(New page: == Part B: The Basics of Linearity == === Linear System Properties === The System takes the input and creates an output following these guidelines: <math> e^{2jt} \rightarrow System \...) |
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=== Linear System's response to Cos(2t) === | === Linear System's response to Cos(2t) === | ||
| − | + | (Modified after seeing Christen Juzeszyn's answer) | |
Following this particular Linear System's properties: | Following this particular Linear System's properties: | ||
| − | <math> cos(2t) \rightarrow System \rightarrow tcos( | + | Using Euler's formula for cos(2t): |
| + | |||
| + | <math> \frac{1}{2}e^{2jt} + \frac{1}{2}e^{-2jt} = \frac{1}{2}cos(2t) + \frac{1}{2}isin(2t) + \frac{1}{2}cos(2t) - \frac{1}{2}isin(2t) = cos(2t) \!</math> | ||
| + | |||
| + | <math> cos(2t) \rightarrow System \rightarrow tcos(2t)\!</math> | ||
Latest revision as of 09:02, 16 September 2008
Part B: The Basics of Linearity
Linear System Properties
The System takes the input and creates an output following these guidelines:
$ e^{2jt} \rightarrow System \rightarrow te^{-2jt}\! $
$ e^{-2jt} \rightarrow System \rightarrow te^{2jt}\! $
Therefore the System modifies the signal accordingly:
$ x(t) \rightarrow System \rightarrow tx(-t)\! $
Linear System's response to Cos(2t)
(Modified after seeing Christen Juzeszyn's answer)
Following this particular Linear System's properties:
Using Euler's formula for cos(2t):
$ \frac{1}{2}e^{2jt} + \frac{1}{2}e^{-2jt} = \frac{1}{2}cos(2t) + \frac{1}{2}isin(2t) + \frac{1}{2}cos(2t) - \frac{1}{2}isin(2t) = cos(2t) \! $
$ cos(2t) \rightarrow System \rightarrow tcos(2t)\! $
