Türfunktion

A Türfunktion, or "door function" in English, is a mathematical concept used primarily in signal processing and control systems. It is a type of step function characterized by its sudden change from one constant value to another. Specifically, it represents a signal that is zero for all time before a certain point, and then instantaneously jumps to a non-zero value (often one) and remains at that value thereafter. The opposite of a Türfunktion is a unit step function which starts at zero and then jumps to one.

The mathematical definition of a standard unit Türfunktion, often denoted as $u(t)$, is as follows: $u(t) = 0$ for $t < 0$, and $u(t) = 1$ for $t \ge 0$. The point at which the jump occurs, $t=0$ in this example, is called the point of discontinuity. Variations exist where the jump occurs at a different time, denoted by $t_0$, making the function $u(t-t_0)$. The amplitude of the jump can also be varied, deviating from the standard unit jump of 1.

Türfunktionen are fundamental in describing the response of systems to an instantaneous input. For example, when a switch is suddenly closed in an electrical circuit, the input voltage or current can be modeled as a Türfunktion. They are also used in analyzing the transient behavior of systems, understanding how a system reacts to a sudden change in its input. Their piecewise constant nature makes them analytically tractable and useful for theoretical studies.