Fourier transforms

Play three piano notes at once and you hear a chord, a single rich sound. Yet a trained ear can pick out the separate notes inside it. The Fourier transform is the piece of maths that does that picking apart, for any signal at all.

The big idea is that almost anything that wiggles, a sound, a radio wave, the brightness along a row of an image, can be built by adding together simple, pure waves: smooth sine waves of different speeds and sizes. A fast wave for the high notes, a slow one for the low, a big one for the loud parts. Stack the right ones and you can reproduce any shape you like.

So a signal has two faces. One is what it looks like over time, the wiggly line you would see on a screen. The other is its recipe: the list of which pure waves are in it and how strong each one is. That recipe is called the spectrum, and the Fourier transform is the machine that converts between the two, in either direction.

This turns out to be one of the most useful tools ever invented. It is how MP3 and JPEG shrink files (throw away the waves your ears and eyes would not notice), how noise is cleaned out of recordings, how a phone separates wifi channels, how doctors turn MRI scanner readings into pictures, and how astronomers read the chemistry of a star from its light. Joseph Fourier dreamt it up around 1822 while studying how heat spreads.

Waves that build anything. Fourier's startling claim was that any repeating wave, however jagged, can be written as a sum of plain sine and cosine waves whose frequencies are whole-number multiples of a base frequency. That sum is a Fourier series. The widget above builds a square wave this way, from the odd harmonics; with a handful of terms it is close, and with many it is almost perfect. The small ripples that cling to the sharp edges never fully vanish, a quirk called the Gibbs phenomenon.

From series to transform. A repeating wave only needs whole-number frequencies. For a one-off signal that never repeats you need every frequency, and the sum becomes an integral. That is the Fourier transform: it takes a signal \(f(t)\) and returns a function \(F(\omega)\) that gives the strength of every frequency \(\omega\) inside it.

\[ F(\omega) = \int_{-\infty}^{\infty} f(t)\,e^{-i\omega t}\,dt. \]

The \(e^{-i\omega t}\) looks exotic but it is just a sine and a cosine of frequency \(\omega\) carried together (that is Euler's formula). Multiplying your signal by it and adding everything up measures how much that one frequency resonates with the signal. Do it for every \(\omega\) and you have the whole recipe, and an inverse transform rebuilds the original from it.

Two domains, one truth. Time and frequency are two views of the same information. A pure tone is a single spike in frequency; a sharp click is spread across all frequencies at once. Squeeze a signal narrow in time and its spectrum spreads wide, and the reverse, which is exactly the trade-off behind the uncertainty principle in physics.

Why it is everywhere. On a computer you use the discrete version, the DFT, and a brilliantly fast way to compute it, the Fast Fourier Transform (FFT, popularised by Cooley and Tukey in 1965). It runs in a whisker of the time a direct calculation would take, and that single speed-up made the modern world: audio and image compression, wifi and 5G, noise filtering, radar, and a standard trick for solving the differential equations of physics and engineering.

Series and convergence. For a periodic function of period \(T\), the complex Fourier series is \(f(t) = \sum_{n=-\infty}^{\infty} c_n\,e^{i 2\pi n t/T}\) with \(c_n = \tfrac{1}{T}\int_0^T f(t)\,e^{-i 2\pi n t/T}\,dt\), built on the orthogonality of the exponentials \(\langle e^{im\theta}, e^{in\theta}\rangle = 2\pi\,\delta_{mn}\). It converges in \(L^2\) for any square-integrable \(f\) (and pointwise under the Dirichlet conditions), while at a jump discontinuity the partial sums overshoot by a fixed fraction, about 9% of the gap, the Gibbs phenomenon. Parseval's theorem equates the energy in the signal with the energy in its coefficients.

The transform as a unitary map. On the line, \(F(\omega) = \int_{-\infty}^{\infty} f(t)\,e^{-i\omega t}\,dt\) with inverse \(f(t) = \tfrac{1}{2\pi}\int F(\omega)\,e^{i\omega t}\,d\omega\). By Plancherel's theorem it extends to a unitary operator on \(L^2(\mathbb{R})\), preserving inner products. Its power comes from how it converts operations: convolution becomes multiplication, \(\widehat{f * g} = \hat f\,\hat g\) (the basis of all filtering and of fast polynomial multiplication), differentiation becomes multiplication by \(i\omega\) (turning linear differential equations into algebra, which was Fourier's route to the heat equation), and translation becomes a phase factor. The transform of a Gaussian is a Gaussian, and the time-bandwidth product is bounded below, \(\sigma_t\,\sigma_\omega \ge \tfrac{1}{2}\), the mathematical heart of the Heisenberg uncertainty principle.

Discrete, fast, and sampled. Computers work with the discrete Fourier transform on \(\mathbb{C}^N\). Computed directly it costs \(O(N^2)\); the Cooley-Tukey FFT factors it recursively into \(O(N\log N)\), one of the most consequential algorithms ever written. The Nyquist-Shannon sampling theorem says a signal band-limited below \(f_s/2\) is recovered exactly from samples at rate \(f_s\); sampling too slowly aliases high frequencies into low ones. Finite windows leak energy between bins, which window functions tame.

The wider family. Fourier analysis is the harmonic analysis of the real line and the circle, the simplest case of Pontryagin duality on locally compact abelian groups. Close relatives extend the toolkit: the Laplace transform adds decay for stability and control theory, and wavelets trade some frequency precision for the time localisation a pure sinusoid cannot give. The same duality is physical: in quantum mechanics a particle's position and momentum wavefunctions are a Fourier pair, and the quantum Fourier transform is the engine inside Shor's factoring algorithm on a quantum computer. It all rests on the machinery of calculus, integrals taken to the limit.