Linear response function - AbsoluteAstronomy.com

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

Susceptibility

*In physics, the susceptibility of a material or substance describes its response to an applied field. There are many kinds of susceptibilities, for example:These two susceptibilities are particular examples of a linear response function;...

, impulse response

Impulse response

In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...

or impedance

Electrical impedance

Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...

, see also transfer function

Transfer function

A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...

. The concept of a Green's function

Green's function

In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...

or fundamental solution

Fundamental solution

In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...

of an ordinary differential equation is closely related.

Mathematical definition

Denote the input of a system by

, and the response of the system by

. 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

This expression is the leading order term of a Volterra-expansion

Volterra Series

The Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series in its ability to capture 'memory' effects. The Taylor series can be used to approximate the response of a nonlinear system to a given input if the output of this system depends...

. 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

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

Example

Consider the damped harmonic oscillator, which gets an external driving by the input

The Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

of the linear response function is given as

From this representation, we see that the Fourier transform

of the linear response function attains a maximum for

: The damped harmonic oscillator acts as a band pass filter.

Mathematical definition

Example

See also