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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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

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