卷积码编码原理的解释

29420078JOURNALOFEEEVol.29No.4Aug.2007,,(,230009):2007-03-21;:2007-05-29:1961-,,,,,(1955-),,,,,DSP(1979-),,,,,:,,,,:;;;:TN911:A:1008-0686(2007)04-0021-04ExplanationaboutthePrincipleofConvolutionalCodingDINGZhi2zhong,JIANGJian2guo,XIANa(SchoolofComputerandInformation,HefeiUniversityofTechnology,Hefei230009,China)Abstract:Inthepublishedtextbooksofthetheoryofinformationandcoding,normallythegeneratingpoly2nomialsofthecoder,orthecodersimplementationbyshift2registerareintroduceddirectly.However,therelationshipofthecoderwiththeconvolutionisnotmentioned.Inthispaper,theshift2registerimple2mentationofconvolutionalcoderisintroducedfromtheviewpointofdiscrete2timesystem.Thisleadstoaneasy2to2understandstatementforthetopicssuchaswhatispolynomialgeneratormatrix,andwhyitisimportant.Amethodoffindingcodesequencefromthecomputationofdiscrete2timeconvolutionisalsopresented.Bydoingso,theprincipleofconvolutionalcodingandthemeaningofconvolutionareexplainedmorerelevantly.Keywords:convolutionalcoder;channelcoding;informationtheory;discrete2timesystem0P.Elias,,,,,,1968[1],1978[2](2002,2003),1991[3](2003)[1][2],[3],,,,();,;,,,,(),,11)11z:Y1(z)=X(z)G1(z)Y2(z)=X(z)G2(z)(1)y1(n)=x(n)3g1(n)y2(n)=x(n)3g2(n)(2)3G1(z)=g1(0)+g1(1)z-1+g1(2)z-2=1+z-2(3)G2(z)=g2(0)+g2(1)z-1+g2(2)z-2=1+z-1+z-2(4)g1(n)={1xn=001}(5)g2(n)={1xn=011}(6)Y1(z)=X(z)G1(z)=X(z)+X(z)z-2Y2(z)=X(z)G2(z)=X(z)+X(z)z-1+X(z)z-222,233,23Y1(z)=X1(z)G11(z)+X2(z)G21(z)Y2(z)=X1(z)G12(z)+X2(z)G22(z)Y3(z)=X1(z)G13(z)+X2(z)G23(z)(7)Y(z)=X(z)G(z)(8)Y(z)=[Y1(z),Y2(z),Y3(z)](9)X(z)=[X1(z),X2(z)](10)G(z)=G11(z)G12(z)G13(z)G21(z)G22(z)G23(z)(11),,G(z)=101+z-101z-1(12)Y1(z)=X1(z)Y2(z)=X2(z)Y3(z)=X1(z)(1+z-1)+X2(z)z-1(13)2229,4,:(1)(G(z)),(2)G(z)z-14232)kk,nn,(n,k),,:(1),,{0,1}2(2),5(a),5(b),1,z-1x(xx),5(3),,-kk,-,nn(4),nk,346763743),,,,(11),,,,(11)zx,,(n,k)G(x)=G11xG12(x)G1n(x)G21xG22(x)G2n(x)wGk1xGk2(x)Gkn(x)(14),(3)(4),6G(x)=[1+x21+x+x2](15)(12),7324,:G(z)=101+x01x(16)(8),X(x),Y(x)Y(x)=X(x)G(x)(17)Y(x),X(x),(9)(10),,104),[4],x(n)={1,1,0,1};g(n)={1,1,1}(n=0),y(n)=x(n)3g(n):x(n):1101g(n):11111011101Ý11011xn=022211:(1);(2)y(n)x(n)g(n),0+0,y(n)0;(3)Ly=Lx+Lg-1,4+3-1=6,26,x(n)=1101(n=0),y1(n)y2(n):x(n):1101g(n):10111010000Ý11011xn=011001x(n):1101g(n):11111011101Ý11011xn=000011-,y(n)=1110100001117,x(n)=01101001(n=0)-,x(n)=[x1(n),x2(n)]=[0110,1001],y(n)=010100100011001,,2,,,:[1]R.Gallager.InformationTheoryandReliableCommunication[M].NewYork:JohnWileyandSons,1968[2]R.J.McEliece.TheTheoryofInformationandCoding(Sec2ondEdition)[M].PublishingHouseofElectronicsIndustry,2003[3]T.M.Cover,andJoyA.Thomas.ElementsofInformationTheory[M].TsinghuaUniversityPress,2003[4],,.[M].:,20034229