Fourier Transform & Series
Decomposing signals into frequencies — the foundation of signal processing
1. Building Intuition
Imagine you hear a chord on a piano — C, E, and G played together. Your ear perceives a single sound, but your brain effortlessly separates it into three distinct notes. The Fourier Transform does exactly this for any signal: it decomposes a complex waveform into its constituent sine waves.
Joseph Fourier's remarkable discovery (1807) was that any periodic function can be expressed as a sum of sines and cosines. This seemingly simple idea underpins everything from MP3 compression to MRI imaging to quantum mechanics.
Watch as individual sine waves (colored) are added one by one. Their sum (purple) progressively approximates a square wave — a classic demonstration of Fourier series convergence.
2. The Mathematics
The Fourier Series expresses a periodic function as an infinite sum of harmonics. Each coefficient measures how much of that frequency is present in the signal.
The Fourier Transform generalizes this to non-periodic functions. Instead of discrete frequencies, we get a continuous spectrum. The transform pairs a time-domain signal with its frequency-domain representation:
The key to understanding why this works is Euler's formula. The complex exponential eiωt traces a circle in the complex plane. Multiplying f(t) by e-iωt and integrating effectively "winds" the signal around the origin at frequency ω — if there's a matching frequency component, the integral is nonzero.
3. Applications
Each circle rotates at a different frequency. The tip of the last circle traces out the output wave — this is literally what the Fourier series does: sum rotating complex exponentials.
Signal Processing
Filtering noise from audio, image compression (JPEG), spectral analysis of signals.
Communications
OFDM in WiFi and 4G/5G, modulation/demodulation, channel equalization.
Medical Imaging
MRI reconstructs images from frequency-space (k-space) data using inverse FFT.
Quantum Mechanics
Position and momentum are Fourier transform pairs — Heisenberg uncertainty principle.
4. Worked Examples
Example 1: Square Wave as Fourier Series
A square wave of amplitude 1 and period 2π can be written as:
Only odd harmonics are present (the square wave has odd symmetry). As you add more terms, the approximation sharpens — but overshoots at discontinuities persist (the Gibbs phenomenon, approximately 9% overshoot that never disappears).
Example 2: Fourier Transform of a Gaussian
One of the most elegant results: the Fourier transform of a Gaussian is another Gaussian.
A narrow Gaussian in time becomes a wide Gaussian in frequency (and vice versa). This is the mathematical basis of the uncertainty principle: you cannot be arbitrarily precise in both time and frequency simultaneously.
Example 3: Signal Filtering
Given a composite signal f(t) = sin(2t) + 0.6sin(5t) + 0.35sin(9t), the Fourier Transform reveals three peaks at ω = 2, 5, and 9 Hz. To remove the high-frequency noise, apply a low-pass filter (zero out F(ω) for |ω| > 6) and inverse-transform back.