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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 signal processing and control theory.

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## Signal processing

Take a complex harmonic signal with a sinusoidal component with amplitude Ain, angular frequency ω and phase pin $x(t) = A_\mathrm{in} e^{i(\omega t + p_\mathrm{in})}$

(where i is 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: $x(t) = A_\mathrm{out} e^{i(\omega t + p_\mathrm{out})}$

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: $G(\omega) = \frac{A_\mathrm{out}}{A_\mathrm{in}} = | H(i\omega) |$

and phase shift: $\theta(\omega) = p_\mathrm{out} - p_\mathrm{in} = \arg( H(i\omega))$.

The group delay (i.e., the frequency-dependent amount of delay introduced by the transfer function) is found by taking the radial frequency derivative of the phase shift, $\tau_{G}(\omega) = \begin{matrix}\frac{d\theta(\omega)}{d\omega}\end{matrix}$.

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.