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 that models the system's output for each possible input.[2][3][4] It is widely used in electronic engineering tools like circuit simulators and control systems. In simple cases, this function can be represented as a two-dimensional graph of an independent scalar input versus the dependent scalar output (known as 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.

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.



Descriptions are given in terms of a complex variable,  . In many applications it is sufficient to set   (thus  ), 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   and output  , dividing the Laplace transform of the output,  , by the Laplace transform of the input,  , yields the system's transfer function  :


which can be rearranged as:




Discrete-time signals may be notated as arrays indexed by an integer   (e.g.   for input and   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   and  ), so a discrete-time system's transfer function can be written as:


Direct derivation from differential equations


A linear differential equation with constant coefficients


where u and r are suitably smooth functions of t, and L is the operator defined on the relevant function space 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   that serves as a right inverse of L, meaning that  .

Solutions of the homogeneous constant-coefficient differential equation   can be found by trying  . That substitution yields the characteristic polynomial


The unhomogeneous case can be easily solved if the input function r is also of the form  . By substituting  ,   if we define


Other definitions of the transfer function are used, for example  [5]

Gain, transient behavior and stability


A general sinusoidal input to a system of frequency   may be written  . The response of a system to a sinusoidal input beginning at time   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:


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   where  , the Laplace transform of a general sinusoid of unit amplitude will be  . The Laplace transform of the output will be  , and the temporal output will be the inverse Laplace transform of that function:


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:


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:


which is the absolute value of the transfer function   evaluated at  . This result is valid for any number of transfer-function poles.

Signal processing


If   is the input to a general linear time-invariant system, and   is the output, and the bilateral Laplace transform of   and   is


The output is related to the input by the transfer function   as


and the transfer function itself is


If a complex harmonic signal with a sinusoidal component with amplitude  , angular frequency   and phase  , where arg is the argument


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


In a linear time-invariant system, the input frequency   has not changed; only the amplitude and phase angle of the sinusoid have been changed by the system. The frequency response   describes this change for every frequency   in terms of gain


and phase shift


The phase delay (the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is


The group delay (the frequency-dependent amount of delay introduced to the envelope of the sinusoid by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency  ,


The transfer function can also be shown using the Fourier transform, a special case of bilateral Laplace transform where  .

Common transfer-function families


Although any LTI system can be described by some transfer function, "families" of special transfer functions are commonly used:

Control engineering


In control engineering and control theory, the transfer function is derived with the Laplace transform. The transfer function was the primary tool used in classical control engineering. A transfer matrix can be obtained for any linear system 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 representation bridging state space and transfer function methods was proposed by Howard H. Rosenbrock, and is known as the Rosenbrock system matrix.



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

Non-linear systems


Transfer functions do not exist for many non-linear systems, such as relaxation oscillators;[6] however, describing functions can sometimes be used to approximate such nonlinear time-invariant systems.

See also



  1. ^ Bernd Girod, Rudolf Rabenstein, Alexander Stenger, Signals and systems, 2nd ed., Wiley, 2001, ISBN 0-471-98800-6 p. 50
  2. ^ M. A. Laughton; D.F. Warne (27 September 2002). Electrical Engineer's Reference Book (16 ed.). Newnes. pp. 14/9–14/10. ISBN 978-0-08-052354-5.
  3. ^ E. A. Parr (1993). Logic Designer's Handbook: Circuits and Systems (2nd ed.). Newness. pp. 65–66. ISBN 978-1-4832-9280-9.
  4. ^ Ian Sinclair; John Dunton (2007). Electronic and Electrical Servicing: Consumer and Commercial Electronics. Routledge. p. 172. ISBN 978-0-7506-6988-7.
  5. ^ Birkhoff, Garrett; Rota, Gian-Carlo (1978). Ordinary differential equations. New York: John Wiley & Sons. ISBN 978-0-471-05224-1.[page needed]
  6. ^ Valentijn De Smedt, Georges Gielen and Wim Dehaene (2015). Temperature- and Supply Voltage-Independent Time References for Wireless Sensor Networks. Springer. p. 47. ISBN 978-3-319-09003-0.
  • ECE 209: Review of Circuits as LTI Systems — Short primer on the mathematical analysis of (electrical) LTI systems.