# Linear shift invariant system description by the step response function

Applying of the step response function in order to characterize the property of the linear shift invariant system is a very often used method.

Definition: Output response of a linear shift invariant system by the Heavyside unit step function is called step response function:
Heavyside unit step function

Consequently the output response function:

is called step response function.

It is possible to see, the relation between the step response function and the transfer function in the extended complex frequency domain is as follow:
, where the transfer function was derived by the step response function.

Furthermore, next question is arisen: step response function of a linear shift invariant system is known in the real parameter space. How is it possible to get the output response for general step function.

As it is known from previously:

Let’s execute Laplace transformation of both sides:

Apply the operation rules of Laplace transformation

,where is the weak derivative.

The output response function can be determined by the step response function as follow:

Apply the convolution rule for the product functions:

, where means the regular derivative.

The output response determined by step response function is called Duhamel-theorem.