Wavelet Definition

Fourier Analysis · Axiom Academy

Understanding the mathematical foundation of wavelets through oscillation, decay, and scaling 1. Definition: Oscillation and Decay A function ψ(t) is considered a wavelet if it satisfies two fundamental properties: Oscillation: The function must wave-like, crossing zero multiple times Decay: The function must rapidly approach zero as time goes to infinity These properties ensure that wavelets are localized in time - they exist only for a finite duration, unlike sine waves that continue forever. 2. The Admissibility Condition For a function to be a valid wavelet, it must satisfy the admissibility condition , which requires that the wavelet has zero mean: This means the area above the horizontal axis must equal the area below it. This condition ensures that the wavelet oscillates around zero and has no DC component, which is essential for proper frequency analysis. A mother wavelet ψ(t) can be translated (shifted in time) and scaled (stretched or compressed) to create a family of wavelets: a (scale parameter): Controls dilation/compression. Smaller a = compressed (high frequency), larger a = stretched (low frequency) b (translation parameter): Controls position in time 1/√a : Normalization factor to preserve energy across scales Different wavelets are suited for different applications. Here are two commonly used wavelets: The second derivative of a Gaussian, symmetric and real-valued. Good for detecting peaks.

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