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Finally,<br>
 
Finally,<br>
<math>cos{(2t)} \,</math> --> '''system''' --> <math> tcos({(2t)}\,</math>
+
<math>cos{(2t)} \,</math> --> '''system''' --> <math> tcos({(2t)})\,</math>

Revision as of 10:05, 13 September 2008

  • I am going to use the definition of Linearity that I learned in class.
  • The definition
 if x1(t) --> system --> y1(t)
x2(t) --> system --> y2(t)
Then ax1(t) + bx2(t) --> system --> ay1(t) + by2(t) , for any complex constants a,b

$ e^{(2jt)} = cos{(2t)} + jsin{(2t)} --> system --> t*{(cos{(2t)} - jsin{(2t)})}\, $
$ e^{(-2jt)} = cos{(2t)} - jsin{(2t)} --> system --> t*{(cos{(2t)} + jsin{(2t)})}\, $

Now, suppose a and b are 1/2.
$ \frac{1}{2}e^{(2jt)} + \frac{1}{2}e^{(-2jt)} = \frac{1}{2}(cos{(2t)} + jsin{(2t)}) + \frac{1}{2}(cos{(2t)} - jsin{(2t)}) = cos{(2t)} $

Then output is
$ \frac{1}{2}(t*{(cos{(2t)} - jsin{(2t)})}) + \frac{1}{2}t*{(cos{(2t)} + jsin{(2t)})} = \frac{1}{2}tcos{(2t)} + \frac{1}{2}tcos{(2t)} = tcos({(2t)} $

Finally,
$ cos{(2t)} \, $ --> system --> $ tcos({(2t)})\, $

Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang