Dispersiekernel

Dispersiekernel, or dispersion kernel, is a function used to describe the spreading of a signal, wave, or probability distribution caused by dispersion in a medium or system. In physics and engineering, it is often the impulse response (or Green's function) of a dispersive linear time-invariant system. The kernel defines how a single impulse input is transformed into a broadened output by the medium’s frequency-dependent behavior, and sometimes by amplitude changes as well.

Mathematically, if x(t) is the input and y(t) is the output, y(t) equals the convolution of x with h, where h(t) is the dispersiekernel. In the frequency domain, Y(ω) = H(ω)X(ω) with H(ω) = |H(ω)| e^{i φ(ω)}; dispersion arises when the phase φ(ω) varies nonlinearly with frequency, causing different frequency components to accumulate different delays and thus temporal or spatial broadening.

Dispersiekernels are used to model optical pulse propagation, seismic wave propagation, and other wave phenomena, as well as to perform deconvolution or smoothing in signal processing. In statistics or data analysis, related kernels can describe spreading of mass or events over time or space, though the term is more common in physics and engineering contexts.

Practically, a dispersiekernel is determined from material properties, experimental measurements, or theoretical dispersion relations, and is often computed via Fourier methods or numerical simulation. Its shape—width, symmetry, and tails—governs the extent and character of the dispersion.

See also: impulse response, Green's function, dispersion relation, kernel density estimation, diffusion kernel, deconvolution.