Continuous Wavelet Transform (CWT)

The Continuous Wavelet Transform (CWT) is a powerful mathematical tool used for analyzing localized variations of power within a time series. Unlike traditional Fourier analysis, which provides only frequency information, CWT provides time-frequency localization, making it ideal for analyzing non-stationary signals whose frequency content varies over time.

Overview

CWT decomposes a signal into wavelets—small waves that are localized in both time and frequency. By stretching and shifting a mother wavelet function, the CWT captures signal features at various scales and positions, enabling detailed multi-resolution analysis.

Mathematical Definition

Given a signal ( x(t) \in L^2(\mathbb{R}) ) and a mother wavelet ( \psi(t) ), the Continuous Wavelet Transform ( W_x(a,b) ) is defined as:

[ W_x(a,b) = \frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} x(t) , \psi^* \left( \frac{t - b}{a} \right) dt ]

where:

• ( a \in \mathbb{R}^+ ) is the scale parameter, controlling the dilation (stretching or compressing) of the wavelet.
• ( b \in \mathbb{R} ) is the translation parameter, controlling the shift along the time axis.
• ( \psi^*(t) ) denotes the complex conjugate of the mother wavelet.
• The factor ( \frac{1}{\sqrt{|a|}} ) ensures energy preservation across scales.

Properties

• Time-Frequency Localization: CWT provides information about the signal's frequency content at each time instant.
• Multi-Resolution: Analysis at different scales allows examination of both short-lived high-frequency components and long-duration low-frequency components.
• Invertibility: Under certain admissibility conditions on ( \psi ), the original signal can be reconstructed from its wavelet coefficients.

Choice of Mother Wavelet

The mother wavelet ( \psi(t) ) must satisfy the admissibility condition:

[ C_\psi = \int_{0}^{\infty} \frac{|\hat{\psi}(\omega)|^2}{\omega} d\omega < \infty ]

where ( \hat{\psi}(\omega) ) is the Fourier transform of ( \psi(t) ). Common choices include:

• Morlet Wavelet: A complex sinusoid modulated by a Gaussian, suitable for time-frequency analysis.
• Mexican Hat Wavelet: Second derivative of a Gaussian, useful for edge detection.
• Haar Wavelet: The simplest wavelet, representing step functions.

Applications

• Signal Processing: Feature extraction, denoising, and time-frequency analysis.
• Physics: Analysis of transient phenomena, such as turbulence or seismic signals.
• Biomedical Engineering: ECG and EEG signal analysis.
• Geophysics and Climate Science: Detecting localized events and oscillations, such as El Niño cycles.
• SDKP Framework: CWT can be applied to analyze scale-density variations and extract macro-causal kernels within complex physical datasets.

Summary

The Continuous Wavelet Transform is a versatile technique that extends Fourier analysis by providing localized frequency information, enabling the study of complex, non-stationary signals across multiple scales. Its mathematical rigor and adaptability make it an essential tool across physics, engineering, and data science.

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References:

• Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM.
• Torrence, C., & Compo, G. P. (1998). A Practical Guide to Wavelet Analysis. Bulletin of the American Meteorological Society, 79(1), 61–78.