Wavelet-based analysis of signals is an interesting, and relatively recent, new tool [Rio91
]. 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, ck
at different scales, are calculated using the following convolution-like expressions: