(New page: =Basics of Linearity= ==Question== A linear system's response to exp(2jt) is t exp(-2jt) and its response to exp(-2jt) is t exp(2jt). What is the system's response to cos(2t)? ==Answer=...) |
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==Question== | ==Question== | ||
| − | A linear system's response to | + | A linear system's response to <math>e^{j2t} \ </math> is <math>te^{-j2t} \ </math> and its response to <math>e^{-2jt}\ </math> is <math>t e^{2jt}\ </math>. |
| − | What is the system's response to cos(2t)? | + | What is the system's response to <math>cos(2t)\ </math>? |
==Answer== | ==Answer== | ||
| + | Accord to Euler's formula, <math>cos(t) = \frac{e^{jt}+e^{-jt}}{2}</math> | ||
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| + | Hence, the response to <math>cos(2t)\ </math> would be <math>\frac{e^{j2t}+e^{-j2t}}{2}\ </math>. | ||
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| + | With that in mind, the response to <math>e^{j2t}\ </math> is <math>t e^{-j2t}\ </math> and its response to <math>e^{-j2t}\ </math> is <math>t e^{j2t}\ </math>. | ||
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| + | Thus the output of the system will be: | ||
| + | |||
| + | <math>=t\frac{e^{j2t}+e^{-j2t}}{2}\,</math> | ||
| + | <math>=t\cos(2t)\,</math> | ||
Latest revision as of 08:59, 19 September 2008
Basics of Linearity
Question
A linear system's response to $ e^{j2t} \ $ is $ te^{-j2t} \ $ and its response to $ e^{-2jt}\ $ is $ t e^{2jt}\ $.
What is the system's response to $ cos(2t)\ $?
Answer
Accord to Euler's formula, $ cos(t) = \frac{e^{jt}+e^{-jt}}{2} $
Hence, the response to $ cos(2t)\ $ would be $ \frac{e^{j2t}+e^{-j2t}}{2}\ $.
With that in mind, the response to $ e^{j2t}\ $ is $ t e^{-j2t}\ $ and its response to $ e^{-j2t}\ $ is $ t e^{j2t}\ $.
Thus the output of the system will be:
$ =t\frac{e^{j2t}+e^{-j2t}}{2}\, $ $ =t\cos(2t)\, $
