  Data from gravitational wave detectors are recorded as time series that
include contributions from myriad noise sources in addition to any
gravitational wave signals. When regularly sampled data are available, such as
for ground based and future space based interferometers, analyses are typically
performed in the frequency domain, where stationary (time invariant) noise
processes can be modeled very efficiently. In reality, detector noise is not
stationary due to a combination of short duration noise transients and longer
duration drifts in the power spectrum. This non-stationarity produces
correlations across samples at different frequencies, obviating the main
advantage of a frequency domain analysis. Here an alternative time-frequency
approach to gravitational wave data analysis is proposed that uses discrete,
orthogonal wavelet wavepackets. The time domain data is mapped onto a uniform
grid of time-frequency pixels. For locally stationary noise - that is, noise
with an adiabatically varying spectrum - the time-frequency pixels are
uncorrelated, which greatly simplifies the calculation of quantities such as
the likelihood. Moreover, the gravitational wave signals from binary systems
can be compactly represented as a collection of lines in time-frequency space,
resulting in a computational cost for computing waveforms and likelihoods that
scales as the square root of the number of time samples, as opposed to the
linear scaling for time or frequency based analyses. Key to this approach is
having fast methods for computing binary signals directly in the wavelet
domain. Multiple fast transform methods are developed in detail.
