Linear Time Invariant System: Difference between revisions

Revision as of 22:18, 6 January 2010

$= = = L i n e a r T i m e I n v a r i a n t S y s t e m s (L T I S y s t e m s) = = = A l i n e a r t i m e i n v a r i a n t s y s t e m i s o n e t h a t i s l i n e a r (s u p e r p o s i t i o n a n d p r o p o r t i o n a l i t y a p p l y) a n d o n e t h a t d o e s n^{'} t c h a n g e w i t h t i m e . F o r e x a m p l e a c i r c u i t w i t h f i x e d c a p a c i t o r s, r e s i s t o r s, a n d i n d u c t o r s h a v i n g a n i n p u t a n d a n o u t p u t i s l i n e a r a n d t i m e i n v a r i a n t . I f a c a p a c i t o r c h a n g e d v a l u e w i t h t i m e, t h e n i t w o u l d n o t b e t i m e i n v a r i a n t . = = = E i g e n f u n c t i o n s a n d E i g e n v a l u e s o f a n L T I S y s t e m s = = = I t i s a n i n t e r e s t i n g e x e r c i s e t o s h o w t h a t < m a t h > e^{j ω t}$ are eigenfunctions of any LTI system. The eigenvalues are $ω$ .

LTI System
Input	Output
$δ (t)$	$h (t)$
$δ (t = λ)$	$h (t - λ)$

</math>

@@ Line 1: / Line 1: @@
-===Linear Time Invariant Systems (LTI Systems)===
+<math>===Linear Time Invariant Systems (LTI Systems)===
 A linear time invariant system is one that is linear (superposition and proportionality apply) and one that doesn't change with time.  For example a circuit with fixed capacitors, resistors, and inductors having an input and an output is linear and time invariant.  If a capacitor changed value with time, then it would not be time invariant.
@@ Line 5: / Line 5: @@
 It is an interesting exercise to show that <math>e^{j\omega t}</math> are eigenfunctions of any LTI system.  The eigenvalues are <math>\omega</math>.
+{| class="wikitable" border="1"
+|+ LTI System
+! Input !! Output
+|-
+| <math>\delta (t)</math> || <math>h(t)</math>
+|-
+| <math>\delta (t= \lambda )</math> || <math>h(t-\lambda )</math>
+|}
+</math>

Linear Time Invariant System: Difference between revisions

Revision as of 22:18, 6 January 2010

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