数字信号试卷解答

7DigitalFilterDesign1四种基本类型的滤波器（低通，高通，带通，带阻）2滤波器设计指标的描述3IIR数字滤波器设计方法从模拟滤波器设计IIR数字滤波器：1）脉冲响应不变法2）双线性变换法4FIR数字滤波器设计方法1）线性相位概念2）具有线性相位的FIR滤波器的特点：（4种类型！）3）设计方法有：窗函数法；频率采样法1Determinetheperiod（page57-58，2.21）（1）12121122121212Hence,w1.1,w0.7.FromEq(2.44a),wethusget2r2r2020NandNbeperiodic,1.3130.332020wemusthaveNN.Thisimplies133Thisequalityholdsfor13and3,andrrTorrrr12henceNNN20（2）N=18（3）aperiodic（4）22cos(n)isaperiodicsequence,and1n-3isanaperiodicsequence.Hence,cos(n)n-3isanaperiodicsequence.2Determinewhetherornotthesystemislinear，casual，stable，andshift-invariable（page67-20，2.26）（1）iii31231212Foraninputx[],theoutputisyexp[3x[n]]iThus,foraninputx[][][],theoutputisgivenbyy[]exp[3([][])]y[]y[].Hencethesystemisnonzlinear.sincnanAxnBxnnaAxnBxnAnBn=12.，，xx1ethereisnooutputbeforetheinputhencethesystemiscausal.Ifx[n]By[n]=exp[3x[n]]exp[3B],henceBIBOstable.Lety[n]betheoutputforaninputx[n],andlety[]betheoutputforaan，110100inputx[].ifx[n]=x[n-n]theny[n]=exp[3x[n-n]]y[n-n].Hencethesystemisshift-invariant.na（2）（Page108T2.26（a））linear,causal,stable,shift-variant（3）（Page108T2.26（e））linear,notcausal,stable,shift-variant3DeterminetheinverseDTFTofeachthefollowingDTFTs（page120，3.15）（1）19339h[][,0,0,,0,1,0,,0,0,],-5n5.2j222jn（2）2133331h[][,0,0,0,0,0,,0,0,],-6n6.4j2j2j2j2j4jn，，4Computations（Page149，3.64）（1）[1,4,5,6,1,1,2]6Lnz0（2）[2,6,5,6,1]4Cnz0（3）%computeLzx=[1,3,0,0,1];y=[1,1,2];L=length(x)+length(y)-1;X=fft(x,L);Y=fft(y,L);ZL=ifft(X,Y)；%ORZL=cconv（x，y，L）；%computeZCx=[1,3,0,0,1];y=[1,1,2];L=max（length(x)，length(y)）;X=fft(x,L);Y=fft(y,L);ZC=ifft(X,Y)；%ORZC=cconv（x，y，L）；5ApplicationsofMATLBfunctions（1）page129Easilyknown，a=[1],b=[0.2,0.06,0.01,0.3].MatlabProgrammea=[1];b=[0.2,0.06,0.01,0.3];[h,w]=freqz(b,a,512);subplot(2,2,1)plot(w/pi,real(h));gridsubplot(2,2,2)plot(w/pi,imag(h));gridsubplot(2,2,3)plot(w/pi,abs(h));gridsubplot(2,2,4)plot(w/pi,angle(h));gridfreqz(b,a,512)(2)page85,173Y=FILTER(B,A,X)filtersthedatainvectorXwiththefilterdescribedbyvectorsAandBtocreatethefiltereddataY.ThefilterisaDirectFormIITransposedimplementationofthestandarddifferenceequation:a(1)*y(n)=b(1)*x(n)+b(2)*x(n-1)+...+b(nb+1)*x(n-nb)-a(2)*y(n-1)-...-a(na+1)*y(n-na)1）directlyb=[0.8,-0.4,0.36,0,0.01]a=[1,0.26,-0.35,0,0,0,0,-0.6]2)Z-transformb=[0.8,-0.4,0.36,0,0.01]a=[1,0.26,-0.35,0,0,0,0,-0.6]Accordingtopage85：b=[0.8,-0.4,0.36,0,0.01]a=[1,0.26,-0.35,0,0,0,0,-0.6]N=41;x=[1zeros(1,N-1)];y=filter(b,a,x);subplot(2,1,1)k=0:1:N-1;stem(k,y)xlabel(‘Timeindexn’)；ylabel(‘Amplitude’)；x=[ones(1,N)];y=filter(b,a,x);subplot(2,1,2)stem(k,y)xlabel(‘Timeindexn’)；ylabel(‘Amplitude’)；（3）Thisfunctionfindsthezerosandpolesofthetransferfunctionrepresentedbynumeratorcoefficientsbanddenominatorcoefficientsa,anddisplaysthepolesandzeros;eachzeroisrepresentedwitha'o'andeachpolewitha'x'ontheplot.6()()(1)1()()2()()(2)1()()2ddddjtjtjwjwnjwjwnXjxtedtxtXjedXexnexnXeedw()()()()()1()FT(())FT(())212=()*()21()ssssnnssnsnssnsxtxttnTxnTtnTXjxttnTXjnTXjjnT考虑xd(n)xd(n)=x(t)|t=nTs=x(nTs)t=nTs代入（1）中得到：1()()2sjnTsxnTXjed（3）比较（3）与（2），它们在数值上相等，但由于等式右边的积分式的积分区间不同，无法直接得到()Xj与()jwXe之间的关系。可以将式（3）表示为无限个积分和，每个积分区间为s2Tssssss(21)T(21)2TsT2sT2TsTss()1()()22,T12()()2T112()()2TT11()(2TsssdrjnTsrrjnTjrndrjrnjnTdrdnxxnTXjedrxnXjreedexnXjjredwxnX令得：（）因，得：）已知sssss()2)TT12()TTTjwdjwnrrXewjjredwwXjjr））()xtt当2sc：()dxn51510t5sT10sT15sT()Xjc-c1()sxtn若2sc：()jwdXewssT-csT1/Tsw()jwdXessT-ssTcsT-csT1/Ts-ssc-c()sXj1/Ts()sXj-cc-ss1/Ts