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# Transfer function

A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. It is mainly used in (digital) signal processing and control theory.

### Background

Take a complex harmonic signal with a sinusoidal component with amplitude $A_{in}$, angular frequency $\omega$ and phase $p_{in}$

$x(t) = A_{in} e^{i(\omega t + p_{in})}$

(where i represents the imaginary unit) and use it as an input to a linear time-invariant system. The corresponding component in the output will match the following equation:

$x(t) = A_{out} e^{i(\omega t + p_{out})}$

Note that the fundamental frequency ω has not changed, only the amplitude and the phase of the response changed as it went through the system. The transfer function H(z) describes this change for every frequency ω in terms of 'Gain':

$\frac{A_{out}}{A_{in}} = | H(i\omega) |$

and 'Phase shift':

$p_{out} - p_{in} = \arg( H(i\omega))$.

The transfer function can also be derived by using the Fourier transform.

In control engineering and control theory the transfer function is derived using the Laplace transform.

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