(New page: ==Definition== A system is linear if two specific inputs yield two specific outputs, and if you multiply the inputs by constants and then run them through the system and sum the result it...) |
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<math> 2+2t^2 + 3e^{t^2} </math> | <math> 2+2t^2 + 3e^{t^2} </math> | ||
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| + | Now, | ||
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| + | <math>y1(t)</math> -> [SYSTEM] -> A*result -> <math>(1+t^2)*2</math> | ||
| + | |||
| + | <math>y2(t)</math> -> [SYSTEM] -> B*result -> <math>(e^{t^2})*3</math> | ||
| + | |||
| + | sum these and you get | ||
| + | |||
| + | <math>2+2t^2+3e^{t^2}</math> | ||
| + | |||
| + | which is the same output as before, so the system is linear. | ||
Revision as of 14:40, 11 September 2008
Definition
A system is linear if two specific inputs yield two specific outputs, and if you multiply the inputs by constants and then run them through the system and sum the result it yields the same result as running them through the system, multiplying the outputs by those same constants, then adding the result together.
Linear System
I'll choose the system $ y(t) = x(t^2) $ Let's choose $ A=2 $, $ B=3 $, $ y1(t)=1+t $, and $ y2(t)=e^t $
$ A*y1(t) $ -> [SYSTEM] -> $ 2(1+t^2) $
$ B*y2(t) $ -> [SYSTEM] -> $ 3(e^{t^2}) $
sum these and you get
$ 2+2t^2 + 3e^{t^2} $
Now,
$ y1(t) $ -> [SYSTEM] -> A*result -> $ (1+t^2)*2 $
$ y2(t) $ -> [SYSTEM] -> B*result -> $ (e^{t^2})*3 $
sum these and you get
$ 2+2t^2+3e^{t^2} $
which is the same output as before, so the system is linear.
