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(Chapter 14): Trellis based decoding of linear block codes

(Chapter 14): Trellis based decoding of linear block codes. Previous discussion on trellis based decoding of convolutional codes applies directly to any code with a (moderately complex) trellis Viterbi SOVA BCJR MAP Log-MAP Max-log-MAP etc.

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(Chapter 14): Trellis based decoding of linear block codes

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  1. (Chapter 14): Trellis based decoding of linear block codes • Previous discussion on trellis based decoding of convolutional codesapplies directly to any code with a (moderately complex) trellis • Viterbi • SOVA • BCJR • MAP • Log-MAP • Max-log-MAP etc. • Irregular trellises: May be more difficult to design HW decoder

  2. Branch complexity Example: (8,4) RM Na = 44 Nc = 11 • Let Ii=2 if there is an information symbol corresponding to output bit i, Ii =1 otherwise. • Then the branch complexity is the number of one-symbol branches= the number of additions in the Viterbi algorithm= Na =  i=0...n-1Ii2i • Let Ji=0 if there is only one branch entering each state at time i, Ji= 1 otherwise • Then the number of comparisons in the Viterbi algorithm= Nc = i=0...n-1Ji2i+1

  3. Trellis sectionalization • Sectionalization combines adjacent bits and provides a trellis with fewer time instances, in order to simplify the decoders. • Let ={t0, t1,…,t}, for   n. Delete all state spaces (and their adjacent branches) at time instances not in , and connect every pair of states, one in tj and one in tj+1, iff there were a path between these states in the original trellis, label the new branches by the old path labels. • A set of parallel paths forms a composite branch

  4. Sectionalization: Example Three trellis sections

  5. Decoding of Sectionalized trellis • For each composite branch, • find the best single path among those that form the composite branch. Make a note of who is the winner. • The branch metric of the composite branch is by definition equal to the branch metric of this winning branch. • The metrics of the composite branch is added to the path metrics of the state it starts in. • The rest proceeds as the ordinary Viterbi algorithm

  6. Complexity of sectionalized trellis decoding • Depends on choice of section boundaries • Optimal sectionalization: Minimum number of additions+comparisons (Lafourcade & Vardy) • (x,y)= number of computations to calculate section from time x to time y • min(x,y)= minimum number of computations to calculate section from time x to time y • min(0,y)=min{ (0,y) , min0<x<y{min(0,x) + (x,y)} } • Algorithm: • Calculate (x,y) for 0x<yn • Use these values to calculate min(0,y) for successive values of y

  7. Example

  8. Recursive ML decoding • Sectionalized trellis • Recursive combining of the path metrics • Start with minimal trellis for an (n,k) block code • Consider state sx at time x and state sy at time y • L(sx,sy) is the set of parallel branches from sx to sy (forming one composite branch) • L(sx =zero state,sy =zero state) is a linear block code Cx,y • Each L(sx,sy) is a coset of Cx,y , that is a coset in px,y(C) /Cx,y • The number of distinct such sets is = 2k(px,y(C))-k(Cx,y) • So each coset appears (Chapter 9)= 2k - k(C0,x) - k(Cy,n) - k(px,y(C)) times as a composite path

  9. Composite Path Metric Table • For each distinct L(sx,sy), store in CPMTx,y • Label: Best path within composite branch, l(L(sx,sy)) • Metric: The metric of the best path, m(L(sx,sy)) • Eventually, CPMT0,nwill contain just one path • Construct CPMTx,y by • For small y-x: Compute metrics for each path in composite path.This requires 2k(Cx,y)(y-x-1) additions 2k(Cx,y)-1 comparisons= 2k(Cx,y)(y-x) – 1 operations. • For larger y-x : Recursively, i. e. from CPMTx,z and CPMTz,y

  10. Recursive MLD (continued) • L(sx,sy) = szL(sx,sz) L(sz,sy) • m(L(sx,sy)) = maxsz { m( L(sx,sz)) +m( L(sz,sy)) } • l(L(sx,sy)) = l( L(sx,s*z)) l( L(s*z,sy)) • This computation requires z additions and z -1 comparisons, • where z = the number of states at time z, = 2k(Cx,y) - k(Cx,z) - k(Cz,y) • Forming the CPMTx,y requires 2k(px,y(C))-k(Cx,y) (2z -1) operations

  11. Comparison • Per L(sx,sy) : • Direct: 2k(Cx,y)(y-x) – 1 • Recursively: (2z -1) = 2(2k(Cx,y) - k(Cx,z) - k(Cz,y)-1) – 1) • Recursive computation significantly faster if k(Cx,z) and k(Cz,y) are large.

  12. Computation • Instead of forming the trellis section Tx,y from the complete trellis: • Construct the special trellis T({x,z,y,nx,y}): Each state at time x

  13. An RMLD-IV algorithm • Divide code into very short sections • Apply MakeCPMT procedure to these. • MakeCPMT-I: ”brute-force” • Apply CombCPMT to the smaller sections • CombCPMT-V: Use Viterbi for the comparisons

  14. Optimum Sectionalization • Lafourcady-Vardy algorithm applied to RMLD

  15. Complexity comparisons

  16. Humble remark • First such algorithm devised in [14,15] • But... • Similar algorithm for convolutional codes in • Marianne Fjelltveit and Øyvind Ytrehus. On Viterbi decoding of high rate convolutional codes. In Abstracts of Papers of the 1994 IEEE International Symposium on Information Theory, Trondheim, 1994. • Marianne Fjelltveit and Øyvind Ytrehus. Two-step trellis decoding of partial unit memory convolutional codes. IEEE Transactions on Information Theory, IT-43:324–330, January 1997.

  17. Suggested exercises • 14.1-14.5

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