Fourier Uncertainty Principle

Localizing a signal in time broadens its frequency spectrum — and vice versa

Window type

Width σ_t σ_t = 30

Center freq ν₀ ν₀ = 10

Center time t₀ t₀ = 50%

Time domain — f(t)

Frequency domain — |F̂(ν)|

Short-time Fourier spectrogram

σ_t (time spread)

—

σ_f (freq spread)

—

σ_t · σ_f

—

Bound (≥ 1/4π)

≥ 0.0796

The Heisenberg-Gabor uncertainty principle: σ_t · σ_f ≥ 1/(4π). A narrow pulse in time has a broad spectrum; a pure tone (infinite duration) has a delta-function spectrum. The Gaussian achieves the lower bound exactly — it is the minimum-uncertainty wavepacket. The spectrogram (bottom) shows time-frequency content via windowed FFT — notice the trade-off: shorter windows give better time resolution but worse frequency resolution. Adjust σ_t and watch σ_f change inversely.