第四章_信道及信道容量

1、第四章第四章 离散信道及其信道容量离散信道及其信道容量)|()|(2121NNxxxyyypxyp)(21NXXXX),(21riaaaAx)(21Nxxxx其取值)(21NYYYY)(21Nyyyy其取值),(21sibbbBy),|(,)|()|(1YxypXxypxypnnNnnn数学模型：如果有如果有)|()|(ixjypixjypmmnn ),|(,2121YabpXbbbByYaaaAxXijsr取值于，输出符号取值于，输入符号)|()|()|(ijijabpaXbYpxyp sjijrsrrssppppppppppP12122221112111且的转置矩阵为 PPapapapPb

2、pbpbpTrTs)()()()()()(212101a12a10b21bp1p1ppXYppppP111010 qqppP10011?01001a12a10b31bp1q1pqX2?bY)|(log)()|(jijibapbapYXH 信道信道H(X|Y)H(X|Y) Y Y I(X;Y)I(X;Y) 110PX表示信道错误传递概率信道矩阵ppppppP1 信道信道 X （0，1） （0，1）);(YXI)|()();(XYHYHYXI)|(log)|()()(ijijijixypxypxpYHijijijixypxypxpYH)|(log)|()()(ijijijixypxypxpYH)|

3、(1log)|()()(iippppxpYH)1log1log)()()()(pHYHiiippapapYp)1 ()|0()()0(iiippapapYp)1 ()|1 ()()1()(log)()(ypypYHpppppppp)1 (1log)1 ()1 (1log)1 ()1 (ppH)()1 ();(pHppHYXI01)(1pH01)(H5 . 0 )(1);(pHYXI3/ 13/210PX4/ 12/ 14/ 102/ 12/ 1P12/ 12/ 112/5210PY159. 0)|()();(166. 1)|(log)|()()|(325. 1)(log)()(213131XY

4、HYHYXIxypxypxpXYHypypYHijijijijjjN N次扩展信道次扩展信道NXNY),(21NrNX),(21NsNY)|(khpNkNkkkrkaaa, 2 , 1,21NhNhhhshbbb, 2 , 1,21ixX取值为iyY取值为)|(ijabp信道信道XY),(21riaaaAx),(21sibbbByNikihikhabpp1)|()|(NNXYkhkhhkNNppppYXI)()()(log)();(01a12a10b21bp1p1ppXY222222222ppppppppppppppppppppppppP);(NNYXI NikihikhNiiiabppxyp

5、xyp11)|()|()|()|(或NiiiYXIYXI1);();(则存在NikikNiippxpxp11)()()()(或NiiiYXIYXI1);();(则存在：NiiiYXIYXI1);();();();(1iiNiiiYXINYXI);();(YXINYXINN);();(iiYXINYXI则： X YZp(y|x)p(y|x)p(z|xy)p(z|xy)iNiNPPPPP121b1a1a2c1b2c2b3c3XYZ1/31/31/31/21/21/31/32/32/312/102/13/13/13/11P3/23/103/13/200012P3/16/12/13/13/13/13/

6、23/103/13/200012/102/13/13/13/1总P12NXY1XNY2Y1YNX2X 信道信道1 1信道信道2 2信道信道N NXYXNY2Y1YXX 信道信道1 1信道信道2 2信道信道N NXY NPPPP21 ondbitYXItRtsec/);(1 ondbitYXItCtsec/);(max1signalbitYXIC/);(maxsignalbitYXHXHYXIR/)|()();(110PX表示信道错误传递概率信道矩阵ppppppP1)()1 ();(pHppHYXI 1b2b3b4b5bsb1a2araXY1B2BrBijijjiBbBbbap10)|(0)|(

7、log)()|(yxpxypYXH符号比特/log)(max);(max)()(rXHYXICxpxp)()();()|()()|()();(YHXHYXIYXHXHXYHYHYXI则：1b2bsbXY1a2a3a4a5ara1A2AsAjijiijAaAaabp10)|( 0)|(log)()|(xypxypXYH)()();()|()()|()();(XHYHYXIYXHXHXYHYHYXI则：sYHYXICxpxplog)(max);(max)()(1a2ararb2b1bXY rXHYXICxpxplog)(max);(max)()( 对称信道对称信道3131616161613131P

8、216131312161613121P对称信道对称信道prprprprprpprprprprppP1111111111rp)(log21spppHsCsppp,21 式中式中 为信道矩阵中任一行的元素为信道矩阵中任一行的元素。若一个离散对称信道具有若一个离散对称信道具有r r个输入符号，个输入符号，s s个输个输出符号，则当输入为等概率分布时，达到信道容量出符号，则当输入为等概率分布时，达到信道容量C CssspppppppH1log1log)(1121)|()();(XYHYHYXI)|(1log)|()()(xypxypxpXYHXXYxXYHxpxypxypxp)|()()|(1log)

9、|()()|(xXYHsspppp1log1log11)(21spppH)()();(21spppHYHYXI)()(max);(max21)()(sxpxppppHYHYXIC)(log21spppHs216131312161613121P signalbitHpppHsCs/126. 061log6131log3121log213log )61,31,21(3log )(log21 。prprprprprpprprprprppP111111111pp1为正确传递概率为正确传递概率p为错误传递概率为错误传递概率) 1log(logloglog)1()1log1(1loglog)1,1,1,(

10、log)(log21rppppprrprrppprrprprppHrpppHrCr)(1pHC riiapYXIF1)();(,21raaa,21sbbb)|(ijabPriiap11)(riapFi, 2 , 1,0)(riip11NNNNNPP)21 (1)21 (1)21 (1)21 (1210 12NXY110P)(1pHCN)21 (12log)21 (1 21)21 (12log)21 (1 211NNNN2/ 12/ 12/ 12/ 1)21 (1)21 (1)21 (1)21 (121limlimNNNNNNPN时，信道信道1信道信道2信道信道NXYXNY2Y1YXX);(ma

11、x21NYYYXIC)(max)(XHCxp2Y12NXY1XNY1YNX2XXYNnnnYXIC1);(max NnCnC12log个信道的信道容量为第nCn NCNiiNiiixpxpNCYXIYXIC11)()();(max);(max);(max)(iixpiYXIC CNCNNiiiYXIYXI1);();();(iiYXI )1log()(0BNSBSCS：0NBNS0例例4-10:4-10: 已知信道的带宽已知信道的带宽B B为为3kHz,3kHz,信号在信道传输中受到单边功率谱信号在信道传输中受到单边功率谱密度密度 为为 的加性白高斯噪声的干扰的加性白高斯噪声的干扰, ,信号的

12、平均功信号的平均功率率S S为为9W.9W. (1)(1)求信道的容量求信道的容量; ; (2) (2)若信道带宽增加到原来的若信道带宽增加到原来的1010倍倍, ,并保持信道容量不变并保持信道容量不变, ,那么那么信号平均功率要改变多少信号平均功率要改变多少dB?dB?0NHzW /106bpsBNSBC36202107 .34)30001091 (log3000)1 (logSdBdBSSBNSBNSBNSBNSBNSBBNSBC9 .23,9 .2390368. 0lg10lg100368. 0 1)30001091(30001010 1)1(10)1 (101)101 (log10)1 (log1016610100101000202分贝为则信号的功率可降低的由于则有：)(logXHrrXHlog)(1rXHHHlog)(11max 1000.980.020.980.02198.002.002.098.