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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>''' | '''<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)</math>''' | '''<math>W(t) = Z(t)</math>''' | ||
== TIME-VARIANT SYSTEM == | == TIME-VARIANT SYSTEM == | ||
Revision as of 15:42, 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) $
