convolution: x(t) = u(t)-2u(t-2)+u(t-5), h(t) = e^(2t) u(1-t

dimon

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Jul 9, 2006
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I am having a problem convolving when a function is constructed of several separate functions.

x(t) = u(t)-2u(t-2)+u(t-5)
h(t) = e^(2t) u(1-t)

Find x(t)*h(t)

Can someone show me how to set this up?
Any help is appreciated
 
Re: convolution: x(t) = u(t)-2u(t-2)+u(t-5), h(t) = e^(2t) u

\(\displaystyle \L (x*h) (t) = \int_0^t x(s) h(t-s) ds\)
\(\displaystyle \L = \int_0^t \left( u(s)-2u(s-2)+u(s-5) \right) \left( e^{2s} u(1-s) \right) ds\)

It looks complicated if you attempt to evaluate this integral. Since I see the step function u there, I bet you are expected to use Laplace transforms. Remember,

\(\displaystyle \L \mathcal{L} ( x * h) (s) = X(s) H(s)\)
where X and H are the transforms of x and h. You can find those easily, multiply, and use inverse Laplace transform to get x*h.
 
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