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== TIME INVARIANCE == | == TIME INVARIANCE == | ||
| + | Let " '''<math>\Rightarrow</math>''' " represent a system. | ||
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| + | If for any signal '''<math>X(t)\Rightarrow Y(t)</math>''' implies that '''<math>X(t - t_o)\Rightarrow Y(t - t_o)</math>''' then the system is time invariant. | ||
== TIME-INVARIANT SYSTEM == | == TIME-INVARIANT SYSTEM == | ||
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| + | '''<math>X(t)\Rightarrow Y(t) = a*X(t)</math>''' where '''<math>a \in \mathbb{{C}}</math>''' is a time invariant system. | ||
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| + | ---- | ||
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| + | '''PROOF''' | ||
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| + | '''<math>X(t)\Rightarrow Y(t) = a*X(t) \to [time delay] \to Z(t) = Y(t - t_o) = a*X(t - t_o)</math>''' | ||
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| + | '''<math>X(t)\to [time delay] \to Y(t) = X(t - t_o) \Rightarrow W(t) = a*Y(t) = a*X(t - t_o)</math>''' | ||
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| + | '''<math>W(t) = Z(t) \Rightarrow</math>''' The system is time-invariant | ||
== TIME-VARIANT SYSTEM == | == TIME-VARIANT SYSTEM == | ||
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| + | ECE 301 lectures are a time variant system. If I show up to class on time on X day, I will listen to the entire lecture (signal). If my friend on the other hand, shows up 10 minutes late to lecture, he will have missed 10 minutes of lecture (signal). Therefore, ECE 301 lectures are a "time-variant" system. | ||
Latest revision as of 16:51, 12 September 2008
TIME INVARIANCE
Let " $ \Rightarrow $ " represent a system.
If for any signal $ X(t)\Rightarrow Y(t) $ implies that $ X(t - t_o)\Rightarrow Y(t - t_o) $ then the system is time invariant.
TIME-INVARIANT SYSTEM
$ X(t)\Rightarrow Y(t) = a*X(t) $ where $ a \in \mathbb{{C}} $ is a time invariant system.
PROOF
$ X(t)\Rightarrow Y(t) = a*X(t) \to [time delay] \to Z(t) = Y(t - t_o) = a*X(t - t_o) $
$ X(t)\to [time delay] \to Y(t) = X(t - t_o) \Rightarrow W(t) = a*Y(t) = a*X(t - t_o) $
$ W(t) = Z(t) \Rightarrow $ The system is time-invariant
TIME-VARIANT SYSTEM
ECE 301 lectures are a time variant system. If I show up to class on time on X day, I will listen to the entire lecture (signal). If my friend on the other hand, shows up 10 minutes late to lecture, he will have missed 10 minutes of lecture (signal). Therefore, ECE 301 lectures are a "time-variant" system.
