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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.
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Encyclopedia
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.
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 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
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 .
An example
Consider the damped harmonic oscillator, which gets an external driving
by the input
.
The Fourier transform 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.
See also
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