BCJR-trellis

# BCJR-trellis - S 1 = c 1 h 1 ; namely, • ‚ and • 1...

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1 BCJR Trellis A commonly used trellis for linear block codes, known for its simplicity and its optimality [1][2], is the so-called BCJR trellis . To derive the BCJR trellis for an arbitrary binary linear code, we start with the equation cH T = 0 , where H is m × n . If we let h j represent the j th column of H , we have c 1 h 1 + c 2 h 2 + ··· + c n h n = 0 . This equation leads directly to a trellis, for its solutions yield the list of codewords just as the paths through a code’s trellis gives the codeword list. Thus, the possible states at the th trellis stage can be computed as S = c 1 h 1 + c 2 h 2 + ··· + c h . Further, since the h j are m -vectors, there are at most 2 m states for trellis stages = 1 , 2 ,...,n - 1, and of course S 0 = S n = 0 . Example 1.1: Following [1], let H = 1 1 0 1 0 0 1 1 0 1 . Then at stage 1 there are two possible states derived from

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Unformatted text preview: S 1 = c 1 h 1 ; namely, • ‚ and • 1 ‚ , corresponding to c 1 = 0 and c 1 = 1 , respectively. At stage 2, there are four possible states derived from S 2 = c 1 h 1 + c 2 h 2 ; and so on. The resulting trellis is given in Figure 1.1. 2 Chapter 1. BCJR Trellis Figure 1.1 BCJR trellis for example code (from [1]). REFERENCES [1] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory , pp. 284-287, Mar. 1974. [2] R. McEliece, “On the BCJR trellis for linear block codes,” IEEE Trans. Inf. Theory , pp. 1072-1092, July1996....
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## This note was uploaded on 10/10/2009 for the course ECE 637 taught by Professor Staff during the Spring '08 term at Arizona.

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BCJR-trellis - S 1 = c 1 h 1 ; namely, • ‚ and • 1...

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