| (2 intermediate revisions by the same user not shown) | |||
| Line 5: | Line 5: | ||
<math>e^{2jt}\!</math> through a system produces <math>e^{-2jt}\!</math>, and <math>e^{-2jt}\!</math> produces <math>e^{2jt}\!</math>. what is the output of <math>cos(2t)\!</math> | <math>e^{2jt}\!</math> through a system produces <math>e^{-2jt}\!</math>, and <math>e^{-2jt}\!</math> produces <math>e^{2jt}\!</math>. what is the output of <math>cos(2t)\!</math> | ||
| + | Answer: | ||
| − | <math>cos(2t)\!</math> can also be written as <math>(e^{2jt} + e^{-2jt})/2\!</math> | + | <math>cos(2t)\!</math> can also be written as <math>(e^{2jt} + e^{-2jt})/2\!</math> which can also be written as <math>1/2*[(e^{2jt} + e^{-2jt})]\!</math> so therefore the linear system's response is <math>t/2*[(e^{-2jt} + e^{2jt})]\!</math> which equals <math>t*cos(2t)\!</math>. |
| + | |||
| + | (Note: The star in this case is the multiplication operator, not the convolution operator) | ||
Latest revision as of 17:18, 18 September 2008
Basics of Linearity
Given:
$ e^{2jt}\! $ through a system produces $ e^{-2jt}\! $, and $ e^{-2jt}\! $ produces $ e^{2jt}\! $. what is the output of $ cos(2t)\! $
Answer:
$ cos(2t)\! $ can also be written as $ (e^{2jt} + e^{-2jt})/2\! $ which can also be written as $ 1/2*[(e^{2jt} + e^{-2jt})]\! $ so therefore the linear system's response is $ t/2*[(e^{-2jt} + e^{2jt})]\! $ which equals $ t*cos(2t)\! $.
(Note: The star in this case is the multiplication operator, not the convolution operator)
