Polynomial Wigner–Ville distribution

Many signals in nature and in engineering applications can be modeled as $z(t)=e^\{j2\backslash pi\backslash phi(t)\}$, where $\backslash phi(t)$ is a polynomial phase and $j=\backslash sqrt\{-1\}$.

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.

The polynomial Wigner–Ville distribution $W^g\_z(t,\; f)$ is defined as

:$W^g\_z(t,\; f)=\backslash mathcal\{F\}\_\{\backslash tau\backslash to\; f\}\backslash left[K^g\_z(t,\; \backslash tau)\backslash right]$

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

:$K^g\_z(t,\; \backslash tau)=\backslash prod\_\{k=-\backslash frac\{q\}\{2\}\}^\{\backslash frac\{q\}\{2\}\}\; \backslash left[z\backslash left(t+c\_k\backslash tau\backslash right)\backslash right]^\{b\_k\}$

where $z(t)$ is the input signal and $q$ is an even number.

The above expression for the kernel may be rewritten in symmetric form as

:$K^g\_z(t,\; \backslash tau)=\backslash prod\_\{k=0\}^\{\backslash frac\{q\}\{2\}\}\; \backslash left[z\backslash left(t+c\_k\backslash tau\backslash right)\backslash right]^\{b\_k\}\backslash left[z^*\backslash left(t+c\_\{-k\}\backslash tau\backslash right)\backslash right]^\{-b\_\{-k\}\}$

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

:$K^g\_z(n,\; m)=\backslash prod\_\{k=0\}^\{\backslash frac\{q\}\{2\}\}\; \backslash left[z\backslash left(n+c\_\{k\}m\backslash right)\backslash right]^\{b\_k\}\backslash left[z^*\backslash left(n+c\_\{-k\}m\backslash right)\backslash right]^\{-b\_\{-k\}\}$

where $n=t\{f\}\_s,\; m=\{\backslash tau\}\{f\}\_\{s\},$ and $f\_s$ is the sampling frequency.

The conventional Wigner–Ville distribution is a special case of the polynomial Wigner–Ville distribution with $q=2,\; b\_\{-1\}=-1,\; b\_1=1,\; b\_0=0,\; c\_\{-1\}=-\backslash frac\{1\}\{2\},\; c\_0=0,\; c\_1=\backslash frac\{1\}\{2\}$

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

:$b\_1=-b\_\{-1\}=2,\; b\_2=b\_\{-2\}=1,\; b\_0=0$

:$c\_1=-c\_\{-1\}=0.675,\; c\_2=-c\_\{-2\}=-0.85$

The resulting discrete-time kernel is then given by

:$K^g\_z(n,\; m)=\backslash left[z\backslash left(n+0.675m\backslash right)z^*\backslash left(n-0.675m\backslash right)\backslash right]^2z^*\backslash left(n+0.85m\backslash right)z\backslash left(n-0.85m\backslash right)$

Given a signal $z(t)=e^\{j2\backslash pi\backslash phi(t)\}$, where $\backslash phi(t)=\backslash sum\_\{i=0\}^p\; a\_i\; t^i$ is a polynomial function, its instantaneous frequency (IF) is $\backslash phi\text{'}(t)\; =\; \backslash sum\_\{i=1\}^p\; ia\_it^\{i-1\}$.

For a practical polynomial kernel $K^g\_z(t,\; \backslash tau)$, the set of coefficients $q,\; b\_k$ and $c\_k$ should be chosen properly such that

:$\backslash begin\{align\}$

K^g_z(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}^pia_it^{i-1}\tau)

\end{align}

:$\backslash begin\{align\}$

W_z^g(t,f) &= \int_{-\infin}^{\infin} \exp(-j2\pi(f - \sum_{i=1}^p i a_i t^{i-1}) \tau)d\tau\\

&\cong \delta (f - \sum_{i=1}^p i a_i t^{i-1})

\end{align}

• When $q=2,\; b\_\{-1\}=-1,\; b\_0=0,\; b\_1=1,\; p=2$,

:$z\backslash left(t+c\_1\backslash tau\backslash right)z^*\backslash left(t+c\_\{-1\}\backslash tau\backslash right)=\backslash exp(j2\backslash pi\; \backslash sum\_\{i=1\}^2\; i\; a\_i\; t^\{i-1\}\backslash tau)$

:$a\_2(t+c\_1)^2\; +\; a\_1(t+c\_1)\; -\; a\_2(t\; +\; c\_\{-1\})^2\; -\; a\_1(t\; +\; c\_\{-1\})\; =\; 2a\_2t\backslash tau\; +\; a\_1\backslash tau$

:$\backslash Rightarrow\; c\_1\; -\; c\_\{-1\}\; =\; 1,\; c\_1\; +\; c\_\{-1\}\; =\; 0$

:$\backslash Rightarrow\; c\_1=\backslash frac\{1\}\{2\},\; c\_\{-1\}=-\backslash frac\{1\}\{2\}$

• When $q=4,\; b\_\{-2\}=b\_\{-1\}=-1,\; b\_0=0,\; b\_2=b\_1=1,\; p=3$

:$\backslash begin\{align\}$

&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_3t^2\tau + 2a_2t\tau + a_1\tau

\end{align}

:$\backslash Rightarrow$

\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}

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.

• {{cite journal|last1=Boashash|first1=B.|last2=O'Shea|first2=P.|title=Polynomial Wigner-Ville distributions and their relationship to time-varying higher order spectra|journal=IEEE Transactions on Signal Processing|volume=42|issue=1|year=1994|pages=216–220|issn=1053-587X|doi=10.1109/78.258143|bibcode=1994ITSP...42..216B |url=https://eprints.qut.edu.au/1849/1/Boashash_et.al.%2C_1994%2C_Polynomial_Wigner-Ville_distributions_and_their_relationship_to_time-varying_higher_order_spectra.pdf}}
• {{cite conference|last1=Luk|first1=Franklin T.|last2=Benidir|first2=Messaoud|last3=Boashash|first3=Boualem|title=Polynomial Wigner-Ville distributions|journal=Proceedings|volume=2563|date=June 1995|pages=69–79|issn=0277-786X|doi=10.1117/12.211426|location=San Diego, CA|conference=SPIE Proceedings}}
• “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.

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