A Brief Introduction to the Wigner Distribution

This report is a 5-page quick summary on the most fundamental properties of the Wigner distribution. This document is structured as follows: in the first section the context is given: we introduce the fundamental issues of the time-frequency analysis and the role of joint time-frequencies distributions; then we enumerate the ideal requirements for such distributions. In the second section we specifically define the Wigner distribution and examine which properties it exhibits, which of the above requirements it satisfies and how it compares to the spectrogram. 1 Densities The fundamental problem of the time-frequency analysis is to discover a good mathematical device able to simultaneously represent a given signal s(t) in terms of its intensity in time and frequency. What we mean by good will be clear later. One of the possible forms in which such a desired mathematical device could be expressed is a density. Roughly said, a density (or a distribution) P (x) is a function which expresses how a given quantity distributes in relation to a given variable x per unit of x, such that P (x)∆x is the amount that falls in an interval∆x at x, while the total amount is given by ∫ +∞ −∞ P (x) dx , which is often normalized to unity. Since most quantities in nature can be expressed as a function of two or more variables, it is wise to introduce two-dimensional (and more-than-two-dimensional) densities. Again, we constrain that a given density P (x, y) is such that the total amount is given by ∫ +∞ −∞ ∫ +∞ −∞ P (x, y) dx dy. Moreover, one can obtain a one-dimensional density for a multi-dimensional one by simply disregarding one or more variables, that is, by integrating out them, as follows P (x) = ∫ +∞ −∞ P (x, y) dy and P (y) = ∫ +∞ −∞ P (x, y) dx . P (x) and P (y), obtained as above from P (x, y), are said to be the marginal distributions (marginals, for short) of P (x, y). Out of generality, the quantity about which our interest is most concerned is energy, and the variables along which it distributes are time and frequency, thus we drop x and y, and will work explicitly with distributions in the form P (t, ω)where t represents time and ω frequency. It is now clear that a candidate for that good mathematical device we introduced at the beginning of this section is a time-frequency joint distribution. And it is time to define what we mean for a good distribution. A rather comprehensive list of the conditions that a good joint density should satisfy is given below: ∗Author Information: Daniele Paolo Scarpazza, Dipartimento di Elettronica e Informazione, Politecnico di Milano, Via Ponzio 34/5, I-20133 Milano, Italia; e-mail: [email protected], phone: ++39-02-23954.247. This paper is Copyright c ©2003 by Daniele Paolo Scarpazza, and can be freely distributed. All the figures of this paper were generated by the author with MATLAB and the Higher-Order Spectral Analysis Toolbox.