# Practice Question on Computing the Output of an LTI system by Convolution

The unit impulse response h(t) of a DT LTI system is

$h(t)= e^{-3t }u(t). \$

Use convolution to compute the system's response to the input

$x(t)= u(t). \$

$y(t)=x(t)*h(t)=\int_{-\infty}^\infty u(\tau)e^{-3(t-\tau)}u(t-\tau)d\tau=e^{-3t}\int_0^\infty e^{3\tau}u(t-\tau)d\tau=\Bigg(e^{-3t}\int_0^t e^{3\tau}d\tau\Bigg)u(t)=\Bigg(\frac{1}{3}e^{-3t}\bigg(e^{3t}-1\bigg)\Bigg)u(t)$

$y(t)=\Bigg(\frac{1}{3}-\frac{e^{-3t}}{3}\Bigg)u(t)$

--Cmcmican 21:00, 4 February 2011 (UTC)

• Instructor's comments: yes, that it! You really got the hang of it. That's exactly how you should answer a question like this on the test. -pm