In engineering, transfer function (also known as system function, transfer function or network function, and the curve drawn is called transfer curve) is a mathematical expression used to fit or describe the input-output relationship of black box model (system). Usually, it is the input-output relationship of linear time-invariant system (LTI) with space or time frequency as variable under zero initial condition and zero equilibrium point. However, in some data sources, the input and output characteristics of some physical quantities are directly expressed by "transfer function" without using the result of transformation to S plane (for example, the output voltage in a two-port network is taken as a function of the input voltage).

Transfer function is usually used to analyze single-input single-output filtering systems, mainly used in signal processing, communication theory and control theory. This term is usually used exclusively for linear time-invariant systems (LTI) described herein. The actual system basically has nonlinear input-output characteristics, but the running state of many systems is very close to linear within the range of nominal parameters, so the input-output behavior can be expressed by linear time-invariant system theory in practical application.

Simply put, the following descriptions are all based on complex numbers. In many applications, definition (then) is enough, thus simplifying Laplace transform of complex parameters to Fourier transform of real parameters.

Then, for the simplest continuous-time input signal and output signal, the transfer function reflects the linear mapping relationship between the Laplace transform of the input signal and the Laplace transform of the output signal under the condition of zero state:

or

In discrete-time systems, by applying the Z transform, the transfer function can be similarly expressed as follows

This is often called the pulse transfer function.

Derived directly from differential equations

Consider a linear differential equation with constant coefficients.

Where U and R are truly smooth functions of T, and L is an operator defined in the correlation function space, which transforms U into R. This equation can be used to constrain the output function U with the forced function R as a variable. The transfer function is written in the form of an operator, which is the right inverse of l, because.

The solution of this homogeneous differential equation with constant coefficients can be found by trying. This substitution will produce a characteristic polynomial.

When the form of the input function r is also 0, the non-homogeneous situation can also be easily solved. In this case, it can be found by substitution if and only if.

As a definition of transfer function, we should pay attention to the distinction between real numbers and complex numbers. This is influenced by the convention that abs(H(s)) stands for gain and -atan(H(s)) stands for phase lag. Other definitions of transfer functions include, for example.