1、BEIJING JIAOTONG UNIVERSITYThe Course Group of Signals and Systems, Beijing Jiaotong University. P. R. CHINA. Copyright 2020Signals and Systems Complex frequency-domain analysis for systemsz-domain description for D-T LTI systemsTransfer function and system propertiesImplementation structure for LTI
2、 systemsz-domain analysis for LTI system responsehkH(z)xk yzs k = xk*hkX(z) Yzs (z) = X(z)H(z)The relation between H(z) and Yzs(z)z-domain analysis for LTI system responseTime domainz-domaindifference equation response ykresponse Y(z) in z-domainsolving the equationequation in z-domain(convolution o
3、peration)z-domain analysis for LTI system responseunilateralz-transformsolving the equation(algebraic operation)inversion of unilateralz-transformSolution: Y z Xz y y zz Y z y z Y z( ) 3 2 ( )( ) 1 ( ) 1 21 2 1yk3yk1+2yk2 = xkInitial conditions: y1=1, y2=3, input xk=(4)k uk, determine yk.According t
4、o the time shift property of unilateral z-transformZ y k u k z Y z y 1 ( ) 11Z y k u k z Y z z y y 2 ( ) 1 221Converting the difference equation into z-domain equation Y z z z y z y y X z( )(1 3 2 ) 3 1 2 1 2 2 ( )1 2 1Example 7.23: The difference equation of a causal LTI system is z z z zY z X zy z
5、 y y1 3 2 1 3 2( ) ( )3 1 2 1 2 2 11 2 1 21Yzi(z) Yzs(z) Y z z z y z y y X z( )(1 3 2 ) 3 1 2 1 2 2 ( )1 2 1X z zHzYz(z) 1 3 2()1()12zsSolution:yk3yk1+2yk2 = xkInitial conditions: y1=1, y2=3, input xk=(4)k uk, determine yk.Example 7.23: The difference equation of a causal LTI system is|z|2zzY z X zz
6、32( ) ( )2zs2 z z zz z z1 2 4/ 5 2 / 3 8 /15 y k u kk k k3 15 5 ( 2) (4) ( 1) 2 8 1zszzYzXz1 3 2()()12zs ZzX z u kzk4( ) (4) z z zzz( 1)( 2) 42Solution:yk3yk1+2yk2 = xkInitial conditions: y1=1, y2=3, input xk=(4)k uk, determine yk.Example 7.23: The difference equation of a causal LTI system is|z|4Co
7、mplete response: z z z zY z zz z z3 2 1 2()2 9 7 162zi y k u kkk 7( 1) 16( 2) ziy1=1, y2=3 zzYzy z y y1 3 2()3 1 2 1 2 212zi1 uky k y k y k k k k 53 15 ( 2) (4) ( 1) 46 8 34zs ziSolution:yk3yk1+2yk2 = xkInitial conditions: y1=1, y2=3, input xk=(4)k uk, determine yk.Example 7.23: The difference equat
8、ion of a causal LTI system is|z|2(1) transfer function H(z), impulse response hk.(2) zero-input response yzik, zero-state response yzsk.(3) the structure diagram of the system H(z).Input xk = 3kuk, and y1=3, y2=1.Example 7.24: The difference equation of a causal LTI systemDetermine:yk3yk1+2yk2 xk+xk
9、1(1) transfer function H(z), impulse response hkSolution: As the system is causal, we can use unilateral z-transformY(z)3(z1Y(z) y1) + 2(z2Y(z) z1y1 + y2) X(z) + z1X(z) z z z zY z X zy z y y z1 3 2 1 3 2( ) ( )3 1 2 1 2 2 (1 )1 2 1 211 zzzzX z z zHzzzYz)32( ) ( 1)( 2()( +1)()1, | | 2(1 )11z2sz1=0, z
10、2= 1; p1=1, p2=2yk3yk1+2yk2 xk+xk111)z( mI20)z( eRSolution: According to the transfer function H(z) z z z zH z zz z z( 1)( 2) ( 1) ( 2)( ) , | | 2z( 1) 2 3As the ROC of H(z) does not include the unit circle in z-plane, the causal LTI system is unstable. h k u k u kk 2 3 2 11)z( mI20)z( eRBy PFE, the
11、 impulse response is(1) transfer function H(z), impulse response hkSolution: z z z zY z X zy z y y z1 3 2 1 3 2( ) ( )3 1 2 1 2 2 11 2 1 211zyzy yzYz 223 11()3 2 2112zi1 z z z zz z z1 3 2 1 29 6 2 8121 y k u k u kk 8(2) ziy1=3, y2 =1(2) zero-input yzik, zero-state response yzsk z z z zY z X zy z y y
12、 z1 3 2 1 3 2( ) ( )3 1 2 1 2 2 11 2 1 211 z z z z zY z X zzz1 3 2 1 3 2 1 3( ) ( )1 1 11 2 1 2 1zs11 z z zz z zz1 1 2 1 31 6 6(1 )(1 2 ) 1 3111 1 11 1 11 y k u k u k u kkk 6(2) 6(3) zsxk = 3kuk(2) zero-input yzik, zero-state response yzskSolution:11)z(Y- -)z(X321-z1-z z z z zHzz z z( 1)( 2) 1 3 2()
13、( 1) 1121Rewrite H(z) in the form z1(3) the structure diagram of the system H(z)Solution:Given the zero-state response of a causal LTI system y k u kkk 3 3(0.5) (1/ 3) zsInput signal is xk=uk. Determine:(1) transfer function H(z) and its pole-zero plot.(2) the difference equation describing the LTI
14、system.(3) system impulse response hk.(4) the system stability.z-domain analysis for LTI system responsepole-zero plot zzzXzHzYz(2)3)()11(2zsz21 y k y k y k x k66 1 2 51 h k u k u kkk23 3 2 11The causal LTI system is stable.z-domain analysis for LTI system response2jj1 1/21/3)zIm (10 )zRe (AcknowledgmentsMaterials 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.z-domain analysis for LTI system response
