Lecture40-41_ErrorControlCoding - a39 a38 a36 a37 E r r o r...

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Unformatted text preview: a39 a38 a36 a37 E r r o r C o n t r o l C o d i n g • C h a n n e l i s n o i s y • C h a n n e l o u t p u t p r o n e t o e r r o r • ⇒ w e n e e d m e a s u r e t o e n s u r e c o r r e c t n e s s o f t h e b i t s t r e a m t r a n s m i t t e d E r r o r c o n t r o l c o d i n g a i m s a t d e v e l o p i n g m e t h o d s f o r c o d i n g t o c h e c k t h e c o r r e c t n e s s o f t h e b i t s t r e a m t r a n s m i t t e d . T h e b i t s t r e a m r e p r e s e n t a t i o n o f a s y m b o l i s c a l l e d t h e c o d e w o r d o f t h a t s y m b o l . D i ff e r e n t e r r o r c o n t r o l m e c h a n i s m s : • L i n e a r B l o c k C o d e s • R e p e t i t i o n C o d e s • C o n v o l u t i o n C o d e s a39 a38 a36 a37 L i n e a r B l o c k C o d e s A c o d e i s l i n e a r i f t w o c o d e s a r e a d d e d u s i n g m o d u l o- 2 a r i t h m e t i c p r o d u c e s a t h i r d c o d e w o r d i n t h e c o d e . C o n s i d e r a ( n , k ) l i n e a r b l o c k c o d e . H e r e , 1 . n r e p r e s e n t s t h e c o d e w o r d l e n g t h 2 . k i s t h e n u m b e r o f m e s s a g e b i t 3 . n- k b i t s a r e e r r o r c o n t r o l b i t s o r p a r i t y c h e c k b i t s g e n e r a t e d f r o m m e s s a g e u s i n g a n a p p r o p r i a t e r u l e . W e m a y t h e r e f o r e r e p r e s e n t t h e c o d e w o r d a s c i = b i , i = , 1 , · · · , n- k- 1 m i + k- n , i = n- k , n- k + 1 , · · · , n- 1 ( 1 ) a39 a38 a36 a37 T h e ( n- k ) p a r i t y b i t s a r e l i n e a r s u m s o f t h e k m e s s a g e b i t s . b i = p i m + p 1 i m 1 + · · · + p k- 1 , i m k- 1 ( 2 ) w h e r e t h e c o e ffi c i e n t s a r e p i j = 1 , i f b i d e p e n d s o n m j , o t h e r w i s e ( 3 ) W e d e fi n e t h e 1- b y- k m e s s a g e v e c t o r , o r i n f o r m a t i o n v e c t o r , m , t h e 1- b y- ( n- k ) p a r i t y v e c t o r b , a n d t h e 1- b y- n c o d e v e c t o r c a s f o l l o w s : m = [ m , m 1 , · · · , m k- 1 ] b = [ b , b 1 , · · · , b n- k- 1 ] n = [ n , n 1 , · · · , n n- 1 ] a39 a38 a36 a37 W e m a y t h u s w r i t e s i m u l t a n e o u s e q u a t i o n s i n m a t r i x e q u a t i o n f o r m a s b = m P w h e r e P i s a k b y n- k m a t r i x d e fi n e d b y P =...
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This note was uploaded on 10/16/2011 for the course ELECTRICAL EE251202 taught by Professor Rejaei during the Spring '10 term at Sharif University of Technology.

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Lecture40-41_ErrorControlCoding - a39 a38 a36 a37 E r r o r...

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