Latest revision as of 07:59, 19 September 2008

Basics of Linearity

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)\$ ?

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:

$=t\frac{e^{j2t}+e^{-j2t}}{2}\,$ $=t\cos(2t)\,$

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 ==Question==
-A linear system's response to exp(2jt) is t exp(-2jt) and its response to exp(-2jt) is t exp(2jt).
+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==
+Accord to Euler's formula, <math>cos(t) = \frac{e^{jt}+e^{-jt}}{2}</math>
+Hence, the response to <math>cos(2t)\ </math> would be <math>\frac{e^{j2t}+e^{-j2t}}{2}\ </math>.
+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>.
+Thus the output of the system will be:
+<math>=t\frac{e^{j2t}+e^{-j2t}}{2}\,</math>
+<math>=t\cos(2t)\,</math>