Emäaaltofunktiot

Emäaaltofunktiot, also known as mother wavelets, form the foundation of wavelet analysis. They are a class of functions that are used to decompose a signal into different frequency components. The concept of mother wavelets was introduced by Jean Morlet in the early 1980s.

A mother wavelet is a function that is localized in both time and frequency. This property allows it to capture transient features in a signal that are not well represented by traditional Fourier analysis. The mother wavelet is then scaled and translated to create a set of daughter wavelets. These daughter wavelets are used to analyze the signal at different scales and positions.

The choice of mother wavelet depends on the specific application and the characteristics of the signal being analyzed. Some common mother wavelets include the Haar wavelet, the Daubechies wavelets, and the Morlet wavelet. Each of these wavelets has unique properties that make them suitable for different types of signal processing tasks, such as image compression, noise reduction, and feature detection.

The mathematical definition of a mother wavelet typically involves properties such as being square-integrable, having a zero average, and satisfying certain admissibility conditions. These conditions ensure that the wavelet transform is well-defined and can effectively reconstruct the original signal. Emäaaltofunktiot are fundamental to wavelet transforms, which are powerful tools in signal processing and applied mathematics.