Frequency Domain

3 min read#fourier

The second of the two equivalent views of a signal — describing it by which frequencies it contains rather than by how it varies in time.

Contents

Frequency Domain

Every signal can be looked at in two ways. In the time domain you plot amplitude against the clock: the raw wiggle as it happens. In the frequency domain you plot how much of each pure frequency the signal holds. The Fourier Transform carries you from the first to the second and back, and — crucially — no information is lost in the trip. They are two photographs of the same statue taken from perpendicular angles.

One signal, two pictures

Why bother with a second view? Because facts that are hidden in one become obvious in the other. A noisy recording looks like chaos in time, but in frequency the steady hum of a power line is a single sharp spike you can simply delete. A chord sounds like one complex pressure wave in time, but in frequency it is plainly three notes. The frequency domain is where structure that is smeared across all of time collapses into a few clean peaks.

Below, your mouse sets a frequency. The top trace is that pure tone in the time domain; the bottom mark is the very same tone in the frequency domain — a single spike whose position is the frequency. One wiggly curve, one lone dot: identical content.

Move the mouse left↔right to change frequency. Top: the tone in time. Bottom: the same tone as a single spike in frequency.

Reading the two domains

A few translations are worth memorizing, because they recur everywhere:

  • A slow variation in time sits at the left (low frequency); a fast one sits at the right (high frequency).
  • A sharp edge or spike in time spreads across many frequencies — see the Gibbs Phenomenon and the Uncertainty Principle.
  • A pure repeating signal collapses to a few sharp peaks — its Spectrum.

See also

Related topics

  • Fourier TransformThe generalization of the Fourier series to non-periodic signals — an integral that resolves any signal into a continuous band of frequencies.
  • SpectrumThe magnitude of a signal's frequency content — the map of where its energy lives, read as a landscape of peaks.
  • Uncertainty PrincipleA signal cannot be sharply localized in both time and frequency at once — the tighter one view, the broader the other.
  • Fourier AnalysisAny signal is a sum of pure sinusoids — and reading it as such turns the tangled language of time into the clean language of frequency.
  • HarmonicsThe integer multiples of a fundamental frequency whose relative strengths give every sound its characteristic color.
  • Gibbs PhenomenonThe persistent ~9% overshoot that a truncated Fourier sum makes near a jump discontinuity — and that never goes away no matter how many terms you add.

Referenced by