Scale
Controls how broad or fine the wavelet is.
Analysis & Signals · Accessible first encounter
Wavelet Analysis:
Finding Where a Signal Changes
Wavelet analysis decomposes data using localized building blocks that can be shifted and rescaled. It is especially effective for edges, transients, compression, denoising, and multiscale structure.
01 · Opening mystery
Two signals can have similar frequencies but very different events.
Imagine a recording with a brief click. A Fourier spectrum can reveal the high frequencies created by the click, but a global spectrum does not directly say when it occurred.
A wavelet is concentrated in a short region. Sliding and resizing it across the signal produces coefficients that respond to both position and scale.
Before exploringCan a representation keep the important shape while discarding most of its numbers?
Make a prediction. The laboratory is designed to challenge or refine it.
02 · Interactive laboratory
Compress an eight-sample signal with the Haar transform.
The Haar transform repeatedly replaces neighboring values by an average and a difference. Raise the threshold to discard weak detail coefficients, then compare the reconstruction.
Coefficients retained8 of 8
Reconstruction RMSE0
Haar coefficients
Reconstructed samples
03 · The big idea
Average describes the coarse picture; difference describes detail.
For two values x and y, the Haar transform records (x + y)/√2 and (x − y)/√2. These normalized average and difference coordinates preserve total squared energy: x² + y² equals the sum of the squared coefficients.
The averages can be paired again, creating a hierarchy. One coefficient describes the overall level; others describe details at coarse, medium, and fine scales.
Central definition
A wavelet family is formed by translating and scaling a localized prototype function, often called the mother wavelet.
ψa,b(t) = 1/√|a| · ψ((t − b)/a)
Location
Moves the wavelet to a particular part of the signal.
Detail coefficient
Measures how strongly the signal matches a local change pattern.
04 · A beautiful result
The Haar transform is perfectly reversible.
The average and difference formulas can be solved backward: x = (average + difference)/√2 and y = (average − difference)/√2. Repeating this reversal reconstructs the full signal exactly when all coefficients are kept.
Compression becomes possible when many coefficients are small. Removing them does not preserve the signal exactly, but it may preserve the visually or scientifically important structure with far fewer active numbers.
- 1
Compute an average coefficient a = (x + y)/√2 and a detail coefficient d = (x − y)/√2.
- 2
Add the equations: a + d = √2x, so x = (a + d)/√2.
- 3
Subtract them: a − d = √2y, so y = (a − d)/√2.
- 4
Apply the inverse pair operation from coarse scale back to the finest scale.
05 · Why this subject matters
Edges and local events are everywhere.
Wavelets have been used in image compression, denoising, numerical differential equations, turbulence analysis, seismic data, astronomical signals, and biomedical measurements. Their multiscale structure also influenced modern sparse representations.
Different wavelets trade simplicity, smoothness, compact support, symmetry, and vanishing moments. The Haar wavelet is the simplest doorway, not the endpoint.
Compression
Keeps dominant coefficients and discards weak detail.
Medical Imaging
Separates structures and noise across scales.
Numerical Analysis
Creates adaptive methods that focus effort where solutions change rapidly.
06 · Friendly assessment
Check the central ideas without pressure.
The questions focus on the main insights, not obscure details. Each response receives an explanation immediately.
Where this idea leads
Continue through the mathematical atlas.
Connected field
Fourier Analysis
Compare global frequency with localized scale.
Explore →Connected fieldSignal Processing
Detect events, denoise data, and build filter banks.
Explore →Connected fieldMedical Imaging
Analyze and reconstruct multiscale image features.
Explore →Connected fieldFunctional Analysis
Study wavelet bases in infinite-dimensional spaces.
Explore →You have now experienced
You have transformed a signal into coarse and detailed information, discarded coefficients, and reconstructed an approximation.
This is an invitation to continue, not a compressed substitute for a full university course.