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Concept

Spectrum

The list of how much of each frequency a signal holds, which describes it as completely as its shape over time does.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

-111 term-113 terms-117 terms-1121 terms

A square wave built entirely out of round ones

Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.

analysis · fourier series −ππ-11× sin 1xarea = 4.000coefficient 1.273−ππ-11× sin 2xarea = 0no such harmonic in it−ππ-11× sin 3xarea = 1.333coefficient 0.424−ππ-11× sin 4xarea = 0no such harmonic in iteach panel is the target multiplied by one harmonic, with the area above the axis in one colour and the area below in the otheragainst its own harmonic the two do not cancel; against any other they cancel exactly, which is what makes one coefficient extractable without disturbing the rest

Where the coefficients come from

The recipe for a square wave has a four over pi in front and a one over three on the second term, and the first rung of this ladder used them without saying where they came from. They come from multiplying by one harmonic and taking the area.

analysis · fourier series period 2lines every 0.500frequencyperiod 4lines every 0.250frequencyperiod 8lines every 0.125frequencya pulse 1 wide, repeated every 2, then 4, then 8 — the same pulse each time, and only the gap between repeats changingthe lines crowd together as 1/T while the curve they sit on stays exactly where it is; at infinite period the lines are denseand the curve is the whole of the answer

When the period grows without bound

A repeating signal has a spectrum of separate lines. Stretch the gap between repeats and the lines crowd together while the curve they sit on stays exactly where it is — and at infinite period the lines are gone and the curve is the whole answer.

analysis · fourier series −ππ-11position around the ringtemperatureat the startt = 0.005t = 0.05t = 0.4a square profile on a ring, left to spread: harmonic m fades by exp(−m²t), so the 41th term is gone 1,681 times fasterthan the firstthe corners disappear immediately and the shape that remains is a single sine — which is why running the flowbackwards is hopeless: the information in the corners has been divided by a number this large

The corners go first

Fourier was not decomposing waves for the pleasure of it. He was solving the flow of heat, and the whole apparatus exists because each harmonic fades at a rate set by the square of its frequency — which is why a sharp profile smooths instantly and why the flow cannot be run backwards.

analysis · fourier series

Named alongside it

The objects these essays reach for when they reach for this one.

Fourier analysisConvergenceHarmonicsOrthogonalityPeriodicityContinuityDiffusionEigenfunctionExponential decayFourier transformGibbs' phenomenonHeat equation

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