1 / 44

REED-SOLOMON CODES

REED-SOLOMON CODES. 数字通信系统模型. 纠错码的发展概况. 通信的数学理论， Shannon(1948) 汉明码， Hamming (1950) 级连码， Forney(1966) 卷积码及有效译码 , (60 年代 ) RS 码及 BCH 码的有效译码 (60 年代 ) TCM ， Ungerboeck (1982),Forney(1984) Turbo 码， Berrou (1993) LDPC 码， Gallager (1963), Macky (1996) 空时编码 , Tarokh (2000)

walden
Download Presentation

REED-SOLOMON CODES

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. REED-SOLOMON CODES

  2. 数字通信系统模型

  3. 纠错码的发展概况 • 通信的数学理论，Shannon(1948) • 汉明码，Hamming (1950) • 级连码，Forney(1966) • 卷积码及有效译码, (60年代) • RS码及BCH码的有效译码(60年代) • TCM，Ungerboeck(1982),Forney(1984) • Turbo码，Berrou(1993) • LDPC 码，Gallager(1963),Macky(1996) • 空时编码,Tarokh(2000) • 协作与网络编码(2000;2002)

  4. 码距与检错和纠错能力的关系

  5. 二进制本源BCH码 • 对于任意正整数m(m≥3)和t(t< )，存在GF(2)具有如下参数的二进制本原BCH码(n, k) ： • 分组长度 • 奇偶校验位的数目 • 最小距离 • 该码字(n, k)能够纠正t个或少于t个差错的任意组合。

  6. 非二进制本源BCH码 • 存在 具有如下参数的非二进制本原BCH码： • 分组长度 • 奇偶校验位的数目 • 最小距离

  7. Reed-solomon码 • m = 1 的q进制BCH码是q进制BCH码中最重要的一个子类 • 分组长度 • 奇偶校验位的数目 • 最小距离

  8. BCH 和 RS 二进制BCH 非二进制RS RS码

  9. 编码的最小码距直接关系到这种码的检错和纠错能力编码的最小码距直接关系到这种码的检错和纠错能力

  10. 硬判决译码 • 接收端解调器将含有噪声的信号用匹配滤波器硬判决，得到0或者1，这就是接收序列r • 处理这种硬判决接收序列的译码算法叫做硬判决译码 • 硬判决丢失了接受信号中含有的一些信息，影响译码性能 判决 译码 解调

  11. 软判决译码 • 若解调器对信号没有量化或量化为多于两个电平，得到软判决接收序列。利用软判决接收序列进行译码叫做软判决译码 • 一般比硬判决译码多3dB增益 解调 译码

  12. 主要译码算法 • hard-decision decoding (HDD): • Berlekamp-Messay algorithm (BMA) • iBM, RiBM, Eucild • soft-decision decoding: • KV • GMD ->BGMD • Chase -> Low-complexity chase (LCC)

  13. RiBM • 信码 • 生成多项式 • 编码 • 发送码字 • 过信道加噪声 c(x) + e(x) • 接收码字

  14. RIBM • 校验子syndrome V是码字错误个数，可见校验的值只与错误位置和值有关， 若传输无错，校验子为零

  15. RiBM • 定义错误位置Xk和错误值Yk: • 校验子和Xk，Yk

  16. RiBM • 定义错误位置多项式 • 错误估值多项式 • 求解关键方程 • Forney算法

  17. algebraic soft-decision decoding • multiplicity assignment • interpolation • factorization

  18. LCC 译码过程

  19. Multiplicity Assignment • The error-correcting capacity and complexity of ASD algorithms are mainly determined by the multiplicity assignment step. • KV • BGMD • LCC

  20. Multiplicity Assignment • KV multiplicity assignment • BGMD multiplicity assignment

  21. LCC multiplicity assignment • In the LCC multiplicity assignment scheme the reliability of each code position is first determined by the log-likelihood ratio (LLR). • Here and are the most likely and second most likely symbols transmitted in the j-th position, respectively.

  22. LCC multiplicity assignment • η<n-k most unreliable code positions • While other n-ηcode positions

  23. Modified-LCC multiplicity assignment • a modified LCC (MLCC) decoding is proposed by adding erasures to the test vectors. With the same η, the proposed algorithm can achieve much better performance than the original LCC decoding. • Implementation of ASD algorithms for a long RS code over the EPR4 channel with 100% AWGN. • BCJR algorithm is used as the channel detector to output the reliability of each received bit.

  24. Modified-LCC multiplicity assignment

  25. Re-encoding and Coordinate Transformation • The complexity of the ASD algorithms can be significantly reduced by applying the re-encoding and coordinate transformation • k most reliable code position • ηmost unreliable code position • rest n-k-ηmost reliable code position

  26. Re-encoding and Coordinate Transformation Coordinate Transformation Re-encoder or erasure-only decoder

  27. Re-encoder and Coordinate Transaction

  28. Re-encoder and Coordinate Transaction

  29. interpolation

  30. interpolation

  31. Backward interpolation for LCC decoding

  32. Interpolation • Backward interpolation for LCC decoding • Unified Backward-forward LCC Interpolation • Reduced-complexity Multi-interpolator Scheme for the LCC decoding

  33. Backward interpolation for LCC decoding

  34. Unified Backward-forward LCC Interpolation

  35. Reduced-complexity Multi-interpolator Scheme for the LCC decoding

  36. Reduced-complexity Multi-interpolator Scheme for the LCC decoding

  37. Comparison with Backward-forward and unified Backward-forward

  38. Factorization

  39. Factorization-Free

  40. Factorization-Free

  41. Hard-Decision based LCC

  42. Hard-Decision based LCC

  43. 几个重要性能指标 • coding gain • FER or BER

  44. 几个重要性能指标 • Throughput • Systematical clock • Latency • Critical path • Area : the number of XOR gates • Power consumption

More Related