(New page: To convolve two functions we have the following: <math> y(t) = h(t) * x(t) = \int_{-\infty}^\infty x(t)h(t-\tau)d\tau </math> Plugging in functions for <math>x(t)= e^{-t}u(t)</math> and...)
 
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To convolve two functions we have the following:
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<math>
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y(t) = h(t) * x(t) = \int_{-\infty}^\infty x(t)h(t-\tau)d\tau
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</math>
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Plugging in functions for <math>x(t)= e^{-t}u(t)</math> and <math>h(t)=u(t-1)</math> we get:
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<math>
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= \int_{-\infty}^\infty e^{-\tau}u(\tau)u(t-1-\tau)d\tau
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</math>
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We now flip and shift x(t) and for t>1 we find
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<math>
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= \int_{1}^t e^{-(t-\tau)}d\tau
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</math>
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Evaluating this for t>1
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<math>
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\begin{align}
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&= \int_{1}^{t} e^{-\tau}d\tau \\
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&= -e^{-t} - (-e^{0}) \\
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&= 1 - e^{-(t-1)}
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\end{align}
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</math>
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Latest revision as of 22:21, 1 July 2008

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett