Editing Transfer function (section)

=== 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.