I have two convolution problems, that I would like to be sure are right. If someone wouldn't mind the fun job of checking these, that would be great.(adsbygoogle = window.adsbygoogle || []).push({});

QUESTION:

Compute the convolution of [itex] x(t) [/itex] and [itex] h(t) [/itex] where:

1:

[tex] x(t) = u(t) [/tex]

[tex] h(t) = e^{-at}u(t), \,\,\, a>0 [/tex]

2.

[tex] x(t) = u(t) - u(t-T_1) [/tex]

[tex] h(t) = u(t) - u(t-T_2), \,\,\, T_1 > T_2 [/tex]

"ANSWER":

Let [itex] y(t) [/itex] be the convolution of x(t), h(t)

(1):

[tex] t < 0 [/tex]:

y(t) =0

[tex] t \geq 0 [/tex]:

[tex] y(t) = \int_{-\infty}^t e^{-a(t-\tau)}d\tau = \frac{1}{a}[/tex]

(2):

[tex] t < 0 [/tex] or [tex] t-T_2 > 0 [/tex]:

y(t) =0

[tex] t \geq 0 [/tex] and [tex] t-T_2 < 0 [/tex]:

[tex] y(t) = \int_0^t d\tau = t [/tex]

[tex] t-T_2 > 0 [/tex] and [tex] t<T_1 [/tex]:

[tex] y(t) = \int_{t-T_2}^t d\tau = T_2 [/tex]

If I need to show more steps, please let me know. Thanks!

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# Homework Help: Basic Convolution - if someone wouldn't mind checking

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