# Transfer function

A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. The transfer function is commonly used in the analysis of single-input single-output analog circuits. It is mainly used in linear, time-invariant system theory, signal processing, communication theory, and control theory.

In its simplest form for continuous-time signals, the function is often written as

[itex] H(s) = \frac{Y(s)} {X(s)} [itex]

where H(s) is the symbol for the transfer function, Y(s) is the output function, and X(s) is the input function (see Laplace transform). In discrete-time systems, the function is similarly written as [itex]H(z) = {Y(z)}/{X(z)}[itex] (see Z transform).

## Signal processing

Let [itex] x(t) [itex] be the input to a general linear time-invariant system, and [itex] y(t) [itex] be the output, and the Laplace transform of [itex] x(t) [itex] and [itex] y(t) [itex] be

[itex] \mathcal{L}\left \{ x(t) \right \} \equiv \int_{-\infty}^{\infty} x(t) e^{-st}\, dt = X(s) [itex]
[itex] \mathcal{L}\left \{ y(t) \right \} \equiv \int_{-\infty}^{\infty} y(t) e^{-st}\, dt = Y(s) [itex].

Then the output is related to the input by the transfer function [itex] H(s) [itex]:

[itex] Y(s) = H(s) X(s) [itex]
[itex] H(s) = \frac{Y(s)} {X(s)} [itex] .

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

[itex] x(t) = |X|e^{j(\omega t + \arg(X))} = Xe^{j\omega t} [itex]
where [itex] X = |X|e^{j\arg(X)} [itex]

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

[itex]y(t) = |Y|e^{j(\omega t + \arg(Y))} = Ye^{j\omega t} [itex]
and [itex] Y = |Y|e^{j\arg(Y)} [itex].

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

[itex]G(\omega) = \frac{|Y|}{|X|} = | H(j \omega) |[itex]

and phase shift:

[itex]\theta(\omega) = \arg(Y) - \arg(X) = \arg( H(j \omega))[itex].

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

[itex]\tau_{\phi}(\omega) = -\begin{matrix}\frac{\theta(\omega)}{\omega}\end{matrix}[itex].

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,

[itex]\tau_{g}(\omega) = -\begin{matrix}\frac{d\theta(\omega)}{d\omega}\end{matrix}[itex].

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 [itex] s = j \omega [itex].

## 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.

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