Transfer function

A transfer function is a mathematical representation of the relation between the input and output of a (linear time-invariant) system.

Explanation
The transfer function is commonly used in the analysis of single-input single-output analog electronic circuits, for instance. It is mainly used in signal processing, communication theory, and control theory. The term is often used exclusively to refer to linear, time-invariant systems (LTI), as covered in this article. Most real systems have non-linear input/output characteristics, but many systems, when operated within nominal parameters (not "over-driven") have behavior that is close enough to linear that LTI system theory is an acceptable representation of the input/output behavior.

In its simplest form for continuous-time input signal $$x(t)$$ and output $$y(t)$$, the transfer function is the linear mapping of the Laplace transform of the input, $$X(s)$$, to the output $$Y(s)$$:


 * $$ Y(s) = H(s)\;X(s) $$

or
 * $$ H(s) = \frac{Y(s)}{X(s)} = \frac{ \mathcal{L}\left\{y(t)\right\} }{ \mathcal{L}\left\{x(t)\right\} } $$

where $$H(s)$$ is the transfer function of the LTI system.

In discrete-time systems, the function is similarly written as $$H(z) = \frac{Y(z)}{X(z)}$$ (see Z transform).

Signal processing
Let $$ x(t) \ $$ be the input to a general linear time-invariant system, and $$ y(t) \ $$ be the output, and the Laplace transform of $$ x(t) \ $$ and $$ y(t) \ $$ be


 * $$ X(s) =  \mathcal{L}\left \{ x(t) \right \} \ \stackrel{\mathrm{def}}{=}\  \int_{-\infty}^{\infty} x(t) e^{-st}\, dt  $$


 * $$ Y(s) =   \mathcal{L}\left \{ y(t) \right \} \ \stackrel{\mathrm{def}}{=}\  \int_{-\infty}^{\infty} y(t) e^{-st}\, dt $$.

Then the output is related to the input by the transfer function $$ H(s) \ $$ as


 * $$ Y(s) = H(s) X(s) \, $$

and the transfer function itself is therefore


 * $$ H(s) = \frac{Y(s)} {X(s)} $$.

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


 * $$ x(t) = Xe^{j\omega t} = |X|e^{j(\omega t + \arg(X))} $$


 * where $$ X = |X|e^{j\arg(X)} $$

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


 * $$ y(t) = Ye^{j\omega t} = |Y|e^{j(\omega t + \arg(Y))} $$


 * and $$ Y = |Y|e^{j\arg(Y)} $$.

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


 * $$G(\omega) = \frac{|Y|}{|X|} = |H(j \omega)| \ $$

and phase shift:


 * $$\phi(\omega) = \arg(Y) -  \arg(X) = \arg( H(j \omega))$$.

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


 * $$\tau_{\phi}(\omega) = -\begin{matrix}\frac{\phi(\omega)}{\omega}\end{matrix}$$.

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 computing the derivative of the phase shift with respect to angular frequency $$ \omega \ $$,


 * $$\tau_{g}(\omega) = -\begin{matrix}\frac{d\phi(\omega)}{d\omega}\end{matrix}$$.

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

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

The transfer function was the primary tool used in classical control engineering. However, it has proven to be unwieldy for the analysis of multiple-input multiple-output (MIMO) systems, and has been largely supplanted by state space representations for such systems. In spite of this, a transfer matrix can be always obtained for any linear system, in order to analyze its dynamics and other properties: each element of a transfer matrix is a transfer function relating a particular input variable to an output variable.