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1、BEIJING JIAOTONG UNIVERSITYSignals and Systems Complex frequency-domain analysis for systemsz-domain description for D-T LTI systemsTransfer function and system propertiesImplementation structure for LTI systemsz-domain analysis for LTI system responseInterconnection of D-T LTI systemsCascade connec
2、tionParallel connectionFeedback connectionStructure diagram of D-T LTI systemsDirect form structureCascade form structureParallel form structureImplementation structure for LTI systemsCascade connectionParallel connectionFeedback connectionInterconnection of D-T LTI systems)z(2H)z(1HY zHz W z2()()()
3、zHXHzz ()()()21)z(2H)z(1HH zY zX zH z Hz()()/()()()12Interconnection of D-T LTI systemsCascade connection of systemsX zH zY z()()()(zX)(zY)(zW)(zY)(zX)z(2H)z(1H)z(2H+)z(1HY zH z X zHz X z12()()()()()zX zH zH()()()12H zY zX zH zHz()()/()()()12Parallel connection of systemsInterconnection of D-T LTI s
4、ystems)(zX)(zY)(zX)(zY)z(2H)z(1HY zE z H z()()()1E zX zHz Y z()()()()2Y zX zH z HzH z()()()1()()121H z HzH zY zX zH z1()()()()/()()121Feedback connection of systemsInterconnection of D-T LTI systems)(zX)(zY)(zESolution:Example 7.19:Determine the transfer function H(z).The output is identical to the
5、input for a all-pass system.H zH zHz Hz Hz()()1()()()1234)z(4H+)z(1H)z(3H)z(Y)z(X)z(2HIn the diagram,there is a all-pass system.Therefore,the transfer function HAP(z)of a all-pass system is HAP(z)=1Solution:The intermediate variables X1(z)and X2(z)are introduced.Y zHz XzHz H z X zHz H zX zHz Y z()()
6、()()()()()()()()()22211213+)z(1H)z(3H)z(Y)z(X)z(2HX z()1Xz()2Y zH z Hz HzH z Hz X z()1()()()()()()12312X zH zY z()()()H z Hz HzH z Hz1()()()()()12312Example 7.20:Determine the transfer function H(z).The relation between G(z)and Y(z)isSolution:From the diagram,we can obtain1zG zzG zX z()(/3)()()1 G z
7、YzG zz()()()(/4)zs1X zzH zzYz()1(/3)()1(/4)()1zs1zzzXG)/3)(1()1Example 7.21:Determine the transfer function H(z),and find the making the causal LTI system stable.p33 X z Y z34()G zInterconnection of D-T LTI systemsCascade connectionParallel connectionFeedback connectionStructure diagram of D-T LTI s
8、ystemsDirect form structureCascade form structureParallel form structureImplementation structure for LTI systemsDirect form structureCascade form structureParallel form structureStructure diagram of D-T LTI systemsa zH zb zjjjniiin1()10a zb zjjijniin1110H1(z)H2(z)The transfer function H(z)for n-orde
9、r LTI systemsLet m=n,H(z)is represented as the cascade connectionStructure diagram of D-T LTI systemsa za zaza zH zbb zbzb zb zjjnnjnnnnmiinmim11()1111(1)01101(1)aWzzX zH zjjjn()1()()111W zHzb zY ziiin()()02W za W z zX zjjjn()()()1Y zbW z ziiin()()0Direct form structureStructure diagram of D-T LTI s
10、ystemsThe difference equations corresponding to H1(z)and H2(z)a zaza zH znmbb zbzb znnnnnnnn ()(),1111(1)0111(1)Direct form structureStructure diagram of D-T LTI systemsW zX za W z zjjjn()()()1Y zbW z ziiin()()0Forward pathFeedback path)z(W1+n z)z(Wn z)z(W1 z1-nb1b0b1-z1-z1-z1-znb)z(W)z(Y)z(X)z(W1+n
11、 z)z(Wn z)z(W1 z-1-z1-na1ana1-z1-z1-znb)z(W)z(Y)z(X)z(W1+n z)z(Wn z)z(W1 z1-nb1b0b-1-z1-na1ana1-z1-z1-znb)z(W)z(Y)z(XH(z)=H1(z)H2(z).Hn(z)z(Y)z(X)z(nH)z(2H)z(1HEach sub-system is implemented in direct form structure.H(z)is represented as the cascade connection of one-order or second-order sub-system
12、s with real coefficients.Cascade form structureStructure diagram of D-T LTI systemsH(z)=H1(z)+H2(z)+.+Hn(z)z(nH)z(2H)z(1H)z(Y)z(XParallel form structureH(z)is represented as the parallel connection of one-order or second-order sub-systems with real coefficients.Each sub-system is implemented in dire
13、ct form structure.Structure diagram of D-T LTI systems4/1-8/11z1)z(Y1-2-)z(X2/11z1zzzzH zzz1 0.50.250.125()1212312Solution:(1)Direct form structureExample 7.22:Depict the structure diagram of H(z).zzzzzH zzzzz10.50.250.12510.510.25()1211212312121211121z4/1-1z)z(Y)z(X2/11zzzzH zzz1 0.50.250.125()1212
14、312Example 7.22:Depict the structure diagram of H(z).Solution:(2)Cascade form structurezzz10.510.254.55.5 1.751214/71z4/1-1z)z(Y2/112/92/11z)z(XzzzH zzz1 0.50.250.125()1212312zzzH zzz1 0.50.250.125()1212312Example 7.22:Depict the structure diagram of H(z).Solution:(3)Parallel form structureAcknowledgmentsMaterials used here are accumulated by authors for years with helpfrom colleagues,media or other sources,which,unfortunately,cannotbe noted specifically.We gratefully acknowledge those contributors.Implementation structure for LTI systems