Wigner-Ville Distribution

The Wigner-Ville distribution (WVD) $\rho(t,f)$ is defined by the relation,

$$\rho(t,f) = \int_{-\infty}^\infty h\left(t - \frac{\tau}{2}\right) h^*\left(t + \frac{\tau}{2}\right) e^{2\pi i f\tau} d\tau,$$ (10.1.219)

where $h(t)$ is the time series data. In practice we use the discrete analog of the previous equation,

$$\rho_{jk} = \sum_{\ell=-N/2}^{N/2}~h_{(j-\ell/2)}~h_{(j+\ell/2)} ~e^{2\pi i k \ell/N}.$$ (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.

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].