Spectrum

2 min read#fourier

The magnitude of a signal's frequency content — the map of where its energy lives, read as a landscape of peaks.

Contents

Spectrum

The spectrum of a signal is the magnitude of its Fourier Transform: for each frequency, how much of it is present, with the phase set aside. Where the full transform X(f) is a complex number, the spectrum keeps only |X(f)| (or its square, the power spectrum). It answers the most common question we ask of a signal — what frequencies is it made of, and how strong are they? — as a single readable picture.

Reading a signal as a spectrum means reading it as a landscape of peaks. A tall peak at some frequency says "there is a strong, steady oscillation right here." Broad, low hills say "noise, energy spread everywhere." The whole craft of spectral analysis is learning to read that terrain.

A line spectrum

A periodic signal has a discrete spectrum: energy lives only at the fundamental and its Harmonics, so the picture is a row of sharp lines. Here is the spectrum of a sawtooth-like tone, whose harmonics fall off as 1/n:

A 1/n line spectrum (harmonic number : amplitude)

Each bar marks one harmonic; the steady decay is the spectral signature of a signal with sharp corners. Smooth signals decay fast; jagged ones decay slowly.

What the spectrum tells you

  • Pitch is the lowest strong peak — the fundamental.
  • Timbre is the pattern of the remaining peaks (see Harmonics).
  • Noise is a raised, featureless floor between the peaks.
  • A pure tone is a single spike; a click is a flat spectrum touching every frequency at once.

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.
  • Frequency DomainThe second of the two equivalent views of a signal — describing it by which frequencies it contains rather than by how it varies in time.
  • HarmonicsThe integer multiples of a fundamental frequency whose relative strengths give every sound its characteristic color.
  • 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.
  • Uncertainty PrincipleA signal cannot be sharply localized in both time and frequency at once — the tighter one view, the broader the other.
  • WavesFrom a single swinging oscillator to ripples that travel, interfere, and carry frequency through space — the bridge between Oscillations and Fourier Analysis.

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