Multi-Carrier and Spread Spectrum Systems: From OFDM and MC ...

Z n1184 Implementation IssuesReceived yLLR LLR(x 1 ) LLR(x 2 )..... LLR(x 8 )n 1 n 2 n 3 n 4 n 5 n 6 n 7 n 8Bit Nodesy 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8y = x + noisen = 8Zn1m3 = LLR (x 1 )m1 = LLR (x 1 )CheckNodesm 1 m 2 m 3 m 4m = n−k =4Figure 4-43Tanner graph; decoding initializationassigning the channel LLR(x i ) to all of the outgoing edges of every bit node as follows(see Figure 4-43):z ni →m j= LLR(x i ), i = 1,...,N, j = 1,...,deg(bit node n i ), (4.75)where the number of edges corresponding to a node is called the ‘deg’ of that node.4.4.4.2 Tanner Graph’s Check Node RefreshThe outgoing values from the check node m j back to adjacent bit nodes n i is computedas follows (see Figure 4-44):wherew mj →n i= g(z n1 →m j,z n2 →m j,...,z ndeg →m j), (4.76)g(a,b) = sign(a) × sign(b) × min(|a|, |b|) + log(1 + e −|a+b| ) − log(1 + e −|a−b| ).(4.77)Note that if the log terms in the above equation are neglected, the expression of g(a,b)results in the well known simple ‘sum-min’ algorithm used, for instance, by the softoutput-Viterbidecoder to add two soft values.Bit Nodesn 1 n 2 n 3 n 4 n 5 n 6 n 7 n 8m 1W m1 n 4W m1 n 5W m1 n 8Wm 1 n 1CheckNodesZ n4 m 1Z n5Z n8 m 1Z n1 m1m 1 m 2 m 3 m 4Figure 4-44Tanner graph; check node refresh