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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.

 Table of contents 1 Background 2 Signal Processing 3 Control Engineering

## Signal Processing

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

(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:

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':

and 'Phase shift':

.

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

## Control Engineering

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