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 signal processing and control theory.

Signal Processing

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

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

(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:

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

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:

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

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,

$\tau _{G}={\begin{matrix}{\frac {d(\arg(H(i\omega )))}{d\omega }}\end{matrix}}$.

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.

See also

• frequency response