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Soft Decision Decoding Algorithms of Reed-Solomon Codes


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  • Title
  • Most of the algebraic hard decision decoders are of similar complexity, which is about o(d^2)
  • Recording channels are the most important application, perhaps!
  • Two forms of representations are equivalent!
  • Basic properties
  • The motivation of an efficient SISO decoder for RS codes
  • outline
  • Generalized Minimum Distance (GMD) Decoding (Forney 1966) Chase decoding (Chase 1972) Related works
  • Algebraic beyond half dmin decoding, which is the kernel of KV algorithm
  • Outline: 1) curve fitting! 2) multiplicity assignment, let (x,y) pass through (alpha, beta) m times 3) the larger the m, the larger the decoding radius 4) when beta fails in the decoding radius, (y-f(x)) can factorize Q(x,y) 5) smart way to construct Q(x,y) and factorize it 6) when soft information is available, can do weighted multiplicity assignment
  • KV 1) proportional assignment 2) asymptotically optimal
  • KV 1) performance analysis becomes interesting 2) no longer a fixed radius 3) sufficient condition
  • KV 1) failure is interesting, can we do better?
  • KV 1) failure is interesting, can we do better?
  • Life becomes much easier when we go to bit level
  • When turbo code comes, people realize convolutional codes are inherently bad codes, but how about algebraic codes?
  • Asymptotically be optimal, but practically suffer a loss!
  • More significant difference
  • Motivation for bit level decomposition
  • Maximum-likelihood SISO decoding and variations Trellis based decoding using the binary image expansions of RS codes over GF(2 m ) (Vardy & Be’ery 1991) Reduced complexity version (Ponnampalam & Vucetic 2002) SISO (Ponnampalam & Grant 2003)
  • 1) BCH subcode and glue vectors 2) Useful to construct sparse parity check matrices for short codes
  • Efficient for general linear block codes
  • The nice property of ldpc codes is that the parity check matrix is sparse, thus, the probability that two erased bits participate in another check diminishes.
  • Even in the optimistic case, erasure channel, it won’t give good results.
  • It is impossible to have a sparse representation of RS codes
  • Iterative decoding for general linear block codes (tough problem!)
  • Since the parity check matrix is of full rank, we are guaranteed to get (n-k) LRB
  • Define syndrome as binary summation of all participating bits Define the soft syndrome product of the channel reliabilities J is minimized iff the decoding converges to a valid codeword
  • A variation of standard gradient descent
  • Think about erasure channel. This does happen! J is not minimized but we get zero gradient!
  • Not going deep to details!
  • KV 0.65dB BMA 1dB With extreme huge complexity, we gain 1.6dB gain at FER = 10^-4.
  • In practice, there is a huge difference!
  • Case study!
  • Coding gain may shrink in practical systems
  • No really successful bit level soft decoder which can handle long constraint length codes
  • Acknowledgements
  • Transcript

    • 1. Soft Decision Decoding Algorithms ofReed-Solomon CodesSarbjeet SinghNITTTRSector 26 CHD
    • 2. Historical Review of Reed Solomon CodesDate of birth: 40 years ago (Reed and Solomon 1960)Related to non-binary BCH codes (Gorenstein and Zierler 1961)Efficient decoder: not until 6 years later (Berlekamp 1967)Linear feedback shift register (LFSR) interpretation (Massey 1969)Other algebraic hard decision decoder:Euclid’s Algorithm (Sugiyama et al. 1975)Frequency-domain decoding (Gore 1973 and Blahut 1979)
    • 3. Wide Range of Applications of Reed Solomon CodesNASA Deep Space: CC + RS(255, 223, 32)Multimedia Storage:CD: RS(32, 28, 4), RS(28, 24, 4) with interleavingDVD: RS(208, 192, 16), RS(182, 172, 10) product codeDigitial Video Broadcasting: DVB-T CC + RS(204, 188)Magnetic Recording: RS(255,239) etc. (nested RS code)
    • 4. Basic Properties of Reed Solomon Codes∏−+=−==12)()(where,)()()(:formpolynomialGenerator(2)tbbiixxgxmxgxc α=−+−−+−)12)(1(12)1(11:matrixcheckparityThetbNtbbNbHαααα1Nwhere),(overdefinedK)RS(N, −= mmqqGF))(),...,(),((),...,,()(:formevaluationPolynomial(1) 11021 −== NN fffxxxxc ααα1110 ...)(where −−+++= KK xfxffxf)(inelementsnonzerodistinctares mi qGFα
    • 5. Properties of BM algorithm:Decoding region:Decoding complexity: UsuallyBasic Properties of Reed Solomon Codes (cont’d)min2 dfe <+Properties of RS code:Symbol level cyclic (nonbinary BCH codes)Maximum distance separable (symbol level): 112min +−=+= KNtd)( 2No2mind∝
    • 6. Motivation for RS Soft Decision DecoderHard decision decoder does not fully exploit the decoding capabilityEfficient soft decision decoding of RS codes remains an open problemRS Coded Turbo Equalization System-+a prioriextrinsicinterleavinga prioriextrinsicΠΣsourceRS EncoderinterleavingPR Encodersinkhard decision+AWGN+RSDecoderChannelEqualizerde-interleavingΠ1Π−ΣSoft input soft output (SISO) algorithm is favorable
    • 7. Presentation OutlineIterative decoding for RS codesSymbol-level algebraic soft decision decodingSimulation resultsBinary expansion of RS codes and soft decoding algorithmsApplications and future works
    • 8. Symbol-level Algebraic Soft Decision Decoding
    • 9. Reliability Assisted Hard Decision DecodingGeneralized Minimum Distance (GMD) Decoding (Forney 1966):New distance measure: generalized minimum distanceSuccessively erase the least reliable symbols and run the hard decision decoderGMD is shown to be asymptotically optimalChase Type-II decoding (Chase 1972):Exhaustively flip the least reliable symbols and run the hard decision decoderChase algorithm is also shown to be asymptotically optimalRelated works:Fast GMD (Koetter 1996)Efficient Chase (Kamiya 2001)Combined Chase and GMD for RS codes (Tang et al. 2001)Performance analysis of these algorithms for RS codes seems still open
    • 10. Bounded distance + 1 decoding (Berlekamp 1996)Beyond decoding for low rate RS codes (Sudan 1997)Decoding up errors (Guruswami and Sudan 1999)A good tutorial paper (JPL Report, McEliece 2003)Algebraic Beyond Half dmin List Decoding2/mind NKN −
    • 11. Outline of Algebraic Beyond Half Distance DecodingComplexity:Interpolation (Koetter’s fast algorithm):Factorization (Roth and Ruckenstein’s algorithm):)( 42mNO)(NKOFactorization Step:generate a list of y-roots, i.e.:Pick up the most likely codeword from the list L}))(deg(),,(|))((:][)({ KxfyxQxfyxFxfL <−∈=)(ˆ xfDecoding:Basic idea: find f(x), which fits as many points in pairs))(),...,(),((:codeworddTransmitte 21 Nfff ααα),...,,(:vectorReceived 21 Nβββ)),(( iif βα),( yxQInterpolation Step:Construct a bivariate polynomial of minimum (1,K-1) degree,which has a zero of order at , i.e.:m Nlll ,...,1),,( =βαmjiyxQ ll =+−− thanlessdegreeoftermnoinvolves),(if βα
    • 12. Algebraic Soft Interpolation Based List DecodingKoetter and Vardy algorithm (Koetter & Vardy 2003)Based on the Guruswami and Sudan’s algebraic list decodingUse the reliability information to assign multiplicitiesKV is optimal in multiplicity assignment for long RS codesReduced complexity KV (Gross et al. submitted 2003)Re-encoding technique: largely reduce the cost for high rate codesVLSI architecture (Ahmed et al. submitted 2003)
    • 13. Basic idea: interpolating more symbols using the soft informationThe interpolation and factorization is the same as GS algorithmSufficient condition for successful decoding:The complexity increases with , maximum number of multiplicitySoft Interpolation Based Decoding)()1(2)( MCKcSM −≥Definition:Reliability matrix:Multiplicity matrix:Score:Cost:cMcSM ,)( =∑∑= = +=qinjjimMC1 1,21)(Nq×Π)(Π= gM2)(MC
    • 14. Recent Works and RemarksThe ultimate gain of algebraic soft decoding (ASD) over AWGNchannel is about 1dBComplexity is scalable but prohibitively huge for large multiplicityThe failure pattern of ASD algorithm and optimal multiplicityassignment scheme is of interestRecent works on performance analysis and multiplicity assignment:Gaussian approximation (Parvaresh and Vardy 2003)Exponential bound (Ratnakar and Koetter 2004)Chernoff bound (El-Khamy and McEliece 2004)Performance analysis over BEC and BSC (Jiang and Narayanan 2005)
    • 15. Performance Analysis of ASD over Discrete Alphabet ChannelsPerformance Analysis over BEC and BSC (Jiang andNarayanan, accepted by ISIT2005)The analysis gives some intuition about the decoding radius of ASDWe investigate the bit-level decoding radius for high rate codesFor BEC, bit-level radius is twice as large as that of the BM algorithmFor BSC, bit-level radius is slightly larger than that of the BM algorithmIn conclusion, ASD is limited by its algebraic engine
    • 16. Binary Image Expansion of RSCodes and Soft Decision Decoding
    • 17. Binary Image Expansion of RS Codes over GF(2m)bmxGFx torbinary vecdim-manasexpressedbecan)2(known thatisIt ∈∀],...,,[ 1101 −× = NN cccC ],...,,,...,,...,,[ )1(1)1(1)0(1)1(0)1(0)0(0)1(−−−−−×= mNNNmNmb ccccccCKm)(Nm,RSexpansionbinaryahas)2(overcodeK)RS(N,ly,Consequent bmGF=−−−−−−−×−NKNKNKNNNKNHHHHHHH),1(1),1(0),1(1,01,00,0)(=−−−−−−−−×−1,1)(1,1)(0,1)(1,01,00,0)(NmmKNmKNmKNNmNmmKNbhhhhhhH
    • 18. Bit-level Weight Enumerator“The major drawback with RS codes (for satellite use) is that the presentgeneration of decoders do not make full use of bit-based soft decisioninformation” (Berlekamp)How does the binary expansion of RS codes perform under ML decoding?Performance analysis using its weight enumeratorAveraged ensemble weight enumerator of RS codes (Retter 1991)It gives some idea about how RS codes perform under ML decoding
    • 19. Performance Comparison of RS(255,239)
    • 20. Performance Comparison of RS(255,127)
    • 21. RemarksRS codes themselves are good codeHowever, ML decoding is NP-hard (Guruswami and Vardy 2004)Are there sub-optimal decoding algorithms using the binary expansions?
    • 22. Trellis based Decoding using BCH Subcode ExpansionMaximum-likelihood decoding and variations:Partition RS codes into BCH subcodes and glue vectors (Vardy and Be’ery 1991)Reduced complexity version (Ponnampalam and Vucetic 2002)Soft input soft output version (Ponnampalam and Grant 2003)
    • 23. Subfield Subcode DecompositionRemarks:Decomposition greatly reduces the trellis size for short codesImpractical for long codes, since the size of the glue vectors is very largeRelated work:Construct sparse representation for iterative decoding (Milenkovic and Vasic2004)Subspace subcode of Reed Solomon codes (Hattori et al. 1998)BCH subcodesGlue vector=4321000000000000~gluegluegluegluebGGGGBBBBG
    • 24. Reliability based Ordered Statistic DecodingReliability based decoding:Ordered Statistic Decoding (OSD) (Fossorier and Lin 1995)Box and Match Algorithm (BMA) (Valembois and Fossorier 2004)Ordered Statistic Decoding using preprocessing (Wu et al. 2004)Basic ideas:Order the received bits according to their reliabilitiesPropose hard decision reprocessing based on the most reliable basis (MRB)Remarks:The reliability based scheme is efficient for short to medium length codesThe complexity increases exponentially with the reprocessing orderBMA algorithm trade memory for time complexity
    • 25. Iterative Decoding Algorithms for RS Codes
    • 26. How does the panacea of modern communication, iterativedecoding algorithm work for RS codes?Note that all the codes in the literature, for which we can usesoft decoding algorithms are sparse graph codes with smallconstraint length.A Quick Question
    • 27. How does standard message passing algorithm work?bit nodes…………. ………... . . . . . . . . …………….check nodes…………….erased bits? If two or more of the incoming messages are erasures the check is erasedFor the AWGN channel, two or more unreliable messages invalidate the check
    • 28. A Few Unreliable Bits “Saturate” the Non-sparse Parity Check Matrix[ ]000000000000000000000=bcIterative decoding is stuck due to only a few unreliable bits“saturating” the whole non-sparse parity check matrix=011110010001111101100001111101100011110010111101100011110010001001011111110101010100100001011111110101010011111110101010100001bHBinary image expansion of the parity check matrix of RS(7, 5) over GF(23)Consider RS(7, 5) over GF(23) :[ ] −−−−−=r
    • 29. 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 Each row has weight at least (K+1)Typically, the row weight is much higher
    • 30. Iterative Decoding for RS CodesRecent progress on RS codes:Sub-trellis based iterative decoding (Ungerboeck 2003)Stochastic shifting based iterative decoding (Jiang and Narayanan,2004)Sparse representation of RS codes using GFFT (Yedidia, 2004)Iterative decoding for general linear block codes:Iterative decoding for general linear block codes (Hagenauer et al.1996)APP decoding using minimum weight parity checks (Lucas et al. 1998)Generalized belief propagation (Yedidia et al. 2000)
    • 31. Recent Iterative TechniquesSub-trellis based iterative decoding (Ungerboeck 2003)Self concatenation using sub-trellis constructed from the parity check matrix:Remarks:Performance deteriorates due to large number of short cyclesWork for short codes with small minimum distances=011110010001111101100001111101100011110010111101100011110010001001011111110101010100100001011111110101010011111110101010100001bHBinary image expansion of the parity check matrix of RS(7, 5) over GF(23)
    • 32. Recent Iterative Techniques (cont’d) Stochastic shifting based iterative decoding (Jiang and Narayanan, 2004) Due to the irregularity in the H matrix, iterative decoding favors some bits Taking advantage of the cyclic structure of RS codes],,,,,,[ 4321065 rrrrrrr ],,,,,,[ 6543210 rrrrrrr Stochastic shift prevent iterative procedure from getting stuckBest result: RS(63,55) about 0.5dB gain from HDD However, for long codes, the performance deterioratesShift by 2=101100101100110001111H
    • 33. Proposed Iterative Decoding for RS Codes
    • 34. Iterative Decoding Based on Adaptive Parity Check Matrixtransmitted codeword [ ]0011010=cIdea: reduce the sub-matrix corresponding to unreliable bits to asparse nature using Gaussian eliminationFor example, consider (7,4) Hamming code:parity check matrix=101011011001011101010H=101100101100110001111H=101100101100110001111HWe can make the (n-k) less reliable positions sparse!received vector [ ] −−−−−−=r
    • 35. Adaptive Decoding Procedurebit nodes…………. ………... . . . . . . . . …………….check nodes…………….unreliable bitsAfter the adaptive update, iterative decoding can proceed
    • 36. Gradient Descent and Adaptive Potential FunctionThe decoding problem is relaxed as minimizing J using gradientdescent with the initial value T observed from the channelJ is also a function of H. It is adapted such that unreliable bits areseparated in order to avoid getting stuck at zero gradient points: ),( )0(THH ψ←Geometric interpretation (suggested by Ralf Koetter)Define the tanh domain transform as:The syndrome of a parity check can be expressed as:Define the soft syndrome as:Define the cost function as:∑⊕==1ijHji rs∏∏ ====11)(ijij HjHji LTS ν∑−=−=kniiSTHJ1),()2/tanh()( jjj LLT ==ν
    • 37. Two Stage Optimization ProcedureProposed algorithm is a generalization of the iterative decoding schemeproposed by Lucas et al. (1998), two-stage optimization procedure:The damping coefficient serves to control the convergent dynamics)))(()((1 },1|{)(1)(1)1(∑ ∏−= ≠=∈−−++←kni mjHjjtmtmtmijTTT νανν),( )()0()( ttTHH ψ←
    • 38. Avoid Zero Gradient PointAdaptive schemechanges the gradientand prevents itgetting stuck at zerogradient pointsZero gradientpoint
    • 39. Variations of the Generic AlgorithmConnect unreliable bits as deg-2Incorporate this algorithm with hard decision decoderAdapting the parity check matrix at symbol levelExchange bits in reliable and unreliable part. Run the decoder multiple timesReduced complexity partial updating scheme
    • 40. Simulation Results
    • 41. AWGN Channels
    • 42. AWGN Channels (cont’d)Asymptotic performanceis consistent with the MLupper-bound.
    • 43. AWGN Channels (cont’d)
    • 44. AWGN Channels (cont’d)
    • 45. Interleaved Slow Fading Channel
    • 46. Fully Interleaved Slow Fading Channels
    • 47. Fully Interleaved Slow Fading Channels (cont.)
    • 48. Turbo Equalization Systems
    • 49. Embed the Proposed Algorithm in the Turbo Equalization SystemRS Coded Turbo Equalization System-+a prioriextrinsicinterleavinga prioriextrinsicΠΣsourceRS EncoderinterleavingPR Encodersinkhard decision+AWGN+RSDecoderBCJREqualizerde-interleavingΠ1Π−Σ
    • 50. Turbo Equalization over EPR4 Channels
    • 51. Turbo Equalization over EPR4 Channels
    • 52. RS Coded Modulation
    • 53. RS Coded Modulation over Fast Rayleigh Fading Channels
    • 54. Applications and Future Works
    • 55. Potential Problems in ApplicationsRespective problems for various decoding schemes:Reliability assisted HDD: Gain is marginal in practical SNRsAlgebraic soft decoding: performance is limited by the algebraic natureReliability based decoding: huge memory, not scalable with SNRSub-code decomposition: only possible for very short codesIterative decoding: adapting Hb at each iteration is a huge costGeneral Problems:Coding gain may shrink down in practical systemsConcatenated with CC: difficult to generate the soft informationPerformance in the practical SNRs should be analyzed“In theory, there is no difference between theory and practice.But, in practice, there is…” (Jan L.A. van de Snepscheut)
    • 56. A Case Study (System Setups)Forward Error Control of a Digital Television Transmission Standard:Modulation format: 64 or 16 QAM modulation (semi-set partitioning mapping)Inner code: convolutional code rate=2/3 or 8/9Bit-interleaved coded modulation (BICM)Iterative demodulation and decoding (BICM-ID)The decoded bytes from inner decoder are interleaved and fed to outer decoderOuter code: RS(208,188) using hard decision decodingWill soft decoding algorithm significantly improve the overall performance?
    • 57. A Case Study (Simulation Results)
    • 58. Future WorksHow to incorporate the proposed ADP with other soft decodingschemes?Taking advantage of the inherent structure of RS codes at bit levelMore powerful decoding tool, e.g., trellisExtend the idea of adaptive algorithms to demodulation and equalizationApply the ADP algorithm to quantization or to solve K-SAT problems
    • 59. Thank you!