Presentation on theme: "Jing Jiang and Krishna R. Narayanan Wireless Communication Group"— Presentation transcript:
1 Soft Decision Decoding of RS Codes Using Adaptive Parity Check Matrices Jing Jiang and Krishna R. NarayananWireless Communication GroupDepartment of Electrical EngineeringTexas 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 codewordRS codes are MDS (dmin = n-k+1)The dual code is also MDS
3 Introduction Advantages Drawback source RS Encoder interleaving PR Encodersinkhard decision+AWGNRS DecoderBCJR Equalizerde-interleavingRS Coded Turbo Equalization System-+a prioriextrinsicinterleavingAdvantagesGuaranteed minimum distanceEfficient bounded distance hard decision decoder (HDD)Decoder can handle errors and erasuresDrawbackPerformance loss due to bounded distance decodingSoft input soft output (SISO) decoding is not easy!
4 Presentation Outline Existing soft decision decoding techniques Iterative decoding based on adaptive parity check matricesVariations of the generic algorithmApplications over various channelsConclusion and future workExisting Reed Solomon (RS) codes soft decision decoding techniquesIterative decoding of RS codes based on adaptive parity check matricesVariations of the generic algorithmSimulation results over different channel modelsConclusion and future works
6 Enhanced Algebraic Hard Decision Decoding Generalized Minimum Distance (GMD) Decoding (Forney 1966):Basic Idea:Erase some of the least reliable symbolsRun algebraic hard decision decoding several timesDrawback: GMD has a limited performance gainChase decoding (Chase 1972):Exhaustively flip some of the least reliable symbolsRunning algebraic hard decision decoding several timesDrawback: Has an exponentially increasing complexityEnhanced algebraic hard decision decoding algorithms:Generalized Minimum Distance (GMD) Decoding (Forney 1966)Chase decoding (Chase 1972)Combined Chase & GMD(Tang et al. 2001)Reliability based decoding:Ordered Statistic Decoding (OSD) algorithm (Fossorier & Lin 1995)Variations, e.g., Box and Match (BMA) decoding (Valembois & Fossorier 2003)Algebraic SIHO decoding:Algebraic interpolation based decoding (Koetter & Vardy 2003)Reduced complexity KV algorithm (Gross et al. submitted 2003)Combined Chase & GMD(Tang et al. 2001).
7 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 decodingConvert the reliability information into a set of interpolation pointsGenerate a list of candidate codewordsPick up the most likely codeword from the codeword listDrawback:The complexity increases with , maximum number of multiplicity.
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 reliabilitiesMake hard decisions on a set of independent reliable bits (MR Basis)Re encode to obtain a list of candidate codewordsDrawback:The complexity increases exponentially with the reprocessing orderBMA must trade memory for complexity
9 Trellis based Decoding using the Binary Image Expansion Maximum-likelihood decoding and variationsTrellis based decoding using binary image expansion (Vardy & Be’ery ‘91)Reduced complexity version (Ponnampalam & Vucetic 2002)Basic ideas:Binary image expansion of RSTrellis structure construction using the binary image expansionDrawback:Exponentially increasing complexityWork only for very short codes or codes with very small distanceMaximum-likelihood SISO decoding and variationsTrellis based decoding using the binary image expansions of RS codes over GF(2m) (Vardy & Be’ery 1991)Reduced complexity version (Ponnampalam & Vucetic 2002)Iterative SISO algorithmsSub-trellis structure self-concatenation and iterative decoding of RS codes using their binary image (Ungerboeck 2003)Stochastic shifting based iterative decoding of RS codes using their binary image (Jing & Narayanan, submitted 2003)Iterative decoding via sparse factor graph representations of RS codes based on a fast Fourier transform (FFT) (Yedidia, MERL report 2003)
11 Consider the (7,5) RS codeBinary image expansion of the parity check matrix of RS(7, 5) over GF(23)
12 Recent Iterative Techniques Sub-trellis based iterative decoding (Ungerboeck 2003)Self-concatenation structure based on sub-trellis constructed from the parity check matrixBinary image expansion of the parity check matrix of RS(7, 5) over GF(23)Drawbacks:Performance deteriorates due to large number of short cyclesWork for short codes with small minimum distancesPotential 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 bitsTaking advantage of the cyclic structure of RS codesShift by 2Stochastic shift prevent iterative procedure from getting stuckBest result: RS(63,55) about 0.5dB gain from HDDHowever, 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 gainMoreover, SIHO decoders cannot generate soft output directlyTrellis-based decoders have exponentially increasing complexityIterative decoding algorithms do not work for long codes, since the parity check matrices of RS codes are not sparseMost SIHO algorithms are either too complex to implement or have only marginal gain.Moreover, SIHO decoders cannot generate soft output efficiently.Trellis-based ML decoder has exponentially increasing complexity.Iterative decoding algorithms do not work for long codes. Since the parity check matrices of RS codes are not sparse. There’s no known technique to generate a sparse parity check matrix for long RS codes.“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 QuestionsQ: 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?Most SIHO algorithms are either too complex to implement or have only marginal gain.Moreover, SIHO decoders cannot generate soft output efficiently.Trellis-based ML decoder has exponentially increasing complexity.Iterative decoding algorithms do not work for long codes. Since the parity check matrices of RS codes are not sparse. There’s no known technique to generate a sparse parity check matrix for long RS codes.
16 How does standard message passing algorithm work? erased bitsbit nodes………….………..?…………….check nodesIf two or more of the incoming messages are erasures the check is erasedOtherwise, check to bit message is the value of the bit that will satisfy the check
17 How does standard message passing algorithm work? bit nodes………….………..…………….check nodesSmall values of vj can be thought of as erasures and hence more than twoedges with small vj’s saturate the check
18 A Few Unreliable Bits “Saturate” the Non-sparse Parity Check Matrix Consider RS(7, 5) over GF(23)Binary image expansion of the parity check matrix of RS(7, 5) over GF(23)Unfortunately, RS codes usually possess no sparse parity check matrices as the codeword length increases. There are no known operation to make the whole parity check matrix sparse.Thus, as the codeword length goes to be large, Consequently, iterative decoding is stuck at some “pseudo-codes” due to only a few unreliable bits.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 codeThe dual of an RS code is also an MDS CodeEvery row has weight at least (N-K)!
20 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:transmitted codewordreceived vectorparity check matrixNeed animation of stopping pattern of iterative decoding.After the adaptive update, iterative decoding can proceed.
22 More Details about the Matrix Adaptive Scheme Consider the previous example: (7,4)Hamming codetransmitted codewordreceived vectorparity check matrixWe can guaranteed reduce some (n-k)m columns to degree 1We attempt to chose these to be the least reliable independent bitsLeast Reliable Basis
23 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:Bit reliabilities updating stage (gradient descent)Iterative decoding is applied to generate extrinsic informationExtrinsic information is scaled by a damping coefficient and fed to update the bit-level reliabilitiesParity check matrix update (change direction)All bit-level reliabilities are sorted by their absolute valuesSystemize the sub-matrix corresponding to LRB in the parity check matrixThe damping coefficient serves to control the convergent dynamics.
24 A Hypothesis Adaption help gradient descent to converge Stuck at pseudo-equilibrium pointAdaption help gradient descent to converge
25 Complexity Analysis Binary Floating Point Operation Check Node UpdateOverall ComplexityVariable Node UpdateMatrix AdaptionReliability OrderingBinaryFloating PointOperationThe complexity is in polynomial time with orComplexity can be even reduced when implemented in parallel
27 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.least reliable symbolsThis step can be efficiently realized using Forney’s algorithm (Forney 1965)binary mapping
28 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.bit nodes………….………..…………….check nodesunreliable bitsweakly connected
29 Variation2: Degree-2 sub-graph in the unreliable part (cont’d) Q: How to adapt the parity check matrix?
30 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:Group1Group2…….A list of candidate codewords are generated using different groups. Pick up the most likely from the list.
31 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.Complexity in floating point operation part is reduced to beascending reliabilityMaximum-likelihood SISO decoding and variationsTrellis based decoding using the binary image expansions of RS codes over GF(2m) (Vardy & Be’ery 1991)Reduced complexity version (Ponnampalam & Vucetic 2002)Iterative SISO algorithmsSub-trellis structure self-concatenation and iterative decoding of RS codes using their binary image (Ungerboeck 2003)Stochastic shifting based iterative decoding of RS codes using their binary image (Jing & Narayanan, submitted 2003)Iterative decoding via sparse factor graph representations of RS codes based on a fast Fourier transform (FFT) (Yedidia, MERL report 2003)
32 ApplicationsQ: How do the proposed algorithm and its variations perform?Simulation results:Proposed algorithm and variations over AWGN channelPerformance over symbol level fully interleaved slow fading channelRS coded turbo equalization (TE) system over EPR4 channelRS coded modulation over fast fading channelSimulation 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.The performance of the proposed algorithm and its variations are verified via computer simulation.Proposed algorithm is implemented in conjunction with a “genie aided” hard decision decoding over AWGN and fading channels to speed up simulation.In the turbo equalization system, no hard decision is assumed, however, when all bits converges to the correct value, iteration stops.
38 RemarksProposed 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?Generic algorithm provides nearSymbol level adaptive updating scheme provides comparable performance with Chase and KV algorithms.Partial updating scheme incurs little penalty while greatly reduces the complexity.For long codes, e.g., RS(255,239), the proposed algorithms, though provides significant gain over hard decision decoding, are still quite away ML performance.
47 RS Coded Modulation over Fast Rayleigh Fading Channels
48 RS Coded Modulation over Fast Rayleigh Fading Channels (cont’d)
49 RemarksMore 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.For EPR4 channels, the proposed turbo equalization scheme can mitigate the effect ISI and performs almost identically as the performance over AWGN channels, while other SIHO algorithms are not efficient to generate soft information.For symbol-level fully interleaved slow fading channels, more noticeable gain is observed for RS codes of practical length, especially for symbol-level adaptive scheme.For RS coded modulation over fast Rayleigh fading channels, proposed algorithm also outperforms hard decision decoding via more efficiently utilizing bit level soft information as long as Gray mapping is applied.For long codes, an “error floor” phenomenon of the proposed algorithm is observed. Simulation down to even lower FER is impossible. Asymptotic performance analysis is 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.Iterative decoding of RS codes based on adaptive parity check matrix works favorably for practical codes both over AWGN channels and in TE systems.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 (mutual information of ordered statistics).Improving the generic algorithm using more sophisticated optimization schemes, e.g., conjugate gradient method.