Revision as of 15:18, 11 September 2008 by Cztan (Talk)

Time Invariance

A time-invariant system is a system in which the output gets time-shifted when the input is time-shifted.


$ x(t - t_0) \rightarrow system \rightarrow y(t - t_0) $


Time-invariant System

An example of a time-invariant system would be the system I used for my linearity problem. Therefore the system is a linear, time-invariant system.


$ x(t) \rightarrow system \rightarrow y(t) = 2x(t) $


Proof:

$ x(t) \rightarrow system \rightarrow 2x(t) \rightarrow time-delay \rightarrow 2x(t-t_0) $


$ x(t) \rightarrow time-delay \rightarrow x(t-t_0) \rightarrow system \rightarrow 2x(t-t_0) $


Since the output is the same for both configurations the system is time-invariant.


Time-variant System

An example for a time-variant system would be $ x(t) \rightarrow 2tx(t) $

Proof:

$ x(t) \rightarrow system \rightarrow 2tx(t) \rightarrow time-delay \rightarrow 2(t-t_0)x(t-t_0) $


$ x(t) \rightarrow time-delay \rightarrow x(t-t_0) \rightarrow system \rightarrow 2tx(t-t_0) $

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