Revision as of 18:11, 10 September 2008

Time Invariance

A system is called time invariance if and only if:

$ x(t) --> [system] --> [time delay] --> y(t)\, $

yields the same result as

$ x(t) --> [time delay] --> [system] --> y(t) \, $

Example of a time invariance system

$ y(t) = x(t) \, $

$ x(t) --> [system] --> x(t) --> [delay] --> <math>x_1(t) \, $ undergoes a time delay before it is inputted into the system : $ x_1(t) = x(t - \delta) \, $

It is then inputted into the system, $ y_1(t) = 2x_1(t) = 2x(t - \delta) \, $

If the system is shifted after the signal undergoes the transformation, then $ y_2(t) = y(t - \delta) = 2x(t - \delta) = y_1(t)\, $

The signal doesn't depend on the time when the signal is being inputted, thus it's time invariant

System is: $ f(x) = 23x \, $

$ X_1(t) = t^2 \, $

$ X_2(t) = 2t^2 \, $

$ f(aX_1 + bX_2) = af(X_1) + bf(X_2) \, $

$ f(at^2 + 2bt^2) = af(t^2) + bf(2t^2) \, $

$ f(at^2 + 2bt^2) = a*23t^2 + b*46t^2 \, $

$ f(at^2 + 2bt^2) = 23(at^2 + 2bt^2) \, $

$ f(x) = 23x \, $

Example of a non time invariance system

System is: $ f(x) = 23x + 1\, $

$ X_1(t) = t^2 \, $

$ X_2(t) = 2t^2 \, $

$ f(aX_1 + bX_2) \neq af(X_1) + bf(X_2) \, $

$ f(at^2 + 2bt^2) \neq af(t^2) + bf(2t^2) \, $

$ f(at^2 + 2bt^2) \neq a(23t^2+1) + b(23*(2t^2)+1) \, $

$ f(at^2 + 2bt^2) \neq 23 at^2 + 1 + 46 bt^2 + b \, $

$ f(at^2 + 2bt^2) \neq 23 (at^2 + 2bt^2) + a + b \, $

$ f(x) \neq 23x + 1 \, $

Reference

http://kiwi.ecn.purdue.edu/ECE301Fall2008mboutin/index.php/Concepts_and_Formulae