Editing Transfer function (section)

== Linear time-invariant systems ==
Transfer functions are commonly used in the analysis of systems such as [[single-input single-output]] [[Filter (signal processing)|filter]]s in [[signal processing]], [[communication theory]], and [[control theory]]. The term is often used exclusively to refer to [[linear time-invariant]] (LTI) systems. Most real systems have [[non-linear]] input-output characteristics, but many systems operated within nominal parameters (not over-driven) have behavior close enough to linear that [[LTI system theory]] is an acceptable representation of their input-output behavior.

=== Continuous-time ===
Descriptions are given in terms of a [[complex variable]], <math>s = \sigma + j \cdot \omega</math>. In many applications it is sufficient to set <math>\sigma=0</math> (thus <math>s = j \cdot \omega</math>), which reduces the [[Laplace transform]]s with complex arguments to [[Fourier transform]]s with the real argument ω. This is common in applications primarily interested in the LTI system's steady-state response (often the case in [[signal processing]] and [[communication theory]]), not the fleeting turn-on and turn-off [[transient response]] or stability issues.

For [[continuous-time]] input signal <math>x(t)</math> and output <math>y(t)</math>, dividing the Laplace transform of the output, <math>Y(s) = \mathcal{L}\left\{y(t)\right\}</math>, by the Laplace transform of the input, <math>X(s) = \mathcal{L}\left\{x(t)\right\}</math>, yields the system's transfer function <math>H(s)</math>:
:<math> H(s) = \frac{Y(s)}{X(s)} = \frac{ \mathcal{L}\left\{y(t)\right\} }{ \mathcal{L}\left\{x(t)\right\} } </math>
which can be rearranged as:
:<math> Y(s) = H(s)\;X(s) \, . </math>

=== Discrete-time ===
{{See also|Z-transform#Linear constant-coefficient difference equation}}
[[Discrete-time]] signals may be notated as arrays indexed by an [[integer]] <math>n</math> (e.g. <math>x[n]</math> for input and <math>y[n]</math> for output). Instead of using the Laplace transform (which is better for continuous-time signals), discrete-time signals are dealt with using the [[z-transform]] (notated with a corresponding capital letter, like <math>X(z)</math> and <math>Y(z)</math>), so a discrete-time system's transfer function can be written as:

<math display="block">H(z) = \frac{Y(z)}{X(z)} = \frac{\mathcal{Z}\{y[n]\}}{\mathcal{Z}\{x[n]\}}.</math>

=== Direct derivation from differential equations ===
A [[linear differential equation]] with constant coefficients

:<math> 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) </math>

where ''u'' and ''r'' are suitably smooth functions of ''t'', has ''L'' as 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 <math>F[r] = u </math> that serves as a right inverse of ''L'', meaning that  <math>L[F[r]] = r</math>.

Solutions of the homogeneous [[Linear differential equation#Homogeneous equations with constant coefficients|constant-coefficient differential equation]] <math>L[u] = 0</math> can be found by trying <math>u = e^{\lambda t}</math>. That substitution yields the [[Characteristic equation (calculus)|characteristic polynomial]]

:<math> p_L(\lambda) = \lambda^n + a_1\lambda^{n-1} + \dotsb + a_{n-1}\lambda + a_n\,</math>

The inhomogeneous case can be easily solved if the input function ''r'' is also of the form <math>r(t) = e^{s t}</math>. By substituting <math>u = H(s)e^{s t}</math>, <math>L[H(s) e^{s t}] = e^{s t}</math> if we define

:<math>H(s) = \frac{1}{p_L(s)} \qquad\text{wherever }\quad p_L(s) \neq 0.</math>

Other definitions of the transfer function are used, for example <math>1/p_L(ik) .</math><ref>{{cite book |title= Ordinary differential equations|last= Birkhoff |first= Garrett|author2=Rota, Gian-Carlo |year=1978|publisher=John Wiley & Sons |location= New York|isbn= 978-0-471-05224-1}}{{page needed|date=April 2013}}</ref>

=== Gain, transient behavior and stability ===

A general sinusoidal input to a system of frequency <math> \omega_0 / (2\pi)</math> may be written <math>\exp( j \omega_0 t )</math>. The response of a system to a sinusoidal input beginning at time <math>t=0</math> 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 [[differential equation]]. The transfer function for an LTI system may be written as the product:

:<math>
H(s)=\prod_{i=1}^N \frac{1}{s-s_{P_i}}
</math>

where ''s<sub>P<sub>i</sub></sub>'' are the ''N'' roots of the characteristic polynomial and will be the [[Pole (complex analysis)|poles]] of the transfer function. In a transfer function with a single pole <math>H(s)=\frac{1}{s-s_P}</math> where <math>s_P = \sigma_P+j \omega_P</math>, the Laplace transform of a general sinusoid of unit amplitude will be <math>\frac{1}{s-j\omega_i}</math>. The Laplace transform of the output will be <math>\frac{H (s)}{s-j \omega_0}</math>, and the temporal output will be the inverse Laplace transform of that function:

:<math>
g(t)=\frac{e^{j\,\omega_0\,t}-e^{(\sigma_P+j\,\omega_P)t}}{-\sigma_P+j (\omega_0-\omega_P)}
</math>

The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity if ''σ<sub>P</sub>'' is positive. 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:

:<math>
g(\infty)=\frac{e^{j\, \omega_0\,t}}{-\sigma_P+j (\omega_0-\omega_P)}
</math>

The [[frequency response]] (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:

:<math>
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}},
</math>

which is the absolute value of the transfer function <math> H(s) </math> evaluated at <math> j\omega_i </math>. This result is valid for any number of transfer-function poles.