1. Memory-efficient Turbo decoding architecture for LDPC codes
Mohammad M. Mansour and Naresh R. Shanbhag

2. Outline Introduction LDPC Codes Alternating View
Turbo Decoding Message-Passing Algorithm
Decoding Architecture
Ramanujan Graphs
Simulation results
Conclusion

3. Introduction
LDPC codes were introduced by Gallager in1961 and rediscovered by Mackay-Neal and Wiberg
Irregular LDPC codes yielding bit-error rates that theoretically surpass the performance of the best codes known so far
This paper attempts to promote LDPC codes as practical serious competitor to Turbo codes

4. Introduction (continue)
TPMP: Two-phase message-passing algorithm
Random and Stringent memory
Interconnections complexity and Parallelism
TDMP: Turbo-decoding messaging-passing

5. TPMP versus TDMP
TDMP algorithm exhibits a faster convergence behavior and improvement in coding gain over the TPMP algorithm
TDMP reduce > 75% memory overhead of the TPMP

6. LDPC codes LDPC is linear block code
If all bit node have degree c and all check nodes have degree r, the code is call regular (c,r)-LDPC codes

7. Alternating View
Submatrix Hi having size n/r by n define super-code Ci 1 C2 C3
Code C can be view as intersection of ci.
Punching c2 and c3 at blue 1’s to 0’s results in irregular code. C1 C

8. Construction H Hi are pseudo random permutation of the columns of H1
Irregular LDPC codes can be similarly be defined by puncturing the super-code

9. Turbo Decoding Message Passing Algorithm
Previous definition is reminiscent of parallel concatenated turbo codes
SISO decoder
Super-code 1
Apply the turbo decoding technique to decode LDPC code.
Gallager’s message update equation & BCRJ algorithm
Iteration
Super-code 2
Super-code 3
Converged

10. Turbo Decoding Message Passing Algorithm (Continue)
Reduce memory requirement
Fast convergence (20%~50% iteration)
Reduce decoding latency
Doesn’t require the saving of multiple check-bit message as is the case for the TPMP

11. Decoding Architecture
Newly generated bit messages from preceding super-codes are directly used to generate bit messages of succeeding super-codes on the same iteration

12. Decoding Architecture (continue)
Bit message are stored in circular buffer
Channel value are stored in MEMλ
Parameter L vs. Throughput & Complexity
Interconnection complexity of mux/demux

13. Ramanujan Graph
Non-trivial eigenvalues of adjacency matrix ARG(q,p) <=
p is node-degree, q is number of vertices
p and q are distinct odd primes
If p is a quadratic non-residue modulo q, biregular bipartite graph having vertices in eachpartition and uniform node-degrees of (p+1)

14. Ramanujan Transformations
Node Spitting
Edge Splitting
Node Replacement + Edge Splitting
Edge Merging
Edge Merging + Edge splitting

15. Ramanujan Transformations (Continue)

16. Simulations Over AWGN channel with BPSK modulation
Code1: p=7, q=15, n=1440, R=1/2, using node splitting
Code2: p=7, q=15, n=2880, R=1/2, using edge merging
Code3: p=11, q=21, n=6048, R=1/3, using edge merging
Code4:p=7, q=33, n=15840 R=0.504, using node splitting

17. Simulation Result 1

18. Simulation Result 2

19. Simulation Result 3

20. Conclusion
If we want to use this method we must study architecture of Turbo Code
Study Algebra and Number Theory (for code search or code design)
Build a simulation model to evaluate our code and architecture