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== LINEARITY == | == LINEARITY == | ||
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| + | For a system to be called Linear the following two scenarios must yield output signals that are equal to each other. | ||
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| + | 1) Signals '''<math>X_1</math>''' and '''<math>Y_1</math>''' are first multiplied by constants <math>C_1 \in \mathbb{C}</math> and <math>C_2\in \mathbb{C}</math> respectively, then added together and passed through a system that yields a signal '''<math>Z(t)</math>'''. | ||
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| + | and | ||
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| + | 2) Signals '''<math>X_1</math>''' and '''<math>Y_1</math>''' each pass through a system, their results are multiplied by constants <math>C_1 \in \mathbb{C}</math> and <math>C_2\in \mathbb{C}</math> respectively, and then added together yielding a signal '''<math>W(t)</math>'''. | ||
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| + | For this system to be linear, signals '''<math>Z(t)</math>''' and '''<math>W(t)</math>''' must be equal to each other. | ||
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| + | '''<math>Z(t) = W(t)</math>''' | ||
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| + | == LINEAR SYSTEM == | ||
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| + | '''<math>X(t) \to Y(2t)</math>''' | ||
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| + | '''PROOF''' | ||
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| + | let '''<math>a \in \mathbb{{C}}</math>''' and '''<math>b \in \mathbb{{C}}</math>''', | ||
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| + | '''<math>X_1(t) \Rightarrow Y_1(t) = X_1(2t), a*X_1(2t) \downarrow</math>''' | ||
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| + | .........................................................................'''<math>\bigoplus \to Z(t) = a*X_1(2t) + b*X_2(2t)</math>''' | ||
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| + | '''<math>X_2(t) \Rightarrow Y_2(t) = X_2(2t), b*X_2(2t) \uparrow</math>''' | ||
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| + | ---- | ||
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| + | '''<math>a*X_1(t) \downarrow</math>''' | ||
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| + | ..................'''<math>\bigoplus \to a*X_1(t) + b*X_2(t) \Rightarrow W(t) = a*X_1(2t) + b*X_2(2t)</math> | ||
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| + | '''<math>a*X_2(t) \uparrow</math>''' | ||
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| + | '''<math>Z(t) = W(t) \Rightarrow</math>''' Non-Linear System | ||
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| + | == NON-LINEAR SYSTEM == | ||
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| + | '''<math>X(t) \to Y(t)^3</math>''' | ||
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| + | '''PROOF''' | ||
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| + | let '''<math>a \in \mathbb{{C}}</math>''' and '''<math>b \in \mathbb{{C}}</math>''', | ||
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| + | '''<math>X_1(t) \Rightarrow Y_1(t) = X_1(t)^3, a*X_1(t)^3 \downarrow</math>''' | ||
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| + | ........................................................................'''<math>\bigoplus \to Z(t) = a*X_1(t)^3 + b*X_2(t)^3</math>''' | ||
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| + | '''<math>X_2(t) \Rightarrow Y_2(t) = X_2(t)^3, b*X_2(t)^3 \uparrow</math>''' | ||
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| + | ---- | ||
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| + | '''<math>a*X_1(t) \downarrow</math>''' | ||
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| + | ..................'''<math>\bigoplus \to a*X_1(t) + b*X_2(t) \Rightarrow W(t)^3 = (a*X_1(t) + b*X_2(t))^3</math> | ||
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| + | '''<math>a*X_2(t) \uparrow</math>''' | ||
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| + | '''<math>Z(t) \ne W(t) \Rightarrow</math>''' Non-Linear System | ||
Latest revision as of 15:12, 12 September 2008
LINEARITY
For a system to be called Linear the following two scenarios must yield output signals that are equal to each other.
1) Signals $ X_1 $ and $ Y_1 $ are first multiplied by constants $ C_1 \in \mathbb{C} $ and $ C_2\in \mathbb{C} $ respectively, then added together and passed through a system that yields a signal $ Z(t) $.
and
2) Signals $ X_1 $ and $ Y_1 $ each pass through a system, their results are multiplied by constants $ C_1 \in \mathbb{C} $ and $ C_2\in \mathbb{C} $ respectively, and then added together yielding a signal $ W(t) $.
For this system to be linear, signals $ Z(t) $ and $ W(t) $ must be equal to each other.
$ Z(t) = W(t) $
LINEAR SYSTEM
$ X(t) \to Y(2t) $
PROOF
let $ a \in \mathbb{{C}} $ and $ b \in \mathbb{{C}} $,
$ X_1(t) \Rightarrow Y_1(t) = X_1(2t), a*X_1(2t) \downarrow $
.........................................................................$ \bigoplus \to Z(t) = a*X_1(2t) + b*X_2(2t) $
$ X_2(t) \Rightarrow Y_2(t) = X_2(2t), b*X_2(2t) \uparrow $
$ a*X_1(t) \downarrow $
..................$ \bigoplus \to a*X_1(t) + b*X_2(t) \Rightarrow W(t) = a*X_1(2t) + b*X_2(2t) $
$ a*X_2(t) \uparrow $
$ Z(t) = W(t) \Rightarrow $ Non-Linear System
NON-LINEAR SYSTEM
$ X(t) \to Y(t)^3 $
PROOF
let $ a \in \mathbb{{C}} $ and $ b \in \mathbb{{C}} $,
$ X_1(t) \Rightarrow Y_1(t) = X_1(t)^3, a*X_1(t)^3 \downarrow $
........................................................................$ \bigoplus \to Z(t) = a*X_1(t)^3 + b*X_2(t)^3 $
$ X_2(t) \Rightarrow Y_2(t) = X_2(t)^3, b*X_2(t)^3 \uparrow $
$ a*X_1(t) \downarrow $
..................$ \bigoplus \to a*X_1(t) + b*X_2(t) \Rightarrow W(t)^3 = (a*X_1(t) + b*X_2(t))^3 $
$ a*X_2(t) \uparrow $
$ Z(t) \ne W(t) \Rightarrow $ Non-Linear System
