Wavelet Decomposition Examples

Fourier Analysis · Axiom Academy

EXAMPLE Haar Wavelet Decomposition Step-by-step decomposition of a discrete signal using Haar wavelets Excellent work! You've completed the Haar wavelet decomposition. Here's what we learned: Hierarchical Decomposition: The Haar wavelet transform decomposes a signal into approximation (averages) and detail (differences) coefficients at multiple resolution levels. Approximation Coefficients: These capture the low-frequency information by averaging adjacent values, preserving the overall trend of the signal. Detail Coefficients: These capture the high-frequency information by computing differences between adjacent values, representing local changes. Perfect Reconstruction: The decomposition is reversible - we can exactly reconstruct the original signal from the wavelet coefficients using inverse operations. Multi-Resolution Analysis: Each level of decomposition provides information about the signal at a different scale, enabling time-frequency localization. This technique is fundamental to signal compression, denoising, and feature extraction in applications ranging from audio processing to image analysis!

This is the written version of the interactive lesson above. See the full Fourier Analysis course.