Wavelet Frames

To cover the time frequency plane with wavelet Heisenberg boxes, a regular grid is not used; time steps which are inverse proportional to the frequecy step are used instead, the latter being itself proportional to the scale.

The wavelet is assumed to satisfy the reconstruction condition

which garantees the invertibility of the wavelet transform. Daubechies gives necessary conditions for the previous tiling to yield a frame:

Sufficient conditions also exist.

The following differences with the windowed Fourier frames should be emphasized:

Translation invariance

In both cases (Fourier or wavelets), the frame representation has the drawback of not being translation invariant with respect to time or frequency. Now, most interesting signal patterns are not naturally synchronized with frame intervals. In particular, the structure of a signal may be degraded at the lower resolutions.

This motivates the study of the dyadic wavelet transform, which is discrete in scale but not in time (in practice, this means that signals are oversampled when switching to coarser resolutions).

Another time invariant represenation is the representation by dyadic wavelet maxima. It is less redundant, but is not complete.

Why wavelet bases are studied nonetheless

In practice, the dyadic wavelet transform is implemented by perfect reconstruction filter banks. These fast filter banks correspond to wavelet bases which are built from multiresolution approximations.

Multiresolution Approximations and Wavelet Bases