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). \ $

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Answer 1

$ 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

Answer 2

Write it here.

Answer 3

Write it here.

Back to ECE301 Spring 2011 Prof. Boutin

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva