(New page: == Linearity == == Background == === Language Definition === A system is considered linear if 2 separate inputs, multiplied by 2 different constants, can produce 2 separate outputs mult...) |
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A system is called linear if: | A system is called linear if: | ||
| − | For any inputs <math>x_1(t)</math> and <math>x_2(t)</math> yielding outputs of <math>y_1(t)</math> and <math>y_2(t)</math>, | + | For any inputs <math>x_1(t)</math> and <math>x_2(t)</math> yielding outputs of <math>y_1(t)</math> and <math>y_2(t)</math>, |
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| + | <math>ax_1(t)+bx_2(t)=ay_1(t)+by_2(t)\,\!</math> | ||
== Example of Linear system == | == Example of Linear system == | ||
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| + | The easiest way to determine linearity is using standard definition: | ||
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| + | Lets take the system <math>y(t)=8x(t)</math> , so lets get 2 y's and 2 x's out of that: | ||
| + | <math>y_1(t)=8x_1(t)</math> for <math>x_1(t)=t</math> | ||
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| + | <math>y_2(t)=16x_2(t)</math> for <math>x_2(t)=2t</math> | ||
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| + | Now testing the theory: | ||
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| + | <math>ax_1+bx_2=a+2b</math> and | ||
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| + | <math>ay_1+by_2=a8+b16</math> , which can be reduced to | ||
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== Example of Non-Linear system == | == Example of Non-Linear system == | ||
Revision as of 14:53, 11 September 2008
Contents
Linearity
Background
Language Definition
A system is considered linear if 2 separate inputs, multiplied by 2 different constants, can produce 2 separate outputs multiplied by those same constants.
Mathematical Definition
A system is called linear if: For any inputs $ x_1(t) $ and $ x_2(t) $ yielding outputs of $ y_1(t) $ and $ y_2(t) $,
$ ax_1(t)+bx_2(t)=ay_1(t)+by_2(t)\,\! $
Example of Linear system
The easiest way to determine linearity is using standard definition:
Lets take the system $ y(t)=8x(t) $ , so lets get 2 y's and 2 x's out of that: $ y_1(t)=8x_1(t) $ for $ x_1(t)=t $
$ y_2(t)=16x_2(t) $ for $ x_2(t)=2t $
Now testing the theory:
$ ax_1+bx_2=a+2b $ and
$ ay_1+by_2=a8+b16 $ , which can be reduced to
