The discrete wavelet transform for infinite signals

The two-way lattice decomposition equations from any level y to y - 1 for a discrete infinite signal are computed using the same equations, that is 

In the simplest case, we can perform the lattice decomposition over the low-pass branch keeping the high frequencies intact, leading to the wavelet transform (see Fig. 1). This division of the frequency axis gives good frequency resolution at low frequencies, and acceptable resolution at high frequencies, a trade-off which works in many practical cases. 

However, for signals/spectra rich in high-frequency components, this decomposition scheme may not be satisfactory. A full decomposition over the high frequency components may be preferable. The one-dimensional 2-band decomposition scheme can be viewed as a full two-way frequency-time tree. 

From the full WPT we can generate a large number of possible redundant subtrees, or arbitrary WP trees (called wavelet bases). In fact, the total number of two-way (dyadic) bases is at least (2-) for a tree depth equal to niev. For example, a dyadic WPT with tree depth niev = 12 has a library of at least 4.2 xlO bases The WPT has an important advantage compared with the WT because fast algorithms exist for the efficient search of the best wavelet basis, based on the minimisation of a cost function such as an entropy criterion. 

For the case of time-varying signals (as encountered in speech, music and video or in applications with time transients), it is possible to segment the time axis into disjoint intervals and construct wavelet bases on each interval -called spatial segmentation. This allows the WPT to adapt to each time interval. This is referred to as a multi-tree WPT or spatially adaptive WPT (see Fig. 3 for an example two-way segmentation of the time axis for the case of a dyadic WPT). 

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