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| − | [[ | + | '''[[Signals_and_systems_practice_problems_list|Practice Question on "Signals and Systems"]]''' |
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| + | [[Signals_and_systems_practice_problems_list|More Practice Problems]] | ||
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| + | Topic: Stability of a System | ||
| + | </center> | ||
| + | ---- | ||
| + | ==Question== | ||
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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. | ||
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| + | :<span style="color:red">OOPS, I actually meant to put a "t" on top of the fraction (now in red). -pm</span> | ||
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---- | ---- | ||
| − | ==Share your answers below== | + | |
| − | 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! | + | == Share your answers below == |
| + | |||
| + | 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! | ||
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---- | ---- | ||
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| − | + | === Answer 1 === | |
| − | I'm not sure | + | This system is stable. I'm actually not sure how to show this, does the following logic work? |
| − | --[[User:Cmcmican|Cmcmican]] 17:44, 24 January 2011 (UTC) | + | <math>\lim_{x(t) \to 0}\frac{1}{1+x^2(t)} = 1</math> and <math>\frac{1}{1+x^2(t)} < 1 </math> for all x(t), thus the system is stable. |
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| + | I'm not sure that the justification works here... | ||
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| + | --[[User:Cmcmican|Cmcmican]] 17:44, 24 January 2011 (UTC) | ||
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| + | :<span style="color:green">Unfortunately no. Here is how you should go about answering such questions. If you think it is stable,</span> | ||
| + | |||
| + | :<span style="color:green"> 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 ).</span> | ||
| + | |||
| + | :<span style="color:green"> 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> | ||
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| + | <br> | ||
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| + | :<span style="color:green"> Hint for this case: Look at the constant signal x(t)=1. -pm </span> | ||
| + | |||
| + | === Answer 2 === | ||
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| + | Now that it has a t on top, it's not bounded. | ||
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| + | If you consider the constant signal x(t)=1, then <math>y(t) = \frac{{t }}{1+1^2} = \frac{{t }}{2}</math>, which is not bounded. | ||
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| + | --[[User:Cmcmican|Cmcmican]] 19:26, 24 January 2011 (UTC) | ||
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| + | :<span style="color:green">Good! And what if there was no t on top? -pm </span> | ||
| + | |||
| + | === Answer 3 === | ||
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| + | If there is not a t on top (i.e it is back to being a '1'), then the signal is bounded*. | ||
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| + | Considering the case where <math>|x(t)| \le \infty</math> then <math>0<\frac{{1}}{1+x^2(t)}\le1</math>. | ||
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| + | <math>\therefore y(t)</math> is bounded by <math>M = \pm 1</math> | ||
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| + | '''*Addendum''': This only works for <math>x(t) \in \Re</math> as there are imaginary values that cause it to be unstable. | ||
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| + | --[[User:Darichar|Darichar]] 14:05, 26 January 2011 (UTC) | ||
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| + | :<span style="color:green">TA's comment: BIBO stability requires that the response doesn't diverge for any bounded input signal, including complex signals. Therefore, we just say that this system is unstable.</span> | ||
| + | --[[User:Ahmadi|Ahmadi]] 22:00, 27 January 2011 (UTC) | ||
| + | <br> <br> <br> | ||
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---- | ---- | ||
| − | [[ | + | |
| + | [[2011 Spring ECE 301 Boutin|Back to ECE301 Spring 2011 Prof. Boutin]] | ||
| + | |||
| + | [[Category:ECE301Spring2011Boutin]] [[Category:Problem_solving]] | ||
Latest revision as of 16:20, 26 November 2013
Practice Question on "Signals and Systems"
Topic: Stability of a System
Question
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.
- Hint for this case: Look at the constant signal x(t)=1. -pm
Answer 2
Now that it has a t on top, it's not bounded.
If you consider the constant signal x(t)=1, then $ y(t) = \frac{{t }}{1+1^2} = \frac{{t }}{2} $, which is not bounded.
--Cmcmican 19:26, 24 January 2011 (UTC)
- Good! And what if there was no t on top? -pm
Answer 3
If there is not a t on top (i.e it is back to being a '1'), then the signal is bounded*.
Considering the case where $ |x(t)| \le \infty $ then $ 0<\frac{{1}}{1+x^2(t)}\le1 $.
$ \therefore y(t) $ is bounded by $ M = \pm 1 $
*Addendum: This only works for $ x(t) \in \Re $ as there are imaginary values that cause it to be unstable.
--Darichar 14:05, 26 January 2011 (UTC)
- TA's comment: BIBO stability requires that the response doesn't diverge for any bounded input signal, including complex signals. Therefore, we just say that this system is unstable.
--Ahmadi 22:00, 27 January 2011 (UTC)
