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Monday, 25 March 2013

Convolution Product


One of the most common mistake students will commit is
displaymath44
Although it is tempting to assume that this is true, one may easily check that it is wrong by taking f(t) = t, and g(t) = t. Therefore, a natural question is: what would tex2html_wrap_inline50 be equal to ?
The answer to this question is given by the formula
displaymath52
where
displaymath54,
is called the convolution product of f(t), and g(t). The following are some of the basic properties of the convolution product:
(1)
tex2html_wrap_inline60 ;
(2)
tex2html_wrap_inline62 ;
(3)
tex2html_wrap_inline64;
(4)
tex2html_wrap_inline66 ;
(5)
tex2html_wrap_inline68.
Example: Find the solution to
displaymath70
Solution: Apply the Laplace transform to get
displaymath72
where tex2html_wrap_inline74 . Hence,
displaymath76
We then rewrite Y(s) to get
displaymath80
Therefore, we have
displaymath82
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