(New page: = Time Invariance = A system is called time invariance if and only if: <math>f(S_{t_0}(x))=S_{t_0}(f(x))\ \,</math> == Example of a linear system == System is: <math> f(x) = 23x \,</mat...) |
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<math>f(S_{t_0}(x))=S_{t_0}(f(x))\ \,</math> | <math>f(S_{t_0}(x))=S_{t_0}(f(x))\ \,</math> | ||
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| + | With a definition of <math>S_{t_0}</math> as the shifting operator <math>S_{t_0}(x(t))=x(t-t_0).</math> (In other words, <math>S_{t_0}</math> introduces a time delay of <math>t_0</math> onto the function/signal x(t).) | ||
== Example of a linear system == | == Example of a linear system == | ||
Revision as of 16:54, 10 September 2008
Contents
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
