信号与系统奥本海姆第4版11章答案.pdf

1 11.11.1 F(z) = E(z)H1(z) E(z) = F(z)G(z) + X(z) Y(z) = X(z)H0(z) + F(z) Y(z) X(z) = H0(z) + H1(z) 1+G(z)H1(z) 11.211.2 E(s) = X(s) Y(s)G2(s) F(s) = E(s)H2(s) Y(s)G1(s) Y(s) = F(s)H1(s) () () = H2(s)H1(s) 1+G2(s)H2(s)H1(s)+G1(s)H1(s) 1 11.31.3 Q(s) = () 1+()() = 1 1 1+ 1 1() = 1 21 要稳定必有：-1-b0 即 b0 稳定，对于 K = 1 6 j1 (j+1)(j+2) = 5jj3+422 (22)2+92 5 3= (5 + )(5 ) = 0 (与实轴交点) = 0, 则 1 K = 1 2 = 5, 则 1 K = 1 3 42 2 = 2(22 1) = 2(2 + 1)(2 1) = 0 (与虚轴交点) 1 K 1 3 1 K K-3 11.911.9 K(s + 1)(s + 3) + (s + 2)(s + 4) = (K + 1)2+ (4 + 6) + (3 + 8) = 0 2 + 13 + 8 4 + 6 03 + 8 K-1 = 0 2+ 5 + 7 = 0 无分支/聚集点，必在实轴上。 1 11.101.10 一个二阶极点 s=-1，一个二阶零点 s=1. 11.1111.11 2 1 4 + = 0 21 1 4 + 20 0 1 4 + |K 1 4| 3 4 1 11.1711.17 Gk(j) = (+1)4 = (1)4 (1+2)4 = (1462+43+4) (1+2)4 虚部为 0,有：4 + 43= 0 = 0,1 Gk(j0) = K 或Gk(j) = K 4 p = 0 要稳定，即 Z=0，必有 R=0 K 1零点：z0= 2,z1= 2 G(s)H(s) = (s+2)(s2) s(s+3)(s+6) = 43027272 ? = 0 令f(s) = 4 302 72 72 s (6,3) = 5() = 163 = 4() = 8 = 4.2() = 12.4 = 4.1() = 1.476 = 4.05() = 3.43 = 4.075() = 1.022 = 4.0875() = 0.22 = 4.081() = 0.43 = 4.085() = 0.033 = 4.086() = 0.066 s0= 4.085 s (2,) = 4() = 584 = 10() = 6208 = 7() = 355 = 5.5() = 460 = 6.25() = 167 = 6.62() = 57.2 = 6.44() = 59 = 6.53() = 3.14 = 6.57() = 23 = 6.55() = 9.9 = 6.54() = 3.38 = 6.535() = 0.1156 s1= 6.535 G(j)H(j) = (j+2)(j2) j(j+3)(j+6) = 4 (182)92 (18 2) = 0= 4.24 (c) a=-4,b=2 极点：p0= 0,p1= 3,p2= 6;零点：z0= 4,z1= 2 G(s)H(s) = (s+4)(s2) s(s+3)(s+6) = 4+43242150144 ? 令() = 4+ 43 242 150 144 s (3，0) = 2() = 44 = 1() = 21 = 1.5() = 18.56 = 1.25() = 0.63 = 1.125() = 9.7 = 1.23() = 0.96 = 1.24() = 0.16 s1= 1.24 s (2，) = 5() = 369 = 6() = 252 = 5.5() = 114 = 5.75() = 53.57 = 5.625() = 34.08 = 5.687() = 8.461 = 5.656() = 13.04 = 5.672() = 1.998 = 5.680() = 3.57 = 5.676() = 0.78 = 5.674() = 0.610 = 5.675() = 0.085 s2= 5.675 G(j)H(j) = (j+4)(j2) j(j+3)(j+6) = (144835)+74+1082 (9 2)2+(183)2 144 83 5= 0= 2.941 (d) a=-7,b=2 极点：p0= 0,p1= 3,p2= 6;零点：z0= 7,z1= 2 G(s)H(s) = (s+7)(s2) s(s+3)(s+6) = 4+103152252252 ? = 0 令：f(s) = 4+ 103 152 252 252 s (,7) () 7252 10768 8.5114.9 9.25201.97 8.87516.6 8.68855.2 8.7725.3 8.825.88 8.842.16 8.831.88 8.8350.14 8.8330.67 8.8340.27 s0= 8.835 s (6,3) () 58 4132 6144 5.568.4 5.2529.8 5.12510.7 5.06251.3 5.0313.4 5.0461.16 5.05430.079 5.0500.56 s1= 5.0543 s (3,0) () 0252 124 1.563.6 1.2522.47 1.1250.12 1.130.81 1.1270.25 1.1260.065 s2= 1.126 s (2,+) () 2720 1015728 61152 4604 512 5.175.7 5.0531.1 5.022.49 5.0313.7 5.0237.65 5.0215.9 5.02055.5 s3= 5.02 G(j)H(j) = (j+7)(j2) j(j+3)(j+6) = (4+412252)+44+2162 (183)2+(92)2 (4+ 412 252) = 0= 2.33 (e) a=-1,b=-2 极点：p0= 0,p1= 3,p2= 6;零点：z0= 1,z1= 2 G(s)H(s) = (s+1)(s+2) s(s+3)(s+6) = 4+63+152+36+36 ? = 0 令f(s) = 4+ 63+ 152+ 36 + 36 s (6,3) () 5106 44 318 3.513.4 3.756.71 3.8751.9 3.93750.9 3.906250.54 3.9218750.17 3.91406250.187 3.9180.007 3.9200.08 3.9190.0386 s0= 3.918 s (2,1) () 28 110 1.50.5625 1.72.9759 1.61.2 1.550.3 1.5250.1 1.5370.1 1.5310.006 s1= 1.531 (f) a=-4,b=-2 极点：p0= 0,p1= 3,p2= 6;零点：z0= 4,z1= 2 G(s)H(s) = (s+4)(s+2) s(s+3)(s+6) (g) a=-7,b=-2 极点：p0= 0,p1= 3,p2= 6;零点：z0= 7,z1= 2 G(s)H(s) = (s+7)(s+2) s(s+3)(s+6) = 4+183+1052+252+252 ? 令f(s) = 4+ 183+ 1052+ 252 + 252 s (,7) () 7140 10232 8.5137 9.2520 9.62586 9.546 9.418 9.38 9.354.6 9.3251.7 9.33751.4 9.331250.19 9.3343750.6 9.3330.26 9.3320.004 s0= 9.332 s (6,3) () 672 336 428 58 4.76.2 4.850.54 4.7752.9 4.811.3 4.830.397 4.840.07 s1= 4.84 (h) a=-5,b=-4 极点：p0= 0,p1= 3,p2= 6;零点：z0= 5,z1= 4 G(s)H(s) = (s+5)(s+4) s(s+3)(s+6) = 4+183+1232+360+360 ? 令f(s) = 4+ 183+ 1232+ 360 + 360 s (5,4) () 510 48 4.50.5625 4.253.8 4.3751.66 4.4380.5 4.4690.005 s0= 4.469 s (3,0) () 0360 318 24 2.38.3519 2.10.8 2.051.5 2.070.55 2.090.37 2.080.084 2.0850.14 2.0820.007 s1= 2.082 (i) a=-7,b=-4 极点：p0= 0,p1= 3,p2= 6;零点：z0= 7,z1= 4 G(s)H(s) = (s+7)(s+4) s(s+3)(s+6) = 4+223+1652+504+504 ? 令f(s) = 4+ 223+ 1652+ 504 + 504 s (,7) () 784 1036 137104 11284 10.27.5 10.115.3 10.170.44 10.154.17 10.161.88 10.1650.7 10.16750.1 10.1680.02 s0= 10.168 s (3,0) () 336 0504 24 1.724.3 1.858.7 1.950.055 s1= 1.95 (j) a=-7,b=-8 极点：p0= 0,p1= 3,p2= 6;零点：z0= 7,z1= 8 G(s)H(s) = (s+7)(s+8) s(s+3)(s+6) = 4+303+2852+1008+1008 ? 令f(s) = 4+ 303+ 2852+ 1008 + 1008 s (,8) () 880 9288 15612 16496 15.627 15.792 15.637.7 15.6116 15.624.1 15.6230.56 15.6240.6 15.62350.029 s0= 15.6235 s (8,7) () 880 728 7.36.56 7.42.66 7.370.2 7.380.7 7.3740.16 7.3720.027 s1= 7.372 s (6,3) () 636 3180 4128 532 5.59 5.36 5.41.79 5.380.266 5.370.5 5.3770.036 s2= 5.377 s (3,0) () 01008 3180 292 1.541 1.721 1.68.47 1.630.7 1.622.3 1.6270.2 1.6280.08 s3= 1.628 1 11.271.27 H(s)G(s) = (+2) 2+2+4 要无振荡s 4 或 s 0K 6 或 K 2 零点：Z0= 2;极点：P0,1= 1 3 = (+4) ? = 0s0= 0,s1= 4 1 11.281.28 (a) G(s)H(s) = 1 1 (b) G(s)H(s) = 1 2 1 (c) G(s)H(s) = 1 ( + 1)2 (d) G(s)H(s) = 1 ( + 1)3 (e) G(s)H(s) = 1 ( + 1)2 (f) G(s)H(s) = + 1 ( 1)2 (g) G(s)H(s) = + 1 2 4 (h) G(s)H(s) = 1 2+ 2 + 2 (i) G(s)H(s) = + 1 2 2 + 2 (j) G(s)H(s) = + 1 ( + 100)( 1)2 (k) G(s)H(s) = 2 ( + 1)3 1 11.291.29 (a) H(s) = 10+1 2+1 ,G(s) = 1 (b) H(s) = 10+1 2+1 ,G(s) = 1 (c) H(s) = 1 (+1)2(+10) ,G(s) = 100 (d) H(s) = 1 (+1)3 ，G(s) = 1 +1 (e) H(s) = 1 (+1)(+10) ,G(s) = 1 (f) H(s) = 1/100 (+1)2 ,G(s) = 10s+1 10+1 (g) H(s) = 1 (+1) ，G(s) = 1 +1 11.3011.30 (a) G(z)H(z) = 1 1/2 (b) G(z)H(z) = z2 (c) G(z)H(z) = 1/z (d) G(z)H(z) = 1/z2 (e) G(z)H(z) = 1 (+1/2)(3/2) (f) G(z)H(z) = 3 (+1/3) G(e)H(e) = 3cos(2) + 1/3 + 3sin(2) 4 3 + 23 3 cos() (g) G(z)H(z) = 1 (2+1/3) G(e)H(e) = cos(2) cos() + 1 3 (sin(2) sin() (cos(2) cos()+ 1 3) 2 + (sin(2) sin()2 (h) G(z)H(z) = 1/2 (2) G(e)H(e) = cos(2) 4cos() + 4 (sin(2) 4sin() 2(cos() 2)2+ (sin()2) (i) G(z)H(z) = (+1)2 3 G(e)H(e) = cos + 2cos2 + cos3 (sin + 2sin2 + sin3) 1 11.311.31 (a) H(z) = z1 ,G(z) = 1 2 20lg|H(e j)G(ej)| = 20lg2 ()() = (b) H(z) = 1 11 2 1 ,G(z) = 1 2 20lg|H(ej)G(ej)| = 20log102 10log10(cos 1/2)2+ (sin)2 ()() = tan1 (sin) (cos 1/2) (c) H(z) = 1 (11 2 1)(1+1 2 1) ,G(z) = 2 20lg|H(ej)G(ej)| = 10log10(2 1/4)2+ (2)2 ()() = tan1 (sin2) (cos2 1/4) (d) H(z) = 2 2 ,G(z) = 1 20lg|H(ej)G(ej)| = 20log102 10log10(cos 2)2+ (sin)2 ()() = tan1 (sin) (cos 2) (e) H(z) = 1 +1 2 ,G(z) = 1 3 2 20lg|H(ej)G(ej)| = 10log10(cos + 1 2) 2 + (sin)2 10log10(cos 3 2) 2 + (sin)2 ()() = tan1 (sin sin2) (cos2 cos 3/4) (f) H(z) = 1 +1 2 ,G(z) = 1 3 2z 1 20lg|H(ej)G(ej)| = 10log10(cos + 1 2) 2 + (sin)2) + 10log10(1 3 2 cos) 2 + (3 2 sin) 2 ()() = tan1 (3 2sin2 1 4sin) (1 4cos 3 2cos2 + 1 2) (g) H(z) = 1 2 2+1 3 ,G(z) = 1 20lg|H(ej)G(ej)| = 10log10(cos2 cos + 1 3) 2 + (sin2 sin)2 ()() = tan1 (sin3 sin2 + 1 3 sin) (cos3 cos2 + 1 3cos) (h) H(z) = 1 1 ,G(z) = 1 4 1 20lg|H(ej)G(ej)| = 20log104 10log10(cos 1)2+ (sin)2 ()() = tan1(sin2 sin) (cos2 cos) 11.3211.32 a Q(s) = () 1 + ()() = 1()2() 1()2+ 1()2() b Q(s)|=0= () = 1() 1() c Q(s) = () () () 1 + () () = () () 1() () 1() () 1 + 1() () 1() () 2() () 2() () = () 1 + ()() Q(s)的极点为q(s) （1 + KG (s)H(s) = 0的解。 d 式子H(s)G(s) = +1 (+4)(+2) +2 +1的零点为: = 1 = 2极点为： = 4 = 2 = 1 e H(z)G(z) = + 1 1/2 1 + 1 零点 z=-1,极点 = 1 = 1/2 f H(z)G(z) = 2 ( 2)( + 2)2 零点 = 0 = 0极点 = 0 = 0 = 2 = 2 1 11.331.33 a b c 1 11.341.34 bi r = 1, s + 0= 1 0= 1 r = 2, s2+ 1s + 0 = ( 1)( 2 ) = s2 (1+ 2)s + 12 1= (1+ 2 ) 假设 r=k,时1= =1 则 r=k+1 时： s+1+ s+ + 0 = ( +1)(s+ 1s1+ + 0 ) = s+1 +1s+ 1s = 1 +1= =1 +1= +1 =1 bii 直接除 biii H(s)G(s) = 1 + 1 ()() = K + s+ (1 1 ) 1 + + 0= 0 b 1 1 = c1 K 0:i= 2l + 1 n m ,= 3 , 3 3 , 5 3 K 0 K + 3 0 2 3 离散时间系统 1 0 aK 2 0可以同时满足，及可稳定 d (s + 1)(s 2) + K(s + 2) = s2+ s + 2 1 = 2 2 = 2 = 3 = 2 e i i (s + 2)(s 2)(s + 1) + K(s + 1 2) = 0 s3+ s2 4s 4 + Ks + K 2 = 0 s3+ s2+ (K 4)s 4 + K 2 = 0 s31K 4 s21 K 2 4 s K 2 s0 K 2 4 K 2 0 K 2 4 0 K 8 只要 K 8，系统则稳定 ii ii (s + 2)(s 2)(s + 1) + K(s + 3) = 0 s3+ s2+ (K 4)s 4 + 3K = 0 s31K 4 s213K 4 s2K s03K 4 2K 0 3K 4 0 不能同时满足 1 11.461.46 Q(s) = () 1 + ()() (a) ()() = 10 + 100 0.00014+ 0.02023+ 1.04012+ 2.02 + 1 1 11.481.48 a G()() = (cos(2) cos() + (sin() sin (2) (cos(2) cos()2+ (sin(2) sin ()2 b e 1 0 当 = 9.8 2 = 0.5 有： () = 1 2+ 2 + 1 = 1 2+ 2 + 2 2= 9 = 1 1= 14.3 2= 2 2= 3 1 11.571.57 a () = ()()() () = () () () = ()