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Wigner-Ville Distribution

The Wigner-Ville distribution (WVD) $\rho(t,f)$ is defined by the relation,
\begin{displaymath}
\rho(t,f) = \int_{-\infty}^\infty h\left(t - \frac{\tau}{2}\right)
h^*\left(t + \frac{\tau}{2}\right) e^{2\pi i f\tau} d\tau,
\end{displaymath} (10.1.219)

where $h(t)$ is the time series data. In practice we use the discrete analog of the previous equation,
\begin{displaymath}
\rho_{jk} = \sum_{\ell=-N/2}^{N/2}~h_{(j-\ell/2)}~h_{(j+\ell/2)}
~e^{2\pi i k \ell/N}.
\end{displaymath} (10.1.220)

This appears to presents a minor dilemma, since ([*]) contains expressions of the form $h(t-\tau/2)$, which when discretized become $h_{j-k/2}$. This implies that the original data has to be oversampled by at least a factor of 2. We resample our simulated data so that accordingly. Among the various algorithms to generated a TF map, we find that the WVD is most suitable for our purpose. We show in figures [*] and [*] WVD distributions of timeseries data containing only noise and timeseries data containing noise and an injected signal.

Figure: The WVD of simulated initial LIGO noise.

The signal has been injected at an optimal filter signal to noise ration of 8.

Figure: The WVD of simulated initial LIGO noise and a signal embedded at an SNR or 8.
\begin{figure}\begin{center}
\epsfig{file=Figures/tfmap8.eps,angle=-90,width=3.2in}\end{center}\end{figure}

Two versions of the Wigner-Ville transform are implemented. Equation ([*]) contains expressions of the form $h(t-\tau/2)$, which when discretized become $h_{j-k/2}$. This implies that for $j=0$ we require array points with negative indices. There are two ways of handling this problem. One can assume that these array points have a value of zero or one can assume that these array points refer to data points before $h_0$. We implement both versions of the Wigner transform. Please note that we have used the former one in [33].


next up previous contents
Next: Windowed Fourier transform Up: Construction of the TF Previous: Construction of the TF   Contents
Bruce Allen 2000-11-19