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 linear system theory, signal processing, communications theory, and control theory.

Signal processing

Let <math> x(t) <math> be the input to a general linear time-invariant system, and <math> y(t) <math> be the output, and the Laplace Transform of <math> x(t) <math> and <math> y(t) <math> be

<math> \mathcal{L}\left \{ x(t) \right \} \equiv \int_{-\infty}^{\infty} x(t) e^{-st}\, dt = X(s) <math>

<math> \mathcal{L}\left \{ y(t) \right \} \equiv \int_{-\infty}^{\infty} y(t) e^{-st}\, dt = Y(s) <math>.

Then the output is related to the input by the transfer function <math> H(s) <math>:

<math> H(s) = \frac{Y(s)} {X(s)} <math> .

In particular, if a complex harmonic signal with a sinusoidal component with amplitude <math>|X|<math>, angular frequency <math>\omega<math> and phase <math>\arg(X)<math>

<math> x(t) = |X|e^{j(\omega t + \arg(X))} = Xe^{j\omega t} <math>

where <math> X = |X|e^{j\arg(X)} <math>

is input to a linear time-invariant system, then the corresponding component in the output is:

<math>y(t) = |Y|e^{j(\omega t + \arg(Y))} = Ye^{j\omega t} <math>

and <math> Y = |Y|e^{j\arg(Y)} <math>.

Note that, in a linear time-invariant system, the input frequency <math> \omega \ <math> has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system. The frequency response <math> H(j \omega) \ <math> describes this change for every frequency <math> \omega \ <math> in terms of gain:

<math>G(\omega) = \frac{|Y|}{|X|} = | H(j \omega) |<math>

and phase shift:

<math>\theta(\omega) = \arg(Y) - \arg(X) = \arg( H(j \omega))<math>.

The phase delay (i.e., the frequency-dependent amount of delay to the sinusoid introduced by the transfer function) is:

<math>\tau_{\phi}(\omega) = -\begin{matrix}\frac{\theta(\omega)}{\omega}\end{matrix}<math>.

The group delay (i.e., the frequency-dependent amount of delay to the envelope of the sinusoid introduced by the transfer function) is found by taking the radian frequency derivative of the phase shift,

<math>\tau_{g}(\omega) = -\begin{matrix}\frac{d\theta(\omega)}{d\omega}\end{matrix}<math>.

The transfer function can also be shown using the Fourier transform which is a special case of the bilateral Laplace transform for the case where <math> s = j \omega <math>.

Control engineering

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

Transfer function

Signal processing

Control engineering

See also