# Basic Definations

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```					                INFORMATION THEORY

 Communication theory deals with systems for transmitting information
from one point to another.

 Information theory was born with the discovery of the fundamental
laws of data compression and transmission.

INFORMATION THEORY                             1
 Introduction

Information theory answers two fundamental questions:

•   What is the ultimate data compression?
•   What is the ultimate transmission rate?

But its reach is beyond Communication Theory. In early days it was
thought that increasing transmission rate over a channel increases the
error rate. Shannon showed that this is not true as long as rate is
below Channel Capacity.

Shannon has further shown that random processes have an irreducible
complexity below which they can not be compressed.

INFORMATION THEORY                                2
Information Theory (IT) relates to other fields:

•   Computer Science: shortest binary program for computing a string.
•   Probability Theory: fundamental quantities of IT are used to estimate
probabilities.
•   Inference: approach to predict digits of pi. Infering behavior of stock
market.
•   Computation vs. communication: computation is communication limited
and vice-versa.

INFORMATION THEORY                            3
It has its beginning at the start of the century but it really took of after
WW II.

•   Weiner: extracting signals of a known ensemble from noise of a
predictable nature.

•   Shannon: encoding messages chosen from a known ensemble so that
they can be transmitted accurately and rapidly even in the presence of
noise.

IT: The study of efficient encoding and its consequences in the form of
speed of transmission and probability of error.

INFORMATION THEORY                                4
 Historical Perspective

•   Follows S. Verdu, “Fifty years of Shannon Theory,” IT-44, Oct. 1998,
pp. 2057-2058.

•   Shannon published “ A mathematical theory of communication” in
1948. It lays down fundamental laws of data compression and
transmission.

•   Nyquist (1924): transmission rate is proportional to the log of the
number of levels in a unit duration.
- Can transmission rate be improved by replacing Morse by an
„optimum‟ code?

•   Whitlaker (1929): loseless interpolation of bandlimited function.

•   Gabor (1946): time-frequency uncertainty principle.

INFORMATION THEORY                             5
•   Hartley (1928): muses on the physical possibilities of transmission
rates.

- Introduces a quantitative measure for the amount of information
associated with n selection of states.

H=n log s

where s = symbols available in each selection.
n = # of selections.

- Information = outcome of a selection among a finite number of
possibilities.

INFORMATION THEORY                            6
 Data Compression

• Shannon uses the definition of entropy
n
H   pi log pi
i 1

as a measure of information.
Rationals: (1) continuous in prob.
(2) increasing with n for equiprobable r.v.
(3) additive – entropy of a sum of r.v. is equal to the
sum of entropies of the individual r.v.
• Entropy satisfies for memoryless sources:
Shannon Theorem 3: Given any  0 and   0 , we can find No
such that sequences of any length N  N 0 fall into two classes:
(1) A set whose probability is less than 
(2) The reminder set, all of whose members have probabilities
{p} satisfying                     1
log p
H             
N

INFORMATION THEORY                          7
 Reliable Communication

•   Shannon: …..redundancy must be introduced to combat the particular
noise structure involved … a delay is generally required to approach
the ideal encoding.
•   Defines channel capacity
C  max( H ( X )  H ( X / Y ))
• It is possible to send information at the rate C through the channel
with as small a frequency of errors or equivocation as desired by
proper encoding. This statement is not true for any rate greater than C.
•   Defines differential entropy of a continuous random variable as a
formal analog to the entropy of a discrete random variable.
•   Shannon obtains the formula for the capacity of:
- power-constrained
- white Gaussian channel
- flat transfer function
 PN 
C  W log      
 N 

INFORMATION THEORY                              8
•   The minimum energy necessary to transmit one bit of information is
1.6 dB below the noise psd.

•   Some interesting points about the capacity relation:
- Since any strictly bandlimited signal has infinite duration, the rate of
information of any finite codebook of bandlimited waveforms is equal
to zero.
- Transmitted signals must approximate statistical properties of white
noise.

•   Generalization to dispersive/nonwhite Gaussian channels by Shannon‟s
“water-filling” formula.

•   Constraints other then power constraints are of interest:
- Amplitude constraints
- Quantized constraints
- Specific modulations.

INFORMATION THEORY                                  9
 Zero-Error Channel Capacity

•   Example of typing a text: a non-zero probability of making an error
with the prob. = 1 as the length increases.

•   By designing a code that takes into account the statistics of the typist‟s
mistakes, the prob. of error can be made  0.

•   Example: consider mistakes made by mistyping neighboring letters.
The alphabet { b, I, t, s} has no neighboring letters, hence will have
zero probability of error.

•   Zero-error capacity: the rate at which information can be encoded with
zero prob. of error.

INFORMATION THEORY                               10
 Error Exponent

•   Rather than focus on the channel capacity, study the error probability
(EP) as a function of block length.

•   Exponential decrease of EP as a function of blocklength in Gaussian,
discrete memoryless channel.

•   The exponent of the minimum achievable EP is a function of the rate
referred to as reliability function.

•   An important rate that serves as lower bound to the reliability function
is the cutoff rate.

•   Was long thought to be the “practical” limit to transmission rate.

•   Turbo codes refuted that notion.

INFORMATION THEORY                               11
 ERROR CONTROL MECHANISMS

 Error Control Strategies

•   The goal of „error-control‟ is to reduce the effect of noise in order to
reduce or eliminate transmission errors.
•   „Error-Control Coding‟ refers to adding redundancy to the data. The
redundant symbols are subsequently used to detect or correct
erroneous data.

INFORMATION THEORY                                12
•   Error control strategy depends on the channel and on the specific
application.

- Error control for one-way channels are referred to as forward error
control (FEC). It can be accomplished by:
* Error detection and correction – hard detection.
* Reducing the probability of an error – soft detection
- For two-way channels: error detection is a simpler task that error
correction. Retransmit the data only when an error is detected:
automatic request (ARQ).

•   In the course, we focus on wireless data communications, hence we
will not delve in error concealment techniques such as interpolation,
used in audio and video recording.
•   Error schemes may be priority based, i.e., providing more protection to
certain types of data that others. For example, in wireless cellular
standards, the transmitted bits are divided in three classes: bits that
get double code protection, bits that get single code protection, and
bits that are not protected.
INFORMATION THEORY                               13
 Block and Convolutional Codes

•   Error control codes can be divided into two large classes: block codes
and convolutional codes.
•   Information bits encoded with an alphabet Q of q distinct symbols.

•   Designers of early digital communications system tried to improve
reliability by increasing power or bandwidth.

INFORMATION THEORY                                14
•   Shannon taught us how to buy performance with a less expensive
resource: complexity.

•   Formal definition of a code C: a set of 2k n-tuples.

•   Encoder: the set of 2k pairs (m,c), where m is the data word and c
is the code word.

•   Linear code: the set of codewords is closed under modulo-2 addition.

•   Error detection and correction correspond to terms in the Fano
inequality:

H ( X / Y )  H (e)  P(e) log(2k  1)
- Error detection reduces     H (e)
- Error correction reduces   P (e) log(2k  1)

INFORMATION THEORY                             15
 BASIC DEFINITIONS

Define Entropy, Relative Entropy, Mutual Information

 Entropy, Mutual Information
A measure of uncertainty of a random variable.
Let x be a discrete random variable (r.v.) with alphabet
A ( x  A) and probability mass p(x) = Pr {X=x}.

•   (D1) The entropy H(x) of a discrete r.v. x is defined
H ( X )   p ( x) log p ( x) bits
x A

where log is to the base 2.
•   Comments: (1) simplest example: entropy of a fair coin = 1bit.
(2) Adding terms of zero probability does not change
entropy (0log 0 = 0).
(3) Entropy depends on probabilities of x, not on actual
values.
(4) Entropy is H(x) = E [ log 1/p(x) ]

INFORMATION THEORY                         16
 Properties of Entropy

•   (P1)    H(x)  0
0  p(x)  1    log [ 1/p(x) ]  0
•   [E]    x=1       p
0 1-p
H(x) = - p log p – (1-p) log(1-p)
= H(p)

INFORMATION THEORY   17
•   [E]      x=a      1/2
b      1/4
c      1/8
d      1/8

H(x) = ½log½ - ¼log¼ - 1/8log 1/8 - 1/8log 1/8 = 1.75 bits

 Another interpretation of entropy

Use minimum number of questions to determine value of X:
Is X=a
 no
Is X=b
no

Is X=c
It turns out that the expected number of binary questions is 1.75.

INFORMATION THEORY                          18
•   (D2) The joint entropy H(X,Y) is defined
H ( X , Y )     p ( x, y ) log p ( X , Y )
x A yB

or
H ( X , Y )   E log p ( X , Y )
where ( X , Y )       p ( x, y )

•   (D3) Conditional entropy H(Y|X)

H (Y | X )         p ( x) H (Y
x A
| X  x)

   p ( x )  p ( y | x ) log p ( y | x )
x A          y B

    p ( x, y ) log p ( y | x )
x A yB

  E[log p ( y | x )]

INFORMATION THEORY                        19
•   (P2)        (chain  rule)
H ( X , Y )  H ( X )  H (Y | X )

H ( X , Y )   p ( x, y ) log p ( x, y )
x   y

  p ( x, y ) log p ( x ) p ( y | x )
x   y

  p ( x, y ) log p ( x )   p ( x, y ) log p ( y | x )
x   y                         x   y

  p ( x) log p ( x )   p ( x, y ) log p ( y | x )
x                     x   y

H ( X , Y )  H ( X )  H (Y | X )
Entropy: A measure of uncertainty of a r.v.
The amount of information required on the average to
describe the r.v.
Relative entropy: A measure of the distance between two
distributions.

INFORMATION THEORY                             20
•   (D4) Relative entropy or Kullback Leibler distance between two
probability mass functions p(x) and q(x) is defined
p ( x)
D( p       q)       x
p ( x ) log
q( x)

Relative entropy is 0 iff p=q
Mutual information: A measure of the amount of information one r.v.

•   (D5) Given two r.v. , X , Y p( x, y) and marginal distributions p(x),
p(y), the mutual information is the relative entropy between the joint
distribution p(x,y) and the product distribution p(x)p(y):
p ( x, y )
I ( X ;Y )      x, y
p ( x, y ) log
p ( x) p( y )
 D ( p ( x, y )        p ( x ) p ( y ))

INFORMATION THEORY                            21
A  {0,1
•    (E)           }
p (0)  1  r , p (1)  1  s, q (0)  1  s, q (1)  s

1 r         r
D( p       q )  (1  r ) log         r log
1 s         s
1       1
r        ,s    
2       4
D( p       q )  0.2075 bits
D(q        p )  0.1887 bits
In general D( p q)  D(q               p)
 Properties of MI:
p ( x, y )
I ( X ;Y )     p ( x, y ) log
x, y                 p( x) p( y )
p( x / y)
    p ( x, y ) log
x, y                  p( x)
   p ( x, y ) log p ( x)   p ( x, y ) log p ( x / y )
x, y                           x, y

                                
   p ( x ) log p ( x )     p ( x, y ) log p ( x / y ) 
x                        x, y                           

INFORMATION THEORY                           22
•   (P1) I(X,Y) = H(X) – H(X,Y)

Interpretation: Mutual Information (MI) is the reduction in the
uncertainty of X due to the knowledge of Y.

•   (P2) I(X,Y) = H(Y) – H(Y|X) = I(Y,X)

•   (P3) I(X,X) = H(X)    (no reduction of certainly)

Since H(X,Y) = H(X) + H(Y|X) (chain rule), it follows that
H(Y|X) = H(X,Y) – H(X), hence

•   (P4) I(X,Y) = H(X) + H(Y) – H(X,Y)

INFORMATION THEORY                         23
 Multiple Variables – Chain Rules

In this section, some of the results of the previous section are
extended to multiple variables.
•   (T1) Chain Rule for Entropy:
Let X 1 , X 2 ,..., X n p ( x1 , x2 ,..., xn )
then
n
H ( X 1 , X 2 ,..., X n )     H (X
i 1
i   | X i 1 ,..., X 1 )

H ( X1, X 2 )  H ( X1 )  H ( X 2 | X1 )
H ( X1, X 2 , X 3 )  H ( X1 )  H ( X 2 , X 3 | X1 )
 H ( X1 )  H ( X 2 | X1 )  H ( X 3 | X 2 , X1 )
•   (D6) The conditional mutual information of random variables X
and Y given Z is defined by
I(X;Y|Z) = H(X,Z) – H(X|Y,Z)

INFORMATION THEORY                                   24
•   (T2) Chain rule for mutual information:
n
I ( X 1 ,..., X n )   I ( X 1 ; Y | X i 1 ,..., X 1 )
i 1

Proof :
p ( x1 , x2 , y )
I ( X1, X 2 )      
x1, x2 , y
p ( x1 , x2 , y ) log
p ( x1 , x2 ) p ( y )
use
p ( X 1 , X 2 ; Y )  p ( x1 , x2 | y ) p ( y ), then
p ( x1 , x2 | y )
I ( X1, X 2 ;Y )      
x1, x2 , y
p ( x1 , x2 , y ) log
p ( x1 , x2 )
                                            
                p ( x1 , x2 , y ) log p ( x1 , x2 )     p ( x1 , x2 , y ) log p( x1 , x2 | y ) 
 x x ,y                                     
x1, x2 , y                                          1, 2                                       
                                            
   p ( x1 , x2 ) log p ( x1 , x2 )     p ( x1 , x2 , y ) log p( x1 , x2 | y ) 
 x x ,y                                     
x1, x2                               1, 2                                       
 H ( X1, X 2 )  H ( X1, X 2 | Y )
 H ( X 1 )  H ( X 2 | X 1 )  [ H ( X 1 / Y )  H ( X 2 | X 1 , Y )]
I ( X 1 , X 2 | Y )  I ( X 1; Y )  I ( X 2 ; Y | X1 )

This can be generalized to arbitrary n.

INFORMATION THEORY                                                              25
•   (D7) The conditional relative entropy D(p(y|x)||q(y|x)) is the relative
entropy between the corresponding conditional distributions averaged
over x:
p( y | x)
D ( p ( y | x) || q ( y | x))   p ( x) p ( y | x) log
x         y                  q( y | x)

•   (T3) Chain rule for relative entropy

D ( p ( x, y ) || q ( x, y ))  D ( p ( x) || q ( x))  D( p ( y | x) || q( y | x))
Proof:
p ( x, y )
D ( p ( x, y ) || q ( x, y ))   p ( x, y ) log
x, y                 q ( x, y )
p ( x)                    p( y | x)
  p ( x, y ) log               p ( x, y ) log
x, y                 q ( x) x , y              q( y | x)
 D ( p ( x) || q ( x))  D( p ( y | x) || q ( y | x))

INFORMATION THEORY                                         26
 Jensen’s Inequality

•    (D8) f(x) is convex over interval (a,b) if

x1 , x2  (a, b), 0    1
f ( x1  (1   ) x2 )   f ( x1 )  (1   ) f ( x2 )

Strictly convex if the strict inequality holds.

•    (D9) f(x) is concave if –f is convex.

Convex function always lies below any chord (straight line connecting
two points on the curve). Convex function are very important in I.T..

INFORMATION THEORY                    27
Simple results for convex functions:
• (T4) If f "( x)  0 the function is convex
Proof:                                                           1
f ( x)  f ( x0 )  f ' ( x0 )( x1  x0 )  f '' ( x* )( x  x0 ) 2
Taylor Expansion:                                                 2
where        x0  x*  x
let      x0   x1  (1   ) x2
Since the last term in the Taylor expansion is non-negative,

x  x1  f ( x1 )  f ( x0 )  f ' ( x0 )( x1  x2 )
 f ( x0 )  f ' ( x0 )[(1   )( x1  x2 )]
Similarly x  x2  f ( x2 )  f ( x0 )  f ( x0 )[ ( x2  x1 )]
'

 f ( x1 )  f ( x0 )  f ' ( x0 ) (1   )( x1  x2 )
(1   ) f ( x2 )  (1   ) f ( x0 )  f ' ( x0 ) (1   )( x2  x1 )
 f ( x1 )  (1   ) f ( x2 )  f ( x0 )
Using        x0   x1  (1   ) x2 ,

The relation meets the definition of a convex function.

INFORMATION THEORY                                          28
•   (T5) (Jensen’s Inequality)
(1) If f is convex and X is a r.v.
Ef ( X )  f ( EX )
(2) If f is strictly convex and E f(X) = f(EX)
then, X = EX, i.e., X is a constant.
Proof: Let the number of discrete points be 2: X1, X2
From the definition of convex functions:
E[ f ( X )]  p1 f ( X )  p2 f ( X 2 )  f ( p1 X 1  p2 X 2 )  f ( EX )
Induction: suppose the theorem is true for k-1 points.

    pi /(1  pk )
let   pi
, this makes pi a set of probabilities.
k                                           k 1
E ( f ( X ))    
i 1
pi f ( xi )  pk f ( xk )  (1  pk )  pi ' f ( xi )
i 1
k 1
 pk f ( xk )  (1  pk ) f (  pi ' X i )
i 1
k 1
 f ( pk X k  (1  pk )  pi ' X i )
i 1
k
 f (  pi X i )
i 1

From Jensen‟s inequality follow a number of fundamental IT theorems
INFORMATION THEORY                                  29
•   (T6) (Information inequality) p(X), q(x)
With equality iff D( p || q)  0
Proof:           p( x)  q( x) x
p( x)
 D ( p || q )    p ( x ) log
x A            q( x)
q( x)
   
x A
p ( x ) log
p( x)
p( x)
 log            p( x)
x A         q( x)
 log  q ( x )  log1  0
x A

If p( x)  q( x), equality is clearly obtained.
If equality holds, it means that q( x)  p( x) (since the log is strictly
concave).

INFORMATION THEORY                                 30
•   (T7) (Non-negativity of MI)

r.v. X , Y       I ( X ;Y )  0
I(X;Y) = 0 iff X, Y are independent.

Proof: Follows from the relation I ( X ; Y )  D( p( x, y) || p( x) p( y))  0
From the information inequality, the equality holds iff
p(x,y) = p(x)p(y), i.e., X, Y are independent.

Let |A| be the number of elements in the set A.
•   (T8) H ( X ) | A |, H ( X )  log | A | iff
X has a uniform distribution over A.

Proof: Let u(X) = 1/ |A| be the uniform distribution.
p ( x)
D( p || u )   p ( x) log            p ( x) log( A p ( x))
u ( x)
 log A  H ( X )  0

Interpretation: uniform distribution achieves maximum entropy.

INFORMATION THEORY                                     31
•   (T9) (Conditioning reduces entropy)
H(X | Y)  H(X )
H(X|Y) = H(X) iff X and Y are independent
Proof: It follows from 0  I ( X , Y )  H ( X )  H ( X | Y )
Interpretation: on the average, knowing about Y can only reduce the
1
p ( x)   p ( X , Y )  p ( x  1)   p (1, y ) 
y                            y               8
7
p ( x  2)   p (2, y ) 
y               8
1 7
H ( X )  H ( , )  0.544 bits
8 8
3    3
H ( X | Y  1)   p ( x |1) log p ( x |1)  0   log  0.3113
x                              4    4
1    1 1    1 3
H ( X | Y  2)   p ( x | 2) log p ( x | 2)   log  log 
x                             8    8 8    8 4
3                  1
H ( X | Y )  H ( X | Y  1)  H ( X | Y  2)  0.4210
4                  4

The uncertainty of X is decreased if Y=1 is observed, it is increased if Y=2
is observed, and is decreased on the average.

INFORMATION THEORY                                           32
•   (T10) (Independence bound for entropy)

X 1 , X 2 ,..., X n        p ( X 1 , X 2 ,..., X n ), then
n
H ( X 1 , X 2 ,..., X n )       H (X
i 1
i   )

equality iff Xi are independent.
Proof:
H ( X1, X 2 )  H ( X1 )  H ( X 2 | X1 )

Chain Rule

 H ( X1 )  H ( X 2 )

T9

INFORMATION THEORY                        33
•   (T11) (Log Sum Inequality)
ai , ..., an , bi , ..., bn  0
n

n
ai     n                       a     i

 ai log b  ( ai ) log                i 1
n
i 1      i   i 1
b     i
Equality iff ai/bi = const               i 1

Proof:
let
 i  0 and               i   1
f (t )  t log t
is strictly convex since its second
derivative > 0, hence by Jensen‟s inequality
     i   f (ti )  f (   i ti )
set
 i  bi /  b j , t  ai / bi ,
i

then
bi   a     a                bi   a           bi   a
         f (ti )            * i log i                 * i log         * i
 b j bi                     b j bi          b j bi
i
bi
ai     a                      ai             ai
          log i                       log 
i     bj    bi                i    bj     i      bj
i

INFORMATION THEORY                        34
•   (T12) Convexity of Relative Entropy

D( p1  (1   ) p2 ||  q1  (1   )q2 )   D( p1 || q1 )  (1   )( p2 || q2 )
Proof: log sum inequality
 p1  (1   ) p2           p                   (1   ) p2
LHS  ( p1  (1   ) p2 )log                           p1 log 1  (1   ) p2 log
 q1  (1   )q2             q1                 (1   ) q2
  D( p1 || q1 )  (1   ) D( p2 || q2 )
•   (T13) Concavity of Entropy                         H(p) is a concave function of p.

Proof: H(p) = log A  - D(p||u)
since D is convex , H is concave.

•   (D10) The r.v. X, Y, Z form a Markov Chain X Y  Z if
 ( x, y, z)  p( x) p( y | x) p( z | y)
(z is conditionally independent of x)

INFORMATION THEORY                                          35
•   (T14) Data Processing Inequality:
If X Y  Z, then
I(X;Y)  I(X;Z)
Proof: Chain Rule for information (T2 slide 46)
I(X;Y,Z) = I(X;Z) + I(X;Y|Z)
also   I(X;Y,Z) = I(X;Y) + I(X;Z|Y)
since X, Z are independent given Y , I(X;Z|Y) = 0
It follows
I ( X ;Y )  I ( X ;Y , Z )  I ( X ; Z )  I ( X ;Y | Z )
 I(X ,Z)
Equality iff I(X;Y|Z) = 0 i.e. X  Y Z also form a Markov Chain.

In particular if Z = g(Y) we have
I ( X ; Y )  I ( X , g (Y ))
A function of the data Y can not increase the information about X.

INFORMATION THEORY                              36
 Application – Sufficient Statistic

Use data processing inequality to clarify idea of sufficient statistic.

•   (D11) A function T(X) is a sufficient statistic relative to the family
{ f ( x)} if
X is independent of  given T(X), i.e. ,   T ( X )  X
T(X) provides all info on  .
In general, we have             X T(X )
{ f ( x)} a family of distributions, X a sample from a dist.
T(X)         a function of the sample.
Hence, by the data processing inequality

I ( ; T ( X ))  I ( ; X )
For a sufficient statistic I ( ; X )  I ( ; T ( X )) which means that MI is
preserved.

INFORMATION THEORY                               37
•   Example
(1) X 1 ,..., X n , X i  {0,1} the distribution

parameter is   Pr( X i  1)
define T ( X 1 ,..., X n )   X i
n

i 1

How to show independence of X and  ? Show that given T, all
sequences with k ones are equally likely, independent of  .
                          n

Pr ( X 1 ,..., X n )  ( x1 ,..., xn ) |  X i  k 
                                      i 1      
 1
 n 

                   X i  k prob. of k out of n
 k 
0
                     otherwise

thus
      X
T
i    ( X 1 ,..., X n )

forms a Markov chain and T is a sufficient statistic.

INFORMATION THEORY                          38
 Fano’s Inequality

Suppose we know r.v. Y and wish to guess the value of correlated r.v.
X. Intuition says that is if H(X|Y) = H(X), knowing Y will not help.
Conversely, if H(X|Y) = 0, then X can be estimated with no error. We
now consider all the cases in between.

Let   X   p( X ) . Observe Y, p(y|x). From Y calculate

g (Y )  X

X    Y  X form a Markov chain (X hat is
conditionally independent of X). Probability of error is defined.

P  Pr{ X
e                      X}

INFORMATION THEORY                          39
•   (T15) Fano‟s Inequality

H ( P )  P log( A  1)  H ( X | Y )
e        e
weaker inequality
1  P log( A)  H ( X | Y )
e

H (X | Y ) 1
P 
e
log A

where |A| is the set size.

Proof:     Define error event E = 1   XX

0 XX
H(E,X|Y) = H(X|Y) + H(E|X,Y)                (*)
               =0
chain rule no error if X is known.

INFORMATION THEORY           40
 Alternative Expansion

H ( E, X | Y )  H ( E | Y )  H ( X | E, Y )

conditioning     H ( E | Y )  H ( E )  Pe log Pe  (1  Pe )log(1  Pe )  H ( Pe )
reduces entropy

H ( X | E , Y )  Pr( E  0) H ( X | Y , E  0)  Pr( E  1) H ( X | Y , E  1)
 (1  Pe )0  Pe log(| A | 1)
(**)

      
Given E=1  X  X , then H(X|Y,E = 1) is bound by the number of
       
remaining outcomes log (|A|-1) (T8 on slide 50)
From (*) and (**) we get Fano‟s inequality

H ( X | Y )  H ( Pe )  Pe log(| A | 1)

INFORMATION THEORY                                       41

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