# Transfer function

In engineering, a **transfer function** (also known as **system function**^{[1]} or **network function**) of a system, sub-system, or component is a mathematical function which theoretically models the system's output for each possible input.^{[2]}^{[3]}^{[4]} They are widely used in electronics and control systems. In some simple cases, this function is a two-dimensional graph of an independent scalar 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 block diagram technique, in electronics and control theory.

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 two-port electronic circuit like 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 electrical 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; here it means the amplitude of the output as a function of the frequency of the input signal. For example, the transfer function of an electronic filter is the voltage 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 (hence a function of spatial frequency).

Transfer functions are commonly used in the analysis of systems such as single-input single-output filters in the fields of 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, when operated within nominal parameters (not "over-driven") have behavior close enough to linear that LTI system theory is an acceptable representation of the input/output behavior.

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. 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:

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:

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

The phase delay (i.e., the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is:

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:

In control engineering and control theory the transfer function is derived using the Laplace transform.

The transfer function was the primary tool used in classical control engineering. However, it has proven to be unwieldy for the analysis of multiple-input multiple-output (MIMO) systems, and has been largely supplanted by state space representations for such systems.^{[citation needed]} In spite of this, a transfer matrix 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 state space and transfer function methods was proposed by Howard H. Rosenbrock and is referred to as Rosenbrock system matrix.

In optics, modulation transfer function 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

where modulation (M) is computed from the following image or light brightness:

In imaging, transfer functions are used to describe the relationship between the scene light, the image signal and the displayed light.

Transfer functions do not properly exist for many non-linear systems. For example, they do not exist for relaxation oscillators;^{[6]} however, describing functions can sometimes be used to approximate such nonlinear time-invariant systems.