Transfer function

Dimensions and units of the transfer function model the output response of the device for a range of possible inputs. The transfer function of a two-port electronic circuit, such as an amplifier, 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 actuator might be the mechanical displacement of the movable arm as a function of electric current applied to the device; the transfer function of a photodetector might be the output voltage as a function of the luminous intensity of incident light of a given wavelength.

The term transfer function is also used in the frequency domain analysis of systems using transform methods, such as the Laplace transform; it is the amplitude of the output as a function of the frequency of the input signal. The transfer function of an electronic filter is the amplitude at the output as a function of the frequency of a constant amplitude sine wave applied to the input. For optical imaging devices, the optical transfer function is the Fourier transform of the point spread function (a function of spatial frequency).

Linear time-invariant systems

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, ${\displaystyle s=\sigma +j\cdot \omega }$. In many applications it is sufficient to set ${\displaystyle \sigma =0}$ (thus ${\displaystyle s=j\cdot \omega }$), 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 ${\displaystyle x(t)}$ and output ${\displaystyle y(t)}$, dividing the Laplace transform of the output, ${\displaystyle Y(s)=\mathcal{L}\left\{y(t)\right\}}$, by the Laplace transform of the input, ${\displaystyle X(s)=\mathcal{L}\left\{x(t)\right\}}$, yields the system's transfer function ${\displaystyle H(s)}$:

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

which can be rearranged as:

${\displaystyle Y(s)=H(s)X(s).}$

Discrete-time

Discrete-time signals may be notated as arrays indexed by an integer ${\displaystyle n}$ (e.g. ${\displaystyle x[n]}$ for input and ${\displaystyle y[n]}$ 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 ${\displaystyle X(z)}$ and ${\displaystyle Y(z)}$), so a discrete-time system's transfer function can be written as:

$${\displaystyle H(z)=\frac{Y(z)}{X(z)}=\frac{\mathcal{Z}\{y[n]\}}{\mathcal{Z}\{x[n]\}}.}$$

Direct derivation from differential equations

A linear differential equation with constant coefficients

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

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 ${\displaystyle F[r]=u}$ that serves as a right inverse of L, meaning that ${\displaystyle L[F[r]]=r}$.

Solutions of the homogeneous constant-coefficient differential equation ${\displaystyle L[u]=0}$ can be found by trying ${\displaystyle u=e^{\lambda t}}$. That substitution yields the characteristic polynomial

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

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

${\displaystyle H(s)=\frac{1}{p_L(s)}\text{wherever }{p}_L(s)\ne 0.}$

Other definitions of the transfer function are used, for example ${\displaystyle 1/p_L(ik).}$

Gain, transient behavior and stability

A general sinusoidal input to a system of frequency ${\displaystyle {\omega }_0/(2\pi )}$ may be written ${\displaystyle \exp (j{\omega }_{0}t)}$. The response of a system to a sinusoidal input beginning at time ${\displaystyle t=0}$ 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:

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

where s_Pi 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 ${\displaystyle H(s)=\frac{1}{s-s_P}}$ where ${\displaystyle s_P={\sigma }_P+j{\omega }_P}$, the Laplace transform of a general sinusoid of unit amplitude will be ${\displaystyle \frac{1}{s-j{\omega }_0}}$. The Laplace transform of the output will be ${\displaystyle \frac{H(s)}{s-j{\omega }_0}}$, and the temporal output will be the inverse Laplace transform of that function:

${\displaystyle g(t)=\frac{e^{j{\omega }_{0}t}-e^{({\sigma }_P+j{\omega }_P)t}}{-{\sigma }_P+j({\omega }_0-{\omega }_P)}}$

The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity if σ_P 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:

${\displaystyle g(\mathrm{\infty })=\frac{e^{j{\omega }_{0}t}}{-{\sigma }_P+j({\omega }_0-{\omega }_P)}}$

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:

${\displaystyle G({\omega }_i)=\left|\frac{1}{-{\sigma }_P+j({\omega }_0-{\omega }_P)}\right|=\frac{1}{\sqrt{{\sigma }_P^2+({\omega }_P-{\omega }_0{)}^2}},}$

which is the absolute value of the transfer function ${\displaystyle H(s)}$ evaluated at ${\displaystyle j{\omega }_i}$. 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

${\displaystyle \sum _{i=0}^n{a}_i{y}^{(i)}+\sum _{j=0}^m{b}_j{u}^{(j)}=0}$

for sinusoidal excitation ${\displaystyle u(t)=\sin (\omega t)}$ can be expressed in terms of its transfer function

${\displaystyle g(s)=\frac{b_m{s}^m+...+b_0}{a_n{s}^n+...+a_0},}$

evaluated at ${\displaystyle s=j\omega }$, i.e. with real part ${\displaystyle \sigma =0}$:

${\displaystyle y(t)=|g(j\omega )|\sin (\omega t+\arg (g(j\omega ))).}$

To show this, use the ansatz function

${\displaystyle y(t)=c{e}^{j\omega t},}$

plug it into the above given differential equation, solve for ${\displaystyle c}$, and note that ${\displaystyle c=g(j\omega )}$.

From the complex identity ${\displaystyle z=|z|e^{j\arg (z)}}$, the argument follows.