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arXiv:0810.2559

Mutual space-frequency distribution of Gaussian signal

Yura Kozlovskii

Published Oct 14, 2008• 0 citations
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Abstract

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 $(x;p)$ on an angle proportional to the mutual parameter $t$. The results of direct calculations of mutual distribution for Gaussian signal in the mutual domain are presented.

DOI
10.48550/arxiv.0810.2559

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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.
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 generalized phase-space distributions, including the Wigner distribution, are presented in terms of expected values of generating operators. A generalization of the Weyl correspondence is obtained to provide expressions for generalized Wigner equivalents. Finally, rather simple relationships are obtained connecting the generalized phase-space distributions to the Wigner distribution; and similar relationships are obtained for the generalized Wigner equivalents. In particular, it appears that, among the class considered, there is no reason to use any distribution other than the Wigner for performing any calculations.
Conceptually, the phase space description of quantum states hinges on the completeness of the set of displacement operators, which can be proved in the form of the Fourier-Weyl relation between density matrices and characteristic functions. This relationship constitutes the bridge between phase space and Hilbert space descriptions, which is useful in several applications in the context of quantum information. This chapter shows that the marginal Wigner function along any phase space direction describes the probability distribution of the quantum measurement of the associated quadrature operator. Hence, as far as Gaussian states are concerned, the Wigner function provides one with a local, 'realistic' model to describe quadrature measurements. If one restricts to quadrature measurements, such systems may be mimicked by multivariate classical Gaussian distributions and will never show any signature of quantum non-locality, such as a violation of Bell or CHSH inequalities.