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 802.11-04/948 Victor Stolpman et. al

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

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

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

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

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

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

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

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

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

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

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

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

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

“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

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

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

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

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

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

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

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

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