The impulse response of a linear time-invariant circuit is h(t) 2 e
–t
u(t) and the input to
the circuit is x(t) e
–2t
u(t). The circuit was initially relaxed. Find its output by the convo-
lution x(t)*h(t). Also, find the convolution h(t)*x(t) and verify that it is the same as x(t)*h(t).
SOLUTION
Since the circuit was initially relaxed, the total response is given by the zero-state
response itself. The zero-state response to any input is obtained by convolving the input
source function with the impulse response. Therefore, the output y(t) is obtained as
The limits of integration were changed from –∞to 0 ...
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