(New page: == Definition == If <math>x_{1}(t) \to sys \to *a \to</math> <math>+ \to a x_{1}(t) + b x_{2}(t)</math> <math>x_{2}(t) \to sys \to *b \to</math> And <math>...) |
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Then the system is linear. | Then the system is linear. | ||
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| + | == Example of a Linear System == | ||
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| + | <font size="3">Equation: <math>y[n] = 2 x[n]</math></font> | ||
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| + | <math>x_{1}[n] \to sys \to *a \to</math> | ||
| + | <math>+ \to 2a x_{1}[n] + 2b x_{2}[n]</math> | ||
| + | <math>x_{2}[n] \to sys \to *b \to</math> | ||
| + | |||
| + | <math>x_{1}[n] \to *a \to</math> | ||
| + | <math>+ \to sys \to 2a x_{1}[n] + 2b x_{2}[n]</math> | ||
| + | <math>x_{2}[n] \to *b \to</math> | ||
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| + | <font size="3">Since <math>2a x_{1}[n] + 2b x_{2}[n]</math> and <math>2a x_{1}[n] + 2b x_{2}[n]</math> are equal, the system is linear.</font> | ||
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| + | == Example of a Non-Linear System == | ||
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| + | <font size="3">Equation: <math>y[n] = x[n]^2</math></font> | ||
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| + | <math>x_{1}[n] \to sys \to *a \to</math> | ||
| + | <math>+ \to a x_{1}[n]^2 + b x_{2}[n]^2</math> | ||
| + | <math>x_{2}[n] \to sys \to *b \to</math> | ||
| + | |||
| + | <math>x_{1}[n] \to *a \to</math> | ||
| + | <math>+ \to sys \to (a x_{1}[n] + b x_{2}[n])^2</math> | ||
| + | <math>x_{2}[n] \to *b \to</math> | ||
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| + | <font size="3">Since <math>a x_{1}[n]^2 + b x_{2}[n]^2</math> and <math>(a x_{1}[n] + b x_{2}[n])^2</math> are not equal, the system is not linear.</font> | ||
Latest revision as of 12:19, 10 September 2008
Definition
If
$ x_{1}(t) \to sys \to *a \to $ $ + \to a x_{1}(t) + b x_{2}(t) $ $ x_{2}(t) \to sys \to *b \to $
And
$ x_{1}(t) \to *a \to $ $ + \to sys \to a x_{1}(t) + b x_{2}(t) $ $ x_{2}(t) \to *b \to $
And $ a $ and $ b $ are any complex number,
Then the system is linear.
Example of a Linear System
Equation: $ y[n] = 2 x[n] $
$ x_{1}[n] \to sys \to *a \to $ $ + \to 2a x_{1}[n] + 2b x_{2}[n] $ $ x_{2}[n] \to sys \to *b \to $
$ x_{1}[n] \to *a \to $ $ + \to sys \to 2a x_{1}[n] + 2b x_{2}[n] $ $ x_{2}[n] \to *b \to $
Since $ 2a x_{1}[n] + 2b x_{2}[n] $ and $ 2a x_{1}[n] + 2b x_{2}[n] $ are equal, the system is linear.
Example of a Non-Linear System
Equation: $ y[n] = x[n]^2 $
$ x_{1}[n] \to sys \to *a \to $ $ + \to a x_{1}[n]^2 + b x_{2}[n]^2 $ $ x_{2}[n] \to sys \to *b \to $
$ x_{1}[n] \to *a \to $ $ + \to sys \to (a x_{1}[n] + b x_{2}[n])^2 $ $ x_{2}[n] \to *b \to $
Since $ a x_{1}[n]^2 + b x_{2}[n]^2 $ and $ (a x_{1}[n] + b x_{2}[n])^2 $ are not equal, the system is not linear.
