Latest revision as of 18:15, 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

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

Remember: delay --> for only every function of t, change the t into t with the offset

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

$x(t) --> [system] --> x(t) --> [timedelay] --> x(t-1) \,$

it yields the same result as:

$x(t) --> [timedelay] --> x(t-1) --> [system] --> x(t-1) \,$

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

$x(t) --> [system] --> t * x(t) --> [timedelay] --> t * x(t-1) \,$

it yields not the same result as:

$x(t) --> [timedelay] --> x(t-1) --> [system] --> (t-1) x(t-1) \,$

@@ Line 8: / Line 8: @@
 <math>x(t) --> [time delay] --> [system] --> y(t) \,</math>
+Remember: delay --> for only every function of t, change the t into t with the offset
 == Example of a time invariance system ==
-System is: <math> f(x) = 23x \,</math>
-<math>X_1(t) = t^2 \,</math>
-<math>X_2(t) = 2t^2 \,</math>
+<math>y(t) = x(t) \,</math>
+<math>x(t) --> [system] --> x(t) --> [timedelay] --> x(t-1) \,</math>
-<math>f(aX_1 + bX_2) = af(X_1) + bf(X_2) \,</math>
+it yields the same result as:
-<math>f(at^2 + 2bt^2) = af(t^2) + bf(2t^2) \,</math>
-<math>f(at^2 + 2bt^2) = a*23t^2 + b*46t^2 \,</math>
-<math>f(at^2 + 2bt^2) = 23(at^2 + 2bt^2) \,</math>
-<math> f(x) = 23x \,</math>
+<math>x(t) --> [timedelay] --> x(t-1) --> [system] --> x(t-1) \,</math>
 == Example of a non time invariance system ==
-System is: <math> f(x) = 23x + 1\,</math>
-<math>X_1(t) = t^2 \,</math>
-<math>X_2(t) = 2t^2 \,</math>
-<math>f(aX_1 + bX_2) \neq af(X_1) + bf(X_2) \,</math>
-<math>f(at^2 + 2bt^2) \neq af(t^2) + bf(2t^2) \,</math>
-<math>f(at^2 + 2bt^2) \neq a(23t^2+1) + b(23*(2t^2)+1) \,</math>
+<math>y(t) = t * x(t) \,</math>
-<math>f(at^2 + 2bt^2) \neq 23 at^2 + 1 + 46 bt^2 + b \,</math>
+<math>x(t) --> [system] --> t * x(t) --> [timedelay] --> t * x(t-1) \,</math>
-<math> f(at^2 + 2bt^2) \neq 23 (at^2 + 2bt^2) + a + b \,</math>
+it yields not the same result as:
-<math> f(x) \neq 23x + 1 \,</math>
+<math>x(t) --> [timedelay] --> x(t-1) --> [system] --> (t-1) x(t-1) \,</math>
 == Reference ==