waveletanalyse

Wavelet analysis is a mathematical technique used for analyzing signals and data by breaking them down into components of varying frequencies and scales. Unlike traditional Fourier analysis, which decomposes signals into sine and cosine waves of fixed frequencies, wavelet analysis employs wavelets—short, oscillatory functions that can be scaled and translated to represent different frequency bands and temporal resolutions.

Developed in the 1980s by Yves Meyer and Jean Morlet, wavelets provide a time-frequency representation that is particularly useful for signals with non-stationary properties, such as those exhibiting sudden changes or transients. The process involves expressing a signal as a sum of wavelet functions at different positions and scales, allowing for detailed examination of both frequency content and temporal localization.

Applications of wavelet analysis span multiple fields, including signal processing, image compression, seismology, finance, and biomedical engineering. For example, in seismology, wavelets help identify and analyze earthquake waves by isolating specific frequency components, while in medical imaging, they assist in detecting abnormalities in MRI or ultrasound scans. In financial time series analysis, wavelets can reveal patterns and anomalies in stock market data.

Mathematically, wavelet analysis relies on the wavelet transform, which produces a coefficient vector that represents the signal’s decomposition. Common types include continuous wavelet transforms (CWT), which provide a smooth time-frequency representation, and discrete wavelet transforms (DWT), which offer computational efficiency by using discrete scales and translations. The choice between these methods depends on the specific requirements of the analysis, such as the need for real-time processing or high-resolution detail.

Overall, wavelet analysis offers a flexible and powerful tool for analyzing complex, dynamic signals, making it indispensable in both theoretical research and practical applications.