Fourier Uncertainty Principle

Δt · Δf ≥ 1/(4π) — Gaussian wavepackets saturate the bound; time-frequency tradeoff is fundamental
0.10
8.0
20
Δt = –
Δf = –
Δt·Δf = –
Bound 1/(4π) ≈ 0.0796
Heisenberg-Kennard (1927): σ_t·σ_ω ≥ ½, i.e., Δt·Δf ≥ 1/(4π). The Gaussian e^{−t²/(2σ²)} is the unique minimizer. In QM: σ_x·σ_p ≥ ℏ/2 — same mathematics. Chirp signals spread in frequency without widening in time, showing the tradeoff is about the spread of energy, not bandwidth.