Every signal is a sum of sine waves.
One of the deepest single facts in mathematics — and the silent engine of MP3, JPEG, Wi-Fi and MRI. The Explorer lets you add harmonics one at a time and watch a square wave appear out of pure sines.
Press middle-C on a piano. The string vibrates at 261.6 Hz (the fundamental), and at the same time it vibrates at 523.3 Hz, 785 Hz, 1046.5 Hz — integer multiples called harmonics, each with its own loudness. That mix of overtones is what makes the piano sound like a piano and not a flute, even though the note has the same name.
A periodic function f(t), under mild conditions (Dirichlet, 1829), can be written as a sum over discrete frequencies: f(t) = Σₖ aₖ sin(kωt + φₖ). The frequencies are integer multiples of one fundamental ω; the amplitudes aₖ and phases φₖ are uniquely determined by f. The space of well-behaved signals splits into independent sine-shaped axes, and analysing a signal means writing it in that basis.
Each k is one independent axis. The amplitudes aₖ are the coordinates of the signal in that basis; the phases φₖ shift each axis along itself.
A square wave is the cleanest test of the principle: pure +1 and −1, with infinitely sharp jumps. Its Fourier series is 4/π · Σ sin((2k−1)·2πt)/(2k−1), summed over k = 1, 2, 3, … At one harmonic the answer is a single rolling sine. At three, the corners begin to appear. At seven, you can read it as a square — except for a stubborn 8.95% overshoot at every edge. That overshoot never disappears: Gibbs described in 1899 how at a jump discontinuity the partial sums always overshoot by exactly this constant, however many harmonics you add. The corner is the cost of representing 'instant' in a basis of smooth waves.
White: the target. Violet: each individual harmonic. Amber: their sum. Push N up and the corners sharpen, yet the overshoot at the discontinuities never quite goes away (Gibbs phenomenon, ≈ 8.95%).
The constant is exact and independent of N. Sharpness in time costs spread in frequency, and the next section turns that trade-off into a law.
The Fourier transform is a change of coordinates. The same signal has two equivalent representations: a graph against time (the waveform you'd see on an oscilloscope) and a graph against frequency (the spectrum you'd see on an analyser). A spectrogram slides a short window along the recording, takes the spectrum inside each window, and stacks them — turning a melody into a glowing piano-roll of frequency-vs-time. Sliders here let you sculpt one and watch the other rebuild itself live.
Top: the waveform you just synthesised. Bottom: the same signal in frequency-space, one bar per harmonic. The two pictures carry the same information.
f(t) = amplitude
vs. timeWhat an oscilloscope or a microphone records.
f̂(ξ) = amplitude
vs. frequencyWhat a spectrum analyser or your inner ear sees.
On paper the transform is an integral, f̂(ξ) = ∫ f(t) e^(−2πi ξ t) dt — defined for any integrable function. On a computer we work with samples and the discrete Fourier transform: N input numbers, N output frequencies. Naïvely that costs N² multiplications — for a 4-second CD recording (≈ 200 000 samples) about 40 billion. In 1965 Cooley and Tukey rediscovered (Gauss already had it in 1805) a recursive trick that brings the cost down to N log N — about 3 million for the same recording. Without that single algorithm there would be no real-time MP3, no JPEG, no smartphone radio, no medical imaging as we know it.
| N | N² · naïve DFT | N log N · FFT | speed-up |
|---|---|---|---|
| 1 024 | 1.05 × 10⁶ | 1.02 × 10⁴ | ≈ 100× |
| 65 536 | 4.29 × 10⁹ | 1.05 × 10⁶ | ≈ 4 000× |
| 1 048 576 | 1.10 × 10¹² | 2.10 × 10⁷ | ≈ 50 000× |
The FFT is not faster mathematics, it is the same DFT computed by reusing every partial product instead of redoing it. The exponent drops from 2 to 1, and real-time signal processing opens up.
Squeeze a signal into a narrow time window and its Fourier transform necessarily smears across many frequencies — and vice versa. Make the variance in time σ_t and the variance in frequency σ_f. They satisfy σ_t · σ_f ≥ 1/(4π) (or, with the angular convention, σ_t · σ_ω ≥ 1/2), and the bound is reached only by the Gaussian function — which is, beautifully, its own Fourier transform. Multiply both sides by ℏ and you have Heisenberg's position-momentum uncertainty: the quantum principle is the same inequality in a different guise. Time and frequency are dual coordinates, and the trade-off between them is mathematics, not engineering.
The Gaussian minimises both. Position and momentum are Fourier duals, and the quantum bound is the same theorem on a different axis.
Almost every signal humans store, transmit, or process today passes through a Fourier transform on the way. The camera in your phone runs a discrete cosine transform on every 8×8 block before saving the JPEG. The microphone driver does a thousand short-time FFTs per second on your voice. The Wi-Fi chip uses OFDM, breaking each packet into thousands of carrier frequencies that are turned back into bits with one inverse FFT per symbol. MRI machines literally collect samples in frequency-space — k-space — and apply an inverse Fourier transform to make the image of you. One theorem, two centuries old, holds the modern world together.
The same theorem, wearing different clothes in every corner of modern technology.
Add harmonics one at a time, watch a square wave emerge, sweep the phase and listen to the spectrum.
How and where this technique lives in the world today.
Topics in the same vein.