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9) Obviously, P {a|r} = ΛK (µK ). Based on the trellis representation of the underlying FSM, the partial metrics, associated with the trellis states, can be computed recursively. , M = 2. The path metrics associated to states µk and (2) (1) (2) µk are indicated as Λk (µk ) and Λk (µk ), respectively. The VA associates (1) (1) (1) to state µk+1 (the common ending state of both transitions tk = (µk , ak ) (2) (2) (2) and tk = (µk , ak )) the following path metric: (1) (1) (1) (2) (2) (2) Λk+1 (µk+1 ) = max Λk (µk ) + λk (µk , ak ), Λk (µk ) + λk (µk , ak ) .
The following matrix gives an example of a possible LDPC code parity check M. Franceschini, G. Ferrari and R. 1007/978-3-540-69457-1_3, © Springer-Verlag Berlin Heidelberg 2009 37 38 matrix 0 1 0 0 0 0 1 0 1 0 CHAPTER 3. LOW-DENSITY PARITY-CHECK CODES with dv = 3 and dc = 6: 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 0 0 0 0 .
2, an illustrative representation of a folding of a tree into a trellis is shown. In the remainder of this chapter, the number of states of the en- 24 CHAPTER 2. TRELLIS-BASED DETECTION TECHNIQUES coder/modulator FSM will be indicated by Sc . Denoting by tk (µk , ak ) a transition in the trellis diagram, a branch metric associated with this transition can be defined as follows: λk (µk , ak ) = λk (tk ) ln p(rk |µk , ak ) + ln P {ak }. 7), the a posteriori sequence probability can be written as follows: K−1 P {a|r} ∼ λk (µk , ak ).