reed-solomon codes
DESCRIPTION
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) - PowerPoint PPT PresentationTRANSCRIPT
REED-SOLOMON CODES
数字通信系统模型
信息源
信源编码器
信道编码器
数字调制器
信
道
数字解调器
信道译码器
信源译码器
受信者
噪声源
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)
码距与检错和纠错能力的关系
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。
12m
2 1mn
n k mt
min 2 1d t
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
( )mGF q
1mn q
2n k mt
min 2 1d t
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
1n q 2n k t
min 2 1d t
( )mGF q
BCH 和 RS
(2)GF
3(4 )
4, 3
GF
q m
(64)
64, 1
GF
q m
( )mGF q
二进制 BCH
非二进制 RS
RS 码
编码的最小码距直接关系到这种码的检错和纠错能力
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
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
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)
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 1 1 1
0 1 2 1( ) ... nnr x r r x r x r x
1 1 10 1 2 1( ) ... n
nc x c c x c x c x
1 1 10 1 2 1( ) ... k
km x m m x m x m x
1
1 1 10 1 2 1
( ) ( )
...
n km i
i
n k n kn k n k
g x x
g g x g x g x g x
( ) ( ) ( )c x m x g x
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
RIBM
定义错误位置 Xk 和错误值 Yk :
校验子和 Xk , Yk
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
1
( ) (1 )v
kk
x xX
( )x
( ) ( ) ( )mod( )n kx S x x x
'
( )1,2...
( )
i
i ie i v
ALGEBRAIC SOFT-DECISION DECODING
multiplicity assignment interpolation factorization
LCC 译码过程
MULTIPLICITY ASSIGNMENT
The error-correcting capacity and complexity of ASD algorithms are mainly determined by the multiplicity assignment step.
KV BGMD LCC
MULTIPLICITY ASSIGNMENT
KV multiplicity assignment
BGMD multiplicity assignment
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.
'jj
LCC MULTIPLICITY ASSIGNMENT
η<n-k most unreliable code positions
While other n-η code positions
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.
MODIFIED-LCC MULTIPLICITY ASSIGNMENT
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
( , ) ( ,0)i i ir
0, 0,
1, 1,
( , ) ( , )
( , ) ( , )
j j i j
j j i j
r
r
0, 0,( , ) ( , )j j i jr
RE-ENCODING AND COORDINATE TRANSFORMATION
k
Re-encoder or erasure-only
decoder
Coordinate Transformation
RE-ENCODER AND COORDINATE TRANSACTION
RE-ENCODER AND COORDINATE TRANSACTION
INTERPOLATION
INTERPOLATION
BACKWARD INTERPOLATION FOR LCC DECODING
INTERPOLATION
Backward interpolation for LCC decoding Unified Backward-forward LCC Interpolation Reduced-complexity Multi-interpolator
Scheme for the LCC decoding
BACKWARD INTERPOLATION FOR LCC DECODING
UNIFIED BACKWARD-FORWARD LCC INTERPOLATION
REDUCED-COMPLEXITY MULTI-INTERPOLATOR SCHEME FOR THE LCC DECODING
REDUCED-COMPLEXITY MULTI-INTERPOLATOR SCHEME FOR THE LCC DECODING
COMPARISON WITH BACKWARD-FORWARD AND UNIFIED BACKWARD-FORWARD
Backward-forward
Unified whole unified(×4) Proposed unified for-only-back-ward
GF Mult. 14 21 84 59
GF Adder 12 19 76 39
GF Inv. 0 1 4 1
Mux (bit) 6q 15q 60q 77q
Ram (bit) 4(n – k + 1)q 4(n – k + 1)q 16(n – k + 1)q 4(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 dx + 5
FACTORIZATION
FACTORIZATION-FREE
FACTORIZATION-FREE
HARD-DECISION BASED LCC
HARD-DECISION BASED LCC
几个重要性能指标 coding gain FER or BER 0/bE N
几个重要性能指标 Throughput Systematical clock Latency Critical path Area : the number of XOR gates Power consumption