A transfer function (also known as the network function) is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a (linear timeinvariant) system. With optical imaging devices, for example, it is the Fourier transform of the point spread function (hence a function of spatial frequency) i.e. the intensity distribution caused by a point object in the field of view.
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The transfer function is commonly used in the analysis of singleinput singleoutput filters, for instance. It is mainly used in signal processing, communication theory, and control theory. The term is often used exclusively to refer to linear, timeinvariant systems (LTI), as covered in this article. Most real systems have nonlinear input/output characteristics, but many systems, when operated within nominal parameters (not "overdriven") have behavior that is close enough to linear that LTI system theory is an acceptable representation of the input/output behavior.
In its simplest form for continuoustime input signal x(t) and output y(t), the transfer function is the linear mapping of the Laplace transform of the input, X(s), to the output Y(s):
or
where H(s) is the transfer function of the LTI system.
In discretetime systems, the function is similarly written as (see Z transform) and is often referred to as the pulsetransfer function.
Consider 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, 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, written as an operator F[r] = u, is the right inverse of L, since L[F[r]] = r.
Solutions of the homogeneous equation L[u] = 0 can be found by trying u = e^{λt}. That substitution yields the characteristic polynomial
The inhomogeneous case can be easily solved if the input function r is also of the form r(t) = e^{st}. In that case, by substituting u = H(s)e^{st} one finds that L[H(s)e^{ st}] = e^{st} if and only if
Taking that as the definition of the transfer function^{[1]} requires to carefully disambiguate between complex vs. real values, which is traditionally influenced by the interpretation of abs(H(s)) as the gain and atan(H(s)) as the phase lag.
Let be the input to a general linear timeinvariant system, and be the output, and the bilateral Laplace transform of and be
Then the output is related to the input by the transfer function as
and the transfer function itself is therefore
In particular, if a complex harmonic signal with a sinusoidal component with amplitude , angular frequency and phase
is input to a linear timeinvariant system, then the corresponding component in the output is:
Note that, in a linear timeinvariant system, the input frequency has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system. The frequency response describes this change for every frequency in terms of gain:
and phase shift:
The phase delay (i.e., the frequencydependent amount of delay introduced to the sinusoid by the transfer function) is:
The group delay (i.e., the frequencydependent 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 which is only a special case of the bilateral Laplace transform for the case where s = jω.
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. Typical infinite impulse response filters are designed to implement one of these special transfer functions.
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 multipleinput multipleoutput (MIMO) systems, and has been largely supplanted by state space representations for such systems. In spite of this, a transfer matrix can be always 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.
