第三章离散信道及信道容量课件.ppt

1、XY,),(11rNiaaAxXXXX ,),(11sNibbByYYYY )|(xyP1)|(yxyP )(0)(1)|(xfyxfyxyP NiiiNNxyPxxxyyyPxyP12121)|()|()|()|(ijabPXY)|()|()|(ijijabPaxbyPxyP ),2,1(),2,1(sjri 1,01,0 BYAXp 1p 1ppppabP 1)|(11ppabP 1)|(22pabP)|(21pabP)|(121)|(21 jijabP ppppppbb1110211021 aaX2,1,01,0 BYAXp 1q 1pqXY10 qqpp1001120XY )|()|(

2、)|()|()|()|()|()|()|(21222211121121rsrrsssabPabPabPabPabPabPabPabPabPbbbraaa21ijijpabP)|(rsrrsspppppppppP2122221112110 ijp1)|(1 sjijabP)(jibaP)|()()(),(ijijijiabPaPbaPbyaxP )(iaP)|(ijabP)(jbP rijiijriijbaPabPaPbP11)()|()()()|(jibaP riijiijijjijiabPaPabPaPbPbaPbaP1)|()()|()()()()|(riiiaPaPXH1)(1log)(

3、)(XxPxP)(log)()|(jbXH)|(jbxP rijijijbaPbaPbXH1)|(log)|()|(XjjbxPbxP)|(log)|()|()()|()|(1jsjjjbXHbPbXHEYXH )|(log)|()(11jirijisjjbaPbaPbP )|(1log)(11jirisjjibaPbaP YXyxPxyP,)|(1log)(0)|(1)|(YXHbaPji，YXXyxPxyPxPxPYXHXHYXI,)|(1log)()(1log)()|()();(YXYXyPxPxyPxyPxPyxPxyP,)()()(log)()()|(log)();()()|(log

4、)(,XYIyPxyPxyPYX )|()();(YXHXHYXI )()()(log)()|(log)()|(log);(yPxPxyPyPxyPxPyxPyxI );(YXI);(yxI)|()();(YXHXHYXI )()()(XYHYHXH )|()(XYHYH );()()|(YXIXHYXH )|(1log)(,yxPxyPYX );()()|(YXIYHXYH YXxyPxyP,)|(1log)()(YH)(XYH)(XH);(YXI)|(YXH)|(XYH )(0)(1)|(xfyxfyxyP)()();(YHXHYXI )(0)(1)|(xfyxfyyxP0)|(0)|(Y

5、XHXYH)()|()()|(xPyxPyPxyP 0);(YXI)()|()()|(XHYXHYHXYH ZzYyXx ,)|()|(log)|;(zxPyzxPzyxI)|()|()|(log)|()|(logzyPzxPzxyPzyPxzyP )|;()()|;()|;(zyxIxyzPzyxIEZYXIXYZ )|()|(YZXHZXH )|()|(XZYHZYH )|()|()|(ZXYHZYHZXH )|()|(.)()|(log)()|(log);(yxPyzxPxPyxPxPyzxPyzxI)|;();()|;();(zyxIzxIyzxIyxI );();(yzxIEYZXI

6、)|;();(YZXIYXI )|;();(ZYXIZXI 0);(YXI0);(YXI)|()();(YXHXHYXI )();(XHYXI)(),(min);(0YHXHYXI )();(YHYXI)|()(XYHYH );();(XYIYXI YXyPxyPxyPYXI,)()|(log)();(YXXxyPxPxyPxyPxP,)|()()|(log)|()()|(),();(xyPxPIYXI 110)(xPXp0110ppp);(YXI)(xP)|()();(XYHYHYXI XYxyPxyPxPYH)|(1log)|()()(XppHxPYH),()()(ppppYH1log1l

7、og)()()(pHYH ppxPxPyPX)1()|0()()0(pp ppyP )1()()();(pHYHYXI )()(1log)(pHpppp )(1log)(pppp )()(pHppH )()();(pHppHYXI );(YXI)|(xyP);(YXI);(YXIbit/)|()();(YXHXHYXIR 符符号号/bit);(max)(YXICxP)(xP)()();(pHppHYXI 取最大取最大时时当当)(1);(,21pHYXI )(1);(max)(pHYXICxP )(xfy 已已知知 )(0)(1)|()|(jijibaPabPjiij100010001P信道传输

8、矩阵321131211bbbYaaaX )()();(0)|()|(YHXHYXIXYHYXH srYHXHYXICxPloglog)()();(max)(集集集集集集363254321211 XBbaBbbbaBbbaY2153103101121 1000000101103530000002121654321Pbbbbbb 损损失失熵熵噪噪声声熵熵0)|(0)|(YXHXYH 01)|(jibaP已已知知)|()()|()();(XYHYHYXHXHYXI rXHYXICxPxPlog)(max);(max)()(且且 1000000101103530000002121654321Pbbbb

9、bb 损损失失熵熵噪噪声声熵熵0)|(0)|(YXHXYH 01)|(jibaP已已知知)|()()|()();(XYHYHYXHXHYXI rXHYXICxPxPlog)(max);(max)()(且且 1000000101103530000002121654321Pbbbbbb 01)|(jibaP已已知知362543121babaaabaaYX111111 01)|(ijabP已已知知 100010010010001001321Pbbb)()();()|()()|()();(0)|(0)|(XHYHYXIXYHYHYXHXHYXIYXHXYH 损损失失熵熵噪噪声声熵熵sYHYXICxPx

10、Plog)(max);(max)()(且且 ;)1(21成成集集的的诸诸元元素素不不同同排排列列组组，中中每每一一行行都都是是由由同同一一spppP ;)2(21元元素素不不同同排排列列组组成成集集的的诸诸，每每一一列列都都是是由由rqqq 相相同同。与与，则则的的子子集集；若若是是，则则，若若一一般般)3(iiiipqsrpqsrsr 2161313121616131212P 31316161616131311P 31613161616131313P 7.01.02.01.02.07.04P prprprprprpprprprprppP111111111XxXYHxPXYHXYHYHYXI)

11、|()()|()|()();(推导：)()|()|(21spppHxXYHXYH，)()(max21)(sxPpppHYHC，符符号号，也也一一定定是是等等概概率率分分布布输输出出符符号号等等概概率率分分布布时时，当当/bit)(log,1)()(21spppHsCYrxPxP )()(max21)(sxPpppHYHC，)()1log(log1logloglog)111(log6.3pHrprrpppprrprprppHrCrr ，阶阶矩矩阵阵强强对对称称信信道道例例021)(12 CppHCr时时，而而二二元元对对称称信信道道时时当当 31613161616131311P 7.01.02.

12、02.01.07.02P nkkksMNpppHrC121log)(log，kXikYyikYyxyPMxyPNk )|()|(nkkksMNpppHrC121log)(log，31613161616131311P 6161313131616131321QQQ符符号号，/bit 04085.03log216531log6132log3121log21)61613131(2log HC 7.01.02.02.01.07.02P 1.01.07.02.02.07.021QQ符符号号，/bit 2122.0)2.01.07.0(369.12.0log1.09.0log9.0)2.01.07.0(2l

13、og HHC11212222111211 sjijrsrrssppppppppppP且且 NiiiNNxyPxxxyyyPxyP12121)|()|()|(NssssrrrrNYbbbaaabbbaaabbbaaaXNNN )()()()()()(2112221111111111 长hkhkP )|(NNkhkhshrkP,2,1,2,1)|(令令 NNNNNNsrrrss 212222111211),1(11NshkhrkN，且且 NibbbbbbbNiaaaaaaashhhhhrkkkkkiNiN,1,1,21212121 ，NikhkkkhhhkhkhiiNNabPaaabbbPP1)|

14、()|()|(2121 11,10,01,0011,10,01,0010102222 BYAXpppp)|(khP 2141312211)00|11()|()0|0()0|1()00|10()|()0|1()0|0()00|01()|()0|0()0|0()00|00()|(pPPppPPPPppPPPPpPPPP 22222222pppppppppppppppppppppppp)()|(log)()()|(log)()|()()|()();();(,hkhYXhkkhkYXhkNNNNNNNNPPPPPPXYHYHYXHXHYXIYXINNNN )(21NXXXX)(21NYYYY)1(取取

15、极极大大值值当当信信源源无无记记忆忆时时，);()2(YXI位位随随机机变变量量。第第中中对对应应的的和和是是随随机机序序列列和和式式中中iYXYXii NiiiYXIYXI1);();(NiiiYXIYXI1);();()(21NXXXX)(21NYYYY)1(取取极极小小值值当当信信道道无无记记忆忆时时，);()2(YXI NiiiYXIYXI1);();(位位的的随随机机变变量量。第第中中对对应应和和是是随随机机序序列列和和式式中中iYXYXii NiiiYXIYXI1);();(通过相同的信道通过相同的信道具有同一概率分布具有同一概率分布，取自于同一信源符号集取自于同一信源符号集若若)

16、3()2()1(21riaaaX);();();();(2211YXIYXIYXIYXINN 则则);();(YXINYXIN 息息次次扩扩展展信信道道的的平平均均互互信信 NiiixPxPNNiiiYXIYXICYXIYXI1)()(1);(max);(max);();(NiiNiiixPNCYXICi11)();(max传传输输的的最最大大信信息息量量。通通过过无无记记忆忆信信道道是是某某时时刻刻 iYXICiixPii);(max)(NCCCCNi 则则因因在在同同一一信信道道中中传传输输，NCYXI);(NNYNXYXYX信信道道信信道道信信道道221121)|(11xyP)|(22x

17、yP)|(NNxyP NiiiNNYXIYYYXXXI12121);();();(max);(max)(111)(,2,11iixPiNiiNNxxPNYXICCYYXXICiN NiiNCC1,2,1iXiX tsrcccCZbbbBYaaaAX,212121 )|()|()|()|(jikijbacPxyzPabPxyP 信信道道信信道道列列行行srabPabPabPabPabpabPabPabPabPPrsrrss )|()|()|()|()|()|()|()|()|(212222111211列列行行tsbacPbacPbacPbacPbacPbacPbacPbacPbacPPsitsi

18、siitiiitii )|()|()|()|()|()|()|()|()|(212222111211 YikxyzPxyPxzPacPxzP)|()|()|()|()|(列列行行 tracPacPacPacPacPacPacPacPacPPrtrrtt )|()|()|()|()|()|()|()|()|(212222111211 YxyzPxyPxzP)|()|()|(sjjikijikbacPabPacP1)|()|()|(SjikjijikjiSjijiaPcbaPbaPcbaPaPbaP11)()()()()()()|()()(ikikiacPaPcaP tssrtrxyzPxyPxz

19、P )|()|()|()|()|()|()|()|()|()|()|()|(212222111211stssttbcPbcPbcPbcPbcPbcPbcPbcPbcPP)|()|(yzPxyzP)|()|()|()|()|()|(yzPxyPxzPyzPxyPxzPY );();(ZYIZXYI);();()|()|(ZYIZXYIyzPxyzP );();();();(ZYIZXIYXIZXI )|()|()|()|(zxPyzxPxzPxyzP )|()|(YXHZXH )(0)(1)|(jijizyP)|()|(zxPyxP);();()|()|(ZXIYXIZXHYXH )|()|(z

20、xPyxP );();();()(WXIZXIYXIXHX);(max),()(ZXICxP 串串);(max),()(WXICxP 串串 ppppPP1121 2222)1()1(2)1(2)1(ppppppppP串串)(1);(max)(pHYXICxP )1(21);(max)(,ppHZXICxP 212110)(xPX);();(ZXIYXI);();(WXIZXI 0);(lim nnXXI610 pS)(21NSSS X)(21LXXX Z)(21NZZZ Y)(21LYYY 形成马尔可夫链形成马尔可夫链ZYXS,)|()|()|()|(yzPyxzPxyPsxyP );();();();();();();();(YXIZSIZXIZSIYSIZSIYXIZXI );(max);()(YXICYXIRxP );(YXIC CYXICYXIC);(1);(rXHlog)(1 01HH