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WignerVille Distribution
The WignerVille distribution (WVD) is defined by the relation,

(10.1.219) 
where is the time series data. In practice we use the discrete
analog of the previous equation,

(10.1.220) 
This appears to presents a minor
dilemma, since () contains expressions of the form
, which when discretized become .
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 WignerVille transform are implemented.
Equation () contains expressions of the form
, which when discretized become . This implies
that for 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 . We implement both versions of the
Wigner transform. Please note that we have used the former one in
[33].
Next: Windowed Fourier transform
Up: Construction of the TF
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Bruce Allen
20001119