M E q u a t i o n E q u a t i o n . 3 0 , M i c r o s o f t E q u a t i o n 3 . 0 ( Z h t t p : / / p e s p m c 1 . v u b . a c . b e / A S C / I N F O R M A T I O N . h t m l Z h t t p : / / p e s p m c 1 . v u b . a c . b e / A S C / I N F O R M A T I O N . h t m l ) \ h t t p : / / w w w . m t m . u f s c . b r / ~ t a n e j a / b o o k / n o d e 5 . h t m l \ h t t p : / / w w w . m t m . u f s c . b r / ~ t a n e j a / b o o k / n o d e 5 . h t m l * \ h t t p : / / w w w . m t m . u f s c . b r / ~ t a n e j a / b o o k / n o d e 6 . h t m l \ h t t p : / / w w w . m t m . u f s c . b r / ~ t a n e j a / b o o k / n o d e 6 . h t m l 0 D P r o p e r t i e s o f S h a n n o n ' s E n t r o p y \ h t t p : / / w w w . m t m . u f s c . b r / ~ t a n e j a / b o o k / n o d e 5 . h t m l : S E q u a t i o n E q u a t i o n . 3 0 , M i c r o s o f t E q u a t i o n 3 . 0 8 T E q u a t i o n E q u a t i o n . 3 0 , M i c r o s o f t E q u a t i o n 3 . 0 D P A M a t h e m a t i c a l T h e o r y o f C o m m u n i c a t i o n z h t t p : / / c m . b e l l - l a b s . c o m / c m / m s / w h a t / s h a n n o n d a y / s h a n n o n 1 9 4 8 . p d f [ X E q u a t i o n E q u a t i o n . 3 0 , M i c r o s o f t E q u a t i o n 3 . 0 { ] E q u a t i o n E q u a t i o n . 3 0 , M i c r o s o f t E q u a t i o n 3 . 0 D / 0 | D A r i a l r N e ( 0 ( z[ 0 D T i m e s N e w R o m a n ( 0 ( z[ 0 D W i n g d i n g s R o m a n ( 0 ( z[ 0 0 D A r i a l B l a c k m a n ( 0 ( z[ 0 "@ D C o u r i e r N e w m a n ( 0 ( z[ 0 1 @ .
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p p ^ _ _ _ P P T 9 @ 0 D WT . @ I CSE468 Information Conflict Lecturer: Dr Carlo Kopp, MIEEE, MAIAA, PEng 0 J + E 7 Lecture 02
Introduction to Information Theory Concepts & 8 , " Reference Sources and Bibliography $ There is an abundance of websites and publications dealing with basic information theory.
Examples include:
http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html
http://okmij.org/ftp/Computation/limits-of-information.html
http://www.sveiby.com/articles/information.html
http://pespmc1.vub.ac.be/ASC/INFORMATION.html
http://www.mtm.ufsc.br/~taneja/book/node5.html
http://www.mtm.ufsc.br/~taneja/book/node6.html
R m 1 m 1 m 8 ; / - . . U
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0 o 4 A R D e f i n i n g I n f o r m a t i o n S h a n n o n a n d W e a v e r S h a n n o n & t h a t w h i c h r e d u c e s u n c e r t a i n t y &
S h a n n o n & W e a v e r T h e q u a n t i t y w h i c h u n i q u e l y m e e t s t h e n a t u r a l r e q u i r e m e n t s t h a t o n e s e t s u p f o r i n f o r m a t i o n t u r n s o u t t o b e e x a c t l y t h a t w h i c h i s k n o w n i n t h e r m o d y n a m i c s a s e n t r o p y .
S h a n n o n & W e a v e r - I n f o r m a t i o n i s a m e a s u r e o f o n e s f r e e d o m o f c h o i c e i n s e l e c t i n g a m e s s a g e . T h e g r e a t e r t h i s f r e e d o m o f c h o i c e , t h e g r e a t e r t h e i n f o r m a t i o n , t h e g r e a t e r i s t h e u n c e r t a i n t y t h a t t h e m e s s a g e a c t u a l l y s e l e c t e d i s s o m e p a r t i c u l a r o n e . G r e a t e r f r e e d o m o f c h o i c e , g r e a t e r u n c e r t a i n t y g r e a t e r i n f o r m a t i o n g o h a n d i n h a n d . ( S v e i b y 1 9 9 4 ) < B O Defining Information - Wiener S h a n n o n a n d W i e n e r d e f i n e i n f o r m a t i o n d i f f e r e n t l y S h a n n o n s e e s i t a s m e a s u r e d b y e n t r o p y , W i e n e r b y t h e o p p o s i t e o f e n t r o p y .
W i e n e r T h e n o t i o n o f t h e a m o u n t o f i n f o r m a t i o n a t t a c h e s i t s e l f v e r y n a t u r a l l y t o a c l a s s i c a l n o t i o n i n s t a t i s t i c a l m e c h a n i c s : t h a t o f e n t r o p y . J u s t a s t h e a m o u n t o f i n f o r m a t i o n i n a s y s t e m i s a m e a s u r e o f i t s d e g r e e o f o r g a n i s a t i o n , s o t h e e n t r o p y o f a s y s t e m i s a m e a s u r e o f i t s d e g r e e o f d i s o r g a n i s a t i o n . ( S v e i b y 1 9 9 4 ) > ] ; C R Wiener Continued Z O n e o f t h e s i m p l e s t , m o s t u n i t a r y f o r m s o f i n f o r m a t i o n i s t h e r e c o r d i n g o f c h o i c e b e t w e e n t w o e q u a l l y p r o b a b l e s i m p l e a l t e r n a t i v e s , o n e o r t h e o t h e r i s b o u n d t o h a p p e n - a c h o i c e , f o r e x a m p l e , b e t w e e n h e a d s a n d t a i l s i n t h e t o s s i n g o f a c o i n . W e s h a l l c a l l a s i n g l e c h o i c e o f t h i s s o r t a d e c i s i o n . I f w e t h e n a s k f o r t h e a m o u n t o f i n f o r m a t i o n i n t h e p e r f e c t l y p r e c i s e m e a s u r e m e n t o f a q u a n t i t y k n o w n t o l i e b e t w e e n A a n d B , w h i c h m a y w i t h u n i f o r m a p r i o r i p r o b a b i l i t y l i e a n y w h e r e i n t h i s r a n g e , w e s h a l l s e e t h a t i f w e p u t A = 0 a n d B = 1 , a n d r e p r e s e n t t h e q u a n t i t y i n t h e b i n a r y s c a l e ( 0 o r 1 ) , t h e n t h e n u m b e r o f c h o i c e s m a d e a n d t h e c o n s e q u e n t a m o u n t o f i n f o r m a t i o n i s i n f i n i t e . $ D P " Defining Information - Krippendorf \ K r i p p e n d o r f f - - L i t e r a l l y t h a t w h i c h f o r m s w i t h i n , b u t m o r e a d e q u a t e l y : t h e e q u i v a l e n t o f o r t h e c a p a c i t y o f s o m e t h i n g t o p e r f o r m o r g a n i z a t i o n a l w o r k , t h e d i f f e r e n c e b e t w e e n t w o f o r m s o f o r g a n i z a t i o n o r b e t w e e n t w o s t a t e s o f u n c e r t a i n t y b e f o r e a n d a f t e r a m e s s a g e h a s b e e n r e c e i v e d , b u t a l s o t h e d e g r e e t o w h i c h o n e v a r i a b l e o f a s y s t e m d e p e n d s o n o r i s c o n s t r a i n e d b y a n o t h e r . E . g . , t h e D N A c a r r i e s g e n e t i c i n f o r m a t i o n i n a s m u c h a s i t o r g a n i z e s o r c o n t r o l s t h e o r d e r l y g r o w t h o f a l i v i n g o r g a n i s m . A m e s s a g e c a r r i e s i n f o r m a t i o n i n a s m u c h a s i t c o n v e y s s o m e t h i n g n o t a l r e a d y k n o w n . T h e a n s w e r t o a q u e s t i o n c a r r i e s i n f o r m a t i o n t o t h e e x t e n t i t r e d u c e s t h e q u e s t i o n e r ' s u n c e r t a i n t y . & E Q Krippendorf - Cont & A t e l e p h o n e l i n e c a r r i e s i n f o r m a t i o n o n l y w h e n t h e s i g n a l s s e n t c o r r e l a t e w i t h t h o s e r e c e i v e d . S i n c e i n f o r m a t i o n i s l i n k e d t o c e r t a i n c h a n g e s , d i f f e r e n c e s o r d e p e n d e n c i e s , i t i s d e s i r a b l e t o r e f e r t o t h e m e a n d d i s t i n g u i s h b e t w e e n i n f o r m a t i o n s t o r e d , i n f o r m a t i o n c a r r i e d , i n f o r m a t i o n t r a n s m i t t e d , i n f o r m a t i o n r e q u i r e d , e t c . P u r e a n d u n q u a l i f i e d i n f o r m a t i o n i s a n u n w a r r a n t e d a b s t r a c t i o n . i n f o r m a t i o n t h e o r y m e a s u r e s t h e q u a n t i t i e s o f a l l o f t h e s e k i n d s o f i n f o r m a t i o n i n t e r m s o f b i t s . T h e l a r g e r t h e u n c e r t a i n t y r e m o v e d b y a m e s s a g e , t h e s t r o n g e r t h e c o r r e l a t i o n b e t w e e n t h e i n p u t a n d o u t p u t o f a c o m m u n i c a t i o n c h a n n e l , t h e m o r e d e t a i l e d p a r t i c u l a r i n s t r u c t i o n s a r e t h e m o r e i n f o r m a t i o n i s t r a n s m i t t e r . ( P r i n c i p i a C y b e r n e t i c a W e b ) .
F S Defining Information - Hornung B e r n d H o r n u n g - I n f o r m a t i o n i s t h e m e a n i n g o f t h e r e p r e s e n t a t i o n o f a f a c t ( o r o f a m e s s a g e ) f o r t h e r e c e i v e r . e G T Key Issues in Definition p Information is a means via which the state of uncertainty in an entity can be reduced or changed.
Entropy in thermodynamics is a measure of the state of disorder in a system; entropy in information theory is a measure of the state of disorder in an information processing system.
For information to have effect the entity must understand the message in receives; if the message has no meaning to the entity receiving it, it cannot alter the state of uncertainty in that entity.
If a message which is understood contains information, it will alter the system by changing the state of uncertainty.
Information can be measured. J X
State Changes H U Meaning in Information A k e y i s s u e w h i c h i s o f t e n n o t c o n s i d e r e d i n d e f i n i t i o n s i s t h e m a t t e r o f m e a n i n g c a n t h e m e s s a g e b e u n d e r s t o o d ?
A w o r k o f S h a k e s p e a r e w r i t t e n i n E n g l i s h w i l l b e r i c h i n i n f o r m a t i o n c o n t e n t , b u t o n l y t o a n E n g l i s h s p e a k e r .
A n E n g l i s h s p e a k e r w i t h g o o d k n o w l e d g e o f E l i z a b e t h a n E n g l i s h w i l l p e r c e i v e g r e a t e r i n f o r m a t i o n c o n t e n t t h a n a r e a d e r w i t h o u t ; a r e a d e r w i t h b e t t e r k n o w l e d g e o f p e r i o d h i s t o r y w i l l p e r c e i v e g r e a t e r i n f o r m a t i o n c o n t e n t t h a n a r e a d e r w i t h o u t ; a n d s o o n &
I n m a t h e m a t i c a l t e r m s , t h e r e c e i v e r o f t h e m e s s a g e m u s t b e c a p a b l e o f d e c o d i n g t h e m e s s a g e , t o d e t e r m i n e w h a t i n f o r m a t i o n i t c o n t a i n s .
. f J I V Example 6 D N A e n c o d e s t h e d e f i n i t i o n o f a n o r g a n i s m s s t r u c t u r e a n d f u n c t i o n .
A l t e r t h e D N A c h a i n o f a n e m b r y o a n d t h e r e s u l t i n g o r g a n i s m w i l l b e d i f f e r e n t , p o s s i b l y i n m a n y w a y s .
D o e s t h i s m e a n t h a t w e c a n s p l i c e D N A i n a n y m a n n e r w e c h o o s e ?
F o r t h e D N A t o b e p r o p e r l y d e c o d e d , i t n e e d s t o b e i n s i d e a b i o l o g i c a l e n t i t y w h i c h c a n p r o c e s s ( u n d e r s t a n d ) w h a t t h e D N A t e l l s i t t o d o .
I f t h e s p e c i e s b e t w e e n w h i c h t h e D N A i s b e i n g s p l i c e d a r e t o o d i f f e r e n t , t h e D N A i s n o t l i k e l y t o b e d e c o d e d i n t h e m a n n e r i n t e n d e d , r e s u l t i n g i n a n o n - v i a b l e o r g a n i s m . N Y " S h a n n o n s E n t r o p y x S h a n n o n d e f i n e s e n t r o p y a s a m e a s u r e o f u n c e r t a i n t y , w h e r e H i s e n t r o p y , a n d p i i s t h e p r o b a b i l i t y o f a s y m b o l o r m e s s a g e ( T h e o r e m 2 ) :
T h e l o g a r i t h m i s b a s e 2 .
S h a n n o n s p r o o f i s w e l l w o r t h r e a d i n g r e f e r :
P r o p e r t i e s o f S h a n n o n ' s E n t r o p y
T h i s i s b a s e d o n t h e p a p e r A M a t h e m a t i c a l T h e o r y o f C o m m u n i c a t i o n
# F ;
" F 0 U
0 D U
0 ; ^ e Thermodynamics - Entropy T h e s e c o n d l a w o f t h e r m o d y n a m i c s i e t h e t o t a l e n t r o p y o f a n y i s o l a t e d t h e r m o d y n a m i c s y s t e m t e n d s t o i n c r e a s e o v e r t i m e , a p p r o a c h i n g a m a x i m u m v a l u e i s o f t e n r e p r e s e n t e d a s :
W h e r e k B i s B o l t z m a n n s c o n s t a n t o r k = 1 . 3 8 0 6 5 0 5 x 1 0 "2 3 [ j o u l e s / k e l v i n ] .
T h i s f o r m i s f o r a l l i n t e n t s a n d p u r p o s e s t h e s a m e a s t h a t p r o v e n b y S h a n n o n i n t h e E n t r o p y T h e o r e m .
f Z % p
v X ! % j U
0 Y a Information in a Message G The Entropy Theorem presents the total entropy (information) in the system, across all of the possible messages in the system.
For many problems we are interested in the information in one of the N messages. This can be represented thus:
As is evident, highly probable messages contain little information and vice versa.
H G Z b $ Information vs Likelihood of Message [ c $ Information vs Likelihood of Message O Z H S h a n n o n s C h a n n e l C a p a c i t y M o d e l ( 1 ) P [ H S h a n n o n s C h a n n e l C a p a c i t y M o d e l ( 2 ) T h e m o d e l h a v e f i v e k e y c o m p o n e n t s :
T h e i n f o r m a t i o n s o u r c e w h i c h g e n e r a t e s m e s s a g e s c o n t a i n i n g i n f o r m a t i o n
T h e t r a n s m i t t e r w h i c h s e n d s m e s s a g e s o v e r t h e c h a n n e l .
T h e c h a n n e l a n d a s s o c i a t e d n o i s e s o u r c e , t h i s c o u l d b e a n y n u m b e r o f p h y s i c a l c h a n n e l t y p e s i n c l u d i n g c o p p e r o r o p t i c a l c a b l e , r a d i o l i n k o r a c o u s t i c c h a n n e l .
T h e r e c e i v e r w h i c h d e t e c t s a n d d e m o d u l a t e s m e s s a g e s r e c e i v e d o v e r t h e c h a n n e l .
T h e d e s t i n a t i o n o r i n f o r m a t i o n s i n k w h i c h r e s p o n d s t o m e s s a g e s b y c h a n g i n g i t s i n t e r n a l s t a t e .
I t i s i m p l i c i t l y a s s u m e d t h a t m e s s a g e s s e n t b y t h e i n f o r m a t i o n s o u r c e c a n b e u n d e r s t o o d b y t h e s i n k . j Z j Q \ H S h a n n o n s C h a n n e l C a p a c i t y M o d e l ( 3 ) S h a n n o n d e m o n s t r a t e d t h a t f o r a n o i s y c h a n n e l , i e o n e i n w h i c h r a n d o m n o i s e c o u l d a d d i t i v e l y c o n t a m i n a t e t h e m e s s a g e s f l o w i n g a c r o s s t h e c h a n n e l , t h e c a p a c i t y o f t h e c h a n n e l ( a m o u n t o f i n f o r m a t i o n i t c o u l d c a r r y ) i s d e f i n e d b y ( T h e o r e m 1 7 ) :
W h e r e C i s c h a n n e l c a p a c i t y , W i s c h a n n e l b a n d w i d t h , P i s s i g n a l p o w e r , a n d N i s n o i s e p o w e r .
T h i s e q u a t i o n i s m o s t c o m m o n l y u s e d i n t h e f o l l o w i n g f o r m , a s P / N i s t h e w i d e l y u s e d m e a s u r e o f s i g n a l t o n o i s e r a t i o o r S N R :
O C 1 R ] H S h a n n o n s C h a n n e l C a p a c i t y M o d e l ( 4 ) A s s u m p t i o n ( 1 ) t h e a d d i t i v e n o i s e i s w h i t e t h e r m a l n o i s e i e i t h a s a n o r m a l d i s t r i b u t i o n i n t i m e / p o w e r ;
A s s u m p t i o n ( 2 ) t h e p o w e r o f t h e s i g n a l ( m e s s a g e ) i n t h e c h a n n e l i s t h e a v e r a g e , r a t h e r t h a n p e a k p o w e r ;
A s s u m p t i o n ( 3 ) t h e p o w e r i s n o t l i m i t e d b y t h e t r a n s m i t t e r s p e a k p o w e r r a t i n g ;
M e t r i c s ( 1 ) c h a n n e l c a p a c i t y i s d e f i n e d i n b i t s / s e c ;
M e t r i c s ( 2 ) b a n d w i d t h i s d e f i n e d i n H e r t z ( c y c l e s / s e c ) ;
M e t r i c s ( 3 ) s i g n a l a n d n o i s e p o w e r a r e d e f i n e d i n W a t t s ;
I n n u m e r i c a l a p p l i c a t i o n s w h i c h c o m p u t e c a p a c i t y , i t i s c u s t o m a r y t o u s e t h i s f o r m a s l o g 2 i s o f t e n n o t a v a i l a b l e :
C = B * ( 1 / l o g ( 2 ) ) * l o g ( 1 + S / N ) ; N I Z Z + 0 =
U ^ What Does This Model Tell Us? I t i s p o s s i b l e t o t r a d e b e t w e e n b a n d w i d t h a n d s i g n a l t o n o i s e r a t i o t o a c h i e v e a n i n t e n d e d c a p a c i t y a n e x a m p l e i s i n s p r e a d s p e c t r u m c o m m u n i c a t i o n s ;
C h a n n e l s w i t h s e v e r e l y l i m i t e d b a n d w i d t h b u t v e r y h i g h s i g n a l t o n o i s e r a t i o c a n s t i l l a c h i e v e h i g h c a p a c i t y e x a m p l e a r e v o i c e b a n d m o d e m s r u n n i n g o v e r d i g i t a l s w i t c h e s ;
W h e r e t h e S N R > > 1 , t h e s e c o n d t e r m a p p r o x i m a t e s t h e l o g a r i t h m o f S N R ;
W h e r e t h e S N R < < 1 , t h e s e c o n d t e r m - > 0 , a n d b a n d w i d t h b e c o m e s t h e d o m i n a n t m e a n s o f i m p r o v i n g c a p a c i t y ;
B y m a n i p u l a t i n g t h e b a n d w i d t h a n d S N R o f a c h a n n e l w e c a n m a n i p u l a t e i t s c a p a c i t y , a n d t h u s h o w m u c h i n f o r m a t i o n i t c a n c a r r y . r Z W \ d Channel Capacity Example V _
Key Points What constitutes information in a message depends on the ability of an entity to understand that information.
If a message contains information, an entity receiving it and understanding it will experience a state change which alters its level of uncertainty.
The less likely the message, the greater its information content (Entropy theorem).
The capacity of a channel to carry information depends on the magnitude of interfering noise in the channel, and the bandwidth of the channel (Capacity theorem).
W ` Tutorial a Discussion and debate
Applications of Shannon
(1) review Entropy proof
(2) review Capacity proof
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Wingdings Arial Black Courier New Pixel Microsoft Equation 3.0 J CSE468 Information Conflict Lecturer: Dr Carlo Kopp, MIEEE, MAIAA, PEng # Reference Sources and Bibliography * Defining Information Shannon and Weaver Defining Information - Wiener Wiener Continued # Defining Information - Krippendorf Krippendorf - Cont Defining Information - Hornung Key Issues in Definition State Changes Meaning in Information Example Shannons Entropy Thermodynamics - Entropy Information in a Message % Information vs Likelihood of Message % Information vs Likelihood of Message % Shannons Channel Capacity Model (1) % Shannons Channel Capacity Model (2) % Shannons Channel Capacity Model (3) % Shannons Channel Capacity Model (4) What Does This Model Tell Us? Channel Capacity Example Key Points Tutorial Fonts Used Design Template Embedded OLE Servers
Slide Titles 0 8 @ _PID_HLINKS A 0 < h t t p : / / o k m i j . o r g / f t p / C o m p u t a t i o n / l i m i t s - o f - i n f o r m a t i o n . h t m l 0 h t t p : / / w w w . s v e i b y . c o m / a r t i c l e s / i n f o r m a t i o n . h t m l 9 h t t p : / / c m . b e l l - l a b s . c o m / c m / m s / w h a t / s h a n n o n d a y / p a p e r . h t m l . h t t p : / / p e s p m c 1 . v u b . a c . b e / A S C / I N F O R M A T I O N . h t m l / h t t p : / / w w w . m t m . u f s c . b r / ~ t a n e j a / b o o k / n o d e 5 . h t m l / h t t p : / / w w w . m t m . u f s c . b r / ~ t a n e j a / b o o k / n o d e 6 . h t m l / h t t p : / / w w w . m t m . u f s c . b r / ~ t a n e j a / b o o k / n o d e 5 . h t m l > h t t p : / / c m . b e l l - l a b s . c o m / c m / m s / w h a t / s h a n n o n d a y / s h a n n o n 1 9 4 8 . p d f % _;
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