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The input x(t) and the output y(t) of a system are related by the equation | The input x(t) and the output y(t) of a system are related by the equation | ||
| − | <math>y(t)=\frac{ | + | <math>y(t)=\frac{ {\color{red} t }}{1+x^2(t)}.</math> |
Is the system stable? Answer yes/no and ustify your answer. | Is the system stable? Answer yes/no and ustify your answer. | ||
| + | :<span style="color:red">OOPS, I actually meant to put a "t" on top of the fraction (now in red). -pm</span> | ||
---- | ---- | ||
==Share your answers below== | ==Share your answers below== | ||
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--[[User:Cmcmican|Cmcmican]] 17:44, 24 January 2011 (UTC) | --[[User:Cmcmican|Cmcmican]] 17:44, 24 January 2011 (UTC) | ||
| + | |||
| + | :<span style="color:green">Unfortunately no. Here is how you should go about answering such questions. | ||
| + | ::If you think it is stable, then assume that x(t) is bounded (i.e., |x(t)|<m ) and then try to show that y(t) is also bounded (|y(t)<M ). | ||
| + | ::If you think it is not stable, then try to think of a bounded signal x(t) for which y(t) would not be bounded. </span> | ||
| + | :<span style="color:green"> Hint for this case: Look at the constant signal x(t)=1. -pm </span> | ||
===Answer 2=== | ===Answer 2=== | ||
Revision as of 13:07, 24 January 2011
Contents
Practice Question on System Stability
The input x(t) and the output y(t) of a system are related by the equation
$ y(t)=\frac{ {\color{red} t }}{1+x^2(t)}. $
Is the system stable? Answer yes/no and ustify your answer.
- OOPS, I actually meant to put a "t" on top of the fraction (now in red). -pm
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
Answer 1
This system is stable. I'm actually not sure how to show this, does the following logic work?
$ \lim_{x(t) \to 0}\frac{1}{1+x^2(t)} = 1 $ and $ \frac{1}{1+x^2(t)} < 1 $ for all x(t), thus the system is stable.
I'm not sure that the justification works here...
--Cmcmican 17:44, 24 January 2011 (UTC)
- Unfortunately no. Here is how you should go about answering such questions.
- If you think it is stable, then assume that x(t) is bounded (i.e., |x(t)|<m ) and then try to show that y(t) is also bounded (|y(t)<M ).
- If you think it is not stable, then try to think of a bounded signal x(t) for which y(t) would not be bounded.
Answer 2
Write it here.
Answer 3
Write it here.
