Wavelet-based analysis of signals is an interesting, and relatively recent, new tool [Rio91,

Str96,

Vai93]. Similar to Fourier series analysis, where sinusoids are chosen as the basis function, wavelet analysis is also based on a decomposition of a signal using an orthonormal (typically, although not necessarily) family of basis functions. Unlike a sine wave, a wavelet has its energy concentrated in time. Sinusoids are useful in analyzing periodic and time-invariant phenomena, while wavelets are well suited for the analysis of transient, time-varying signals.

The 1D discrete wavelet transform is calculated using Mallat's algorithm [Mal89]. The transform coefficients, c

_{k} and d

_{k} at different scales, are calculated using the following convolution-like expressions: