(New page: A linear system’s response to '''<big><big><math>e^{2jt}</math></big></big>''' is '''<big><big><math>t*e^{-2jt}</math></big></big>''', and its response to '''<big><big><math>e^{-2jt}</ma...) |
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Latest revision as of 16:38, 19 September 2008
A linear system’s response to $ e^{2jt} $ is $ t*e^{-2jt} $, and its response to $ e^{-2jt} $ is $ t*e^{2jt} $.
What is the response to $ cos(2t) $?
The first thing when solving this problem is to realize the that $ cos(2t) = \frac{e^{2t} + e^{-2t}}{2} = \frac{e^{2t}}{2} + \frac{e^{-2t}}{2} $.
Being a linear system, you can pass each of the two parts shown above individually and add the results together.
Therefore, $ \frac{e^{2t}}{2} \to t*\frac{e^{-2t}}{2} $ and $ \frac{e^{-2t}}{2} \to t*\frac{e^{2t}}{2} $.
Adding the results together yields $ t*\frac{e^{-2t}}{2} + t*\frac{e^{2t}}{2} = t * cos(2t) $
