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 — no matter how jagged or complex — can be expressed as a sum of sines and cosines. This is not obvious. A square wave has sharp corners; a sine wave is perfectly smooth. Yet infinitely many smooth sines can approximate a sharp edge arbitrarily well.
Why does this matter? Because sine waves at a single frequency are easy to work with — they pass through linear systems without changing shape (only amplitude and phase change). By decomposing a complicated signal into sine waves, analyzing each one separately, and recombining, you can analyze arbitrarily complex systems. This idea underpins MP3 compression, MRI imaging, noise-cancelling headphones, WiFi, and quantum mechanics.
There are two related but distinct concepts: the Fourier Series handles periodic functions (repeating patterns) and gives discrete frequency components. The Fourier Transformgeneralizes to non-periodic functions by allowing a continuous spectrum of frequencies.
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
Prerequisites: Complex Numbers & Euler's Formula
To understand Fourier analysis deeply you need one key piece of background: a complex number z = a + ib lives in a 2D plane (real axis horizontal, imaginary axis vertical). Multiplying by i rotates 90°. The complex exponential eiθ is not just a notational trick — it literally traces a unit circle in the complex plane as θ increases:
This means eiωt is a vector of length 1 spinning at angular frequency ω rad/s. Euler's formula is the bridge that lets us write sines and cosines as a single exponential, which makes Fourier math dramatically cleaner.
Fourier Series (Periodic Functions)
For any periodic function f(x) with period 2L, the Fourier Series expresses it as an infinite sum of harmonics. Each term is a sine or cosine at an integer multiple of the fundamental frequency ω = π/L:
The coefficients aₙ and bₙ measure how much of each frequency is present. They come from orthogonality: sine and cosine functions at different frequencies are perpendicular in the sense that their integral product over one period is zero. This lets you isolate each coefficient by multiplying both sides by that frequency's sine or cosine and integrating:
Using Euler's formula, the series takes the compact complex form with a single sum. The complex coefficient cₙ encodes both amplitude and phase:
Fourier Transform (Non-Periodic Functions)
The Fourier Transform generalizes the series to functions that are not periodic — by imagining the period L → ∞, the discrete frequencies become a continuous spectrum. Instead of coefficients cₙ at integer frequencies, we get a continuous function F(ω):
Why does the integral extract frequencies? Multiplying f(t) by e-iωt"winds" the signal around the origin at frequency ω — like wrapping a piece of string around a cylinder. If f(t) contains a component at that frequency, the winding clumps mass away from zero and the integral is nonzero. For any other frequency, positive and negative contributions cancel and the integral is zero. This is exactly what the ear does mechanically in the cochlea.
Key Properties
| Property | f(t) | F(ω) |
|---|---|---|
| Linearity | af(t) + bg(t) | aF(ω) + bG(ω) |
| Time shift | f(t − t₀) | e^{−iωt₀} F(ω) |
| Frequency shift | e^{iω₀t} f(t) | F(ω − ω₀) |
| Differentiation | f '(t) | iω F(ω) |
| Convolution | (f * g)(t) | F(ω)·G(ω) |
| Parseval's Theorem | ∫|f|² dt | (1/2π) ∫|F|² dω |
The Convolution Theorem is particularly powerful: convolution in the time domain (a slow O(n²) operation) becomes pointwise multiplication in the frequency domain (O(n)). This is why the FFT (Fast Fourier Transform) algorithm makes audio filtering and image processing fast.
Fourier Transform Table
| Function f(t) | Fourier Transform F(ω) |
|---|---|
| δ(t) | 1 |
| 1 | 2πδ(ω) |
| e^{-at}u(t), a > 0 | \frac{1}{a + i\omega} |
| e^{-at}u(-t), a > 0 | \frac{1}{a - i\omega} |
| \cos(\omega_0 t) | \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)] |
| \sin(\omega_0 t) | i\pi[\delta(\omega - \omega_0) - \delta(\omega + \omega_0)] |
| e^{-\alpha t^2} | \sqrt{\frac{\pi}{\alpha}} e^{-\omega^2/(4\alpha)} |
| \frac{1}{1 + t^2} | \pi e^{-|\omega|} |
| \text{rect}(t/T) | T \frac{\sin(\omega T/2)}{\omega T/2} |
| \text{sign}(t) | \frac{2}{i\omega} |
This table shows common Fourier transform pairs. The rect function represents a rectangular pulse, and the sign function is the signum function. These pairs are fundamental in signal processing and are used to solve differential equations and analyze systems.
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.