Soft Decision Decoding of RS Codes Using Adaptive Parity Check Matrices

1. Soft Decision Decoding of RS Codes Using Adaptive Parity Check Matrices Jing Jiang and Krishna R. Narayanan Wireless Communication Group Department of Electrical Engineering Texas A&M University

2. Reed Solomon Codes • Consider an (n,k) RS code over GF(2m), n = 2m-1 • Linear block code – e.g. (7,5) RS code over GF(8) •  be a primitive element in GF(8) • Cyclic shift of any codeword is also a valid codeword • RS codes are MDS (dmin = n-k+1) • The dual code is also MDS

3. source interleaving RS Encoder PR Encoder + interleaving AWGN extrinsic + - sink de-interleaving RS Decoder BCJR Equalizer extrinsic a priori hard decision RS Coded Turbo Equalization System a priori + Introduction • Advantages • Guaranteed minimum distance • Efficient bounded distance hard decision decoder (HDD) • Decoder can handle errors and erasures • Drawback • Performance loss due to bounded distance decoding • Soft input soft output (SISO) decoding is not easy!

4. Presentation Outline Existing soft decision decoding techniques Iterative decoding based on adaptive parity check matrices Variations of the generic algorithm Applications over various channels Conclusion and future work

5. Existing Soft Decoding Techniques

6. Enhanced Algebraic Hard Decision Decoding • Generalized Minimum Distance (GMD) Decoding (Forney 1966): • Basic Idea: • Erase some of the least reliable symbols • Run algebraic hard decision decoding several times • Drawback: GMD has a limited performance gain • Chase decoding (Chase 1972): • Exhaustively flip some of the least reliable symbols • Running algebraic hard decision decoding several times • Drawback: Has an exponentially increasing complexity Combined Chase & GMD(Tang et al. 2001).

7. Drawback: • The complexity increases with , maximum number of multiplicity. Algebraic Soft Input Hard Output Decoding • Algebraic SIHO decoding: • Algebraic interpolation based decoding (Koetter & Vardy 2003) • Reduced complexity KV algorithm (Gross et al. submitted 2003) • Basic ideas: • Based on Guruswami and Sudan’s algebraic list decoding • Convert the reliability information into a set of interpolation points • Generate a list of candidate codewords • Pick up the most likely codeword from the codeword list

8. Reliability based Ordered Statistic Decoding • Reliability based decoding: • Ordered Statistic Decoding (OSD) (Fossorier & Lin 1995) • Box & Match Algorithm(BMA) (Valembois & Fossorier to appear 2004) • Basic ideas: • Order the received bits according to their reliabilities • Make hard decisions on a set of independent reliable bits (MR Basis) • Re encode to obtain a list of candidate codewords • Drawback: • The complexity increases exponentially with the reprocessing order • BMA must trade memory for complexity

9. Trellis based Decoding using the Binary Image Expansion • Maximum-likelihood decoding and variations • Trellis based decoding using binary image expansion (Vardy & Be’ery ‘91) • Reduced complexity version (Ponnampalam & Vucetic 2002) • Basic ideas: • Binary image expansion of RS • Trellis structure construction using the binary image expansion • Drawback: • Exponentially increasing complexity • Work only for very short codes or codes with very small distance

10. Binary Image Expansion of RS Codes

11. Binary image expansion of the parity check matrix of RS(7, 5) over GF(23) • Consider the (7,5) RS code

12. Binary image expansion of the parity check matrix of RS(7, 5) over GF(23) Recent Iterative Techniques Sub-trellis based iterative decoding (Ungerboeck 2003) Self-concatenation structure based on sub-trellis constructed from the parity check matrix • Drawbacks: • Performance deteriorates due to large number of short cycles • Work for short codes with small minimum distances • Potential error floor problem in high SNR region

13. Recent Iterative Techniques (cont’d) Stochastic shifting based iterative decoding (Jing & Narayanan, to appear 2004) Due to the irregularity in the H matrix, iterative decoding favors some bits Taking advantage of the cyclic structure of RS codes Shift by 2 Stochastic shift prevent iterative procedure from getting stuck Best result: RS(63,55) about 0.5dB gain from HDD However, for long codes, this algorithm still doesnt provide good improvement

14. Remarks on Existing Techniques Most SIHO algorithms are either too complex to implement or having only marginal gain Moreover, SIHO decoders cannot generate soft output directly Trellis-based decoders have exponentially increasing complexity Iterative decoding algorithms do not work for long codes, since the parity check matrices of RS codes are not sparse “Soft decoding of large RS codes as employed in many standard transmission systems, e.g., RS(255,239), with affordable complexity remains an open problem” (Ungerboeck, ISTC2003)

15. Questions Q: Why doesn’t iterative decoding work for codes with non-sparse parity check matrices? Q: Can we get some idea from the failure of iterative decoder?

16. erased bits bit nodes …………. ……….. ? ……………. ……………. . . . . . . . . . check nodes How does standard message passing algorithm work? If two or more of the incoming messages are erasures the check is erased Otherwise, check to bit message is the value of the bit that will satisfy the check

17. bit nodes …………. ……….. ……………. ……………. . . . . . . . . . check nodes How does standard message passing algorithm work? Small values of vj can be thought of as erasures and hence more than two edges with small vj’s saturate the check

18. Binary image expansion of the parity check matrix of RS(7, 5) over GF(23) A Few Unreliable Bits “Saturate” the Non-sparse Parity Check Matrix Consider RS(7, 5) over GF(23) Iterative decoding is stuck due to only a few unreliable bits “saturating” the whole non-sparse parity check matrix

19. Sparse Parity Check Matrices for RS Codes Can we find an equivalent binary parity check matrix that is sparse? For RS codes, this is not possible! The H matrix is the G matrix of the dual code The dual of an RS code is also an MDS Code Every row has weight at least (N-K)!

20. transmitted codeword received vector parity check matrix Iterative Decoding Based on Adaptive Parity Check Matrix Idea: reduce the sub-matrix corresponding to the unreliable positions to a sparse nature. For example, consider (7,4) Hamming code: After the adaptive update, iterative decoding can proceed.

21. unreliable bits bit nodes …………. ……….. ……………. ……………. . . . . . . . . . check nodes Adaptive Decoding Procedure

22. received vector transmitted codeword More Details about the Matrix Adaptive Scheme Consider the previous example: (7,4)Hamming code parity check matrix We can guaranteed reduce some (n-k)m columns to degree 1 We attempt to chose these to be the least reliable independent bits Least Reliable Basis

23. Bit reliabilities updating stage (gradient descent) • Iterative decoding is applied to generate extrinsic information • Extrinsic information is scaled by a damping coefficient and fed to update the bit-level reliabilities Interpretation as an Optimization Procedure Standard iterative decoding procedure is interpreted as gradient descent optimization (Lucas et al. 1998). Proposed algorithm is a generalization, two-stage optimization procedure: • Parity check matrix update (change direction) • All bit-level reliabilities are sorted by their absolute values • Systemize the sub-matrix corresponding to LRB in the parity check matrix The damping coefficient serves to control the convergent dynamics.

24. Stuck at pseudo-equilibrium point A Hypothesis Adaption help gradient descent to converge

25. Operation Floating Point Binary Reliability Ordering Matrix Adaption Variable Node Update Check Node Update Overall Complexity The complexity is in polynomial time with or Complexity Analysis Complexity can be even reduced when implemented in parallel

26. Complexity Comparison

27. least reliable symbols binary mapping Variation1: Symbol-level Adaptive Scheme Systemizing the sub-matrix involves undesirable Gaussian elimination. This problem can be detoured via utilizing the structure of RS codes. We implement Symbol-level adaptive scheme. This step can be efficiently realized using Forney’s algorithm (Forney 1965)

28. unreliable bits bit nodes …………. ……….. weakly connected ……………. ……………. . . . . . . . . . check nodes Variation2: Degree-2 sub-graph in the unreliable part Reduce the “unreliable” sub-matrix to a sparse sub-graph rather than an identity to improve the asymptotic performance.

29. Variation2: Degree-2 sub-graph in the unreliable part (cont’d) Q: How to adapt the parity check matrix?

30. Group1 Group2 Variation3: Different grouping of unreliable bits (cont’d) Some bits at the boundary part may also have the wrong sign. Run the proposed algorithm several times, each time with an exchange of some “reliable” and “unreliable” bits at the boundary. Consider the received LLR of an RS(7,5) code: ……. A list of candidate codewords are generated using different groups. Pick up the most likely from the list.

31. ascending reliability Complexity in floating point operation part is reduced to be . Variation4: Partial updating scheme (cont’d) The main complexity comes from updating the bits in the high density part, however, only few bits at the boundary part will be affected. In variable node updating stage: update only the “unreliable” bits in the sparse sub-matrix and a few “reliable” bits at the boundary part. In check node updating stage: make an approximation of the check sum via taking advantage of the ordered reliabilities.

32. Applications Q: How do the proposed algorithm and its variations perform? • Simulation results: • Proposed algorithm and variations over AWGN channel • Performance over symbol level fully interleaved slow fading channel • RS coded turbo equalization (TE) system over EPR4 channel • RS coded modulation over fast fading channel • Simulation setups: • A “genie aided” HDD is assumed for AWGN and fading channel. • In the TE system, all coded bits are interleaved at random. A “genie aided” stopping rule is applied.

33. Additive White Gaussian Noise (AWGN) Channel

34. AWGN Channels

35. Asymptotic performance is consistent with the ML upper-bound. AWGN Channels (cont’d)

36. AWGN Channels (cont’d)

37. AWGN Channels (cont’d)

38. Remarks Proposed scheme performs near ML for medium length codes. Symbol-level adaptive updating scheme provides non-trivial gain. Partial updating incurs little penalty with great reduction in complexity. For long codes, proposed scheme is still away from ML decoding. Q: How does it work over other channels?

39. Interleaved Slow Fading Channel

40. Fully Interleaved Slow Fading Channels

41. Fully Interleaved Slow Fading Channels (cont.)

42. Turbo Equalization Systems

43. source interleaving RS Encoder PR Encoder + interleaving AWGN extrinsic + - sink de-interleaving RS Decoder BCJR Equalizer extrinsic a priori hard decision RS Coded Turbo Equalization System a priori + Embed the Proposed Algorithm in the Turbo Equalization System

44. Turbo Equalization over EPR4 Channels

45. Turbo Equalization over EPR4 Channels

46. RS Coded Modulation

47. RS Coded Modulation over Fast Rayleigh Fading Channels

48. RS Coded Modulation over Fast Rayleigh Fading Channels (cont’d)

49. Remarks More noticeable gain is observed for fading channels, especially for symbol-level adaptive scheme. In RS coded modulation scheme, utilizing bit-level soft information seems provide more gain. The proposed TE scheme can combat ISI and performs almost identically as the performance over AWGN channels. The proposed algorithm has a potential “error floor” problem. However, simulation down to even lower FER is impossible. Asymptotic performance analysis is still under investigation.

50. Conclusion and Future work Iterative decoding of RS codes based on adaptive parity check matrix works favorably for practical codes over various channels. The proposed algorithm and its variations provide a wide range of complexity-performance tradeoff for different applications. • More works under investigation: • Asymptotic performance bound. • Understanding how this algorithm works from an information theoretic perspective, e.g., entropy of ordered statistics. • Improving the generic algorithm using more sophisticated optimization schemes, e.g., conjugate gradient method.