欠采样频率估计方法

©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(,710071):(ESPRIT),.,,.,,.CRLB,.:;;;:TN91117:A:037222112(2004)1221978204FrequencyEstimationwithSub2NyquistSamplingWANGHong2yang,LIAOGui2sheng,WUYun2tao(KeyLaboratoryforRadarSignalProcessing,XidianUniversity,Xian,Shaanxi710071,China)Abstract:Basedonanauxiliarytime2delaychannelandESPRITapproach,anewalgorithmforwidebandfrequencyestimationwithsub2Nyquistsamplingwasproposed.Thefrequencyaliasingduetosub2Nyquistsamplingcanberesolvedbytheproposedfrequen2cyaliasingresolvingalgorithmandinformationfromthetime2delaychannel.Tofurtherreducethecomplexityoftheproposedmethod,asimplifiedalgorithmwithlittleperformancedegradationwasalsogiveninthispaper.Atlast,simulationresultsverifytheefficiencyoftheproposedmethod.Keywords:wideband;sub2Nyquistsampling;frequencyaliasing;simplifiedalgorithm1,.,DFT,,.,PisarenkoProny[13],.ML,MUSIC,ESPRIT[46],,.Nyquist,,.,,ADCDSP,Nyquist.,.(FBSLP),[7,8],.MUSIC,[9],,.Pro2ESPRIT[10],[10].,[11]Pro2ESPRIT,.,,.,,,,,.2,ADCFs,FsNyquist,.,Nyquist.x(n),y(n)n,x(n)=Kk=1sk(n)+wx(n)=Kk=1bkej[(2fkn)/Fs]+wx(n)y(n)=Kk=1e-j2fksk(n)+wy(n)=Kk=1bkej2fk[(n/Fs)-]+wy(n)(1),sk(n)k(1FkFK)n(),bkfk;e-j2fkk(1FkFK);wx(n),wy(n).:2003209201;:2004206221:(No.60172028);(No.01163)12200412ACTAELECTRONICASINICAVol.32No.12Dec.2004©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(QK)X(n)=[x(n),x(n+1),,x(n+Q-1)]T=A(f)S(n)+Wx(n)Y(n)=[y(n),y(n+1),,y(n+Q-1)]T=A(f)S(n)+Wy(n)(2),A(f)=[a(f1),,a(fK)]a(fk)=[1,e(j2fk)/Fs,,e[j2(Q-1)fk]/Fs]T(fk)=[1,e-j2f1,,e-j2fK]S(n)=[s1(n),,sK(n)]T(3)Wx(n),Wy(n).,a(fk),.,.,.211Z(n)=[XT(n)YT(n)]T=AAS(n)+Wx(n)Wy(n)=BS(n)+W(n)(4)B.ESPRIT.Z(n)Rzz=E[Z(n)ZH(n)]=BRssBH+2I2Q(5)E[],RssK,2,I2Q2Q.Rzz=2Qi=1iuiuHi=EssEHs+2EwEHw(6),As=diag(1,,K)K,Es,EwK2Q-K(),.T,[12]Es=Es1Es2=AAT=BT(7)Es1,Es2EsQQK,Es1=ATEs2=ATEs2=Es2(EHs1Es1)-1EHs1Es1]AT=Es2E#s1AT]Es2E#s1A=A(8),E#s1=(EHs1Es1)-1EHs1.,Es2E#s1,K1,,Ke-j2f1,,e-j2fK,u1,,uKAa(f1),,a(fK),.fcoarsek=-k/(2)ffinek-kFs=fk=Fs2(Q-1)Q-1l=1uk(l+1)uk(l),lFkFK(9)().fcoarsekk,,;ffinek-kFsk,ffinek,k,|ffinek-kFs|Fs]0kFceilffinekFs]0kFceilfmaxFs(10),ceil().fmax.k,,k=0.ffinekk,ffinekfcoarsekk=argmink|fk+kFs-fcoarsek|0kFceil(fmax/Fs)(11)(11),.(9),ffinek=fk+kFs(12)(11),(12).,N(NQ)^Rzz=1N-Q+1N-Q+1n=1Z(n)ZH(n)(13)(78).(),C1=4(N-Q+1)Q2+KQ(Q+2K)+O(9Q3)+O(K3)(14)N,Q,..212BKB1,2Q-KB2,V[13]VHB1=B2(15),Z=[Z(1),Z(2),,Z(N)],Z=Z1Z2=B[S(1),S(2),,S(N)]=B1B2S(16)Z1,Z2ZK2Q-K,S=[S(1),S(2),,S(N)].Z1VHVHZ1=VHB1S=B2S=Z2(17),,J(V)=argminZ2-VHZ12(18)V.Frobenius,(18)V=(Z1ZH1)-1Z1ZH2(19)979112:©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.ŽVH=IKVH,ŽVHB1=IKVHB1=B1B2=B=AA(20)ŽVH,(5)(6).ŽVH,QŽVH1,QŽVH2,ŽVH1B1=AŽVH2B1=AŽVH2(ŽVH1)#A=A(21)(8),ŽVH2Es2,ŽVH1Es1,.V,C2=K(N+1)(2Q+K)+KQ(K-Q)+O(Q3)+2O(K3)(22),NQK,K(N+1)(2Q+K)+KQ(K-Q)+O(Q3)4Q2(N-Q+1)+KQ(Q+2K)+O(9Q3)(23),.3fmax=500MHz,=1ns,Fs=200MHz.N=64,Q=20.:DFT31125MHz;(11)3;Q[14]34Q3,8%,(11)1/5..1327MHz330MHz,DFT.20dB,200Monter2Carlo..1MHzRMSERMSE327326190196326167115330330102018733012211362200MHz330MHz,030dB,200Monter2Carlo.1.2,,,5dB,,CRLB.4ESPRIT,.,.,.,.:CRLB,(1)L=-Nln(2)-Nln(2)-122Nn=1x(n)-bTcn2-122Nn=1y(n)-bTcn2(24),b=[b1,,bK]T=…b+j‰bcn=[ej2nf1/Fs,,ej2nfK/Fs]T(25)…b,‰bb.2,…b,‰bf=[f1,,fK]T,3K+1.FisherF3K+1,..(1)5L(22)5L52=-N2+124Nn=1wx(n)2+124Nn=1wy(n)2(26)E5L522=N24-N6ENn=1wx(n)2+Nn=1wy(n)2+1482ENn=1Nm=1wx(n)2wx(m)2+2ENn=1Nm=1wx(n)2wy(m)2=-N24+128Nn=1Nm=1mnE[wx(n)2]E[wx(m)2]+Nn=1E[wx(n)4]+N24=-N24+128N(N-1)4+3N4+N24=N4(27)(2)5L(2…b)5L5…b=-122Nn=15(x(n)-bTcn)5…b(x(n)-bTcn)3--122Nn=15(x(n)-bTcn)35…b(x(n)-bTcn)--122Nn=15(y(n)-bTcn)5…b(y(n)-bTcn)3--122Nn=15(y(n)-bTcn)35…b(y(n)-bTcn)08912004©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=12Nn=1Re[cnw3x(n)]+12Nn=1Re[cnw3y(n)](28)E5L525L5…bT=-N4ENn=1Re[cnw3x(n)+cnw3y(n)]+126ENn=1Nm=1wx(n)2+wy(n)2Re[cnw3x(m)+cnw3y(m)]=0(29)0.(3)5L(2‰b)5L5‰b=-12Nn=1ln[cnw3x(n)]-12Nn=1Im[cnw3y(n)]E5L525L5‰bT=0(30)(4)5L(2f)5L5f=-12Nn=1Im2nFsCnbw3x(n)-12Nn=1Im2(nFs-)Cnbw3y(n)(31)E5L525L5fT=0(32),Cn=diag(cn).(5)5L(…b…b)E5L5…b5L5…bT=12Nn=1Re[cncHn]+12Nn=1Re[cncHn3](33)(6)5L(…b‰b)E5L5…b5L5‰bT=-12Nn=1Im[cncHn]-12Nn=1Im[cncHn3](34)(7)5L(…bf)E5L5…b5L5fT=-12Nn=1Im[2nFxcnbHC3n]-12Nn=1Im[2(nFs-)cnbHC3n3](35)(8)5L(‰b‰b)E5L5‰b5L5‰bT=12Nn=1Re[cncHn]+12Nn=1RecncHn3(36)(9)5L(‰bf)E5L5‰b5L5fT=12Nn=1Re2nFscnbHC3n+12Nn=1Re2(nFs-)cnbHC3n3(37)(10)5L(ff)E5L5f5L5fT=12Nn=1Re2nFs2CnbbHC3n+12Nn=1Re2nFs-22CnbbHC3n3(38)10F,F.,CRB[F-1]ii,F-1i,CRLB..:[1]WahlbergBO.ARMAspectralestimationofnarrowbandprocessesviamodelreduction[J].IEEETransASSP,1990,38(7):1144-1154.[2]ChanYT,LangfordRP.SpectralestimationviathehighorderYule2Walkerequations[J].IEEETransASSP,1982,30(5):689-698.[3]LangW,McclellanH.Frequencyestimationwithmaximumentropyspectralestimators[J].IEEETransASSP,1980,28(6):716-723.[4]StoicaP,MosesL.Maximumlikelihoodestimationoftheparametersofmultiplesinusoidsfromnoisymeasurements[J].IEEETransASSP,1989,37(3):378-392.[5]RoyR,PaulrajA,KailahT.ESPRIT2Asubspace