Uncertainty Principle

4 min read#fourier

A signal cannot be sharply localized in both time and frequency at once — the tighter one view, the broader the other.

Contents

Uncertainty Principle

There is a hard limit on how well a signal can be pinned down in both time and frequency at the same time. A signal that is narrow in time — a brief click — is necessarily broad in frequency, splattering energy across a wide band. A signal that is narrow in frequency — a pure, sustained tone — must be broad in time, droning on and on. You cannot have both sharp. Quantitatively, if \Delta t and \Delta f measure the spread in each domain,

\Delta t \,\Delta f \;\gtrsim\; \frac{1}{4\pi}.

This is not a statement about measurement clumsiness or quantum mechanics — it is a theorem about the Fourier Transform itself. The quantum Heisenberg principle is this very inequality applied to a particle's wavefunction.

Squeeze one, the other spreads

The extremal case — the signal that comes closest to equality — is the Gaussian bell. Below, your mouse sets the width of a Gaussian pulse in time (left). Its transform, also a Gaussian, is drawn in frequency (right). Make the time pulse skinny and watch the frequency bell fatten; widen the time pulse and the frequencies collapse toward a single tone. Their product stays roughly fixed.

Move the mouse to set the time-domain width (left). The frequency-domain width (right) moves the opposite way — their product is bounded below.

Why narrow demands wide

To synthesize a feature that is sharp and short-lived, you need sinusoids that cancel everywhere except in that brief window — and cancelling over all of time while reinforcing in one spot requires a wide range of frequencies beating against each other. Conversely, a single frequency, by definition, oscillates identically forever and cannot be confined. The math just makes this bookkeeping exact.

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.
  • Wave PacketA localized burst of oscillation — a carrier wave wrapped in a traveling envelope — whose envelope and crests move at different speeds in a dispersive medium.
  • DispersionWhen wave speed depends on frequency, a pulse made of many frequencies spreads and reshapes as it travels, and phase and group velocities part ways.
  • SpectrumThe magnitude of a signal's frequency content — the map of where its energy lives, read as a landscape of peaks.
  • 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.

Referenced by