Fig.4.8 The connection between the step and impulse response

Let’s consider -function in the form of the following unit impulse which is applied to some system at . We may change this impulse to two step functions:

- applied at ;

- applied with time shift .

Then we obtain the output signal:

(4.36)

If we will increase and simultaneously reduce in order to have the impulse area equal to 1, i.e. , then we may obtain:

(4.37)

Using the Laplace transform we may establish the relation between transfer function of an operational block and its weighting function:

(4.38)

Using the Carson transform we may establish the relation between the transfer function of a operational block and its transient function:

(4.39)

If we have an arbitrary input signal which is applied to the system input at then we may obtain transient functions at the system output using the DuHamel-Carson integral:

(4.40)

(4.41)

where - auxiliary integration time ().