Electrical Engineering ⇒ Topic : Relation between step response and impulse response

RELATION BETWEEN STEP RESPONSE AND IMPULSE RESPONSE

Let the input to the network in Figure (a) be v_i(t), and the output or response of the network be v_o(t). Let the Laplace transform of v_i(t) and v₀(t) be V_i(s) and V₀(s) respectively.

The ratio of the Laplace transform of output to that of the input, putting initial conditions as zero, is known as transfer function, i.e.

                       Figure (a) Impulse response of a network.

                            ............ (1)

Taking the inverse Laplace transform, we have network.
                          

                                             .................. (2)

                                            .................... (2a)

If v_i(t) be an impulse function, i.e., v_i(t) = δ (t), the Laplace transform of v_i(t) is V_i(s) = 1. Putting V_i(s) = 1, Eq. (1) can be written as

                                  ............ (3)

Taking the inverse Laplace transform, we get

                                  .............. (4)

Since h(t) is the response of the impulse function, it is called the impulse response of the network.Since the transfer function characterizes the network, the impulse response is the characteristic of the network.