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Ehavior of error probability with n let us then

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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[error|m, 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[error|m, 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.

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