Concept

Linear response function

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, impulse response or impedance; see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation is closely related. 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 . Consider a damped harmonic oscillator with input given by an external driving force , The complex-valued Fourier transform of the linear response function is given by The amplitude gain is given by the magnitude of the complex number and the phase shift by the arctan of the imaginary part of the function divided by the real one. From this representation, we see that for small the Fourier transform of the linear response function yields a pronounced maximum ("Resonance") at the frequency . The linear response function for a harmonic oscillator is mathematically identical to that of an RLC circuit.

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https://en.wikipedia.org/wiki/Linear_response_function

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Official source

https://en.wikipedia.org/wiki/Linear_response_function

About this result

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An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC. The circuit forms a harmonic oscillator for current, and resonates in a manner similar to an LC circuit. Introducing the resistor increases the decay of these oscillations, which is also known as damping.

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