Linear response function

A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information theory, physics and engineering there exist alternative names for specific linear response functions such as susceptibility, impulse response or impedance, see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation is closely related.

Denote the input of a system by ${\displaystyle h(t)}$ (e.g. a force), and the response of the system by ${\displaystyle x(t)}$ (e.g. a position). Generally, the value of ${\displaystyle x(t)}$ will depend not only on the present value of ${\displaystyle h(t)}$, but also on past values. Approximately ${\displaystyle x(t)}$ is a weighted sum of the previous values of ${\displaystyle h(t')}$, with the weights given by the linear response function ${\displaystyle \chi (t-t')}$:

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