A transfer function (also known as the system function^{[1]} or 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.
Contents
Explanation
The transfer functions are 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.
Direct derivation from differential equations
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.
Full article ▸
