Latest revision as of 19:15, 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

Example of a time invariance system

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

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

it yields the same result as:

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

Example of a non time invariance system

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

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

it yields not the same result as:

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

Reference

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

@@ Line 3: / Line 3: @@
 A system is called time invariance if and only if:
-<math>f(S_{t_0}(x))=S_{t_0}(f(x))\ \,</math>
+<math>x(t) --> [system] --> [time delay] --> y(t)\,</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).)
+yields the same result as
-<math>'x --> [system] --> y' <==> 'y = f(x)' ('x --> [f] --> f(x)') \,</math>
+<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 ==

Difference between revisions of "HW2-D Ronny Wijaya ECE301Fall2008mboutin" - Rhea

Latest revision as of 19:15, 10 September 2008

Contents

Time Invariance

Example of a time invariance system

Example of a non time invariance system

Reference

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