Polynomial Wigner–Ville distribution


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.

Introduction

Many signals in nature and in engineering applications can be modeled as, where is a polynomial phase and.
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 with signals with nonlinear phase.

Definition

The polynomial Wigner–Ville distribution is defined as
where denotes the Fourier transform with respect to, and is the polynomial kernel given by
where is the input signal and is an even number.
The above expression for the kernel may be rewritten in symmetric form as
The discrete-time version of the polynomial Wigner–Ville distribution is given by the discrete Fourier transform of
where and is the sampling frequency.
The conventional Wigner–Ville distribution is a special case of the polynomial Wigner–Ville distribution with

Example

One of the simplest generalizations of the usual Wigner–Ville distribution kernel can be achieved by taking. The set of coefficients and must be found to completely specify the new kernel. For example, we set
The resulting discrete-time kernel is then given by

Design of a Practical Polynomial Kernel

Given a signal, where is a polynomial function, its instantaneous frequency is.
For a practical polynomial kernel, the set of coefficients and should be chosen properly such that
  • When,
  • When

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.