Select mother wavelets, adjust scale and shift parameters, superimpose wavelet components, and observe time-domain waveforms and time-frequency localization in real time
A wavelet is a localized oscillatory function ψ(t) with zero mean and finite energy. Unlike sine waves in Fourier analysis, wavelets are confined in both time and frequency. Four common mother wavelets: (1) Haar — a simple step function, discontinuous but compactly supported, ideal for edge detection. (2) Morlet — a Gaussian-windowed complex exponential ψ(t) = e^(-t²/2)·cos(ω₀t), providing the best frequency localization. (3) Mexican Hat — the second derivative of a Gaussian ψ(t) = (1-t²)·e^(-t²/2), symmetric with good localization in both domains. (4) Gaussian Derivative — the first derivative of a Gaussian ψ(t) = -t·e^(-t²/2), antisymmetric, sensitive to signal slopes. Each wavelet is dilated by scale a and translated by shift b: ψ_{a,b}(t) = (1/√a)·ψ((t-b)/a).
The Heisenberg uncertainty principle states that a signal cannot be simultaneously localized to arbitrary precision in both time and frequency: Δt · Δf ≥ 1/(4π). Wavelets exploit this by adapting resolution: at large scale (low pseudo-frequency), the wavelet is wide in time but narrow in frequency; at small scale (high pseudo-frequency), it is narrow in time but wide in frequency. This multi-resolution property makes wavelets superior to the Short-Time Fourier Transform (STFT) for signals with transient features. The ellipses in the bottom plot visualize this tradeoff in a time-pseudo-frequency view: wider ellipses correspond to larger scales, while taller ellipses correspond to broader frequency spread.
Image compression: JPEG 2000 uses the CWT/DWT for multi-resolution image representation, achieving superior compression at low bitrates compared to JPEG's DCT. Signal denoising: thresholding wavelet coefficients removes noise while preserving edges and transients — widely used in audio, biomedical, and seismic processing. Edge detection: Haar and Gaussian derivative wavelets excel at detecting discontinuities and sharp transitions in signals and images. Biomedical analysis: wavelet decomposition of ECG signals isolates the QRS complex from baseline wander and noise. Seismology: wavelet analysis detects arrival times and frequency content of seismic phases. Music synthesis: wavelet-based additive synthesis creates time-varying timbres by superimposing components at different scales and positions.
Start with the Single Pulse preset: one Morlet wavelet at the center of the time window. Drag the Scale slider to stretch or compress the wavelet, and notice how the ellipse below becomes wider in time and lower in pseudo-frequency. Drag Shift to move it in time. Try the Chirp preset: three Morlet wavelets with decreasing scale (increasing pseudo-frequency) simulate a sweep-like packet sequence. The Multi-Scale preset places three Mexican Hat wavelets at the same position with different scales so you can compare coarse and fine structure. The Beat preset uses two nearby Morlet packets at the same position to show interference. Add components with the button, change their wavelet type, and observe how the synthesized signal (white curve) is the sum of all components (colored curves).