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REED-SOLOMON CODES
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数字通信系统模型
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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)
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码距与检错和纠错能力的关系
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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。
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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
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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
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BCH 和 RS 二进制BCH 非二进制RS RS码
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编码的最小码距直接关系到这种码的检错和纠错能力
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解调 判决 译码 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 解调 判决 译码
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解调 译码 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
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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)
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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
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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
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RiBM 定义错误位置Xk 和错误值 Yk : 校验子和Xk, Yk
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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
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algebraic soft-decision decoding
multiplicity assignment interpolation factorization
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LCC 译码过程
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Multiplicity Assignment
The error-correcting capacity and complexity of ASD algorithms are mainly determined by the multiplicity assignment step. KV BGMD LCC
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Multiplicity Assignment
KV multiplicity assignment BGMD multiplicity assignment
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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.
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LCC multiplicity assignment
η<n-k most unreliable code positions While other n-η code positions
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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.
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Modified-LCC multiplicity assignment
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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 η most unreliable code position rest n-k-η most reliable code position
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Re-encoding and Coordinate Transformation
Re-encoder or erasure-only decoder
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Re-encoder and Coordinate Transaction
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Re-encoder and Coordinate Transaction
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interpolation
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interpolation
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Backward interpolation for LCC decoding
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Interpolation Backward interpolation for LCC decoding
Unified Backward-forward LCC Interpolation Reduced-complexity Multi-interpolator Scheme for the LCC decoding
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Backward interpolation for LCC decoding
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Unified Backward-forward LCC Interpolation
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Reduced-complexity Multi-interpolator Scheme for the LCC decoding
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Reduced-complexity Multi-interpolator Scheme for the LCC decoding
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Comparison with Backward-forward and unified Backward-forward
whole unified(×4) Proposed unified for-only-back-ward GF Mult. 14 21 84 59 GF Adder 12 19 76 39 GF Inv. 1 4 Mux (bit) 6q 15q 60q 77q Ram (bit) 4(n – k + 1)q 16(n – k + 1)q Reg.(bit) 16q 27q 108q 117q iterations (n - k) + 2*(2^η - 1) (n - k) + (2^η - 1) (n - k) + (2^η/4 - 1) Clks of each iter. dx + 4 dx + 5
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Factorization
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Factorization-Free
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Factorization-Free
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Hard-Decision based LCC
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Hard-Decision based LCC
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几个重要性能指标 coding gain FER or BER
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几个重要性能指标 Throughput Systematical clock Latency Critical path
Area : the number of XOR gates Power consumption
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