waveletperheet

Wavelet families, or waveletperheet in Finnish usage, are sets of wavelets generated from a single mother wavelet by stretching (dilations) and shifting (translations). Each member of a family is a scaled and shifted version of the mother wavelet, enabling analysis of signals at multiple resolutions.

Mathematically, if psi(t) is a mother wavelet, a wavelet family is the collection psi_{a,b}(t) = (1/√a) psi((t − b)/a), with a > 0 and b ∈ R. This family provides a time–frequency representation that can adapt to different scales. In practical applications, discrete wavelet transforms use restricted sets of scales and shifts, commonly with dyadic scales a = 2^j and translations b = k 2^j, producing efficient, often orthogonal or biorthogonal bases.

Common examples of wavelet families include the Haar wavelets, which form the simplest family, and the Daubechies, Symlet, and Coiflet families, which offer varying degrees of smoothness and compact support. Properties such as compact support, the number of vanishing moments, and regularity influence suitability for particular tasks, such as detecting sharp edges or smoothing noise.

Applications of wavelet families span signal processing, data compression, denoising, and feature extraction in time–frequency analysis. The framework of wavelet families underpins many algorithms for analyzing non-stationary signals, where different features occur at different time scales.

Historically, wavelet theory developed through contributions from Morlet, Grossmann, and later Mallat and Meyer, who formalized multi-resolution analysis and efficient algorithms for working with wavelet families.