Concept

Fourier transform

The recipe that rewrites a signal as a sum of waves of every frequency, recording how much of each is present. It turns convolution into multiplication and is how sampling, filtering and smoothing are analysed.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The spectrum of a pulse train, as the period grows. The same pulse repeated at three different intervals, with its spectrum below each. The lines move closer together as the period lengthens and the curve they lie on does not move at all.

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

Two waves through the same samples. A sine of 9 cycles per unit sampled 10 times per unit coincides at every sample with minus a sine of 1 cycle per unit. Samples: 0.000, -0.588, -0.951, -0.951, -0.588, 0.000, 0.588, 0.951, 0.951, 0.588, -0.000.

A wave known only at its ticks

Read a sine wave nine cycles a second at ten instants a second and the readings describe a wave of one cycle a second instead. Read a wave with nothing faster than 3.7 cycles a second at more than 7.4 instants a second and the readings determine it exactly. Both facts are the same fold of the frequency axis, and it is why wagon wheels turn backwards on film.

analysis · Fourier series

Named alongside it

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

Periodicity Aliasing Fourier analysis Fourier series Frequency Interpolation Lattice Limit Sampling Sinc Spectrum Uncertainty principle

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