数字信号处理II习题解答-2004(老师版)

2019-12-14No.1of81.Asinusoidalsignal)2/sin()(nnxisappliedtoasecond-orderlinearpredictorasinFig.1.CalculatethetheoreticalACF(Auto-CorrelationFunction)ofthesignalandthepredictioncoefficients.VerifythatthezerosoftheFIRpredictionfilterareontheunitcircleattherightfrequency.UsingtheLMSalgorithmwith1.0,showtheevolutionofthecoefficientsfromtime0nto10n.Howisthatevolutionmodifiedifthesignalgorithmisusedinstead.1z1z++)(nx)(ne)(1na)(2naFig.1.Second-orderpredictionfilter解：a)计算预测系数理论值(滤波器系数的维纳最优解)，由2cos21]2)(sin2sin[21]2)(sin2[sin)]()([)(10kkiiknnEknxnxEkrixx2/1002/1)0()1()1()0(rrrrRx，2/10)2()1()]()([rrnxnyEryx11201optxyxaHRra输出最小均方误差理论值可由下式计算:2min00[()]0.5011/2TToptyxJEynHr其中5.0)0()]([)]([22rnxEnyEb)FIR滤波器零点：jzzzazHiii22111)(,即零点在正弦信号x(n)频率的21,0对应的z平面位置.c)用LMS算法，n＝0～10时系数的近似值LMS：)1()()1()1()1()1()()1(nXnHnynenenXnHnHT在线性预测误差滤波的LMS算法中：2019-12-14No.2of8)1()1()]1(),([)()1()](),([)()(21nxnynxnxnXnXnananAnHTT)()()1()1()1()()()1(nXnAnxnenenXnAnATTAnnxnnx]0,0[)0(0,0)(),2sin()(所以,LMS算法下的预测误差滤波器)1()()1()()()1()1()1()()()()1()1(212121nxnxnenananananxnxnananxne344.000344.00729.027.00027.0081.019.00019.009.01.0001.0010)2(0)2(0)2()2(20)1(0)1(1)1()1(10)0(0)0(0212121aaxenaaxenaand)符号算法)1()()]1([)()()1()1()1()()()()1()1(212121nxnxsignnesignnananananxnxnananxne)()]([0,1)(nxnxsignnx2019-12-14No.3of84.0004.007.03.0003.008.02.0002.009.01.0001.0010)2(0)2(0)2()2(20)1(0)1(1)1()1(10)0(0)0(0212121aaxenaaxenaan2．Asecond-orderadaptiveFIRfilterhastheinputas)2/sin()(nnxand)2(5.0)1()()(nxnxnxnyasreferencesignal.Calculatethecoefficients,startingfromzeroinitialvalues,fromtimen=0ton=10.Calculatethetheoreticalresidualerrorandthetimeconstantandcomparewiththeexperimentalresults.()1.0解：取1.0a)计算n＝0～10的系数)3()1()1()()1()2()1()()1()1()1()2(5.0)1()()(nenXnHnHnXnHnynenxnxnxnyT2cos21)(kkrxx2/1002/1)0()1()1()0(rrrrRx，2/14/1)1(5.0)0()1()2(5.0)1()0(rrrrrrryx①12/11yxxoptrRH2min[()]ToptyxJEynHr2/14/112/1)]2()()2()1()1()(2)2(25.0)1()([222TnxnxnxnxnxnxnxnxnxE08/5)2()1(3)0(25.2rrr2019-12-14No.4of8②Theoreticalresidualerror:0)21()(2minNJJ③Theoreticaltimeconstant：2211111120(,)(0)0.10.5NekxkxkxRrN是特征值注意:由于均方收敛的时间常数比均值收敛的时间常数小,所以实际应用中采用较保守的理论估计值,即采用均值收敛的时间常数作为算法收敛的时间常数的理论估计值.④利用（2）（3）作迭代：)0(0)(nnx，0)0()0(21hh)()1()1()()()1()1()()1()()()1()1(212121nxnxnenhnhnhnhnxnxnhnhnyne)1(5.0)()1()1(nxnxnxny5.01112624.0]0,1[]4095.0,2376.0[1010]4095.0,2376.0[]06561.0,0[6561.0]1,0[]3439.0,2376.0[5.019]3429.0,2376.0[]0,02916.0[2916.0]0,1[]3439.0,2084.0[108]3439.0,2084.0[]0729.0,0[729.0]1,0[]271.0,2084.0[5.017]271.0,2084.0[]0,0324.0[324.0]0,1[]271.0,176.0[106]271.0,176.0[]081.0,0[81.0]1,0[]19.0,176.0[5.015]19.0,176.0[]0,036.0[36.0]0,1[]19.0,14.0[104]19.0,14.0[]09.0,0[9.0]1,0[]1.0,14.0[5.013]1.0,14.0[]0,04.0[4.0]0,1[]1.0,1.0[102]1.0,1.0[]1.0,0[1]1,0[]0,1.0[111.]0,01[]0,1.0[1]0,1[]0,0[000)1()1()1()1()1()()()(TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTnHnXnenenXnHnynxnb)根据实验结果计算残差和时间常数残差=0698.0)11(2e时间常数:通过观察见,在n为偶数时,误差变化波动大,因此应选择在n为奇数时的误差值确定时间常数较合理.2916.0)9()8()9()9(XHyeT26424.0)11()10()11()11(XHyeT)0)(,0)()]([(2994.20)()11()]()9([2222)911(222eJeEeeeee所以在计算时可假设由于所以实验结果和理论值是符合的.2019-12-14No.5of83．Adaptivelineenhancer.Consideranadaptivethird-orderFIRpredictor.Theinputsignalis)()sin()(0nbnnxwhere)(nbisawhitenoisewithpower2b.Calculatetheoptimalcoefficients31,,iaopti.Givethenoisepowerinthesequence31,)()(ioptiinxansaswellasthesignalpower.CalculatetheSNRenhancement.解：a)计算31,,iaopti20002000221cos212cos21cos2121cos212cos21cos2121bbbxR，)1()(nxny由于窄带信号为白噪声，时延参数D选择1。0003cos212cos21cos21yxroptoptoptyxxoptaaarRA3211b)原2021bSNR而3122})]({[)}({iioptinxaEnSE)2(2)1()(2)0()(313221232221raaraaaaraaa031032212322212coscos)(21)(aaaaaaaaa（信号）2232221)(baaa（噪声）2019-12-14No.6of8所以此时])(2coscos)(1[212232221031032212bbaaaaaaaaaSNR))(2coscos)(21(232221031032210aaaaaaaaaSNR))(2coscos)(21(0/tenhancemen23222103103221aaaaaaaaaSNRSNRSNR4．UsetheLevinson-DubinalgorithmtocomputethePARCORcoefficients(反射系数)associatedwiththecorrelationsequence,1)0(r109.0)(nnrGivethediagramofthelatticefilterwiththreesections.Commentonthecase1解：a)计算L－D反射系数31,jkj①初始化1)0(0rE②1j，9.0)0()1(111rrka，2021181.01)1(EkE③31j，121111111)1(1,,2,1])()([1jjjjjjjijijijijjjjEkEjiakaaijrajrEka2j，2211122281.0181.081.0)]1()2([1rarEka22112112112181.0181.09.0)1(akakaa2222122281.0181.09.01)81.01()1(EkE3j，…b)lattice预测器结构2019-12-1