🌊 The Core Idea

Any complex signal can be broken down into simple sine waves of different frequencies, amplitudes, and phases. The Fourier Transform tells us which frequencies are present and how strong each one is.

$$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(t)\, e^{-2\pi i \xi t}\, dt$$

f(t) — signal in time ξ — frequency f̂(ξ) — frequency spectrum

🎛️ Building a Signal from Sine Waves

Watch how adding harmonics constructs a square wave. Toggle components below.

📊 Frequency Spectrum (What the FT Reveals)

The Fourier Transform maps the time-domain signal → frequency domain. Each bar is a frequency component.

⏱ Time Domain

"What does the signal look like over time?"

Shows amplitude vs time
Reveals waveform shape
Hard to see individual frequencies

⟶ FT ⟶

📡 Frequency Domain

"Which frequencies make up the signal?"

Shows amplitude vs frequency
Exposes hidden harmonics
Basis for audio, image compression

🚀 Real-World Applications

🎵
Audio Processing
Equalizers, compression, noise removal

📡
Telecommunications
Radio, WiFi, signal modulation

🖼️
Image Compression
JPEG uses DCT (discrete cosine transform)

🩺
Medical Imaging
MRI reconstruction uses inverse FT

🌍
Seismology
Detecting earthquake frequencies

🔬
Quantum Mechanics
Position ↔ momentum duality