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'''Definition:''' A system is invertible if and only if unique inputs yield distinct outputs. In other words, when different inputs are put into the system, then no output should be the same, they should all be unique. | '''Definition:''' A system is invertible if and only if unique inputs yield distinct outputs. In other words, when different inputs are put into the system, then no output should be the same, they should all be unique. | ||
| − | '''Corollary:''' If a system is invertible, then the system will have an inverse system <math>(S^ | + | '''Corollary:''' If a system is invertible, then the system will have an inverse system <math>(S^{-1})</math>. The original system and the inverse system will allow for the output one of to be used as the input of the other and get the original input. |
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| + | '''Examples:''' | ||
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| + | <math>y(t)=5*x(t)+4</math> ~~ This system is invertible. All inputs will yield distinct outputs and there is an inverse system. | ||
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| + | * <math>x(t)=\frac{(y(t)-4)}{5}</math> ~~ Inverse to system above. | ||
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| + | <math>y(t)=2*x^2(t)+3</math> ~~ This system is non-invertible based on the definition of invertibility. If the test cases of <math>x_1(t)=t</math> and <math>x_2(t)=-t</math> are applied, then <math>y_1(t)</math> would equal <math>y_2(t)</math>, which means the outputs are not distinct. | ||
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| + | * <math>y_1(t)=2*(t)^2+3=2t+3</math> and <math>y_2(t)=2*(-t)^2+3=2t+3</math>, which shows that <math>y_1(t)=y_2(t)</math>, therefore implying that it is non-invertible | ||
[https://kiwi.ecn.purdue.edu/rhea/index.php/LTI_System_Properties Return to LTI System Properties] | [https://kiwi.ecn.purdue.edu/rhea/index.php/LTI_System_Properties Return to LTI System Properties] | ||
Latest revision as of 05:48, 1 July 2009
Invertibility
Definition: A system is invertible if and only if unique inputs yield distinct outputs. In other words, when different inputs are put into the system, then no output should be the same, they should all be unique.
Corollary: If a system is invertible, then the system will have an inverse system $ (S^{-1}) $. The original system and the inverse system will allow for the output one of to be used as the input of the other and get the original input.
Examples:
$ y(t)=5*x(t)+4 $ ~~ This system is invertible. All inputs will yield distinct outputs and there is an inverse system.
- $ x(t)=\frac{(y(t)-4)}{5} $ ~~ Inverse to system above.
$ y(t)=2*x^2(t)+3 $ ~~ This system is non-invertible based on the definition of invertibility. If the test cases of $ x_1(t)=t $ and $ x_2(t)=-t $ are applied, then $ y_1(t) $ would equal $ y_2(t) $, which means the outputs are not distinct.
- $ y_1(t)=2*(t)^2+3=2t+3 $ and $ y_2(t)=2*(-t)^2+3=2t+3 $, which shows that $ y_1(t)=y_2(t) $, therefore implying that it is non-invertible
