Wavelet Analysis

Physics: Wavelet transform: W(a,b) = ∫x(t)·ψ*((t-b)/a)dt, where a=scale (inverse frequency), b=translation. Morlet wavelet: ψ(t) = e^(iω₀t)·e^(-t²/2) — a complex sinusoid modulated by a Gaussian envelope. Time-frequency uncertainty: Δt·Δf ≥ 1/(4π) (Heisenberg bound). Unlike DFT (perfect frequency resolution, zero time resolution), wavelets adapt: low frequencies use wide windows, high frequencies use narrow windows. Daubechies (1988) compact support wavelets enable fast computation; JPEG 2000 uses wavelets; MP3 uses MDCT (related).