Uncertainty principle

The uncertainty principle states that a signal cannot be simultaneously localized in both time and frequency beyond a bound determined by the product of their uncertainties. Most familiar from quantum mechanics, where it governs the complementarity of position and momentum, the principle is in fact a theorem of Fourier analysis: a function and its Fourier transform cannot both be arbitrarily concentrated.

The principle is not merely a quantum curiosity but a fundamental limit of information geometry. It implies that any system that localizes events in time must necessarily spread their spectral content, and vice versa. This trade-off governs everything from the bandwidth of communication channels to the resolution limits of signal processing systems. The uncertainty principle is the reason that instantaneous frequency is an oxymoron.