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<math>e^{(2jt)} = cos{(2t)} + jsin{(2t)} --> system --> t*{(cos{(2t)} - jsin{(2t)})}\,</math><br> | <math>e^{(2jt)} = cos{(2t)} + jsin{(2t)} --> system --> t*{(cos{(2t)} - jsin{(2t)})}\,</math><br> | ||
<math>e^{(-2jt)} = cos{(2t)} - jsin{(2t)} --> system --> t*{(cos{(2t)} + jsin{(2t)})}\,</math><br> | <math>e^{(-2jt)} = cos{(2t)} - jsin{(2t)} --> system --> t*{(cos{(2t)} + jsin{(2t)})}\,</math><br> | ||
| + | |||
| + | Now, suppose a and b are 1/2.<br> | ||
| + | <math>\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)}</math> | ||
Revision as of 10:59, 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)} $
