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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 &amp; Be’ery 1991) Reduced complexity version (Ponnampalam &amp; Vucetic 2002) SISO (Ponnampalam &amp; 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&#xF06E;Date of birth: 40 years ago (Reed and Solomon 1960)&#xF06E;Related to non-binary BCH codes (Gorenstein and Zierler 1961)&#xF06E;Efficient decoder: not until 6 years later (Berlekamp 1967)&#xF06E;Linear feedback shift register (LFSR) interpretation (Massey 1969)&#xF06E;Other algebraic hard decision decoder:&#xF06E;Euclid&#x2019;s Algorithm (Sugiyama et al. 1975)&#xF06E;Frequency-domain decoding (Gore 1973 and Blahut 1979)
• 3. Wide Range of Applications of Reed Solomon Codes&#xF06E;NASA Deep Space: CC + RS(255, 223, 32)&#xF06E;Multimedia Storage:&#xF06E;CD: RS(32, 28, 4), RS(28, 24, 4) with interleaving&#xF06E;DVD: RS(208, 192, 16), RS(182, 172, 10) product code&#xF06E;Digitial Video Broadcasting: DVB-T CC + RS(204, 188)&#xF06E;Magnetic Recording: RS(255,239) etc. (nested RS code)
• 4. Basic Properties of Reed Solomon Codes&#x220F;&#x2212;+=&#x2212;==12)()(where,)()()(:formpolynomialGenerator(2)tbbiixxgxmxgxc &#x3B1;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FB;&#xF8F9;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8F0;&#xF8EE;=&#x2212;+&#x2212;&#x2212;+&#x2212;)12)(1(12)1(11:matrixcheckparityThetbNtbbNbH&#x3B1;&#x3B1;&#x3B1;&#x3B1;&#xF04B;&#xF04D;&#xF04F;&#xF04D;&#xF04D;&#xF04B;1Nwhere),(overdefinedK)RS(N, &#x2212;= mmqqGF))(),...,(),((),...,,()(:formevaluationPolynomial(1) 11021 &#x2212;== NN fffxxxxc &#x3B1;&#x3B1;&#x3B1;1110 ...)(where &#x2212;&#x2212;+++= KK xfxffxf)(inelementsnonzerodistinctares mi qGF&#x3B1;
• 5. &#xF06E;Properties of BM algorithm:&#xF06E;Decoding region:&#xF06E;Decoding complexity: UsuallyBasic Properties of Reed Solomon Codes (cont&#x2019;d)min2 dfe &lt;+&#xF06E;Properties of RS code:&#xF06E;Symbol level cyclic (nonbinary BCH codes)&#xF06E;Maximum distance separable (symbol level): 112min +&#x2212;=+= KNtd)( 2No2mind&#x221D;
• 6. Motivation for RS Soft Decision Decoder&#xF06E;Hard decision decoder does not fully exploit the decoding capability&#xF06E;Efficient soft decision decoding of RS codes remains an open problemRS Coded Turbo Equalization System-+a prioriextrinsicinterleavinga prioriextrinsic&#x3A0;&#x3A3;sourceRS EncoderinterleavingPR Encodersinkhard decision+AWGN+RSDecoderChannelEqualizerde-interleaving&#x3A0;1&#x3A0;&#x2212;&#x3A3;&#xF06E;Soft input soft output (SISO) algorithm is favorable
• 7. Presentation Outline&#xF06E;Iterative decoding for RS codes&#xF06E;Symbol-level algebraic soft decision decoding&#xF06E;Simulation results&#xF06E;Binary expansion of RS codes and soft decoding algorithms&#xF06E;Applications and future works
• 8. Symbol-level Algebraic Soft Decision Decoding
• 9. Reliability Assisted Hard Decision Decoding&#xF06E;Generalized Minimum Distance (GMD) Decoding (Forney 1966):&#xF06E;New distance measure: generalized minimum distance&#xF06E;Successively erase the least reliable symbols and run the hard decision decoder&#xF06E;GMD is shown to be asymptotically optimal&#xF06E;Chase Type-II decoding (Chase 1972):&#xF06E;Exhaustively flip the least reliable symbols and run the hard decision decoder&#xF06E;Chase algorithm is also shown to be asymptotically optimal&#xF06E;Related works:&#xF06E;Fast GMD (Koetter 1996)&#xF06E;Efficient Chase (Kamiya 2001)&#xF06E;Combined Chase and GMD for RS codes (Tang et al. 2001)&#xF06E;Performance analysis of these algorithms for RS codes seems still open
• 10. &#xF06E;Bounded distance + 1 decoding (Berlekamp 1996)&#xF06E;Beyond decoding for low rate RS codes (Sudan 1997)&#xF06E;Decoding up errors (Guruswami and Sudan 1999)&#xF06E;A good tutorial paper (JPL Report, McEliece 2003)Algebraic Beyond Half dmin List Decoding2/mind&#xF8F0; &#xF8FB;NKN &#x2212;
• 11. Outline of Algebraic Beyond Half Distance Decoding&#xF06E;Complexity:&#xF06E;Interpolation (Koetter&#x2019;s fast algorithm):&#xF06E;Factorization (Roth and Ruckenstein&#x2019;s algorithm):)( 42mNO)(NKO&#xF06E;Factorization Step:&#xF06E;generate a list of y-roots, i.e.:&#xF06E;Pick up the most likely codeword from the list L}))(deg(),,(|))((:][)({ KxfyxQxfyxFxfL &lt;&#x2212;&#x2208;=)(&#x2C6; xf&#xF06E;Decoding:&#xF06E;Basic idea: find f(x), which fits as many points in pairs))(),...,(),((:codeworddTransmitte 21 Nfff &#x3B1;&#x3B1;&#x3B1;),...,,(:vectorReceived 21 N&#x3B2;&#x3B2;&#x3B2;)),(( iif &#x3B2;&#x3B1;),( yxQ&#xF06E;Interpolation Step:&#xF06E;Construct a bivariate polynomial of minimum (1,K-1) degree,which has a zero of order at , i.e.:m Nlll ,...,1),,( =&#x3B2;&#x3B1;mjiyxQ ll =+&#x2212;&#x2212; thanlessdegreeoftermnoinvolves),(if &#x3B2;&#x3B1;
• 12. Algebraic Soft Interpolation Based List Decoding&#xF06E;Koetter and Vardy algorithm (Koetter &amp; Vardy 2003)&#xF06E;Based on the Guruswami and Sudan&#x2019;s algebraic list decoding&#xF06E;Use the reliability information to assign multiplicities&#xF06E;KV is optimal in multiplicity assignment for long RS codes&#xF06E;Reduced complexity KV (Gross et al. submitted 2003)&#xF06E;Re-encoding technique: largely reduce the cost for high rate codes&#xF06E;VLSI architecture (Ahmed et al. submitted 2003)
• 13. &#xF06E;Basic idea: interpolating more symbols using the soft information&#xF06E;The interpolation and factorization is the same as GS algorithm&#xF06E;Sufficient condition for successful decoding:&#xF06E;The complexity increases with , maximum number of multiplicitySoft Interpolation Based Decoding)()1(2)( MCKcSM &#x2212;&#x2265;&#xF06E;Definition:&#xF06E;Reliability matrix:&#xF06E;Multiplicity matrix:&#xF06E;Score:&#xF06E;Cost:cMcSM ,)( =&#x2211;&#x2211;= =&#xF8F7;&#xF8F7;&#xF8F8;&#xF8F6;&#xF8EC;&#xF8EC;&#xF8ED;&#xF8EB; +=qinjjimMC1 1,21)(Nq&#xD7;&#x3A0;)(&#x3A0;= gM2)(MC
• 14. Recent Works and Remarks&#xF06E;The ultimate gain of algebraic soft decoding (ASD) over AWGNchannel is about 1dB&#xF06E;Complexity is scalable but prohibitively huge for large multiplicity&#xF06E;The failure pattern of ASD algorithm and optimal multiplicityassignment scheme is of interest&#xF06E;Recent works on performance analysis and multiplicity assignment:&#xF06E;Gaussian approximation (Parvaresh and Vardy 2003)&#xF06E;Exponential bound (Ratnakar and Koetter 2004)&#xF06E;Chernoff bound (El-Khamy and McEliece 2004)&#xF06E;Performance analysis over BEC and BSC (Jiang and Narayanan 2005)
• 15. Performance Analysis of ASD over Discrete Alphabet Channels&#xF06E;Performance Analysis over BEC and BSC (Jiang andNarayanan, accepted by ISIT2005)&#xF06E;The analysis gives some intuition about the decoding radius of ASD&#xF06E;We investigate the bit-level decoding radius for high rate codes&#xF06E;For BEC, bit-level radius is twice as large as that of the BM algorithm&#xF06E;For BSC, bit-level radius is slightly larger than that of the BM algorithm&#xF06E;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 &#x2208;&#x2200;],...,,[ 1101 &#x2212;&#xD7; = NN cccC ],...,,,...,,...,,[ )1(1)1(1)0(1)1(0)1(0)0(0)1(&#x2212;&#x2212;&#x2212;&#x2212;&#x2212;&#xD7;= mNNNmNmb ccccccCKm)(Nm,RSexpansionbinaryahas)2(overcodeK)RS(N,ly,Consequent bmGF&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FB;&#xF8F9;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8F0;&#xF8EE;=&#x2212;&#x2212;&#x2212;&#x2212;&#x2212;&#x2212;&#x2212;&#xD7;&#x2212;NKNKNKNNNKNHHHHHHH),1(1),1(0),1(1,01,00,0)(&#xF04C;&#xF04D;&#xF04F;&#xF04D;&#xF04C;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FB;&#xF8F9;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8F0;&#xF8EE;=&#x2212;&#x2212;&#x2212;&#x2212;&#x2212;&#x2212;&#x2212;&#x2212;&#xD7;&#x2212;1,1)(1,1)(0,1)(1,01,00,0)(NmmKNmKNmKNNmNmmKNbhhhhhhH&#xF04C;&#xF04D;&#xF04F;&#xF04D;&#xF04C;
• 18. Bit-level Weight Enumerator&#xF06E;&#x201C;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&#x201D; (Berlekamp)&#xF06E;How does the binary expansion of RS codes perform under ML decoding?&#xF06E;Performance analysis using its weight enumerator&#xF06E;Averaged ensemble weight enumerator of RS codes (Retter 1991)&#xF06E;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&#xF06E;RS codes themselves are good code&#xF06E;However, ML decoding is NP-hard (Guruswami and Vardy 2004)&#xF06E;Are there sub-optimal decoding algorithms using the binary expansions?
• 22. Trellis based Decoding using BCH Subcode Expansion&#xF06E;Maximum-likelihood decoding and variations:&#xF06E;Partition RS codes into BCH subcodes and glue vectors (Vardy and Be&#x2019;ery 1991)&#xF06E;Reduced complexity version (Ponnampalam and Vucetic 2002)&#xF06E;Soft input soft output version (Ponnampalam and Grant 2003)
• 23. Subfield Subcode Decomposition&#xF06E;Remarks:&#xF06E;Decomposition greatly reduces the trellis size for short codes&#xF06E;Impractical for long codes, since the size of the glue vectors is very large&#xF06E;Related work:&#xF06E;Construct sparse representation for iterative decoding (Milenkovic and Vasic2004)&#xF06E;Subspace subcode of Reed Solomon codes (Hattori et al. 1998)BCH subcodesGlue vector&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FB;&#xF8F9;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8F0;&#xF8EE;=4321000000000000~gluegluegluegluebGGGGBBBBG
• 24. Reliability based Ordered Statistic Decoding&#xF06E;Reliability based decoding:&#xF06E;Ordered Statistic Decoding (OSD) (Fossorier and Lin 1995)&#xF06E;Box and Match Algorithm (BMA) (Valembois and Fossorier 2004)&#xF06E;Ordered Statistic Decoding using preprocessing (Wu et al. 2004)&#xF06E;Basic ideas:&#xF06E;Order the received bits according to their reliabilities&#xF06E;Propose hard decision reprocessing based on the most reliable basis (MRB)&#xF06E;Remarks:&#xF06E;The reliability based scheme is efficient for short to medium length codes&#xF06E;The complexity increases exponentially with the reprocessing order&#xF06E;BMA algorithm trade memory for time complexity
• 25. Iterative Decoding Algorithms for RS Codes
• 26. &#xF06E;How does the panacea of modern communication, iterativedecoding algorithm work for RS codes?&#xF06E;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&#x2026;&#x2026;&#x2026;&#x2026;. &#x2026;&#x2026;&#x2026;... . . . . . . . . &#x2026;&#x2026;&#x2026;&#x2026;&#x2026;.check nodes&#x2026;&#x2026;&#x2026;&#x2026;&#x2026;.erased bits?&#xF06E; If two or more of the incoming messages are erasures the check is erased&#xF06E;For the AWGN channel, two or more unreliable messages invalidate the check
• 28. A Few Unreliable Bits &#x201C;Saturate&#x201D; the Non-sparse Parity Check Matrix[ ]000000000000000000000=bc&#xF06E;Iterative decoding is stuck due to only a few unreliable bits&#x201C;saturating&#x201D; the whole non-sparse parity check matrix&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FB;&#xF8F9;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8F0;&#xF8EE;=011110010001111101100001111101100011110010111101100011110010001001011111110101010100100001011111110101010011111110101010100001bHBinary image expansion of the parity check matrix of RS(7, 5) over GF(23)&#xF06E;Consider RS(7, 5) over GF(23) :[ ]1.15.01.08.02.09.06.01.02.09.05.01.03.04.08.01.10.17.06.19.08.0 &#x2212;&#x2212;&#x2212;&#x2212;&#x2212;=r
• 29. Sparse Parity Check Matrices for RS Codes&#xF06E; Can we find an equivalent binary parity check matrix that is sparse?&#xF06E; For RS codes, this is not possible!&#xF06E; The H matrix is the G matrix of the dual code&#xF06E; The dual of an RS code is also an MDS Code&#xF06E; Each row has weight at least (K+1)&#xF06E;Typically, the row weight is much higher
• 30. Iterative Decoding for RS Codes&#xF06E;Recent progress on RS codes:&#xF06E;Sub-trellis based iterative decoding (Ungerboeck 2003)&#xF06E;Stochastic shifting based iterative decoding (Jiang and Narayanan,2004)&#xF06E;Sparse representation of RS codes using GFFT (Yedidia, 2004)&#xF06E;Iterative decoding for general linear block codes:&#xF06E;Iterative decoding for general linear block codes (Hagenauer et al.1996)&#xF06E;APP decoding using minimum weight parity checks (Lucas et al. 1998)&#xF06E;Generalized belief propagation (Yedidia et al. 2000)
• 31. Recent Iterative Techniques&#xF06E;Sub-trellis based iterative decoding (Ungerboeck 2003)&#xF06E;Self concatenation using sub-trellis constructed from the parity check matrix:&#xF06E;Remarks:&#xF06E;Performance deteriorates due to large number of short cycles&#xF06E;Work for short codes with small minimum distances&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FB;&#xF8F9;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8F0;&#xF8EE;=011110010001111101100001111101100011110010111101100011110010001001011111110101010100100001011111110101010011111110101010100001bHBinary image expansion of the parity check matrix of RS(7, 5) over GF(23)
• 32. Recent Iterative Techniques (cont&#x2019;d)&#xF06E; Stochastic shifting based iterative decoding (Jiang and Narayanan, 2004)&#xF06E; Due to the irregularity in the H matrix, iterative decoding favors some bits&#xF06E; Taking advantage of the cyclic structure of RS codes],,,,,,[ 4321065 rrrrrrr ],,,,,,[ 6543210 rrrrrrr&#xF06E; Stochastic shift prevent iterative procedure from getting stuck&#xF06E;Best result: RS(63,55) about 0.5dB gain from HDD&#xF06E; However, for long codes, the performance deterioratesShift by 2&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FB;&#xF8F9;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8F0;&#xF8EE;=101100101100110001111H
• 33. Proposed Iterative Decoding for RS Codes
• 34. Iterative Decoding Based on Adaptive Parity Check Matrixtransmitted codeword [ ]0011010=c&#xF06E;Idea: reduce the sub-matrix corresponding to unreliable bits to asparse nature using Gaussian elimination&#xF06E;For example, consider (7,4) Hamming code:parity check matrix&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FB;&#xF8F9;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8F0;&#xF8EE;=101011011001011101010H&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FB;&#xF8F9;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8F0;&#xF8EE;=101100101100110001111H&#xF8FA;&#xF8FA;&#xF8FA;&#xF8FB;&#xF8F9;&#xF8EF;&#xF8EF;&#xF8EF;&#xF8F0;&#xF8EE;=101100101100110001111H&#xF06E;We can make the (n-k) less reliable positions sparse!received vector [ ]1.01.02.14.11.06.01.1 &#x2212;&#x2212;&#x2212;&#x2212;&#x2212;&#x2212;=r
• 35. Adaptive Decoding Procedurebit nodes&#x2026;&#x2026;&#x2026;&#x2026;. &#x2026;&#x2026;&#x2026;... . . . . . . . . &#x2026;&#x2026;&#x2026;&#x2026;&#x2026;.check nodes&#x2026;&#x2026;&#x2026;&#x2026;&#x2026;.unreliable bits&#xF06E;After the adaptive update, iterative decoding can proceed
• 36. Gradient Descent and Adaptive Potential Function&#xF06E;The decoding problem is relaxed as minimizing J using gradientdescent with the initial value T observed from the channel&#xF06E;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 &#x3C8;&#x2190;&#xF06E;Geometric interpretation (suggested by Ralf Koetter)&#xF06E;Define the tanh domain transform as:&#xF06E;The syndrome of a parity check can be expressed as:&#xF06E;Define the soft syndrome as:&#xF06E;Define the cost function as:&#x2211;&#x2295;==1ijHji rs&#x220F;&#x220F; ====11)(ijij HjHji LTS &#x3BD;&#x2211;&#x2212;=&#x2212;=kniiSTHJ1),()2/tanh()( jjj LLT ==&#x3BD;
• 37. Two Stage Optimization Procedure&#xF06E;Proposed algorithm is a generalization of the iterative decoding schemeproposed by Lucas et al. (1998), two-stage optimization procedure:&#xF06E;The damping coefficient serves to control the convergent dynamics)))(()((1 },1|{)(1)(1)1(&#x2211; &#x220F;&#x2212;= &#x2260;=&#x2208;&#x2212;&#x2212;++&#x2190;kni mjHjjtmtmtmijTTT &#x3BD;&#x3B1;&#x3BD;&#x3BD;),( )()0()( ttTHH &#x3C8;&#x2190;
• 39. Variations of the Generic Algorithm&#xF06E;Connect unreliable bits as deg-2&#xF06E;Incorporate this algorithm with hard decision decoder&#xF06E;Adapting the parity check matrix at symbol level&#xF06E;Exchange bits in reliable and unreliable part. Run the decoder multiple times&#xF06E;Reduced complexity partial updating scheme
• 40. Simulation Results
• 41. AWGN Channels
• 42. AWGN Channels (cont&#x2019;d)Asymptotic performanceis consistent with the MLupper-bound.
• 43. AWGN Channels (cont&#x2019;d)
• 44. AWGN Channels (cont&#x2019;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 prioriextrinsic&#x3A0;&#x3A3;sourceRS EncoderinterleavingPR Encodersinkhard decision+AWGN+RSDecoderBCJREqualizerde-interleaving&#x3A0;1&#x3A0;&#x2212;&#x3A3;
• 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&#xF06E;Respective problems for various decoding schemes:&#xF06E;Reliability assisted HDD: Gain is marginal in practical SNRs&#xF06E;Algebraic soft decoding: performance is limited by the algebraic nature&#xF06E;Reliability based decoding: huge memory, not scalable with SNR&#xF06E;Sub-code decomposition: only possible for very short codes&#xF06E;Iterative decoding: adapting Hb at each iteration is a huge cost&#xF06E;General Problems:&#xF06E;Coding gain may shrink down in practical systems&#xF06E;Concatenated with CC: difficult to generate the soft information&#xF06E;Performance in the practical SNRs should be analyzed&#xF06E;&#x201C;In theory, there is no difference between theory and practice.But, in practice, there is&#x2026;&#x201D; (Jan L.A. van de Snepscheut)
• 56. A Case Study (System Setups)&#xF06E;Forward Error Control of a Digital Television Transmission Standard:&#xF06E;Modulation format: 64 or 16 QAM modulation (semi-set partitioning mapping)&#xF06E;Inner code: convolutional code rate=2/3 or 8/9&#xF06E;Bit-interleaved coded modulation (BICM)&#xF06E;Iterative demodulation and decoding (BICM-ID)&#xF06E;The decoded bytes from inner decoder are interleaved and fed to outer decoder&#xF06E;Outer code: RS(208,188) using hard decision decoding&#xF06E;Will soft decoding algorithm significantly improve the overall performance?
• 57. A Case Study (Simulation Results)
• 58. Future Works&#xF06E;How to incorporate the proposed ADP with other soft decodingschemes?&#xF06E;Taking advantage of the inherent structure of RS codes at bit level&#xF06E;More powerful decoding tool, e.g., trellis&#xF06E;Extend the idea of adaptive algorithms to demodulation and equalization&#xF06E;Apply the ADP algorithm to quantization or to solve K-SAT problems
• 59. Thank you!