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Impulse response of RC system

We have seen how we could calculate the impulse response from the unit step response of a system. Now we shall calculate the unit step response, or in general the response to any input signal, given its impulse response. We shall use convolution to obtain the required result.

The unit impulse , when fed into the RC system gives the corresponding impulse response h(t), which in this case is given by

We shall now find the response to the input signal u(t).

The convolution of an input signal x(t) and the impulse response of a system h(t) is given by the formula:

Hence we have now shown how to calculate the impulse response given the unit step response and also any response given the impulse response. Moreover we can now say that given either the unit step response or the impulse response we can calculate the response to any other input signals.

But in this case x(t) = u(t), and so the output signal y(t) will be given by :

Now we have if and only if . In all other cases

Hence the given equation for y(t) will now simplify to :

Solving which we get ,

which is the response to the unit step function.