Function specifying the behavior of a component in an electronic or control system
In
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
Transfer functions are commonly used in the analysis of systems such as
LTI system theory
is an acceptable representation of their input-output behavior.
Continuous-time
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 :
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:
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 .
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
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
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:
Butterworth filter – maximally flat in passband and stopband for the given order
Chebyshev filter (Type I) – maximally flat in stopband, sharper cutoff than a Butterworth filter of the same order
Chebyshev filter (Type II) – maximally flat in passband, sharper cutoff than a Butterworth filter of the same order
group delay
for a given order
Elliptic filter – sharpest cutoff (narrowest transition between passband and stopband) for the given order