数字信号处理2010试卷及参考答案.docx

数字信号处理2010试卷及参考答案

第一篇：数字信号处理2010试卷及参考答案《数字信号处理》参考答案1.Determinetheperiodofthesinusoidalsequencex(n)5sin(0.04n).Determineanotherdistinctsinusoidalsequencehavingthesameperiodasx(n).Solution:4N0.02rNselect50;r1502r;r310.122.WithoutcomputingtheDTFT,Determinewhichofthefollowingsequenceshavereal-valuedDTFTandhaveimaginary-valuedDTFT:n,NnN(a)x1(n)0,otherwise2n,NnN(b)x2(n)0,otherwiseSolution:(a):oddsymmetry----imaginary-valuedDTFT(b):evensymmetry----real-valuedDTFT3.Acontinuous-timesignalxa(t)iscomposedofalinearcombinationofsinusoidalsignalsoffrequencies300Hz,500Hz,1.2kHzand2.15kHz.Thesignalxa(t)issampledatan2kHzrate,andthesampledsequenceisthenpassedthroughanideallowpassfilterwithacutofffrequencyof900Hz,generatingacontinuous-timesignalya(t).Whatarethefrequencyya(t)?componentspresentinthereconstructedsignalSolution:kFTFi300Hz----300,1700,2300Hz...500Hz----500,1500,2500Hz...1.2kHz----1200,800,3200Hz...2.15kHz----2150,150,4150Hz...So4.Consideralength-Nsequenceya(t)are150Hz,300Hz,500Hz,800Hz.x(n),0nN1,whichNeven.Define2subsequencesoflength-N/2each:x0(n)x(2n)andx1(n)x(2n1),0nN2.LetX(k),0kN1,denotetheN-pointDFTofx(n),andX0(k)andX1(k),0kN21,denotetheN/2-pointDFTsofx0(n)andx1(n),respectively.ExpressX(k)asafunctionofX0(k)andX1(k).nkX(k)x(n)WNn0N12r0N12r0N1Solution:2rk(2r1)kx(2r)WNx(2r1)WNN12r0N12r0rkkrkx(2r)WNWNx(2r1)WN22N12r0N12r0rkkrkx0(r)WNWNx(r)W1N22X0kWkXkNN12,N20kN15.Usingz-transformmethods,determinetheexplicitexpressionfortheimpulseresponseh(n)ofacausalLTIdiscrete-timesystemthatdevelopsanoutputy(n)2(0.3)nu(n)foraninputx(n)4(0.6)nu(n).Solution:2,10.3z-14X（z）=,-11-0.6zY（z）=z0.3z0.2Y（z）0.5(10.6z-1)H（z）==,-1X（z）10.3z1.5110.3z-1z0.3y(n)(n)1.5(0.3)nu(n).6.Thetransferfunctionofacausalstablefilteris:K(1z-1)（z）=,H-11-zWhere01isrealandKisarealconstant.Determinetheshapeofthefilter.Solution:lowpassfilter.7.Considerthethird-orderIIRblockform:Thengivethetransferfunct