Haar Wavelet

Fourier Analysis · Axiom Academy

The simplest wavelet: understanding the building blocks of wavelet analysis The Haar scaling function, denoted as φ(t), is a simple box function that equals 1 over the interval [0, 1) and 0 elsewhere. This function serves as the fundamental building block for the Haar wavelet system. The Haar wavelet function, ψ(t), captures changes or details in a signal. It takes the value +1 on the first half of its support [0, 0.5) and -1 on the second half [0.5, 1). This allows it to detect discontinuities and edges in signals. 3. Haar Decomposition of Signals Any signal can be decomposed into a combination of scaled and translated versions of the Haar scaling and wavelet functions. The scaling function captures the average (low-frequency) information, while the wavelet captures details (high-frequency) at different scales. 4. Applications and Limitations The Haar wavelet excels at detecting edges and discontinuities due to its step-function nature. It's widely used in image compression (including JPEG 2000 variants), edge detection, and signal denoising. However, its discontinuous nature can introduce artifacts in smooth signals.

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