第四章_信道及信道容量

1、第四章第四章 離散信道及其信道容量離散信道及其信道容量)|()|(2121NNxxxyyypxyp)(21NXXXX),(21riaaaAx)(21Nxxxx其取值)(21NYYYY)(21Nyyyy其取值),(21sibbbBy),|(,)|()|(1YxypXxypxypnnNnnn數(shù)學(xué)模型：如果有如果有)|()|(ixjypixjypmmnn ),|(,2121YabpXbbbByYaaaAxXijsr取值于，輸出符號(hào)取值于，輸入符號(hào))|()|()|(ijijabpaXbYpxyp sjijrsrrssppppppppppP12122221112111且的轉(zhuǎn)置矩陣為 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表示信道錯(cuò)誤傳遞概率信道矩陣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次擴(kuò)展信道次擴(kuò)展信道NXNY),(21NrNX),(21NsNY)|(khpNkNkkkrkaaa, 2 , 1,21NhNhhhshbbb, 2 , 1,21ixX取值為iyY取值為)|(ijabp信道信道XY),(21riaaaAx),(21sibbbByNikihikhabpp1)|()|(NNXYkhkhhkNNppppYXI)()()(log)();(01a12a10b21bp1p1ppXY222222222ppppppppppppppppppppppppP);(NNYXI NikihikhNiiiabppxyp

5、xyp11)|()|()|()|(或NiiiYXIYXI1);();(則存在NikikNiippxpxp11)()()()(或NiiiYXIYXI1);();(則存在：NiiiYXIYXI1);();();();(1iiNiiiYXINYXI);();(YXINYXINN);();(iiYXINYXI則： 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總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表示信道錯(cuò)誤傳遞概率信道矩陣ppppppP1)()1 ();(pHppHYXI 1b2b3b4b5bsb1a2araXY1B2BrBijijjiBbBbbap10)|(0)|(

7、log)()|(yxpxypYXH符號(hào)比特/log)(max);(max)()(rXHYXICxpxp)()();()|()()|()();(YHXHYXIYXHXHXYHYHYXI則：1b2bsbXY1a2a3a4a5ara1A2AsAjijiijAaAaabp10)|( 0)|(log)()|(xypxypXYH)()();()|()()|()();(XHYHYXIYXHXHXYHYHYXI則：sYHYXICxpxplog)(max);(max)()(1a2ararb2b1bXY rXHYXICxpxplog)(max);(max)()( 對(duì)稱信道對(duì)稱信道3131616161613131P

8、216131312161613121P對(duì)稱信道對(duì)稱信道prprprprprpprprprprppP1111111111rp)(log21spppHsCsppp,21 式中式中 為信道矩陣中任一行的元素為信道矩陣中任一行的元素。若一個(gè)離散對(duì)稱信道具有若一個(gè)離散對(duì)稱信道具有r r個(gè)輸入符號(hào)，個(gè)輸入符號(hào)，s s個(gè)輸個(gè)輸出符號(hào)，則當(dāng)輸入為等概率分布時(shí)，達(dá)到信道容量出符號(hào)，則當(dāng)輸入為等概率分布時(shí)，達(dá)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為正確傳遞概率為正確傳遞概率p為錯(cuò)誤傳遞概率為錯(cuò)誤傳遞概率) 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時(shí)，信道信道1信道信道2信道信道NXYXNY2Y1YXX);(ma

11、x21NYYYXIC)(max)(XHCxp2Y12NXY1XNY1YNX2XXYNnnnYXIC1);(max NnCnC12log個(gè)信道的信道容量為第nCn NCNiiNiiixpxpNCYXIYXIC11)()();(max);(max);(max)(iixpiYXIC CNCNNiiiYXIYXI1);();();(iiYXI )1log()(0BNSBSCS：0NBNS0例例4-10:4-10: 已知信道的帶寬已知信道的帶寬B B為為3kHz,3kHz,信號(hào)在信道傳輸中受到單邊功率譜信號(hào)在信道傳輸中受到單邊功率譜密度密度 為為 的加性白高斯噪聲的干擾的加性白高斯噪聲的干擾, ,信號(hào)的

12、平均功信號(hào)的平均功率率S S為為9W.9W. (1)(1)求信道的容量求信道的容量; ; (2) (2)若信道帶寬增加到原來(lái)的若信道帶寬增加到原來(lái)的1010倍倍, ,并保持信道容量不變并保持信道容量不變, ,那么那么信號(hào)平均功率要改變多少信號(hào)平均功率要改變多少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分貝為則信號(hào)的功率可降低的由于則有：)(logXHrrXHlog)(1rXHHHlog)(11max 1000.980.020.980.02198.002.002.098.