Coding and Error Control

差错控制编码也称信道编码

Chapter 8

张宝贤：无线网络技术课程—— 2006 年 10 月 14 日电子邮件： bxzhang@gucas.ac.cn

差错控制的基本概念 数据传输误码率：数据比特在传输过程中的差错概率

线性 PCM: 10-7; 压扩 PCM: 10-5 ; ADPCM: 10-4 如何才能降低传输误码率呢？1. 加大传输功率：但在有些系统中不太适用，如手持终端系统中无

限制地加大电池重量是不可接受的2. 采用分集 (diversity) 技术以对抗信号衰落 (signal fading)

空间分集：采用两个或以上分得足够开的天线进行信号接收 频率分集：采用两个频率传输同样的信息比特——如前面说 过的快跳频技术 时间分集：同样的信息在不同的时间传输多次

3. 采用全双工传输——即：同时双向数据传输 发送端沿一个信道向接收节点发送信息 接收节点沿另外一个反馈信道将收到的信息发回给发送节点 如果发送端发现错误，则重传数据 需要的带宽是单向（单工）传输的两倍——在频谱利用方面

通常是不可接受的

差错控制的基本概念 ( 续 )4. ARQ ( 自动反馈重传： automatic repeat request) ，主

要有以下两种典型技术 停等模式 (stop and wait) 连续 ARQ (Continuous ARQ)

Go-Back-N mode 收到 NAK (Negative ACKnowledge) 或超时的分组，那么该分组及

其以后的分组都要重传 Selective repeat

只重传收到 NAK 或超时的那些分组分组 性重选择 传模式带宽利用效率高但需要的存储空间大

实现 ARQ 必须有差错检测码字的配合！ 检测分组在传输过程中是否出现差错

5. 纠错码，或称前向纠错码 (FEC: forward correction codes)—— 适合比特差错率 (BER: Bit Error Rate) 高的环境中

Designed to detect and correct errors

门限现象

图 8.6 显示 FEC 可以降低比特误码率 - 如图中，当 BER=10-6 时， FEC 带来 2.77 dB 增益 - 而未编码的传输方式的比特差错率为 >10-3

另外：如果 Eb/N0 低于某一门限（在本图中为 5 .4dB) ，编码带来副效应——即：升高了误码率

Eb ：每比特数据能量

比特误码率

BER

Eb/N0 (dB)

差错控制的应用 Reed-Solomon 码

应用于光盘数据存贮中可以有效对抗突发差错 也用于计算机中的数据存贮和提取

通信应用 移动无线通信系统（如 GSM) 中广泛应用 视听系统中常常采用 FEC

因为视听数据时延要求比较严格 用于空间应用 (Aerospace applications) 的控制与

通信系统

Error Detection Probabilities Definitions

Pb : Probability of single bit error (BER) P1 : Probability that a frame arrives with no bit

errors P2 : While using error detection, the probability that

a frame arrives with one or more undetected errors （也称剩余差错率 )

P3 : While using error detection, the probability that a frame arrives with one or more detected bit errors but no undetected bit errors

Error Detection Probabilities With no error detection （如果没有差错

检测技术的话）

F = Number of bits per frame

( )

0

1

1

3

12

1

=−=−=

P

PP

PP Fb

帧越长，单个帧中比特数越多，整个帧中包含差错位的概率越高

差错率计算一例如果单个比特差错率为 0.01,那么传送一个 10 比特的码字中出现

0、 1、 2 个比特差错的概率。假定各比特之间出现传送差错是相互独立的

单个比特成功接收的概率是 1-0.01=0.99 无差错概率 P( 0 个差错 ) =(0.99)10 = 0.904382 单比特差错概率 P( 1 个差错 ) =(0.01)1×(0.99)9 ×10= 0.091352 两比特差错概率 P( 2 个差错 ) =(0.01)2×(0.99)8 ×C10

2 = 0.00415

存在 3 个及以上差错的概率为 1- 0.904382- 0.091352 - 0.00415 =0.000116

差错控制码大全 《数字通信》 - Glover & Grant 　

error=1

Error Detection Process Transmitter

For a given frame, an error-detecting code (check bits) is calculated from data bits

Check bits are appended to data bits Receiver

Separates incoming frame into data bits and check bits Calculates check bits from received data bits Compares calculated check bits against received check

bits Detected error occurs if mismatch

Error Detection Process

Sender

Receiver

合法码字总数 2k

Parity Check (奇偶校验） As an option in ASCII coded data transmission Parity bit appended to a block of data

码长度 n = k + 1 Even parity ( 能检查出来奇数个错）

Added bit ensures an even number of 1s Odd parity ( 能检查出来奇数个错）

Added bit ensures an odd number of 1s Example, 7-bit character [1110001]

Even parity [11100010] Odd parity [11100011]

应对突发差错 (burst error) 能力较差

Parity Check (Cont’d)

二维行列奇偶校验

n1

n2

k2

k1

Collumn checks

Row

checks

Checks on checks

Parity Check (Cont’d)

Modulo-(2n-1) checksums 常用于网络连接中的差错检测 如在 Internet 协议消息的 checksum计算中

，　将一个消息看成是长度１６比特为单位的字

节的组合 ,　消息 :mw-1,…,m0∑ −

=−1

0)65535(mod

w

i im校验 : 的反码

原始消息和计算出来的校验同时传输以使收端可以进行差错检测

循环码循环码的特征 符合 (n,k) 线性分组码特征

线性分组码 : 任意两个合法码字之和仍然是一个合法码字

(n,k) 码字指一个码字由 k位数据和 n-k位校验位组成

任何 (n,k) 码字连续位移 i位后仍然是一个合法码字

Cyclic Redundancy Check (CRC)( 循环冗余校验 )CRC 是最常用的也是功能最强大的差错检测码之一 易于电路实现编解码 检查到出错的分组 / 帧 , 利用 ARQ 方式重传

Transmitter For a k-bit block, transmitter generates an (n-k)-bit

frame check sequence (FCS) Resulting frame of n bits is exactly divisible by

predetermined number Receiver

Divides incoming frame by predetermined number If no remainder, assumes no error

CRC using Modulo 2 Arithmetic

模２运算就是无进位的２进制加法，它恰好就是异或 (XOR) 运算

1 0 1 0

0 01 1

1 1 1 1

−

1 0 1 0

0 01 1

1 1 1 1

+

101011

1 00 1 1

1 00 1 1

1 1

1 00 1 1

×

模 2 加和模 2 减等价

CRC using Modulo 2 Arithmetic

Exclusive-OR (XOR) operation Parameters:

T = n-bit frame to be transmitted D = k-bit block of data; the first k bits of T F = (n–k)-bit FCS; the last (n – k) bits of T P = pattern of n–k+1 bits; this is the predetermined

divisor 　（除数 : 长度 n-k+1 比特）…可以称之为样式

Q = Quotient ( 商） R = Remainder （余数）

CRC using Modulo 2 Arithmetic

For T/P to have no remainder, start with

Divide 2n-kD by P gives quotient and remainder

Use remainder as FCS( 将余数作为 FCS)

FDT kn += −2

P

RQ

P

Dkn

+=−2

RDT kn += −2待传序列

CRC using Modulo 2 Arithmetic

Does R cause T/P have no remainder?

Substituting,

No remainder, so T is exactly divisible by P

P

R

P

D

P

RD

P

T knkn

+=+=−− 22

QP

RRQ

P

R

P

RQ

P

T =++=++=

CRC using Polynomials

All values expressed as polynomials

( )( ) ( ) ( )

( )( ) ( ) ( )XRXDXXT

XP

XRXQ

XP

XDX

kn

kn

+=

+=

−

−

例 8.4: D=1010001101, 那么 D(X) = X9 +X7 + X3+X2+1, 生成多项式 P(X)= X5 +X4 + X2+1. 那么计算结果 R(X) = X3+X2+X

具体过程见下页

CRC 一例

***

CRC using Polynomials Widely used versions of P(X) ( 也称生成多项式 )

CRC–12 X12 + X11 + X3 + X2 + X + 1

CRC–16 X16 + X15 + X2 + 1

CRC – CCITT X16 + X12 + X5 + 1

CRC – 32 X32 + X26 + X23 + X22 + X16 + X12 + X11 + X10 + X8 + X7 + X5 + X4 + X2 + X

+ 1

CRC using Digital Logic

Dividing circuit consisting of: XOR gates

Up to n – k XOR gates Presence of a gate corresponds to the presence of a

term in the divisor polynomial P(X) A shift register

String of 1-bit storage devices Register contains n – k bits, equal to the length of

the FCS

Digital Logic CRC

Wireless Transmission Errors Error detection requires retransmission Detection inadequate for wireless

applications Error rate on wireless link can be high, results

in a large number of retransmissions Long propagation delay compared to

transmission time 如卫星链路 常用的重传策略是重传出错帧及其以后的所有帧

Go-back-N ARQ, as explained later in details

Block Error Correction Codes

Transmitter Forward error correction (FEC) encoder maps

each k-bit block into an n-bit block codeword Codeword is transmitted; analog for wireless

transmission Receiver

Incoming signal is demodulated Block passed through an FEC decoder

Forward Error Correction Process

FEC Decoder Outcomes

Four possible outcomes as follows No errors present

Codeword produced by decoder matches original codeword

Decoder detects and corrects bit errors Decoder detects but cannot correct bit errors;

reports uncorrectable error Decoder detects no bit errors, though errors are

present

Block Code Principles Hamming distance ( 汉明距离 , 或称海明距离 ) – for two n-bit

binary sequences, the number of different bits E.g., v1=011011; v2=110001; d(v1, v2)=3

Minimal Hamming distance Dmin ( 最小汉明距离 ) 所有可能码字对间汉明距离最小值 ,

对于四个 5 位码字 00000,00111,11100,11011 来说最小汉明距离为 3 最小汉明距离表征了一个 block code 的总体性能 线形分组码的最小汉明距离只考虑每个码和全零码字距离即可

纠错检错能力 纠错能力 : t = 检错能力 : Dmin-1 给定 n 和 k, Dmin 越大 , 纠错检错能力越强 n-k 越小越好——降低开销，提高利用率 n-k 越大越好——降低差错率

Redundancy – ratio of redundant bits to data bits Code rate – ratio of data bits to total bits: k/n Coding gain – the reduction in the required Eb/N0 to achieve a specified

BER of an error-correcting coded system

−

2

1minD

两者相互矛盾

Block Code PrinciplesNearest neighbor decoding of block codes

codewords 00000 11100 00111 11011

Single-bit error correctable pattern

1000001000001000001000001

0110010100110001111011101

1011101111000110010100110

0101110011111111100111010

double-bit error detectable pattern 1000110010

0110101110

1011010101

0101001001

最后一行中的码字都和两个可能的码字等距离 ,无法纠正，因而只能检错 , 无法就错

理论基础 : 产生 t 个差错的概率大大高于产生 t+1 个错误的概率 - 见前面的例子

Hamming Code Hamming Code is easy to analyze but is rarely used Designed to correct single bit errors Family of (n, k) block error-correcting codes with

parameters: Block length: n = 2m – 1 Number of data bits: k = 2m – m – 1 Number of check bits: n – k = m Minimum distance: dmin = 3

Single-error-correcting (SEC) code SEC double-error-detecting (SEC-DED) code ，与 SEC 码相比

较而言多一个额外的对整个码块的校验位

Hamming Code Process

Encoding: k data bits + (n -k) check bits Decoding: compares received (n -k) bits

with calculated (n -k) bits using XOR Resulting (n -k) bits called syndrome word Syndrome range is between 0 and 2(n-k)-1 Each bit of syndrome indicates a match (0) or

conflict (1) in that bit position ( 即：症候码的值表示的就是出差错的比特的位置）

汉明码——举例 例 8.7

8 比特数据为 : 0011 1001 校验码为 : 0111

汉明码——举例

校验码的位置在 :1, 2, 4, …, 2n-k

汉明码——举例

Cyclic Codes Can be encoded and decoded using linear

feedback shift registers (LFSRs) For cyclic codes, a valid codeword (c0, c1, …, cn-1),

shifted right one bit, is also a valid codeword (cn-1, c0, …, cn-2)

Takes fixed-length input (k) and produces fixed-length check code (n-k) In contrast, CRC error-detecting code accepts arbitrary

length input for fixed-length check code

Cyclic Codes 传送码字 T(X)=Xn-kD(X)+C(X) 如果没有差错发生的话 ,receiver 收到的码字能够被生成多 式项 P(X) 整除 , 余数为 0

如果发生差错的话 ,receiver 收到的码为 Z(X) =T(X) +E(X), E(X) 是 n-bit 差错多项式 , 发生差错的比特位系数为

1 Z(X)/P(X)=T(X)/P(X)+E(X)/P(X)=S(X)

由于前一项 T(X)/P(X) 余数为 0, 所以症候码 S(X) 仅取决于差错多项式 E(X)

Receiver 端 : 用收到的 n 比特码字除以 P(X), 得到的余数 ( 如果不为 0 的话 ) 就是症候码 (syndrome codes), 通过症候码查表就可以得知那一位或那些位发生差错

BCH Codes(Bose-Chaudhuri-Hocquengham)

For positive pair of integers m and t, a (n, k) BCH code has parameters: Block length: n = 2m – 1 Number of check bits: n – k ≤ mt Minimum distance: dmin ≥ 2t + 1

Correct combinations of t or fewer errors Flexibility in choice of parameters

Block length, code rate

BCH Codes(Bose-Chaudhuri-Hocquengham)

Reed-Solomon Codes Subclass of non-binary BCH codes Data processed in chunks of m bits, called

symbols An (n, k) RS code has parameters:

Symbol length: m bits per symbol Block length: n = 2m – 1 symbols = m(2m – 1) bits Data length: k symbols Size of check code: n – k = 2t symbols = m(2t) bits Minimum distance: dmin = 2t + 1 symbols

Block Interleaving Data written to and read from memory in different

orders Data bits and corresponding check bits are

interspersed with bits from other blocks At receiver, data are deinterleaved to recover

original order A burst error that may occur is spread out over a

number of blocks, making error correction possible

Block Interleaving

Convolutional Codes （卷积码） Generates redundant bits continuously Error checking and correcting carried out

continuously (n, k, K) code

Input processes k bits at a time Output produces n bits for every k input bits, code rate: k/n K = constraint factor k and n generally very small…in most cases, k=1

n-bit output of (n, k, K) code depends on: Current block of k input bits Previous K-1 blocks of k input bits

Convolutional Encoder

例 : 比如前两个比特是 10 (un-1=1, un-2=0)而接下来的比特 un=1, 那么当前处在状态 b下一个状态将是 d(11). 而输出是Vn1 =un ⊕un-1⊕un-2 = 0⊕1⊕1=0vn2= un ⊕un-2 = 0⊕1 =1

状态指历史状态 , 即（ un-1,un-2 ）的值

从状态 a 开始是因为寄存器的初态总是全零

Decoding Trellis diagram – expanded encoder diagram Viterbi code – error correction algorithm

Compares received sequence with all possible transmitted sequences

Algorithm chooses path through trellis whose coded sequence differs from received sequence in the fewest number of places ( 汉明距离最小 )

Once a valid path is selected as the correct path, the decoder can recover the input data bits from the output code bits

Viterbi 算法 Distance of a path =(distance of the last edge) + (distance of the last-but-one state)对于一个 (n,k,K) 码字和一个解码窗口 b, 下面的步骤可以

解出第一个输出块中的 n 比特 , (1) 0 时刻网格图的状态为 0(2) 时刻 i+1, 找到到达每个状态 S 的路径 ( 或路径集

合 )—— 采用第一行给出的式子。并以路径的距离标记 S

(3) 算法在时刻 b 终止。若此时所有的 active 路径的第一条边相同，那么那条边的标记 x0x1x2…xn-1 就是我们要找的第一码块 w0w1w2…wn-1. 如果第一条边不唯一，那么差错不可纠正。

如果 b=5则无法成功解码

采用图 8.9 中的编码器 , 粗实线表示活跃路径 (active path)接收序列为 10 01 01 00 10 11 00…, 解码窗口 b=7

00011011

Automatic Repeat Request

Mechanism used in data link control and transport protocols

Relies on use of an error detection code (such as CRC)

Flow Control Error Control

Flow Control Assures that transmitting entity does not

overwhelm a receiving entity with data Protocols with flow control mechanism allow

multiple PDUs ( 协议数据单元 ) in transit at the same time

PDUs arrive in same order they’re sent Sliding-window （滑动窗口） flow control

Transmitter maintains list (window) of sequence numbers allowed to send

Receiver maintains list allowed to receive

Flow Control

Reasons for breaking up a block of data before transmitting: Limited buffer size of receiver Retransmission of PDU due to error requires

smaller amounts of data to be retransmitted On shared medium, larger PDUs occupy

medium for extended period, causing delays at other sending stations

Flow Control

Error Control

Mechanisms to detect and correct transmission errors

Types of errors: Lost PDU : a PDU fails to arrive Damaged PDU : PDU arrives with errors

Error Control Requirements Error detection

Receiver detects errors and discards PDUs Positive acknowledgement

Destination returns acknowledgment of received, error-free PDUs

Retransmission after timeout Source retransmits unacknowledged PDU

Negative acknowledgement and retransmission Destination returns negative acknowledgment to PDUs

in error

Go-back-N ARQ

Acknowledgments RR = receive ready (no errors occur) REJ = reject (error detected)

Contingencies （可能发生的意外事件） Damaged PDU Damaged RR Damaged REJ

作业 For P(X)=X5+X4+X+1 and D=11100011, 求 CRC （给出过程） Consider a convolutional encoder defined by (vn1=un⊕un-1) and

(un2=un-1⊕un-2) Draw a shift register implementation for this encoder similar

to Figure 8.9(a) Draw a Trellis diagram for this encoder similar to Figure 8.10

For the encoder of the above problem, assume that the shift register is initialized to all zeros and that after the transmission of the last information bit, two zero bits are transmitted

Why are the two extra (all-zero) bits are needed? 假定信息输入为 1101011 ，那么编码后的输出比特流是什么

？ ( 最左边的比特是最先输入到编码器的比特）