(→Linearity) |
(→Linearity) |
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== Linearity == | == Linearity == | ||
| − | If a linear system has a response to exp(2jt) of t*exp(-2jt) and a response to exp(-2jt) of t*exp(2jt), then it's response to cos(2t) must be t*cos(2t) | + | If a linear system has a response to exp(2jt) of t*exp(-2jt) and a response to exp(-2jt) of t*exp(2jt), then it's response to cos(2t) must be <math>t*cos(2t)\!</math> |
To look at this in more detail, we must first understand that <math>cos(2t)\!</math> can be expressed as follows: <math> \frac{1}{2}(e^{-2jt}+e^{2jt})\!</math> | To look at this in more detail, we must first understand that <math>cos(2t)\!</math> can be expressed as follows: <math> \frac{1}{2}(e^{-2jt}+e^{2jt})\!</math> | ||
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| + | Based on the given information, we know that the system must be linear. Since it is linear and give the output (shown in paragraph 1), we can concluded that it must have the output: | ||
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| + | <math> \frac{1}{2}(te^{-2jt}+te^{2jt})\!</math>. | ||
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| + | Upon converting the previous output back into a cosine function, we get the output: | ||
| + | |||
| + | <math>cos(2t)\!</math> | ||
Revision as of 16:10, 18 September 2008
Linearity
If a linear system has a response to exp(2jt) of t*exp(-2jt) and a response to exp(-2jt) of t*exp(2jt), then it's response to cos(2t) must be $ t*cos(2t)\! $
To look at this in more detail, we must first understand that $ cos(2t)\! $ can be expressed as follows: $ \frac{1}{2}(e^{-2jt}+e^{2jt})\! $
Based on the given information, we know that the system must be linear. Since it is linear and give the output (shown in paragraph 1), we can concluded that it must have the output:
$ \frac{1}{2}(te^{-2jt}+te^{2jt})\! $.
Upon converting the previous output back into a cosine function, we get the output:
$ cos(2t)\! $
