Transfer functions are commonly used in the analysis of systems such as
single-input single-output filters 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, s = \sigma + j \cdot \omega. In many applications it is sufficient to set \sigma=0 (thus s = j \cdot \omega), which reduces the
Laplace transforms with complex arguments to
Fourier transforms 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 x(t) and output y(t), dividing the Laplace transform of the output, Y(s) = \mathcal{L}\left\{y(t)\right\}, by the Laplace transform of the input, X(s) = \mathcal{L}\left\{x(t)\right\}, yields the system's transfer function H(s): : H(s) = \frac{Y(s)}{X(s)} = \frac{ \mathcal{L}\left\{y(t)\right\} }{ \mathcal{L}\left\{x(t)\right\} } which can be rearranged as: : Y(s) = H(s)\;X(s) \, .
Discrete-time Discrete-time signals may be notated as arrays indexed by an
integer n (e.g. x[n] for input and y[n] 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 X(z) and Y(z)), so a discrete-time system's transfer function can be written as: H(z) = \frac{Y(z)}{X(z)} = \frac{\mathcal{Z}\{y[n]\}}{\mathcal{Z}\{x[n]\}}.
Direct derivation from differential equations A
linear differential equation 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, 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 F[r] = u that serves as a right inverse of
L, meaning that L[F[r = r. Solutions of the homogeneous
constant-coefficient differential equation L[u] = 0 can be found by trying u = e^{\lambda t}. That substitution yields the
characteristic polynomial : 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}. By substituting u = H(s)e^{s t}, 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. 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
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
sPi are the
N roots of the characteristic polynomial and will be the
poles of the transfer function. In 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_0}. 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
σP 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: : g(\infty)=\frac{e^{j\, \omega_0\,t}}{-\sigma_P+j (\omega_0-\omega_P)} 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: : 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 the absolute value of the transfer function H(s) evaluated at j\omega_i . This result is valid for any number of transfer-function poles.
Steady state behavior for sinusoidal excitation The steady state behavior of a linear system : \sum_{i=0}^n a_i y^{(i)} + \sum_{j=0}^m b_j u^{(j)} = 0 for sinusoidal excitation u(t) = \sin(\omega t) can be expressed in terms of its transfer function : g(s) = \frac{b_m s^m + ... + b_0}{a_n s^n + ... + a_0}, evaluated at s = j \omega, i.e. with real part \sigma = 0 : : y(t) = |g(j \omega)| \sin(\omega t + \arg(g(j \omega))). To show this, use the ansatz function : y(t) = c e^{j \omega t}, plug it into the above given differential equation, solve for c , and note that c = g(j \omega). From the complex identity z = |z| e^{j \arg(z)}, the argument follows. ==Signal processing==