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.

Background

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

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

(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:

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

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'

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