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 concept of a Greens function or fundamental solution of an ordinary differential equation is closely related. The exposition of linear response theory can be found in the seminal paper by Ryogo Kubo.^[1]

1 Mathematical definition
2 An example
3 References
4 See also

[edit] Mathematical definition

Denote the input of a system by $h (t)$ , and the response of the system by $o (t)$ . Generally, the value of $o (t)$ will depend not only on the present value of $h (t)$ , but also on past values. Approximately $o (t)$ is a weighted sum of the previous values of $h (t')$ , with the weights given by the linear response function $χ(t - t')$ , which is understood to be zero for $t < t'$ because of causality

$o(t)\approx\int_{-\infty}^{\infty} dt'\, \chi(t-t')h(t')$ .

This formula 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 adequately be 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.