6 信道编码0001111111

1、6 信道编码信道编码在数字通信系统中的位置信源信源编码 信宿信道编码 调制 发送 信源解码信道解码 解调 接收 信道 信道编码（也称为纠错编码或者差错控制编码）通过增加信号的冗余度，提高系统的可靠性。 作用：在接收信号功率一定的前提下(接收信噪比Eb/No也就固定)，提高系统数据传输的可靠性（降低误码率BER）在系统对数据传输的可靠性要求一定的前提下(BER固定)，降低对信号功率或接收信号信噪比的要求。 为了保证通信内容的可靠性和准确性，每一个数字通信系统对输出信息码的差错概率，即常说的误码率Pe都有一定的要求。例如：声音-pe10-3 图像-DVB-C peC，就不可能有任何一种编码能使差错

2、概率任意小。每码字的码元数6.2 纠错编译码的基本原理与分析方法 从信道编码定理出发，考虑如何使左边的Pe减小？ 从概念上分析，可把纠错能力的获取归结为两条：一条是利用冗余度，另一条是噪声均化(随机化、概率化)；25XNUUYf1fXFYXY12 ,rAa aa12 ,sBb bbX12 ,rAa aaN26XY12 ,rAa aa12 ,sBb bbX12 ,rAa aaN*:(),1,2,.,jjF BAF baAjs27XY12 ,rAa aa12 ,sBb bbX12 ,rAa aaN*(),1,2,.,jjF baAjs *(|)()|jjjjP XaYbP F bb ()|1()|

3、)|(jjjjjP XF bYbP FPbebb 11()() 1()(|)jssejjjjjjPP bP bbFPPbeb28XY12 ,rAa aa12 ,sBb bbX12 ,rAa aaN*(),1,2,.,jjF baAjs11()() 1()(|)jssejjjjjjPP bP bbFPPbeb111(),1()|()ssejjjjjjjPP F bbP F bP bF b *,( ,)( ) (|)eijijiY XaY XaPP a bP a P ba*1,111|()(|)sejjjijY XaPP bF bP barr 290.80.90.20.12b1b1a2a31133

4、22( ):()F baFF ba1()0.4P a1111121( ):()F baFF ba4124421( ):()F baFF ba2122222( ):()F baFF ba0.40.6XP1122|0.80.20.10.9Y XaPbba11220.320.080.060.54XYabbaP1111121()1(),1(,)(,)1 (0.320.08)0.6sejjjP FP F bbP a bP a b 2221221()1(),1(,)(,)1 (0.060.54)0.4sejjjP FP F bbP a bP a b 33112210.()1(),1( ,)(,)1 (0.

5、320.514)4sejjjP FP F bbP a bP a b 44211210.()1(),1(,)( ,)1 (0.080.086)6sejjjP FP F bbP a bP a b 30两种典型的译码规则：两种典型的译码规则：最佳译码规则最佳译码规则、极大似然译码规则极大似然译码规则 1 1、最佳译码规则：最佳译码规则：平均差错率最小的译码规则平均差错率最小的译码规则 。XY12 ,rAa aa12 ,sBb bbX12 ,rAa aaN1() 1()|sejjjjPP bP F bb*(|)()|)(,:,jjjjiijjP abP aF baAbBFaAb *(,( ,)jjij

6、P a bP a b *(|) (|)()jjijjjP bPPbP aba b *(,( ,)(,:,jjijjjjiP a bP aF baAbBFabA 310.80.90.20.12b1b1a2a1()0.4P a0.40.6XP1122|0.80.20.10.9Y XaPbba12120.0800.320.54.06XYbbaaP3113322( ):()F baFF ba1111121( ):()F baFF ba4124421( ):()F baFF ba2122222( ):()F baFF ba1()0.6eP F 2()0.4eP F3()0.14eP F4()0.86eP

7、 F1212|0.12900.15790.84210.8710X YbPbaa0.380.62YP1122( ):()F baFF ba1122( ):()F baFF ba33111220.()1(),1( ,)(,)1 (0.320.514)4sejjjP FP F bbP a bP a b 32XY12 ,rAa aa12 ,sBb bbX12 ,rAa aaN*(),:(|)(|),jjjjjjiiF baAbBFP baP baaA 33|0.50.30.20.20.30.50.30.30.4Y XP 1231|230.50.300.20.20.30.30.30.4.5Y Xbbba

8、Paa11212332( ):()(or,)()F baFF baa aF ba 34*(|)(|)(),:,jjjjjjiiP baP bF baAbaAaBF *(,( ,)(,:,jjijjjjiP a bP aF baAbBFabA *(|)(|)jjjiP baP ba *(,( ,)jjijP a bP a b *()( )jiP aP a*(|)()(|)jjjiijP aP aP baP ba 注：350.99p 0.99p 0.01p 0.01p 10a 21a 10b 21b XY12 ,a a12 ,b bU12 ,u u11010.50.5UUuuP2M ()log1H

9、 UM12120.990.010.010.99XYbbaPa1122( ):()F baFF ba21111|()1(0.990.99)102sejjjPP bF br 36 3X3X3Y128, 128, U12 ,u u12 ,u u33|YXP112345672800000010100111001011101111ffuu 12354678000001010100011101110111331234567832222223132222223|8YXpp pp pppp ppppppPpppppp pppp pp pp 1121314851687888()()()()()()()()FFF

10、FFFFF83222222341111|()1()3 102elllPPFpp pp pp pp pp pp ppr 18000111FF ()loglog2133H UMRNN37 1N loglog211MRN210eP 3N 43 10eP loglog2133MRN515105eNPR7174 107eNPR819109eNPR38 3X3X3Y128, 128, 2U00,01,10,1112 ,u u33|YXP11111 11212342156227800000001010100111010010111110111ffffu uu uu uu u 1234567800000101

11、00111001011101111000F 331234567832222223122322322322233222|5232222327YXpp pp pppp ppppppp ppppp ppppp pppPp pppppppp pp ppppppp pppp ppppp p 1121334355657787()()()()()()()()FFFFFFFF8132211|()11(44)1.99 104elllPPFMpp p 3010F 5100F 7110F loglog4233MRN 39 40 123400011011ssss1212345:1,2,3,4iiiiiiiiifsm

12、ma a a a ai112231241112:iiiiiiiiiiiiamamfammamamm0000000010110110101111111010ffff 00000000010001000100010001000010001000110110101100011110100100101 111011110001110101111011010101100111111100111001101010011010110111100011110100100101001011 11001FFFF000000110110111110105次扩展信道log4255R 5432411(4208)7.86

13、 104ePpp pp p 21.99 10eP2 3R 41 42 yx( , )D x yxy1011111320120( , )3,( ,)21111001220320 xD x yDy 。( , )D x yxy( , )D x yxy( , )0D x y xy( , )( , )D x yD y x( , )( , )( , )D x zD z yD x y 431212,0,1,0,1NkNkxx xxxyy yyy1( , )NkkkD x yxy二元序列二元序列汉明重量汉明重量 ：二元序列：二元序列 中含中含“1 1”的个数。的个数。 ( )W xx( )( ,)NW xD

14、xO4412 ,qCc cc( ,),ijijijD c cccc cC ：minmin( ,),ijijijdD c cccc cC45 1212 , ,fMMs ssc cc 12122, ,NFMc cc NXNXNY122,N 12212, ,NMc cc S12 ,Ms ss12 ,Mc cc|nnYXPpppp10a 21a 10b 21b XY12 ,a a12 ,b b*(),:(|)(|),NjjjNjjjiiFcCBFPcPccCA 46NXNXNY122,N 12212, ,NMc cc S12 ,Ms ss12 ,Mc cc|nnYXP*(),:(|)(|),NjjjN

15、jjjiiFcCBFPcPccCA 1212NNiiiijjjjca aab bb12121122(,)(,)(,)(,)(,)(|)(|)(|) (|)(|)(1)(1)(1)1NNNNijijijijijjijjjiiijijijiD cND cN D cD cND cPcP b bba aaP ba P baP bapppppppppp1 2p (1)1pp ( ,)(|)ijjiD cPc*(,)mi()n( ,),:,jNjjjjijNiFcCD ccDAcBFC 4748*(),njjjFcCB*(,)(,)*111|1jjjjD cN D cejjjjPPcppMM 495051

16、525354 12;0,1immmm12345cccccc31211221241151212: cmmcccmccmmcmcmfcc cmG 123451210 1 1 101 10 1cccccmm12345001110010010110010ccccc T 0cHT 0Hc3121241151212cmmcccmccmmcc12314125000cccccccc55 cmGT 0cHT 0Hc12,0,1Kimmmmm12,0,1Niccccc 5611121212221112NNKKKKNggggggggggggG1122KKmmmGcm=ggg 123Kmmmm= 1,= 01cg57

17、11121212122212100010001rrK KK rKKKrKaaaaaaaaaggAgGI ：T 0cH,1,2,TiiK0g HT 0GHTK rr rHAIK KK rGIATK rTTK KK rK rr rK KK rK rK rr r0AGHIAAIIAAAI58 59 60 0000001Error010Error011Error100Error101Error110Error1111min1ddt min32 1d00000010010001111000101111011111min21cdt min1132d 61 1011101101G00000000101101

18、10101111111010cmGmin3d 47.86 10eP43 10eP 2 5R 1 3R 62 121212NNNyyyccceeey = ce =c = ye = yeT 0cH()TTTs = yHc+e HeH 0s Ts = yHTseHc = ye = ye。出发码的估值准阵列译码表，然后译试求：先构造该码的标；，设收码系统线性码的生成矩阵某C)10101(R1011011101G)2 , 5(解：解：(11010)GmC(10111)GmC(01101)GmC(00000)GmCmGC(11010)C(10111),C(01101),C(00000),C :4mG,C,

19、G(11)(10),(01),(00),m(1)443322114321个许用码字求出根据已知及信息组模模2相加相加100010001110111IPH,PI1011111001G(2)2-5T2阵由系统线性码的生成矩000011)1(010100111)1(1110)1(01)1(1)1)(1(1100011)-0(n10111)-1(n1)0-k-(n1)1-k-(n1)-1)(n-k-(n01 -n01 -k-n1T01 -k-nhhhhhhhhh)e,(eS)S,(SSRH)S,(SS(3)heheheSheheheSheheheSHeHESEHnnnnknknnknnknniiiTT

20、T根据据伴随式0340141234201234012100110100100111),(),( eeeSeeSeeeSeeeeeSSST得种组合伴随式有8223k-n0340141234201234012100110100100111),(),( eeeSeeSeeeSeeeeeSSST得种组合伴随式有8223k-n个最轻的差错图案；个伴随式对应到种个差错种个差错种无差错差错图案中8810C-25C-11-2515全零全零1个；个；1个差错个差错5个；个；2个差错个差错2个；个；(00000)(000)(10000)(111)(01000)(101)(00100)(100)(00010)(01

21、0)(00001)(001)(00011)(011)(00110)(110)jEjS110110S03414234eeeeeeee，则若e1e4e0e3e2Ej1010000011100110111001110110010100110100 据4个码字和8个差错图案，可列出标准阵列译码表如下：C0=0000C1=10111C2=01101C3=11010S0=000E0=0000000000101110110111010S1=111E1=1000010000001111110101010S2=101E2=0100001000111110010110010S3=100E3=00100001001

22、00110100111110S4=010E4=0001000010101010111111000S5=001E5=0000100001101100110011011S6=011E6=0001100011101000111011001S7=110E7=0011000110100010101111100若收码R=(10101),则：);10111()00010()10101(ERC),00010(E,)010(RH)();10111(S,)010(H)10101(RH)();10111(),10111()10101()(444T44TT所以对应的差错图案找到码字搜索找到所在列的最上所在行，再对码表行所以确定在译码输出为，所在列的子集头是找到直接对码表，行列搜索ScSba完备码EG：构造一个m=3的二元(7,4)汉明码；解：解： 1100111111011000010000100001PIGIP10001000110111110011110101011100110111100043T列置换H由于生成矩阵G中包含了单位阵I4 ，矩阵的秩是4 ，G的4行是4个线性无关的基底，可以张成一个包含24=16码字的码空间。相等。矩阵的列数所以二进制恰好和校验其中汉明码中是除全零矢量个麻原组成的列矢量二进制时列，行和码的校验矩阵有12k-nm),12, 12(),( ,12)(),(mmm