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 ${\displaystyle z(t)=e^{j2\pi \phi (t)}}$, where ${\displaystyle \phi (t)}$ is a polynomial phase and ${\displaystyle j={\sqrt {-1}}}$.

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 ${\displaystyle W_{z}^{g}(t,f)}$ is defined as

${\displaystyle W_{z}^{g}(t,f)={\mathcal {F}}_{\tau \to f}\left[K_{z}^{g}(t,\tau )\right]}$

where ${\displaystyle {\mathcal {F}}_{\tau \to f}}$ denotes the Fourier transform with respect to ${\displaystyle \tau }$, and ${\displaystyle K_{z}^{g}(t,\tau )}$ is the polynomial kernel given by

${\displaystyle K_{z}^{g}(t,\tau )=\prod _{k=-{\frac {q}{2}}}^{\frac {q}{2}}\left[z\left(t+c_{k}\tau \right)\right]^{b_{k}}}$

where ${\displaystyle z(t)}$ is the input signal and ${\displaystyle q}$ is an even number. The above expression for the kernel may be rewritten in symmetric form as

${\displaystyle K_{z}^{g}(t,\tau )=\prod _{k=0}^{\frac {q}{2}}\left[z\left(t+c_{k}\tau \right)\right]^{b_{k}}\left[z^{*}\left(t+c_{-k}\tau \right)\right]^{-b_{-k}}}$

The discrete-time version of the polynomial Wigner–Ville distribution is given by the discrete Fourier transform of

${\displaystyle K_{z}^{g}(n,m)=\prod _{k=0}^{\frac {q}{2}}\left[z\left(n+c_{k}m\right)\right]^{b_{k}}\left[z^{*}\left(n+c_{-k}m\right)\right]^{-b_{-k}}}$

where ${\displaystyle n=t{f}_{s},m={\tau }{f}_{s},}$ and ${\displaystyle f_{s}}$ is the sampling frequency. The conventional Wigner–Ville distribution is a special case of the polynomial Wigner–Ville distribution with ${\displaystyle q=2,b_{-1}=-1,b_{1}=1,b_{0}=0,c_{-1}=-{\frac {1}{2}},c_{0}=0,c_{1}={\frac {1}{2}}}$

Example

One of the simplest generalizations of the usual Wigner–Ville distribution kernel can be achieved by taking ${\displaystyle q=4}$. The set of coefficients ${\displaystyle b_{k}}$ and ${\displaystyle c_{k}}$ must be found to completely specify the new kernel. For example, we set

${\displaystyle b_{1}=-b_{-1}=2,b_{2}=b_{-2}=1,b_{0}=0}$

${\displaystyle c_{1}=-c_{-1}=0.675,c_{2}=-c_{-2}=-0.85}$

The resulting discrete-time kernel is then given by

${\displaystyle K_{z}^{g}(n,m)=\left[z\left(n+0.675m\right)z^{*}\left(n-0.675m\right)\right]^{2}z^{*}\left(n+0.85m\right)z\left(n-0.85m\right)}$

Design of a Practical Polynomial Kernel

Given a signal ${\displaystyle z(t)=e^{j2\pi \phi (t)}}$, where ${\displaystyle \phi (t)=\sum _{i=0}^{p}a_{i}t^{i}}$is a polynomial function, its instantaneous frequency (IF) is ${\displaystyle \phi '(t)=\sum _{i=1}^{p}ia_{i}t^{i-1}}$.

For a practical polynomial kernel ${\displaystyle K_{z}^{g}(t,\tau )}$, the set of coefficients ${\displaystyle q,b_{k}}$and ${\displaystyle c_{k}}$should be chosen properly such that

${\displaystyle {\begin{aligned}K_{z}^{g}(t,\tau )&=\prod _{k=0}^{\frac {q}{2}}\left[z\left(t+c_{k}\tau \right)\right]^{b_{k}}\left[z^{*}\left(t+c_{-k}\tau \right)\right]^{-b_{-k}}\\&=\exp(j2\pi \sum _{i=1}^{p}ia_{i}t^{i-1}\tau )\end{aligned}}}$

${\displaystyle {\begin{aligned}W_{z}^{g}(t,f)&=\int _{-\infty }^{\infty }\exp(-j2\pi (f-\sum _{i=1}^{p}ia_{i}t^{i-1})\tau )d\tau \\&\cong \delta (f-\sum _{i=1}^{p}ia_{i}t^{i-1})\end{aligned}}}$

• When ${\displaystyle q=2,b_{-1}=-1,b_{0}=0,b_{1}=1,p=2}$,

${\displaystyle z\left(t+c_{1}\tau \right)z^{*}\left(t+c_{-1}\tau \right)=\exp(j2\pi \sum _{i=1}^{2}ia_{i}t^{i-1}\tau )}$

${\displaystyle a_{2}(t+c_{1})^{2}+a_{1}(t+c_{1})-a_{2}(t+c_{-1})^{2}-a_{1}(t+c_{-1})=2a_{2}t\tau +a_{1}\tau }$

${\displaystyle \Rightarrow c_{1}-c_{-1}=1,c_{1}+c_{-1}=0}$

${\displaystyle \Rightarrow c_{1}={\frac {1}{2}},c_{-1}=-{\frac {1}{2}}}$

• When ${\displaystyle q=4,b_{-2}=b_{-1}=-1,b_{0}=0,b_{2}=b_{1}=1,p=3}$

${\displaystyle {\begin{aligned}&a_{3}(t+c_{1})^{3}+a_{2}(t+c_{1})^{2}+a_{1}(t+c_{1})\\&a_{3}(t+c_{2})^{3}+a_{2}(t+c_{2})^{2}+a_{1}(t+c_{2})\\&-a_{3}(t+c_{-1})^{3}-a_{2}(t+c_{-1})^{2}-a_{1}(t+c_{-1})\\&-a_{3}(t+c_{-2})^{3}-a_{2}(t+c_{-2})^{2}-a_{1}(t+c_{-2})\\&=3a_{3}t^{2}\tau +2a_{2}t\tau +a_{1}\tau \end{aligned}}}$

${\displaystyle \Rightarrow {\begin{cases}c_{1}+c_{2}-c_{-1}-c_{-2}=1\\c_{1}^{2}+c_{2}^{2}-c_{-1}^{2}-c_{-2}^{2}=0\\c_{1}^{3}+c_{2}^{3}-c_{-1}^{3}-c_{-2}^{3}=0\end{cases}}}$

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.

References

• Boashash, B.; O'Shea, P. (1994). "Polynomial Wigner-Ville distributions and their relationship to time-varying higher order spectra". IEEE Transactions on Signal Processing. 42 (1): 216–220. doi:10.1109/78.258143. ISSN 1053-587X.
• Luk, Franklin T.; Benidir, Messaoud; Boashash, Boualem (June 1995). Polynomial Wigner-Ville distributions. SPIE Proceedings. 2563. San Diego, CA. pp. 69–79. doi:10.1117/12.211426. ISSN 0277-786X.
• “Polynomial Wigner–Ville distributions and time-varying higher spectra,” in Proc. Time-Freq. Time-Scale Anal., Victoria, B.C., Canada, Oct. 1992, pp. 31–34.