Page 126 - Phase Space Optics Fundamentals and Applications

CHAPTER4

The Radon-Wigner

Transform in Analysis,

Design, and Processing

of Optical Signals

Walter D. Furlan and Genaro Saavedra
Universitat de Val` encia, Optics Department
Burjassot, Spain

4.1 Introduction
One of the main features of phase space is that its conjugate coor-
dinates are noncommutative and cannot be simultaneously specified
with absolute accuracy. As a consequence, there is no phase-space joint
distribution that can be formally interpreted as a joint probability den-
sity. Indeed, most of the classic phase-space distributions, such as the
Wigner distribution function (WDF), the ambiguity function (AF), or
the complex spectrogram, have difficult interpretation problems due
to the complex, or negative, values they have in general. Besides, they
may be nonzero even in regions of the phase space where either the
signal or its Fourier transform vanishes. This is a critical issue, espe-
cially for the characterization of nonstationary or nonperiodic signals.
As an alternative, the projections (marginals) of the phase-space distri-
butions are strictly positive, and as we will see later, they give informa-
tion about the signal on both phase-space variables. These projections
can be formally associated with probability functions, avoiding all
interpretation ambiguities associated with the original phase-space
distributions. This is the case of the Radon-Wigner transform (RWT),

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