Impulsmenge

Impulsmenge (German for “impulse set”) is a term used in mathematics and engineering to refer to a collection of impulses (Dirac delta functions) or discrete measures that are used to represent signals or distributions as a combination of point-like events.

In continuous-time signal processing, an impulse set is a collection {δ(t − t_i) | i ∈ I} with associated weights a_i; the signal x(t) can be written as x(t) = ∑ a_i δ(t − t_i). In discrete time, impulses are Kronecker deltas δ[n − n_i] and the same idea applies.

In distribution theory, the Dirac delta δ_t is a linear functional; finite or countable linear combinations ∑ a_i δ_{t_i} are discrete measures, called a discrete impulse measure; they are not functions in the ordinary sense but distributions that act on test functions via integration.

Trains of impulses, such as the Dirac comb ∑_k δ(t − kT), are special cases of an impulse set with infinite indices; convergence and topology depend on the space of test functions used.

Applications include system identification, deconvolution, and sparse signal representations. The impulse response of a system is the output for a Dirac input, and an impulse set allows modeling of complex inputs as sums of impulses.

The term is mainly found in German-language literature and appears in contexts tied to distribution theory and signal processing. In English-language literature, the concepts are usually described with Dirac delta functions or impulse inputs rather than the global label “Impulsmenge.”

See also: Dirac delta, impulse response, Dirac comb, distribution (mathematics), Kronecker delta.