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Unformatted text preview: source of the p oor
handle we have on the b ehavior of error
probability with N
Let us then consider another criterion for
decoding: maximum likelihood (ML)
select m for which probability of receiving
y N is maximum
PY N ,X N � �
N xN (m)
y ≥ PY N ,X N � �
N xN (m� )
y ∀m� = m
�
Let Ym b e the set of output vectors y N
whose decoding is the message m
The probability of error when the message
m was transmitted is:
Pe,m = � C
y N ∈Ym PY N X N � �
N xN (m)
y Upper b ound on probability
Theorem:
The average probability of decoding error
given that the message m was sent, aver
aged over all p ossible block codes, is b ounded,
for any choice of ρ ∈ [0, 1] by
Ecodebooks[Pe,m] ≤ (M − 1)ρ ⎡
⎤1+ρ
�
�
�
�1
� ⎢�
NP
N xN 1+ρ ⎥
PX N x
⎣
⎦
Y N X N y
y N xN Proof:
The probability of error given that m was
transmitted averaged over all p ossible codes
is:
Ecodebooks[Pe,m] = � � xN (m) y N
�
�
�
�
N (m) P
N xN (m)
PX N x
Y N X N y
P r[errorm, X N = xN (m), Y N = y N ] Upper b ound on probability
Proof continued
m, xN (m), y N , For a given
b e the event that
PY N X N � �
N xN (m)
y let A ≤ PY N  X N � m�, xN (m), y N � �
N xN (m� )
y an error � ccurs when at least one of the
o
�
� , xN (m), y N , m �= m� takes place
events A m
therefore P r[errorm, X N = xN (m), Y N = y N ]
⎛ ⎞ �...
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This document was uploaded on 03/19/2014 for the course EECS 6.441 at MIT.
 Spring '10
 MaruielMedard

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