Denote the input of a system by (e.g. a force), and the response of the system by (e.g. a position). Generally, the value of will depend not only on the present value of , but also on past values. Approximately is a weighted sum of the previous values of , with the weights given by the linear response function :
The explicit term on the right-hand side is the leading order term of a Volterra expansion for the full nonlinear response. If the system in question is highly non-linear, higher order terms in the expansion, denoted by the dots, become important and the signal transducer cannot adequately be described just by its linear response function.
The complex-valued Fourier transform of the linear response function is very useful as it describes the output of the system if the input is a sine wave with frequency . The output reads
with amplitude gain and phase shift .
Kubo, R., Statistical Mechanical Theory of Irreversible Processes I, Journal of the Physical Society of Japan, vol. 12, pp. 570–586 (1957).
De Clozeaux,Linear Response Theory, in: E. Antončik et al., Theory of condensed matter, IAEA Vienna, 1968