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A system is considered linear if for any constants a, b that exist within the complex domain and for any inputs <math>x_1(t)\!</math> and <math>x_{2}(t)\!</math> yielding outputs <math>y_{1}(t)\!</math> and <math>y_2(t)\!</math> respectively, the response to <math>x_1(t) + bx_{2}(t)\!</math> is <math>y_1(t) + by_{2}(t)\!</math> | A system is considered linear if for any constants a, b that exist within the complex domain and for any inputs <math>x_1(t)\!</math> and <math>x_{2}(t)\!</math> yielding outputs <math>y_{1}(t)\!</math> and <math>y_2(t)\!</math> respectively, the response to <math>x_1(t) + bx_{2}(t)\!</math> is <math>y_1(t) + by_{2}(t)\!</math> | ||
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| + | This is saying that if i have 2 inputs of x(t) and y(t) and put them through a system, then multiply the 2 outputs by a constant, and add them together to get my final signal i should get the same if i multiply the two input signals by the constant first and add them, then send them through the system. | ||
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| + | x(t) <math>\to\!</math> (system) <math>\to\!</math> *a <math>\to\!</math><BR> | ||
| + | .....................(sum) <math>\to\!</math> z(t)<BR> | ||
| + | y(t) <math>\to\!</math> (system) <math>\to\!</math> *b <math>\to\!</math> | ||
Revision as of 05:51, 12 September 2008
Definition of Linear System
A system is considered linear if for any constants a, b that exist within the complex domain and for any inputs $ x_1(t)\! $ and $ x_{2}(t)\! $ yielding outputs $ y_{1}(t)\! $ and $ y_2(t)\! $ respectively, the response to $ x_1(t) + bx_{2}(t)\! $ is $ y_1(t) + by_{2}(t)\! $
This is saying that if i have 2 inputs of x(t) and y(t) and put them through a system, then multiply the 2 outputs by a constant, and add them together to get my final signal i should get the same if i multiply the two input signals by the constant first and add them, then send them through the system.
x(t) $ \to\! $ (system) $ \to\! $ *a $ \to\! $
.....................(sum) $ \to\! $ z(t)
y(t) $ \to\! $ (system) $ \to\! $ *b $ \to\! $
