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Unformatted text preview: lutional encoder with 2 g1 = (7) and g2 = (5).; (b) illustration of trellis termination. We now develop the BCJR algorithm. Assume that the initial state of the encoder (1) is composed of all zeros, so that the trellis at stage j = 0 is constrained to be S0 . (1) Additionally, assume that the state at stage N is constrained to be SN . This latter constraint is equivalent to xj = 0; j = N − (K − 2), · · · , N . That is, the trellis is terminated to the zero state. Figure 5(b) illustrates the trellis diagram under these constraints. To use established notation related to the BCJR algorithm, let λj
(l,m) = P (Sj /r1,N ) (m) (29) be the state m probability at stage j , conditioned on all the data r1,N , for (m) (l) (0,m) (m) (0) Sj ∈ Sj ; l = 0 or 1 (e.g. λj is deﬁned only for Sj ∈ Sj ). Let
(l,m) αj = (m) P (Sj /r1,j ) P (Sj , r1,j ) = P (r1,j ) (m) (30) be the forward state m probability at stage j , conditioned on the data up to stage (m) (l) j , r1,j , for Sj ∈ Sj ; l = 1 or 2. Let
m βj = P (rj +1,N /Sj ) P (rj +1,N /r1,j ) (m) (31) Kevin Buckley - 2007 be a backward data probability ratio given Sj . It...
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- Spring '09