Time Invariance

A system is called time invariance if and only if:

$ f(S_{t_0}(x))=S_{t_0}(f(x))\ \, $

With a definition of $ S_{t_0} $ as the shifting operator $ S_{t_0}(x(t))=x(t-t_0). $ (In other words, $ S_{t_0} $ introduces a time delay of $ t_0 $ onto the function/signal x(t).)

Example of a linear system

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-linear 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