# Transfer function

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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.

### Background

Take a complex harmonic signal with a sinusoidal component with amplitude Ain, angular frequency ω and phase pin

x(t) = Ain * exp(j * (ωt + pin))

(where j 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:

y(t) = Aout * exp(j * (ωt + pout)).

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',

Aout/Ain = | H(jω) |

and 'Phase shift'

pout - pin = arg( H(jω) ).