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Optimizing LDPC Codes for message-passing decoding. Jeremy Thorpe Ph.D. Candidacy 2/26/03 Overview Research Projects Background to LDPC Codes Randomized Algorithms for designing LDPC Codes Open Questions and Discussion Data Fusion for Collaborative Robotic Exploration Developed a version of the Mastermind game as a model for autonomous inference. Applied the Belief Propagation algorithm to solve this problem. Showed that the algorithm had an interesting performance-complexity tradeoff. Published in JPL's IPN Progress Reports. Dual-Domain Soft-in Soft-out Decoding of Conv. Codes Studied the feasibility of using the Dual SISO algorithm for high rate turbo-codes. Showed that reduction in state-complexity was offset by increase in required numerical accuracy. Report circulated internally at DSDD/HIPL S&S Architecture Center, Sony. Short-Edge Graphs for Hardware LDPC Decoders. Developed criteria to predict performance and implementational simplicity of graphs of Regular (3,6) LDPC codes. Optimized criteria via randomized algorithm (Simulated Annealing). Achieved codes of reduced complexity and superior performance to random codes. Published in ISIT 2002 proceedings. Evalutation of Probabilistic Inference Algorithms Characterize the performance of probabilistic algorithms based on observable data Axiomatic definition of "optimal characterization" Existence, non-existence, and uniqueness proofs for various axiom sets Unpublished Optimized Coarse Quantizers for Message-Passing Decoding Mapped 'additive' domains for variable and check node operations Defined quantized message passing rule in these domains Optimized quantizers for 1-bit to 4-bit messages Submitted to ISIT 2003 Graph Optimization using Randomized Algorithms Introduce Proto-graph framework Use approximate density evolution to predict performance of particular graphs Use randomized algorithms to optimize graphs (Extends short-edge work) Achieves new asymptotic performance- complexity mark Bacground to LDPC codes The Channel Coding Strategy Encoder chooses the mth codeword in codebook C m {0,1}k x C X n and transmits it across the Encoder channel Decoder observes the channel output y and Channel generates m’ based on the knowledge of the codebook Decoder C and the channel m'{0,1}k y Y n statistics. Linear Codes A linear code C (over a finite field) can be defined in terms of either a generator matrix or parity-check matrix. Generator matrix G (k×n) C {mG} Parity-check matrix H (n-k×n) C {c : cH' 0} LDPC Codes LDPC Codes -- linear codes defined in terms of H H has a small average number of non-zero elements per row or column. 0 1 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 H 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 Graph Representation of LDPC Codes H is represented by a Variable nodes bipartite graph. v There is an edge from c v to c if and only if: H (v, c) 0 ... ... A codeword is an assignment of v's s.t.: c, xv 0 Check nodes v|c Message-Passing Decoding of LDPC Codes Message Passing (or Belief Propagation) decoding is a low-complexity algorithm which approximately answers the question “what is the most likely x given y?” MP recursively defines messages mv,c(i) and mc,v(i) from each node variable node v to each adjacent check node c, for iteration i=0,1,... Two Types of Messages... Likelihood Ratio Probability Difference p( y | x 1) x, y p( x 1 | y) p( x 0 | y) x , y p( y | x 0) For y1,...yn independent For x1,...xn conditionally on x: independent: x , y x , y n i x , y x , y i i 1 i i i ...Related by the Biliniear Transform Definition: B ( x , y ) B ( p ( y | x 1) ) p ( y | x 0) 1 x B( x) p ( y | x 0) p ( y | x 1) 1 x p ( y | x 0) p ( y | x 1) Properties: p ( y | x 0) p ( y | x 1) 2 p( y) B ( B ( x)) x 2 p( x 0 | y ) p( y ) 2 p( x 1 | y ) p( y ) B ( x , y ) x , y 2 p( y) B( x , y ) x, y p( x 0 | y) p( x 1 | y) x, y Message Domains Probability Difference Likelihood Ratio B() P( y | x 0) P( x 0 | y ) P( x 1 | y ) P( y | x 1) e log() e log() Log Prob. Difference Log Likelihood Ratio B ' () Variable to Check Messages On any iteration i, the message from v to c is: v v B(mc ',v (i ) ( i 1) mv,c ) c c '|v c ... In the additive domain: ... mv,c log( v ) B' (mc ',v ) (i ) ( i 1) c '|v c Check to Variable Messages On any iteration, the message from c to v is: v B' (mv',c ) (i ) (i ) mc,v c v '|c v ... In the additive domain: ... B' ( m (i ) (i ) mc,v v ',c ) v '|c v Decision Rule After sufficiently many iterations, return the likelihood ratio: 0, if x , y B(mc ,v ( i 1) ) 0 x ˆ v v c|v 1, otherwise Theorem about MP Algorithm If the algorithm stops after r iterations, then the algorithm r returns the maximum a posteriori probability estimate of xv given y within radius r of ... v. However, the variables within ... a radius r of v must be v dependent only by the equations within radius r of v, ... Regular (λ,ρ) LDPC codes Every variable node has degree λ, every check node has degree ρ. Best rate 1/2 code is (3,6), with threshold 1.09 dB. This code had been invented by 1962 by Robert Gallager. Regular LDPC codes look the same from anywhere! The neighborhood of every edge looks the same. If the all-zeros codeword is sent, the distribution of any message depends only on its neighborhood. We can calculate a single message distribution once and for all for each iteration. Analysis of Message Passing Decoding (Density Evolution) We assume that the all-zeros codeword was transmitted (requires a symmetric channel). We compute the distribution of likelihood ratios coming from the channel. For each iteration, we compute the message distributions from variable to check and check to variable. D.E. Update Rule The update rule for Density Evolution is defined in the additive domain of each type of node. Whereas in B.P, we add (log) messages: B' ( m B' ( m ( i 1) log( v ) (i ) (i ) (i ) mc,v v ',c ) mv,c c ',v ) v '|c v c '|v c In D.E, we convolve message densities: PM (i ) *P B '( M v ',c (i ) ) PM (i) Plog(v ) * PB '( M ( i 1) ) c ,v v ,c c ',v v '|c v c '|v c Familiar Example: If one die has density function given by: 1 2 3 4 5 6 The density function for the sum of two dice is given by the convolution: 2 3 4 5 6 7 8 9 10 11 12 D.E. Threshold Fixing the channel message densities, the message densities will either "converge" to minus infinity, or they won't. For the gaussian channel, the smallest SNR for which the densities converge is called the density evolution threshold. D.E. Simulation of (3,6) codes Threshold for regular (3,6) codes is 1.09 dB Set SNR to 1.12 dB (.03 above threshold) Watch fraction of "erroneous messages" from check to variable Improvement vs. current error fraction for Regular (3,6) Improvement per iteration is plotted against current error fraction Note there is a single bottleneck which took most of the decoding iterations Irregular (λ, ρ) LDPC codes a fraction λi of variable Variable nodes nodes have degree i. ρi of check nodes have λ2 degree i. ρ4 Edges are connected by λ3 π a single random ... ... permutation. ρm Nodes have become λn specialized. Check nodes D.E. Simulation of Irregular Codes (Maximum degree 10) Set SNR to 0.42 dB (~.03 above threshold) Watch fraction of erroneous check to variable messages. This Code was designed by Richardson et. al. Comparison of Regular and Irregular codes Notice that the Irregular graph is much flatter Note: Capacity achieving LDPC codes for the erasure channel were designed by making this line exactly flat Constructing LDPC code graphs from a proto-graph Consider a bipartite graph G, called a "proto-graph" Generate a graph G α called =G an "expanded graph" replace each node by α α=2 nodes. replace each edge by α edges, permuted at random =G2 Local Structure of G α The structure of the neighborhood of any edge in Gα can be found by examining G The neighborhod of radius r of a random edge is increasingly probably loop-free as α→∞. Density Evolution on G For each edge (c,v) in G, compute: PM (i ) Pv * * PB '( M (i) ) v ,c c ',v c '|v c and: PM c ,v (i ) *P v '|c v B '( M v ',c ( i 1) ) Density Evolution without convolution One-dimensional approximation to D.E, which requires: A statistic that is approximately additive for check nodes A statistic that is approximately additive for variable nodes A way to go between these two statistics A way to characterize the message distribution from the channel Optimizing a Proto Graph using Simulated Annealing Simulated Annealing is an iterative algorithm that approximately minimizes an energy function Requirements: A space S over which to find the optimum point An energy function E(s):S→R A random perturbation function p(s):S→S A "temperature profile" t(i) Optimization Space Graphs with a fixed number of variable and check nodes (rate is fixed) Optionally, we can add untransmitted (state) variables to the code Typical Parameters 32 transmitted variables 5 untransmitted variables 21 parity checks Energy function Ideal: density evolution threshold. Practical: Approximate density evolution threshold Number of iterations to converge to fixed error probability at fixed SNR Perturbations Types of operation Add an edge Delete an edge Swap two edges Note: Edge swapping operation not necessary to span the space Basic Simulated Annealing Algorithm Take s0 = a random point in S For each iteration i, define si' = p(si) if E(si') < E(si) set si+1 = si' E ( s ') E ( s ) i i if E(si ') > E(si) set si+1 = si' w.p. e t (i ) Degree Profile of Optimized Code The optimized graph 12 has a large fraction of 10 degree 1 variables 8 Check variables range Variable from degree 3 to 6 nodes Check degree 8 4 nodes (recall that the graph is 2 not defined by the 0 degree profile) 1 2 3 4 5 6 7 8 Threshold vs. Complexity Designed codes of rate .5 with threshold 8 mB from channel capacity on AWGN channel Low complexity (maximum degree = 8) Improvement vs. Error Fraction Comparison to Regular (3,6) The regular (3,6) code has a dramatic bottleneck. The irregular code with maximum degree 10 is flatter, but has a bottleneck. The optimized proto-graph based code is nearly flat for a long stretch. Simulation Results n=8192, k=4096 Achieves bit error rate of about 4×10-4 at SNR=0.8dB. Beats the performance of n=10000 code in [1] by a small margin. There is evidence that there is an error floor Review We Introduced the idea of LDPC graphs based on a proto-graph We designed proto-graphs using the Simulated Annealing algorithm, using a fast approximation to density evolution The design handily beats other published codes of similar maximum degree Open Questions What's the ultimate limit to the performance vs. maximum degree tradeoff? Can we find a way to achieve the same tradeoff without randomized algorithms? Why do optimizing distributions sometimes force the codes to have low-weight codewords? A Big Question Can we derive the shannon limit in the context of MP decoding of LDPC codes, so that we can meet the inequalities with equality? Free Parameters within S.A. Rate Maximum check, variable degrees Proto-graph size Fraction of untransmitted variables Channel Parameter (SNR) Number of iterations in Simulated Annealing Performance of Designed MET Codes Shows performance competitive with best published codes Block error probability <10-5 at 1.2 dB a soft error floor is observed at very high SNR, but not due to low-weight codewords Multi-edge-type construction Edges of a particular "color" are connected through a permutation. Edges become specialized. Each edge type has a different message distribution each iteration. MET D.E. vs. decoder simulation

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