The Wavelet Transform

A visual, interactive guide to understanding how wavelets decompose signals in both time and frequency.

Intuition Fourier vs Wavelet Mother Wavelets CWT DWT Multiresolution Applications
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Chapter 1

01 The Core Intuition

A wavelet is a short, wave-like oscillation that starts at zero, oscillates, and returns to zero. Unlike infinite sine waves, wavelets are localised in both time and frequency.

The wavelet transform works by sliding a wavelet along a signal and computing the correlation at each point. By stretching (scaling) the wavelet, we capture different frequency components. Short wavelets catch high-frequency detail; stretched wavelets capture the slow-moving trend.

Drag scale to stretch the wavelet · Drag position to translate it along the signal
Key idea: The wavelet transform measures how much a wavelet at a given scale and position resembles the local structure of the signal. High correlation → the signal contains energy at that frequency, at that moment in time.

Chapter 2

02 Fourier vs Wavelet

The Fourier Transform tells you which frequencies exist, but not when they occur. The Wavelet Transform gives you both.

Consider a signal whose frequency changes over time — a chirp. The Fourier spectrum shows a broad band of frequencies with no timing information. The wavelet scalogram shows exactly when each frequency appears.

Chirp signal — frequency sweeps from low to high
Fourier power spectrum — no time information
Wavelet scalogram — time × frequency map
The uncertainty principle: You can't have perfect resolution in both time and frequency simultaneously. Wavelets offer an optimal trade-off — good time resolution at high frequencies, good frequency resolution at low frequencies.

Chapter 3

03 Mother Wavelets

Different wavelet shapes are suited to different signal characteristics. The choice of mother wavelet determines what features the transform is sensitive to.

Click to explore different mother wavelet families
Morlet: A complex sinusoid modulated by a Gaussian. Excellent for time-frequency analysis of smooth, oscillatory signals. Widely used in geophysics and neuroscience.
Haar: The simplest wavelet — a step function. Perfect for detecting sharp transitions and edges. It's the foundation of the Discrete Wavelet Transform.

Chapter 4

04 Continuous Wavelet Transform

The CWT convolves the signal with scaled and translated copies of the mother wavelet at every point, producing a 2D time-scale map.

W(a, b) = (1/√a) · ∫ x(t) · ψ*( (t − b) / a ) dt

Here a is the scale (inversely related to frequency), b is the translation (position in time), and ψ is the mother wavelet.

Watch the wavelet slide across the signal, building the scalogram row by row
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Choose a scale a (stretch the wavelet)

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Slide the wavelet across the signal (vary b)

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At each point, compute the inner product

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Repeat for all scales to build the scalogram

Chapter 5

05 Discrete Wavelet Transform

The DWT samples the CWT at dyadic scales (powers of 2), making it computationally efficient and perfectly invertible — you can reconstruct the original signal exactly.

Instead of continuously varying the scale, the DWT uses a filter bank: a low-pass filter (scaling function) captures the approximation, and a high-pass filter (wavelet function) captures the detail. After filtering, the signal is downsampled by 2.

Animated filter bank decomposition — watch the signal split into approximation + detail at each level
Level j: cA_j = downsample( x * h )   |   cD_j = downsample( x * g )

Where h is the low-pass (scaling) filter and g is the high-pass (wavelet) filter. The process is repeated on the approximation coefficients.

Chapter 6

06 Multiresolution Analysis

Wavelets let you see your signal at multiple resolutions simultaneously — like zooming into a map. Each level reveals structure at a different scale.

Original signal and its wavelet decomposition at 4 levels — approximation + detail coefficients
Perfect reconstruction: Sum the final approximation with all detail levels and you recover the original signal exactly. Nothing is lost — energy is merely redistributed across scales.

Chapter 7

07 Real-World Applications

Wavelets are everywhere — from JPEG 2000 image compression to detecting gravitational waves at LIGO.

Wavelet denoising — transform → threshold small coefficients → inverse transform
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Audio — denoising, compression (e.g. FBI fingerprint standard)

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Images — JPEG 2000, texture analysis, edge detection

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Medical — ECG/EEG analysis, tumour detection in MRI

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Geophysics — seismic data, climate oscillation analysis