# Transfer function

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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 time-invariant 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 single-input single-output 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, time-invariant systems (LTI), as covered in this article. Most real systems have non-linear input/output characteristics, but many systems, when operated within nominal parameters (not "over-driven") 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 continuous-time 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 discrete-time systems, the function is similarly written as $H(z) = \frac{Y(z)}{X(z)}$ (see Z transform) and is often referred to as the pulse-transfer 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.