1. Irregular Structured LDPC Codes and Structured Puncturing
September, 2004
Irregular Structured LDPC Codes and Structured Puncturing
Victor Stolpman, Nico van Waes, Tejas Bhatt, and Charlie Zhang
This presentation accompanies submission
IEEE /948
Victor Stolpman et. al

2. Parity-Check “Seed” Matrix
September, 2004
Parity-Check “Seed” Matrix
Small binary matrix  low storage costs.
Acts as a blueprint to the structure of the expanded LDPC code.
Constructed from an ensemble with good asymptotic properties for the desired channel (e.g. AWGN, BEC, Fading, MIMO, etc.).
Expanded using permutation matrices (e.g. single circular-shift) to construct the LDPC matrix used in the error control system.
After expansion, final LDPC matrix will be of the same ensemble.
Storage is smaller than the storage of exponents.
For example purposes only, a seed matrix:
Victor Stolpman et. al

3. Permutation “Spread” Matrices
September, 2004
Permutation “Spread” Matrices
Finite set of matrices consisting of circular-shift matrices, the identity matrix, and the all zeros matrix.
Indexed via their exponent values.
Victor Stolpman et. al

4. September, 2004
Expanded LDPC Matrix
“Expanded” LDPC matrix whose sub-matrices belong to the finite set of permutation matrices
In matrix notation, we write
Thus, the final exponents are of the finite set:
Victor Stolpman et. al

5. Universal Exponential Matrix
September, 2004
Universal Exponential Matrix
Exponential matrix definition used for all LDPC code constructions despite the ensemble and seed construction dimension.
Because it is “rule-based” and not tied to a particular “seed” matrix construction, it offers forward-compatibility and hardware reuse for different products and standards.
Adapts for multiple block sizes and code rates without additional storage for exponent values.
Applicable to different ensembles designed for different channels.
Victor Stolpman et. al

6. Final Exponential Matrix
September, 2004
Final Exponential Matrix
Constructed via masking the “seed” matrix with the “universal” exponent matrix (Note: operations can be reduced to just the ones locations in the seed parity-check matrix).
Using the following mapping:
Victor Stolpman et. al

7. Small Construction Example
September, 2004
Small Construction Example
Parity
Systematic
Victor Stolpman et. al

8. Simulations Simulated Block Sizes: Permutation sub-matrix dimensions:
September, 2004
Simulations
Simulated Block Sizes:
{576,720, 768, 864, 960, 1008, 1152, 1296, 1344, 1440, 1536, 1584, 1728, 1872, 1920, 2016, 2112, 2160, 2304}
Permutation sub-matrix dimensions:
{12,15,16,18,20,21,24,27,28,30,32,33,36,39,40,42,44,45,48}
Rate 1/2 Seed Matrices of dimension (24x48)
3 seed matrices (designed for AWGN – all 3 from the same ensemble)
Rate 2/3 Seed Matrices of dimension (16x48)
4 seed matrices (designed for AWGN – all 4 from the same ensemble)
Rate 3/4 Seed Matrices of dimension (12x48)
3 Seed matrices (designed for AWGN – all 3 from the same ensemble)
50 iterations of conventional belief propagation
In the pipeline …
Rate 7/8 Seed Matrices
Additional block sizes (forward-compatibility and hardware reuse)
Additional Layered Belief-Propagation decoding
Victor Stolpman et. al

9. September, 2004
Rate 1/2 BLER – AWGN BPSK
Victor Stolpman et. al

10. September, 2004
Rate 2/3 BLER – AWGN BPSK
Victor Stolpman et. al

11. September, 2004
Rate 3/4 BLER – AWGN BPSK
Victor Stolpman et. al

12. Layered v/s Conventional BP (Rate 1/2)
September, 2004
Layered v/s Conventional BP (Rate 1/2)
Layered BP
(15 iterations)
Conventional BP
(50 iterations)
Victor Stolpman et. al

13. Features Supports a wide range of block sizes without shortening
September, 2004
Features
Supports a wide range of block sizes without shortening
Shortening causes some inefficiencies with hardware:
Must shorten a longer codeword in decoding than needed
Power consumption and/or performance may vary with block size
Shortening rules continue to propagate in future systems
Upper triangular seed matrices  linear time encoder
Wide range of block sizes reduces zero-padding inefficiencies
Supports ensembles designed for different channel models
Future compatibility and hardware reuse going forward
Additional block sizes are easily added for advancements in silicon
Additional ensembles are easily added for difference channel models
Layered Belief-Propagation decoding can be done to speed up convergence and reduce decoding latency
Victor Stolpman et. al

14. Structured Puncturing of LDPC Codes
September, 2004
Structured Puncturing of LDPC Codes
Used to offer additional code rates in between and above the basic code rate set {1/2,2/3,3/4,7/8}.
Puncturing of LDPC codes to achieve higher code rates from a mother code of a lower rate while maintaining a given information block size.
Puncturing does not require changing the parity-check connective net at either the encoder or decoder.
Supports link adaptation easily. In MIMO applications, puncturing allows for different spatial streams to have different code rates without using multiple coding blocks.
Approach can be reused in Hybrid-ARQ systems.
Structured approach reduces storage requirements and expands easily to multiple block lengths.
Puncturing can be used in conjunction with other rate-adjustment schemes (e.g. nested matrices and shortening).
Victor Stolpman et. al

15. “Seed” Degree-Sequence for Puncturing
September, 2004
“Seed” Degree-Sequence for Puncturing
Vector containing the variable-degrees corresponding to the variable-nodes of a seed parity-check matrix to be punctured. Acts as a blueprint and reduces puncture sequence storage.
Expanded using the Kronecker product for obtaining the variable-degrees corresponding to the variable-nodes of the expanded LDPC matrix.
Then, mapped to variable-nodes of the expanded LDPC matrix
Victor Stolpman et. al

16. Code Rate and Nested Puncture Sequence
September, 2004
Code Rate and Nested Puncture Sequence
For P punctures, the effective code rate becomes
By using the first P variable-nodes in (i.e ), we can nest the puncturing sequence such that
Victor Stolpman et. al

17. Structured Puncturing Simulations
September, 2004
Structured Puncturing Simulations
Mother codes seed matrices defined as in the original proposal.
(new results to come on block lengths for multiples of 48)
Puncture seed sequences defined as in the original proposal.
BPSK signaling in AWGN.
50 iterations of conventional belief-propagation.
Victor Stolpman et. al

18. Rate 1/2 Puncture Example (Mother Code, N=624)
September, 2004
Rate 1/2 Puncture Example (Mother Code, N=624)
Victor Stolpman et. al

19. Rate 1/2 Puncture Example (Mother Code, N=2340)
September, 2004
Rate 1/2 Puncture Example (Mother Code, N=2340)
Victor Stolpman et. al

20. Rate 2/3 Puncture Example (Mother Code, N=1769)
September, 2004
Rate 2/3 Puncture Example (Mother Code, N=1769)
Victor Stolpman et. al

21. Rate 2/3 Puncture Example (Mother Code, N=2318)
September, 2004
Rate 2/3 Puncture Example (Mother Code, N=2318)
Victor Stolpman et. al

22. Rate 3/4 Puncture Example (Mother Code, N=2352)
September, 2004
Rate 3/4 Puncture Example (Mother Code, N=2352)
Victor Stolpman et. al

23. Summary Irregular Structure LDPC Codes
September, 2004
Summary
Irregular Structure LDPC Codes
Applicable to seed matrices designed for different channels and antenna configurations (i.e. AWGN, BEC, fading, SISO, MIMO, etc.).
“Rule-based” exponent reduces storage requirements because only seed matrices need to be stored and not exponents.
Reusable hardware for different channel models and product lines.
Performance is in line with other structured approaches customized for specific channels.
Structured Puncturing of LDPC Codes
A change in code rate does not require a change of connective nets in either the encoder or decoder.
Can work with different ensembles of different rates.
Allows for simple link adaptation and can easily support different code rates for separate spatial streams in MIMO antenna configurations.
Can reuse hardware for different error control applications.
Can coexist with other code rate adaptation approaches (e.g. nested matrices, shortening, etc.).
Victor Stolpman et. al