WaveletBasis

A WaveletBasis is a set of functions used in wavelet analysis to represent a signal or data. These functions, called wavelets, are localized in both time and frequency, unlike the sinusoids used in Fourier analysis. A wavelet basis is constructed from a single prototype wavelet through dilation and translation. This means that the basis functions are scaled versions of the original wavelet at different frequencies and shifted versions at different time intervals.

The properties of a wavelet basis make them well-suited for analyzing signals with transient features or non-stationary characteristics. By using a set of wavelets that are scaled and shifted, one can effectively capture both the high-frequency components of a signal and its localized behavior. The choice of the specific wavelet function, known as the mother wavelet, influences the characteristics of the resulting basis and its suitability for different applications. Common mother wavelets include the Haar wavelet, Daubechies wavelets, and Morlet wavelets, each with its own mathematical properties and applications.

Decomposition of a signal using a wavelet basis involves projecting the signal onto each wavelet function in the basis. This results in a set of coefficients that represent the signal's content at different scales and positions. This representation is often sparse, meaning that many coefficients are zero or close to zero, which is advantageous for compression and denoising. The process of analyzing and reconstructing signals using wavelet bases forms the foundation of various signal processing techniques.