion of a signal into a discrete set of properly shifted and modulated versions of an elementary signal, which is again another space-frequency signal representation, and which is related to the degrees of freedom of the signal.
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In the general theory of optical coherence, the following problem discussed in the present paper, arises: to determine the statistical properties of a field represented by an analytic signal from the knowledge of the statistical properties of the corresponding real field. It is shown by the use of the characteristic functionals that in order to determine the joint probability distributions of the complex field at N space-time points, the knowledge of the complete statistical description of the real field is required; on the other hand, the moments of the complex field up to that order can be determined from the knowledge of the moments of the real field up to the same order. The results are illustrated by explicit calculations relating to the Gaussian random process, which, as is well known, characterizes the fluctuations of thermal light. A converse of a well-known theorem of Kac and Siegert relating to a Gaussian random process is derived as an immediate consequence of our analysis.