# Transfer function

Transfer function

A transfer function (also known as the system function[1] or network function) is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function (hence a function of spatial frequency) i.e. the intensity distribution caused by a point object in the field of view.

## Explanation

Transfer functions are commonly used in the analysis of systems such as single-input single-output filters, typically within the fields of 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) and is often referred to as the pulse-transfer function.

### Direct derivation from differential equations

Consider a linear differential equation with constant coefficients

$L[u] = \frac{d^nu}{dt^n} + a_1\frac{d^{n-1}u}{dt^{n-1}} + \dots + a_{n-1}\frac{du}{dt} + a_nu = r(t)$

where u and r are suitably smooth functions of t, and L is the operator defined on the relevant function space, that transforms u into r. That kind of equation can be used to constrain the output function u in terms of the forcing function r. The transfer function, written as an operator F[r] = u, is the right inverse of L, since L[F[r]] = r.

Solutions of the homogeneous equation L[u] = 0 can be found by trying u = eλt. That substitution yields the characteristic polynomial

$p_L(\lambda) = \lambda^n + a_1\lambda^{n-1} + \dots + a_{n-1}\lambda + a_n\,$

The inhomogeneous case can be easily solved if the input function r is also of the form r(t) = est. In that case, by substituting u = H(s)est one finds that L[H(s)est] = est if and only if

$H(s) = \frac{1}{p_L(s)}, \qquad p_L(s) \neq 0.$

Taking that as the definition of the transfer function[2] requires careful disambiguation between complex vs. real values, which is traditionally influenced by the interpretation of abs(H(s)) as the gain and -atan(H(s)) as the phase lag.

## Signal processing

Let $x(t) \$ be the input to a general linear time-invariant system, and $y(t) \$ be the output, and the bilateral 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) = Xejωt = | X | ejt + arg(X))
where X = | X | ejarg(X)

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

y(t) = Yejωt = | Y | ejt + arg(Y))
and Y = | Y | ejarg(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:

ϕ(ω) = arg(Y) − arg(X) = arg(H(jω)).

The phase delay (i.e., the frequency-dependent amount of delay introduced to the sinusoid 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 introduced to the envelope of the sinusoid 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 = jω.

### Common transfer function families

While any LTI system can be described by some transfer function or another, there are certain "families" of special transfer functions that are commonly used. Typical infinite impulse response filters are designed to implement one of these special transfer functions.

Some common transfer function families and their particular characteristics are:

## 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.

## Optics

In optics, modulation transfer function indicates the capability of optical contrast transmission.

For example, when observing a series of black-white-light fringes drawn with a specific spatial frequency, the image quality may decay. White fringes fade while black ones turn brighter.

The modulation transfer function in a specific spatial frequency is defined by:

$\mathrm{MTF}(f) = \frac{M(\mathrm{image})} {M(\mathrm{source})}$

Where modulation (M) is computed from the following image or light brightness:

$M = \frac{(L_\mathrm{max} - L_\mathrm{min} )} {(L_\mathrm{max} + L_\mathrm{min})}$

## References

1. ^ Bernd Girod, Rudolf Rabenstein, Alexander Stenger, Signals and systems, 2nd ed., Wiley, 2001, ISBN 0471988006 p. 50
2. ^ The transfer function is defined by 1 / pL(ik) in, e.g., Birkhoff, Garrett; Rota, Gian-Carlo (1978). Ordinary differential equations. New York: John Wiley & Sons. ISBN 0-471-05224-8.

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