White Paper for LDPC Codes CCSDS P1B Houston Meeting Wai Fong - PowerPoint PPT Presentation

White Paper for LDPC Codes CCSDS P1B Houston Meeting Wai Fong NASA/GSFC October 2, 2002

White Paper for LDPC Codes Introduction • Two techniques for code synthesis: 1. Computer Generated Codes- Regular (Gallager) and Irregular (Richardson), 2. Regular Geometry- based (Lin). • Regular and Irregular Computer Generated codes are slow to converge and have small to moderate minimum distance. • Geometry-based codes have a simpler encoder (Cyclic or Quasi-cyclic encoder) because of their structure with many decoding options and are faster than Computer Generated codes to convergence with very large minimum distances.

White Paper for LDPC Codes Considerations: • Many near-Earth missions use Rate (R)=0.43 RS/CC @ SNR of 2.5 dB at 10 -5 BER. Some missions require only R=0.5 CC @ SNR of 4.2 dB at 10 -5 BER. • • Large frame lengths are useful for higher data-rate missions. • Too large of a frame length may impact encoder/decoder size and speed. • Smaller satellites may have limited resources i.e. power, memory • Most sensors are either 8, 12 or 16 bits/sample and packing/unpacking frames is an issue on space/ground processing. • Mission operation centers prefer 8 or 16 bit boundaries for frame lengths. • Geometry-based LDPC codes can be shortened or lengthened to accommodate 8 or 16 bit boundaries with little effect on performance. • Existing receivers have buffer sizes at the CCSDS AOS frame lengths of 255x8xI where I=1 to 8. Data compression requires 10 -10 BER. • • Bandwidth efficiency is a major consideration on near-Earth missions.

White Paper for LDPC Codes Code Requirements: 1. n and/or k must be a multiple of 8 (and/or 16 if possible) with various frame lengths. 2. Fast decoding > 600Mbps to handle higher data-rate missions. Very low error floor, below 10 -10 BER 3. 4. Minimize encoding complexity to help reduce spacecraft power, weight and size requirements. 5. Coding rates >> ½ to help increase bandwidth efficiency.

White Paper for LDPC Codes LDPC Research Results: • Two code candidates: LDPC-EG (4095, 3367) (or shorten to (4088, 3360)) Rate = 0.822 and LDPC-EG (8176, 7156) Rate = 0.875. • d min = 65 for LDPC-EG (4095, 3367) and d min > 7 for LDPC-EG (8176, 7156). Both codes have been simulated to > 10 -10 BER with no error floor. • • LDPC-EG (4095, 3367) is a cyclic code and LDPC-EG (8176, 7156) is a quasi-cyclic code. • Both codes can be encoded with a sequence of shift registers. • Both codes have very fast iterative convergence.

White Paper for LDPC Codes LDPC-EG (4095, 3367) Code Description: • Cyclic code with generator polynomial g(X) of degree 724. • Encoding circuit can be implemented with a feedback shift-register using 728 flip-flops and no more than 728 X-OR gates. Constructed based on 4160 lines and 4095 points of the 2-dimensional EG(2, 2 6 ). • • Each line consists of 64 points. • Two lines are either disjoint or intersect at one and only one point. For each point in EG(2, 2 6 ) there are 65 lines intersecting it. • • Therefore there are 65 lines passing through the origin and 4095 lines not. • If L is a line not passing through the origin, the incidence vector of line L can be defined as a 4095-tuple over GF(2): v L = ( v 1 , v 2 , . . . , v n ), where v i = 1 if and only if the i th non-origin point of EG(2, 2 6 ) is on L , otherwise v i = 0. • Then a parity-check matrix H 1 can be constructed as a 4095 x 4095 square circulant matrix with column and row weights of 64 where the rows (or the columns) of H 1 are simply the incidence vectors of the 4095 lines in EG(2, 2 6 ) not passing through the origin.

White Paper for LDPC Codes LDPC-EG (4095, 3367) BER and FER Performance 0 0 10 10 BPSK uncoded uncoded BPSK EG−LDPC IDBP bit FER MLD EG−LDPC IDBP block −1 BER MLD 10 EG−LDPC BF bit FER BF EG−LDPC one−step majority−logic BER BF −1 10 EG−LDPC weighted OSML bit −2 FER IDBP 10 EG−LDPC weighted BF bit BER IDBP Shannon limit Shannon limit −3 10 block/bit error probability −2 10 −4 10 Error Rate −5 10 −3 10 −6 10 −7 10 −4 10 −8 10 −9 10 −5 10 −10 10 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 E b /N 0 (dB) Eb/No (dB)

White Paper for LDPC Codes LDPC-EG (4095, 3367) Iterative Convergence Uncoded BPSK Max ItNum 1 Max ItNum 2 −1 10 Max ItNum 5 Max ItNum 10 Max ItNum 20 Max ItNum 100 −2 10 Error Rate −3 10 −4 10 −5 10 0 1 2 3 4 5 6 7 8 E b /N 0 (dB)

White Paper for LDPC Codes LDPC-EG (8176, 7156) Code Description: • Quasi-cyclic code--every cyclic shift of 4 bits of one codeword is also another codeword. • Encoding can also be implemented with shift-registers. Constructed based on 512 points and 4672 lines of the 3-dimensional EG(3, 2 3 ) over • GF(2 3 ). • Incidence vectors of the 4577 lines not passing through origin can be partitioned into 9 cyclic classes, Q 1 , Q 2 , . . . , Q 9 , each class consists of 511 incidence vectors. • Each Q i can be obtained by cyclically shifting any vector in Q i 511 times. • A 511 x 511 square circulant matrix A i is formed whose rows are simply the incidence vectors of Q i with column and row weights of 8. (1) , A i (2) , A i (3) , A i (4) • Q i can be partitioned into four 511 x 511 square circulant matrices, A i ( j ) has column and row weights of 2. where each circulant A i (1) , A i (2) , A i (3) , A i (4) ] can be • By using these 4 circulants, a 511x2044 matrix, G i = [ A i formed. • The column and row weights of G i are 2 and 8, respectively. • Then the parity check matrix H 2 is defined as:   G G G G 1 2 3 4 =   H 2   G G G G 5 6 7 8 • The column and row weights of H 2 are 4 and 32, respectively.

White Paper for LDPC Codes LDPC-EG (8176, 7156) BER Performance and Iterative Convergence 0 −1 10 10 uncoded BPSK uncoded BPSK FER (8176,7156) MaxIT=5 BER (8176,7156) −1 −2 MaxIT=10 10 10 Shannon limit MaxIT=20 MaxIT=100 −2 −3 10 10 −3 −4 10 10 block/bit error probability bit error probability −4 −5 10 10 −5 −6 10 10 −6 −7 10 10 −7 10 −8 10 −8 10 −9 10 −9 −10 10 10 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 Eb/No (dB) Eb/No (dB)