Linear response function

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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 or impedance. The exposition of linear response theory can be found in the seminal paper by Ryogo Kubo.^[1]

[edit] Mathematical definition

Denote the input of a system by $i (t)$ , and the output by $o (t)$ . Generally, the value of $o (t)$ will depend not only on the present value of $i (t)$ , but also on past values. Approximately $o (t)$ is a weighted sum of the previous values of $i (t)$ , with the weights given by the linear response function $χ(t)$

$o(t)\approx\int_{-\infty}^0 d\tau\, \chi(\tau)i(t-\tau)$ .

This formulae is actually the leading order term of a Volterra-expansion. If the system in question is highly non-linear, higher order terms become important and the signal transducer can not adequetly described just by its linear response function.

The Fourier transform $\tilde{\chi}(\omega)$ of the linear response function is very useful as it describes the output of the system if the input is a sine wave $i (t) = i 0 s i n (ω t)$ with frequency $ω$ . The output reads $o(t)=|\tilde{\chi}(\omega)| i_0 sin(\omega t+\arg\tilde{\chi}(\omega))$ with amplitude gain $|\tilde{\chi}(\omega)|$ and phase shift $\arg\tilde{\chi}(\omega)$ .

[edit] An example

Consider the damped harmonic oscillator, which gets an external driving by the input $i (t)$

$\ddot{o}(t)+\gamma \dot{o}(t)+\omega_0^2 o(t)=i(t)$ .

The Fourier transform of the linear response function is given as

$\tilde{\chi}(\omega) = \frac{1}{\omega_0^2-\omega^2+i\gamma\omega}.$

From this representation, we see that the Fourier transform $\tilde{\chi}(\omega)$ of the linear response function attains a maximum for $\omega\approx\omega_0$ : The damped harmonic oscillator acts as a band pass filter.