Fourier Transform
Discover how any complex wave can be broken down into simple sine waves. Start from the basics and build deep intuition through interactive visualizations.
What You'll Learn
A wave is a disturbance that transfers energy from one place to another without transferring matter. Think of ripples in a pond - the water molecules don't travel across the pond, but the energy does!
Amplitude (A)
The height of the wave from the center line. Determines intensity (loudness for sound, brightness for light).
Frequency (f)
How many complete cycles per second (measured in Hz). Determines pitch for sound, color for light.
Phase (φ)
The starting position of the wave. Two identical waves with different phases can cancel out!
Play with the controls below to see how these three properties affect a wave:
Height of the wave
Cycles per unit time
Starting position
The fundamental wave equation: Amplitude × sine of (frequency × time + phase)
Key Relationships
Higher frequency = shorter wavelength
Time for one complete cycle
Not all waves look like smooth sine waves! In the real world, we encounter many different wave shapes. Each shape has a unique "recipe" of frequencies - and that's exactly what the Fourier Transform reveals.
Sine Wave
The most fundamental wave - smooth and continuous. It's the 'purest' form of oscillation with only a single frequency.
Real-World Examples
Pure musical tones, AC electricity, pendulum motion, light waves
Mechanical Waves
Need a medium to travel through:
- • Sound waves (through air, water, solids)
- • Ocean waves (through water)
- • Seismic waves (through Earth)
- • Waves on a string (guitar, piano)
Electromagnetic Waves
Can travel through vacuum:
- • Radio waves (communication)
- • Microwaves (cooking, WiFi)
- • Visible light (what we see)
- • X-rays, Gamma rays (medical, nuclear)
Waves are everywhere in nature and technology. Understanding how to analyze and manipulate waves unlocks incredible capabilities:
Music & Audio
Every musical instrument creates unique wave patterns. Synthesizers create sounds by combining sine waves!
Communication
WiFi, 5G, radio, satellite TV - all encode information onto electromagnetic waves.
Medical Imaging
MRI, CT scans, and ultrasound all use wave analysis to see inside the body.
Seismology
Analyzing earthquake waves helps predict disasters and understand Earth's interior.
Image Processing
JPEG compression, Instagram filters, and facial recognition all use wave mathematics.
Quantum Physics
Matter itself behaves like waves at quantum scales - wave functions describe probability!
The Big Insight: Almost any repeating pattern or signal in nature can be understood as a combination of simple waves. The Fourier Transform is the mathematical tool that reveals these hidden components!
Imagine you're listening to a chord on a piano. Your ear hears a single sound, but that sound is actually made up of multiple notes playing together. The Fourier Transform is like having super-powered ears that can separate any complex sound (or signal) into its individual component frequencies.
The Core Idea
Any signal, no matter how complex, can be represented as a sum of simple sine waves at different frequencies.
Time Domain
What we normally see - the signal changing over time. Like watching a wave move across water.
Frequency Domain
What Fourier shows us - which frequencies are present and how strong they are. Like knowing which notes make up a chord.
Here's the beautiful part: we can use rotating circles (called epicycles) to draw any wave! Each circle rotates at a different speed (frequency) and has a different size (amplitude). Watch how adding more circles makes a better approximation of a square wave.
Try this: Start with 1 circle and slowly increase. Notice how with just 1 circle you get a simple sine wave, but with more circles, you start to see the sharp corners of a square wave forming!
Now it's your turn! Add multiple sine waves with different frequencies and amplitudes. Watch how they combine to create complex shapes. This is the reverse of Fourier analysis - it's called Fourier synthesis.
Challenge: Try to create a square-ish wave by adding waves with frequencies 1, 3, 5 (odd numbers) where each subsequent wave has a smaller amplitude!
Let's walk through exactly how Fourier decomposition works. We'll take a square wave and break it down into its component frequencies step by step.
Step 1: Start with the Target Signal
We want to decompose this square wave into simple sine waves. The square wave jumps between +1 and -1.
Drag to add more harmonics and watch the approximation improve
Fourier Coefficients for Square Wave
| Harmonic (n) | Coefficient (bₙ) | Decimal | Contribution |
|---|---|---|---|
| 1f | 4/1π | 1.2732 |
Different wave shapes have different "frequency fingerprints". A pure sine wave has just one frequency. But look at what happens with square, sawtooth, and triangle waves - they need many frequencies working together!
Square Wave
Only odd harmonics (1f, 3f, 5f...). Amplitude decreases as 1/n. This creates those sharp corners!
Sawtooth Wave
Contains all harmonics (1f, 2f, 3f...). Amplitude decreases as 1/n. Commonly used in synthesizers!
Now let's dive into the actual equations. Don't worry - we'll break them down step by step!
1. Fourier Series (Periodic Signals)
For any periodic function with period T, we can write:
DC offset (average value)
Fourier coefficients
Fundamental frequency
2. Finding the Coefficients
The magic question: how much of each frequency is present? We integrate!
💡 The integral "measures" how much the signal correlates with each frequency!
3. Fourier Transform (Any Signal)
For non-periodic signals, we use the continuous Fourier Transform:
Time → Frequency domain
f(t) = (1/2π) ∫ F(ω)·e^(iωt) dω
4. The Secret Weapon: Euler's Formula
This beautiful formula connects exponentials with trigonometry! It's why we can write the Fourier Transform so elegantly using complex exponentials instead of separate sines and cosines.
Example: Square Wave Decomposition
For a square wave oscillating between +1 and -1:
Notice: only odd harmonics, amplitudes decrease as 1/n!
Audio & Music
MP3 compression, audio equalizers, noise cancellation, voice recognition, and music production all rely on Fourier analysis.
Image Processing
JPEG compression, image filtering, edge detection, and medical imaging (MRI, CT scans) use 2D Fourier transforms.
Telecommunications
Cell phones, WiFi, and all digital communication use FFT to encode and decode signals efficiently.
Science & Engineering
Earthquake analysis, quantum mechanics, spectroscopy, structural engineering, and solving differential equations.
Test Your Understanding
Fourier Transform Quiz
Question 1 of 8What is a wave?