Revision as of 17:56, 10 September 2008

Time Invariance

A system is called time invariance if and only if:

$'x --> [system] --> y' <==> 'y = f(x)' ('x --> [f] --> f(x)') \,$

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

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

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

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

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

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

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

$f(x) = 23x \,$

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

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

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

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

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

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

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

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

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

@@ Line 2: / Line 2: @@
 A system is called time invariance if and only if:
-<math>f(S_{t_0}(x))=S_{t_0}(f(x))\ \,</math>
-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).)
 <math>'x --> [system] --> y' <==> 'y = f(x)' ('x --> [f] --> f(x)') \,</math>