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 ω and phase p_in

x(t) = A_in * exp(i * (ωt + 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:

y(t) = A_out * exp(i * (ωt + 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:

A_in/A_out = | H(iω) |

and

p_in - p_out = arg( H(iω) ).

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