Transfer function

In engineering, a transfer function (also known as system function or network function) of an electronic or  is a  which  the device's output for each possible input. In its simplest form, this function is a two-dimensional of an independent  input versus the dependent scalar output, called a transfer curve or characteristic curve. Transfer functions for components are used to design and analyze systems assembled from components, particularly using the technique, in electronics and.

The dimensions and units of the transfer function model the output response of the device for a range of possible inputs. For example, the transfer function of a electronic circuit like an  might be a two-dimensional graph of the scalar voltage at the output as a function of the scalar voltage applied to the input; the transfer function of an electromechanical  might be the mechanical displacement of the movable arm as a function of electrical current applied to the device; the transfer function of a  might be the output voltage as a function of the  of incident light of a given wavelength.

The term "transfer function" is also used in the analysis of systems using transform methods such as the ; here it means the  of the output as a function of the  of the input signal. For example, the transfer function of an is the voltage amplitude at the output as a function of the frequency of a constant amplitude  applied to the input. For optical imaging devices, the is the  of the  (hence a function of ).

Linear time-invariant systems
Transfer functions are commonly used in the analysis of systems such as s in the fields of, , and. The term is often used exclusively to refer to (LTI) systems. Most real systems have input/output characteristics, but many systems, when operated within nominal parameters (not "over-driven") have behavior close enough to linear that  is an acceptable representation of the input/output behavior.

The descriptions below are given in terms of a complex variable, $$s = \sigma + j \cdot \omega$$, which bears a brief explanation. In many applications, it is sufficient to define $$\sigma=0$$ (thus $$s = j \cdot \omega$$), which reduces the s with complex arguments to s with real argument ?. The applications where this is common are ones where there is interest only in the steady-state response of an LTI system, not the fleeting turn-on and turn-off behaviors or stability issues. That is usually the case for and.

Thus, for input signal $$x(t)$$ and output $$y(t)$$, the transfer function $$H(s)$$ is the linear mapping of the Laplace transform of the input, $$X(s) = \mathcal{L}\left\{x(t)\right\}$$, to the Laplace transform of the output $$Y(s) = \mathcal{L}\left\{y(t)\right\}$$:
 * $$ Y(s) = H(s)\;X(s) $$

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

In systems, the relation between an  input signal $$x(t)$$ and output $$y(t)$$ is dealt with using the, and then the transfer function is similarly written as $$H(z) = \frac{Y(z)}{X(z)}$$  and this is often referred to as the pulse-transfer function.

Direct derivation from differential equations
Consider a with constant coefficients


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

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 can be used to define an operator $$F[r] = u $$ that serves as a right inverse of L, meaning that $$L[F[r}} = r$$.

Solutions of the homogeneous, $$L[u] = 0$$ can be found by trying $$u = e^{\lambda t}$$. That substitution yields the


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

The inhomogeneous case can be easily solved if the input function r is also of the form $$r(t) = e^{s t}$$. In that case, by substituting $$u = H(s)e^{s t}$$ one finds that $$L[H(s) e^{s t}] = e^{s t}$$ if we define


 * $$H(s) = \frac{1}{p_L(s)} \qquad\text{wherever }\quad p_L(s) \neq 0.$$

Taking that as the definition of the transfer function requires careful disambiguation between complex vs. real values, which is traditionally influenced by the interpretation of abs(H(s)) as the and -atan(H(s)) as the. Other definitions of the transfer function are used: for example $$1/p_L(ik) .$$

Gain, transient behavior and stability
A general sinusoidal input to a system of frequency $$ \omega_0 / (2\pi)$$ may be written $$\exp( j \omega_0 t )$$. The response of a system to a sinusoidal input beginning at time $$t=0$$ will consist of the sum of the steady-state response and a transient response. The steady-state response is the output of the system in the limit of infinite time, and the transient response is the difference between the response and the steady state response (It corresponds to the homogeneous solution of the above differential equation.) The transfer function for an LTI system may be written as the product:



H(s)=\prod_{i=1}^N \frac{1}{s-s_{P_i}} $$

where sP i  are the N roots of the characteristic polynomial and will therefore be the of the transfer function. Consider the case of a transfer function with a single pole $$H(s)=\frac{1}{s-s_P}$$ where $$s_P = \sigma_P+j \omega_P$$. The Laplace transform of a general sinusoid of unit amplitude will be $$\frac{1}{s-j\omega_i}$$. The Laplace transform of the output will be $$\frac{H (s)}{(s-j \omega_0)}$$ and the temporal output will be the inverse Laplace transform of that function:



g(t)=\frac{e^{j\,\omega_0\,t}-e^{(\sigma_P+j\,\omega_P)t}}{-\sigma_P+j (\omega_0-\omega_P)} $$

The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity if &sigma;P is positive. In order for a system to be stable, its transfer function must have no poles whose real parts are positive. If the transfer function is strictly stable, the real parts of all poles will be negative, and the transient behavior will tend to zero in the limit of infinite time. The steady-state output will be:



g(\infty)=\frac{e^{j\, \omega_0\,t}}{-\sigma_P+j (\omega_0-\omega_P)} $$

The (or "gain") G of the system is defined as the absolute value of the ratio of the output amplitude to the steady-state input amplitude:



G(\omega_i)=\left|\frac{1}{-\sigma_P+j (\omega_0-\omega_P)}\right|=\frac{1}{\sqrt{\sigma_P^2+(\omega_P-\omega_0)^2}} $$

which is just the absolute value of the transfer function $$ H(s) $$ evaluated at $$ j\omega_i $$. This result can be shown to be valid for any number of transfer function poles.

Signal processing
Let $$ x(t) \ $$ be the input to a general, and $$ y(t) \ $$ be the output, and the of $$ x(t) \ $$ and $$ y(t) \ $$ be


 * $$\begin{align}

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. \end{align}$$

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   with a  component with  $$|X| \ $$,  $$\omega \ $$ and  $$\arg(X) \ $$, where arg is the


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


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

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


 * $$\begin{align}

y(t) &= Ye^{j\omega t} = |Y|e^{j(\omega t + \arg(Y))}, \\ Y &= |Y|e^{j\arg(Y)}. \end{align}$$

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 $$ H(j \omega) \ $$ 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 (i.e., the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is:


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

The (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 $$ \omega \ $$,


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

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

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.

Some common transfer function families and their particular characteristics are:


 * – maximally flat in passband and stopband for the given order
 * – maximally flat in stopband, sharper cutoff than a Butterworth filter of the same order
 * – maximally flat in passband, sharper cutoff than a Butterworth filter of the same order
 * – best pulse response for a given order because it has no group delay ripple
 * – sharpest cutoff (narrowest transition between pass band and stop band) for the given order
 * – minimum group delay; gives no overshoot to a step function
 * – minimum group delay; gives no overshoot to a step function

Control engineering
In and  the transfer function is derived using the.

The transfer function was the primary tool used in classical control engineering. However, it has proven to be unwieldy for the analysis of systems, and has been largely supplanted by  representations for such systems. In spite of this, a can always be 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.

A useful representation bridging and transfer function methods was proposed by  and is referred to as.

Optics
In optics, 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:


 * $$ M = \frac{L_{\max} - L_{\min} } {L_{\max} + L_{\min}}. $$

Non-linear systems
Transfer functions do not properly exist for many. For example, they do not exist for s; however, s can sometimes be used to approximate such nonlinear time-invariant systems.