ZeitFrequenzUnschärferelation

Zeit-Frequenz-Unschärferelation, or the time–frequency uncertainty principle, is a fundamental constraint in signal processing and related fields that limits how precisely a signal can be localized in time and in frequency at the same time. For a signal x(t) with Fourier transform X(f), one defines the time center t0 and a time spread Δt, and similarly a frequency center f0 and a frequency spread Δf. A common formulation states that Δt Δf ≥ 1/(4π) when using usual definitions for Δt and Δf. If angular frequency ω is used instead of f, the bound becomes Δt Δω ≥ 1/2. The principle expresses a trade-off: tightening a signal’s temporal localization inevitably broadens its spectral content, and vice versa.

The concept has roots in the mathematical properties of the Fourier transform and is closely related to the Heisenberg uncertainty principle in physics. It is widely applicable in both theoretical and applied contexts. Gaussian pulses are known to minimize the product Δt Δf, achieving equality in the standard bound.

In practice, the uncertainty principle influences time–frequency representations. The short-time Fourier transform uses a fixed window to balance time and frequency resolution, while wavelet transforms provide multiresolution analysis with varying emphasis across scales. Other representations, such as the Wigner–Ville distribution, offer high resolution but can introduce cross-term artifacts.

Applications span audio analysis, communications, radar, and quantum physics as an analytical framework. The exact numerical bound depends on the chosen definitions of Δt and Δf, but the qualitative message remains: simultaneous precise localization in time and frequency is inherently limited by the mathematics of Fourier analysis.