Fourier Series

A Fourier series writes any periodic function as a sum of sines and cosines. Joseph Fourier introduced the technique in 1822 to solve the heat equation, and the idea proved so general that Fourier series and their continuous cousin, the Fourier transform, now sit at the heart of signal processing, quantum mechanics, image compression, audio synthesis, and pure mathematics. The principle is simple: complex periodic patterns are made of simple sinusoidal harmonics — and knowing the harmonics is knowing the pattern.

The Series

For a function \( f(x) \) that’s periodic with period \( 2\pi \), the Fourier series is:

$$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[a_n \cos(nx) + b_n \sin(nx)\right] $$

The Fourier coefficients \( a_n, b_n \) are computed by integrating \( f \) against the corresponding sine and cosine:

$$ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx)\, dx, \quad b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx)\, dx $$

These formulas extract the ‘amount’ of each harmonic frequency in \( f \). For a function with period \( T \) other than \( 2\pi \), replace \( nx \) with \( 2\pi n x / T \).

Complex Form

Using Euler’s formula, the series condenses into a single sum of complex exponentials:

$$ f(x) = \sum_{n=-\infty}^{\infty} c_n e^{in x}, \quad c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) e^{-inx}\, dx $$

This compact form is preferred in physics and engineering, especially when extending to the Fourier transform of non-periodic functions.

Why It Works: Orthogonality

The sines and cosines are orthogonal functions over any full period: \( \int_{-\pi}^{\pi} \sin(mx) \cos(nx)\, dx = 0 \) for all \( m, n \), and \( \int \sin(mx) \sin(nx)\, dx = 0 \) for \( m \neq n \). Orthogonality is what lets the integral formulas above extract one harmonic at a time. It’s exactly analogous to projecting a vector onto orthogonal basis vectors — Fourier coefficients are inner products with the harmonic basis.

Convergence and the Gibbs Phenomenon

Truncating the series at \( N \) terms gives an approximation that’s exact in the limit \( N \to \infty \) for piecewise-smooth functions. Near jump discontinuities (e.g., the edges of a square wave), the truncated series overshoots the target by about 9% — the Gibbs phenomenon. Adding more terms makes the overshoot narrower in width but doesn’t reduce its height. It’s a fundamental feature, not a bug, of representing discontinuities with smooth functions.

Worked Example: Square Wave

The square wave that equals +1 on \( (0, \pi) \) and -1 on \( (-\pi, 0) \) has Fourier coefficients:

$$ b_n = \frac{4}{n\pi} \text{ if } n \text{ is odd}, \quad b_n = 0 \text{ if } n \text{ is even}, \quad a_n = 0 $$

So the series is \( f(x) = \tfrac{4}{\pi}\big[\sin x + \tfrac{1}{3}\sin 3x + \tfrac{1}{5}\sin 5x + \cdots\big] \). Only odd harmonics appear, with amplitudes that fall off like \( 1/n \). This is why a square wave ‘sounds harsh’ compared to a sine wave: it carries a lot of energy at odd harmonics.

From Series to Transform

For non-periodic functions, the discrete Fourier coefficients become a continuous function: the Fourier transform:

$$ \hat{f}(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x}\, dx $$

Discrete versions of this (DFT, FFT) run on every spectrum analyzer, audio app, and image-processing pipeline in the world.

Applications

• Audio. Every digital audio signal can be decomposed into its spectrum. EQ, pitch shifting, noise reduction, vocal tuning, and MP3 compression all work in the frequency domain.
• Image compression. JPEG uses the discrete cosine transform (a Fourier cousin) on 8×8 blocks. High-frequency coefficients carry less perceptually important information and are aggressively quantized to save space.
• Quantum mechanics. A particle’s wave function in position space and its wave function in momentum space are Fourier transforms of each other. The Heisenberg uncertainty principle is a direct consequence of this Fourier duality.
• Heat and wave equations. Fourier introduced the series specifically to solve the heat equation, by separating spatial and temporal parts.
• NMR and MRI. Medical imaging hardware measures Fourier-transformed signals and inverts them to reconstruct images.
• Communications. OFDM (used in Wi-Fi, 4G/5G, DSL) and most modern modulation schemes encode data on the amplitudes and phases of carefully chosen Fourier components.

Related study notes: Euler’s Identity, Imaginary Numbers, Partial Derivatives, Differential Equations.

Frequently Asked Questions