See the Fourier Transform
come alive
Interactive visualizer, FFT calculator, signal tools, and a structured learning path — everything you need to truly understand Fourier analysis.
Interactive Tools
Six powerful, free tools for exploring Fourier analysis — right in your browser.
Interactive Visualizer
Draw a shape and watch its Fourier series reconstruction in real time with spinning epicycles.
FFT Calculator
Paste or generate a signal, compute its FFT, and explore the frequency spectrum interactively.
Signal Generator
Build complex signals by mixing sine, square, sawtooth, and triangle waves with adjustable parameters.
Spectrogram
Visualize how frequency content changes over time with a live, color-mapped spectrogram.
Filter Designer
Design lowpass, highpass, bandpass, and notch filters. See the frequency response and filtered output.
Convolution Demo
Visualize convolution step by step – see how two signals combine to produce a third.
What is the Fourier Transform?
The Fourier Transform decomposes any signal into a sum of simple sine and cosine waves. Instead of looking at a signal in the time domain (amplitude vs. time), you see it in the frequency domain (amplitude vs. frequency).
$$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i \xi x}\, dx$$
The Continuous Fourier Transform — it converts a time-domain function $f(x)$ into its frequency-domain representation $\hat{f}(\xi)$.
Audio & Music
Break sound into component frequencies — the basis of equalizers, compression, and noise cancellation.
Communications
FM/AM radio, Wi-Fi, 5G — all rely on frequency-domain analysis to encode and decode signals.
Medical Imaging
MRI scanners use the Fourier Transform to reconstruct images from raw frequency data.
Structured Learning Path
Go from zero to confident. Each lesson builds on the last, with interactive demos throughout.
Signals & Sinusoids
What is a signal and why sinusoids are the building blocks of everything.
Frequency, Phase & Amplitude
The three numbers that define a sinusoid.
DFT & FFT Basics
How the discrete Fourier transform works and why FFT is fast.
Sampling & Aliasing
Nyquist theorem, what happens when you sample too slowly.
Real-World Use Cases
Step-by-step walkthroughs with live demos. Copy the approach straight into your project.
Quick Reference
Common Fourier pairs, properties, and formulas — bookmark-worthy.
Ready to explore?
Draw a shape and watch the Fourier series reconstruct it with spinning circles. It's the best way to build intuition.
Launch Visualizer →