Sequential Soft Decision Decoding of Reed Solomon Codes

1. Sequential Soft Decision Decoding of Reed Solomon Codes Hari Palaiyanur Cornell University Prof. John Komo Clemson University 2003 SURE Program

2. Abstract The Stack Algorithm for convolutional codes is adapted to Reed-Solomon codes to provide a nearly maximal-likelihood decoder. Symbol error rates are found for a system employing this decoding scheme over an additive white Gaussian noise channel. Similarly, performance of the Bucket Algorithm is investigated. The performance of the Stack and Bucket Algorithms is compared to that of errors only and errors and erasures decoders. Additionally, the algorithms are modified by adding a quick erasure decoding. Computational advantages of this method are shown.

3. Background • Error Control Coding – adding redundancy to improve reliability over noisy channel • Reed Solomon Codes – (n,k) cyclic block codes over GF(qm), n = qm – 1 • (n+1)k possible code words • Erasure – if symbol is unreliable, denote it as an erasure

4. Background • Soft Decision Decoding – takes advantage of “side information”, i.e. quality of received signal • Sequential decoding searches through “tree” of possible code words

5. Motivation and History • Soft decision decoding more reliable • Soft decision decoding more decoding time • Good, efficient errors only and errors and erasures decoders available • Still need good, efficient soft decision decoders • Stack/Bucket Algorithm - 1969

6. System Model • MFSK over AWGN channel • For each symbol, detector outputs mi = As + ni (i sent) mi = ni (i not sent) ni – indep. G. R. V. Data RS Encoder MFSK Modulator AWGN Channel Sequential RS Decoder Coherent MFSK Detector

7. Stack Algorithm Start and Load stack with 2m initial nodes P is Head of Stack No Len P = k-1? Yes Remove P Search 2m forward nodes Update Metrics Push onto Stack Remove P Search 2m forward nodes and encode Update Metrics Push onto Stack No Len P = n? Yes Done. Decoded code word is P

8. Metrics and RS (7,3) Example • Metrics provide information about quality of symbol • Forward metric is sum of previous metric and metric for added symbol • Good metric: mi – max({mj}) Symbol 0 1 2 3 4 5 6 7 Mi 3.3 0.2 –1.5 –0.1 1.1 0.7 –2.0 0.8 Node: 0 0.0 Node: 1 -3.1 Node: 2 -4.8 Node: 3 -3.4 Start Node: 4 -2.2 Node: 5 -2.6 Node: 6 -5.3 Node: 7 -2.5

9. Stack Algorithm • Algorithm goes through many unnecessary code words • Perform quick erasure decoding • Let threshold be this code word’s metric • Only push nodes onto stack if their metric is at least threshold

10. Bucket Algorithm • Instead of one sorted stack, use many “buckets” • Buckets have certain metric gradations • No need to keep code words sorted, if gradations are fine enough • Sacrifice memory for speed Metrics 0.0 to -5.0 Metrics -5.0 to -10.0 Metrics -10.0 to -15.0 Metrics Below -15.0

11. Results – RS (15,9)

12. Results – Number of Searches

13. Results – RS (31,21)

14. Conclusions • Soft decision decoding gives better reliability over hard decision decoding • Stack algorithm adapted to RS codes for sequential soft decision decoding

15. Future Work • Other transmissions schemes including non-coherent MFSK • Other channel models • Quantize metrics to a limited number of bits

16. References [1] S.B. Wicker, Error Control Systems for Digital Communication and Storage, Englewood Cliffs, NJ: Prentice Hall, 1995. [2] Komo, J.J. and L.L. Joiner, "Fast Error Magnitude Evaluations for Reed-Solomon Codes," Proc. 1995 IEEE International Symposium on Information Theory, p. 416, Sep. 1995. [3] F. Jelinek, “A Fast Sequential Decoding Algorithm Using a Stack,” IBM Journal of Research and Development, Vol. 13, pp. 675- 685, Nov. 1969