The eq 33 denominator normalizes j over l and m

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Unformatted text preview: has been shown, [?], that λj (l,m) (m) 15 = αj (l,m) m βj . (32) This suggests the following algorithm: ∗ Forward pass: For each j = 1, 2, · · · , N , compute the forward state probabilities (i,m′ ) M 1 l ′ (l,m) m′ =1 i=0 γ (rj , m , m) αj −1 (33) αj = (i,m′ ) M 1 M 1 l ′ m=1 l=0 m′ =1 i=0 γ (rj , m , m) αj −1 where for each j , m′ denotes the state index for stage j − 1 and m denotes the index for stage j . The γ (rj , m , m) = l ′ 0 (m) (m′ ) 1 P (rj /Sj , Sj −1 ) 2 Sj −1 , Sj f easible otherwise (m′ ) (m) (34) (l,m) are state transition probabilities1 . The Eq (33) denominator normalizes αj over l and m. ∗ Backward pass: For each j = N, N − 1, · · · , 1, compute the backward data probabilities (m) βj = M m=1 M 1 m′ =1 i=0 1 M l=0 m′ =1 γ l (rj +1 , m, m′ ) βj +1 1 i=0 (m′ ) (i,m′ ) γ l (rj +1, m′ , m) αj , (36) and the state probabilities λj (l,m) = αj (l,m) m βj . (37) Eqs(33 - 37) constitute the BCJR algorithm, where as stated earlier λj (m) (l) (l,m) = P (Sj /...
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This document was uploaded on 10/12/2009.

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