In signal processing, the polynomial Wigner–Ville distribution is a quasiprobability distribution that generalizes the Wigner distribution function. It was proposed by Boualem Boashash and Peter O'Shea in 1994.
Contents
Introduction
Many signals in nature and in engineering applications can be modeled as
For example, it is important to detect signals of an arbitrary high-order polynomial phase. However, the conventional Wigner–Ville distribution have the limitation being based on the second-order statistics. Hence, the polynomial Wigner–Ville distribution was proposed as a generalized form of the conventional Wigner–Ville distribution, which is able to deal signals with nonlinear phase.
Definition
The polynomial Wigner–Ville distribution
where
where
The discrete-time version of the polynomial Wigner–Ville distribution is given by the discrete Fourier transform of
where
Example
One of the simplest generalizations of the usual Wigner–Ville distribution kernel can be achieved by taking
The resulting discrete-time kernel is then given by
Applications
Nonlinear FM signals are common both in nature and in engineering applications. For example, the sonar system of some bats use hyperbolic FM and quadratic FM signals for echo location. In radar, certain pulse-compression schemes employ linear FM and quadratic signals. The Wigner–Ville distribution has optimal concentration in the time-frequency plane for linear frequency modulated signals. However, for nonlinear frequency modulated signals, optimal concentration is not obtained, and smeared spectral representations result. The polynomial Wigner–Ville distribution can be designed to cope with such problem.