Improving BER Performance of LDPC Codes Based on Intermediate Decoding Results Esa Alghonaim, M. Adnan Landolsi, Aiman El-Maleh King Fahd University of Petroleum & Minerals Saudi Arabia Esa Alghonaim, M. Adnan Landolsi, Aiman El-Maleh King Fahd University of Petroleum & Minerals Saudi Arabia

2 OutlineOutline n Motivation n Overview of LDPC codes n Belief Propagation (BP) Algorithm n LDPC Decoding Error Patterns Types n Proposed Improvement on BP Algorithm n Experimental Results n Conclusions n Motivation n Overview of LDPC codes n Belief Propagation (BP) Algorithm n LDPC Decoding Error Patterns Types n Proposed Improvement on BP Algorithm n Experimental Results n Conclusions

3 MotivationMotivation n LDPC codes belong to a family of error correction systems with performance close to information- theoretic limits. n Selected for next-generation digital satellite broadcasting standard (DVB-S2), ultra high-speed Local Area Networks (10Gbps Ethernet LANs). n Amenable to efficient parallel hardware implementation. n Built-in Error Checking. n At high SNR, uncorrected error patterns dominated by oscillating patterns Number of bits in error varies considerably between iterations Number of bits in error varies considerably between iterations n LDPC codes belong to a family of error correction systems with performance close to information- theoretic limits. n Selected for next-generation digital satellite broadcasting standard (DVB-S2), ultra high-speed Local Area Networks (10Gbps Ethernet LANs). n Amenable to efficient parallel hardware implementation. n Built-in Error Checking. n At high SNR, uncorrected error patterns dominated by oscillating patterns Number of bits in error varies considerably between iterations Number of bits in error varies considerably between iterations

4 LDPC Codes Overview n LDPC codes: linear block codes decoded by efficient iterative decoding. n An LDPC parity check matrix H represents the parity equations in a linear form codeword c satisfies the set of parity equations H. c = 0. codeword c satisfies the set of parity equations H. c = 0. each column in the matrix represents a codeword bit each column in the matrix represents a codeword bit each row represents a parity check equation each row represents a parity check equation n LDPC codes: linear block codes decoded by efficient iterative decoding. n An LDPC parity check matrix H represents the parity equations in a linear form codeword c satisfies the set of parity equations H. c = 0. codeword c satisfies the set of parity equations H. c = 0. each column in the matrix represents a codeword bit each column in the matrix represents a codeword bit each row represents a parity check equation each row represents a parity check equation c 0  c 1  c 3 = 0 c 1  c 2  c 4 = 0 c 2  c 3  c 5 = 0 c 3  c 4  c 6 = 0

5 LDPC Codes Overview n Code Rate ratio of information bits to total number of bits in codeword. n LDPC codes represented by Tanner Graphs two types of vertices: Bit Vertices and Check Vertices two types of vertices: Bit Vertices and Check Vertices n Performance of LDPC code affected by presence of cycles in Tanner graph. n Code Rate ratio of information bits to total number of bits in codeword. n LDPC codes represented by Tanner Graphs two types of vertices: Bit Vertices and Check Vertices two types of vertices: Bit Vertices and Check Vertices n Performance of LDPC code affected by presence of cycles in Tanner graph

6 BP LDPC Decoding Algorithm n Iterative algorithm n Produces optimum performance in cycle-free graphs n Iterative algorithm n Produces optimum performance in cycle-free graphs BP-LDPC (Conventional) Decoding Initialize variable nodes Loop Update check and variable nodes Update check and variable nodes Compute estimated variable nodes vector Compute estimated variable nodes vector Compute syndrome vector: Compute syndrome vector: Until or maximum iterations reached Return

7 BP LDPC Decoding Algorithm Variable to Check Information Check to variable Information Information bit node i sends to check node j about P(x i =b) Information check node j sends to bit node i about P(x i =b)

8 LDPC Decoding Error Patterns Types n Frame errors can be classified intro three main categories: Oscillation error pattern: with nearly periodic change between maximum & minimum number of bits in errors. Oscillation error pattern: with nearly periodic change between maximum & minimum number of bits in errors. High variation in bit error count as a function of iteration number.High variation in bit error count as a function of iteration number. Nearly-constant error pattern: bit error count becomes constant after few decoding iterations Nearly-constant error pattern: bit error count becomes constant after few decoding iterations Mainly due small size trapping setsMainly due small size trapping sets Random-like error pattern: error count evolution follows a random shape characterized by low variation range. Random-like error pattern: error count evolution follows a random shape characterized by low variation range. n Frame errors can be classified intro three main categories: Oscillation error pattern: with nearly periodic change between maximum & minimum number of bits in errors. Oscillation error pattern: with nearly periodic change between maximum & minimum number of bits in errors. High variation in bit error count as a function of iteration number.High variation in bit error count as a function of iteration number. Nearly-constant error pattern: bit error count becomes constant after few decoding iterations Nearly-constant error pattern: bit error count becomes constant after few decoding iterations Mainly due small size trapping setsMainly due small size trapping sets Random-like error pattern: error count evolution follows a random shape characterized by low variation range. Random-like error pattern: error count evolution follows a random shape characterized by low variation range.

9 LDPC Decoding Error Patterns Types

10 Percentage of Error Patterns Types n Progressive-Edge-Growth (PEG) LDPC code minimizes girth (cycle length) and achieves good performance. n (1024, 512) PEG LDPC code n Progressive-Edge-Growth (PEG) LDPC code minimizes girth (cycle length) and achieves good performance. n (1024, 512) PEG LDPC code Error Pattern Type (PEG code) SNROscillationConstantRandom-like % 0 % 96 % % 0 % 88 % % 1 % 75 % % 3 % 34 %

11 Correlation Between Uncorrected Codeword Bits & Failed Parity Check Equations

12 Proposed Improvement on BP Algorithm BP-LDPC Decoding with Proposed Improvement Initialize variable nodes Set Minimum = number of check nodes Loop Update check and variable nodes Update check and variable nodes Compute estimated variable nodes vector Compute estimated variable nodes vector Compute syndrome vector: Compute syndrome vector: Check Errors = number of non-zero elements in Check Errors = number of non-zero elements in If Check Errors < Minimum then If Check Errors < Minimum then Minimum = Check Errors Minimum = Check Errors Until Check Errors = 0 or maximum iterations reached Return =

13 Experimental Results n Parallel computing simulation platform developed to run LDPC decoding simulations on 130 nodes LAN network. n Simulated LDPC codes PEG (1024, 512) PEG (1024, 512) IEEE e (960,480) IEEE e (960,480) Randomly constructed LDPC codes (free of 4- and 6-cycles) Randomly constructed LDPC codes (free of 4- and 6-cycles) n Parallel computing simulation platform developed to run LDPC decoding simulations on 130 nodes LAN network. n Simulated LDPC codes PEG (1024, 512) PEG (1024, 512) IEEE e (960,480) IEEE e (960,480) Randomly constructed LDPC codes (free of 4- and 6-cycles) Randomly constructed LDPC codes (free of 4- and 6-cycles)

14 BER Improvement for (1024, 512) PEG LDPC Code

15 BER Improvement for IEEE802.16e(960,480)

16 ConclusionsConclusions n A method to improve residual BER level in BP decoding of LDPC codes. n Oscillating error pattern dominant at high SNR for well designed LDPC codes. n Minimized BER using number of failed check equations as an indicator for the number of bits in error. n At SNR=3 dB, BER reduction of 40% achieved. n A method to improve residual BER level in BP decoding of LDPC codes. n Oscillating error pattern dominant at high SNR for well designed LDPC codes. n Minimized BER using number of failed check equations as an indicator for the number of bits in error. n At SNR=3 dB, BER reduction of 40% achieved.