Transfer function

In engineering, a transfer function (also known as system function or network function) of an electronic or control system component is a mathematical function which theoretically models the device's output for each possible input. In its simplest form, 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. In engineering, a transfer function (also known as system function or network function) of an electronic or control system component is a mathematical function which theoretically models the device's output for each possible input. In its simplest form, 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 descriptions below are given in terms of a complex variable, s = σ + j ⋅ ω {displaystyle s=sigma +jcdot omega } , which bears a brief explanation. In many applications, it is sufficient to define σ = 0 {displaystyle sigma =0} (thus s = j ⋅ ω {displaystyle s=jcdot omega } ), which reduces the Laplace transforms with complex arguments to Fourier transforms with real argument ω. The applications where this is common are ones where there is interest only in the steady-state response of an LTI system, not the fleeting turn-on and turn-off behaviors or stability issues. That is usually the case for signal processing and communication theory. Thus, for continuous-time input signal x ( t ) {displaystyle x(t)} and output y ( t ) {displaystyle y(t)} , the transfer function H ( s ) {displaystyle H(s)} is the linear mapping of the Laplace transform of the input, X ( s ) = L { x ( t ) } {displaystyle X(s)={mathcal {L}}left{x(t) ight}} , to the Laplace transform of the output Y ( s ) = L { y ( t ) } {displaystyle Y(s)={mathcal {L}}left{y(t) ight}} :