Wavelets and filter theory

Wavelets are functions that can be used effectively to represent phenomena of a highly transient nature resulting from the time dilation and offset of the original signal. For example, if we consider the propagation of a "pulse" in a layered medium, due to the dispersion caused by the different electrical properties of the layers and the finite propagation velocity, the "pulse" is dilated and delayed.

The theory of the discrete wavelet transform is developed here from the perspective of filter theory. It is shown to be equivalent to filtering the original signal by filters of constant quality factor with bandwidths of one octave, and which do not overlap for ideal filters. This makes wavelets easier to understand and apply.

The authors would like to thank Ms. Marta Salazar DÍaz and Ms. Ana Chiclana, who contributed to the French editing of the initial version of the article.