HowTo:Fourier

The Fourier transform is such a useful fundamental tool, it deserves its own howto (as opposed to a simple Code). The Fourier transform is a transformation of a function into another function. The transformation can (in almost all cases) be reversed.

Before getting started, a few analogies are helpful.

Sound comes from acoustic pressure waves -- so you hear middle C when your ear feels a time-varying pressure at precisely 261.6 Hz (1 Hz = 1 cycle per second). The pressure signal would be <math>P(t) = cos(2 \pi (261.6 t))</math> -- so that 1/261.6 second goes in a complete cycle, from 0 to <math>2\pi</math> radians. Fourier transforming that signal gives a function that is zero everywhere except for <math>k = 261.6</math> -- a delta function. The transform of a pure sin or cos wave is always a delta function -- peaked at one particular frequency. If you shift the phase of the sin or cos wave (say by <math>\Delta t</math>, this shows up in Fourier space as a phase factor, <math>\exp(2\pi i\Delta t)</math>, which does not change the magnitude.

Light comes from electromagnetic waves. In parking lots, you can usually see the grungy yellow/orange color of 589.29 nm light from cheap high-pressure sodium lights. This is because your eye is experiencing a time-varying electric field at a frequency of <math>3\cdot 10^17 nm/s / 589.29 nm \simeq 510 THz</math>.