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

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White Paper for LDPC Codes

CCSDS P1B Houston Meeting Wai Fong NASA/GSFC October 2, 2002

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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.

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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.

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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. 3. Very low error floor, below 10-10 BER 4. Minimize encoding complexity to help reduce spacecraft power, weight and size requirements. 5. Coding rates >> ½ to help increase bandwidth efficiency.

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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.

dmin = 65 for LDPC-EG (4095, 3367) and dmin > 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.

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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, 26).
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, 26) 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): vL = (v1, v2, . . . , vn), where vi = 1 if and only if the ith non-origin point of EG(2, 26) is on L, otherwise vi = 0.

Then a parity-check matrix H1 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 H1 are simply the incidence vectors of the 4095 lines in EG(2, 26) not passing through the

rigin.

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White Paper for LDPC Codes

LDPC-EG (4095, 3367) BER and FER Performance

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10 Eb/No (dB) block/bit error probability uncoded BPSK FER MLD BER MLD FER BF BER BF FER IDBP BER IDBP Shannon limit 1 2 3 4 5 6 7 8 9 10

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10 Eb/N0 (dB) Error Rate BPSK uncoded EG−LDPC IDBP bit EG−LDPC IDBP block EG−LDPC BF bit EG−LDPC one−step majority−logic EG−LDPC weighted OSML bit EG−LDPC weighted BF bit Shannon limit

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White Paper for LDPC Codes

LDPC-EG (4095, 3367) Iterative Convergence

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Eb/N0 (dB) Error Rate Uncoded BPSK Max ItNum 1 Max ItNum 2 Max ItNum 5 Max ItNum 10 Max ItNum 20 Max ItNum 100

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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, 23) over

GF(23).

Incidence vectors of the 4577 lines not passing through origin can be partitioned into 9

cyclic classes, Q1, Q2, . . . , Q9, each class consists of 511 incidence vectors.

Each Qi can be obtained by cyclically shifting any vector in Qi 511 times.
A 511 x 511 square circulant matrix Ai is formed whose rows are simply the incidence

vectors of Qi with column and row weights of 8.

Qi can be partitioned into four 511 x 511 square circulant matrices, Ai

(1), Ai (2), Ai (3), Ai (4)

where each circulant Ai

(j) has column and row weights of 2.

By using these 4 circulants, a 511x2044 matrix, Gi = [Ai

(1), Ai (2), Ai (3), Ai (4)] can be

formed.

The column and row weights of Gi are 2 and 8, respectively.
Then the parity check matrix H2 is defined as:
The column and row weights of H2 are 4 and 32, respectively.

      =

8 7 6 5 4 3 2 1 2

G G G G G G G G H

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White Paper for LDPC Codes

LDPC-EG (8176, 7156) BER Performance and Iterative Convergence

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Eb/No (dB) bit error probability uncoded BPSK MaxIT=5 MaxIT=10 MaxIT=20 MaxIT=100 1 2 3 4 5 6 7 8 9 10

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10 Eb/No (dB) block/bit error probability uncoded BPSK FER (8176,7156) BER (8176,7156) Shannon limit