waveletanalüüs

Wavelet analysis is a signal processing technique that involves decomposing a signal into different scale components, allowing for the analysis of both frequency and time localization. Unlike Fourier transform, which provides frequency information but lacks time localization, wavelet analysis offers a multiresolution approach.

The core idea behind wavelet analysis is to use a family of functions, called wavelets, which are generated from a single prototype wavelet by translations and dilations. The continuous wavelet transform (CWT) of a signal x(t) is defined as:

CWT(a, b) = (1/√a) ∫ x(t) ψ*((t - b)/a) dt

where ψ(t) is the mother wavelet, a is the scale parameter, and b is the translation parameter. The CWT provides a time-scale representation of the signal, capturing both low-frequency components over long time intervals and high-frequency components over short time intervals.

Discrete wavelet transform (DWT) is a sampled version of the CWT, using a discrete set of scales and translations. The DWT is computationally efficient and widely used in various applications, such as image compression, denoising, and feature extraction. The DWT can be implemented using a fast algorithm called the Mallat algorithm, which involves a series of high-pass and low-pass filters.

Wavelet analysis has several advantages over traditional Fourier analysis. It can handle non-stationary signals, providing better time-frequency localization. Additionally, wavelets can capture sharp transitions and discontinuities in signals, making them suitable for analyzing signals with abrupt changes.

However, wavelet analysis also has its limitations. The choice of the mother wavelet is crucial, and different wavelets may yield different results. Moreover, wavelet analysis can be sensitive to noise, and proper denoising techniques are often required. Despite these challenges, wavelet analysis remains a powerful tool in signal processing and has numerous applications in fields such as engineering, physics, and mathematics.