Table Of Content

2.1 (a) E 2T P 0 (b) E 10T 2T 1 2 P0 (c) E  1 P 2 (d) E  A2 P 2 (e) E  3 P 8 (f) E T P0 (g) 1 E  4a P0 (h) A2T3 E  4 P0 (i) E  1 3T 33T1/333 3 3T1/3 P lim  t2/3tdt lim 3lim 0 TT T T T T 3 2.2 (a) 8 (b) 0 (c) 4 2 (d) 1 2.3 (a) 5 X(s) s1 (b) x(t)5t eln(56) etln(5) 1 X(s) sln(5) (c) 5 s1 (d) 5 s1 2.4 (a) x(t)t22u(t)t2 4t4u(t) 2 4 4 24s4s2 X(s)    s3 s2 s s3 (b) x(t)t22e3tu(t)t2 4t4e3tu(t) 2 4 4 2620s4s2 X(s)    s33 s32 s3 s33 (c)   x(t)cos t u(t)    0 4      cos costsin sint u(t)         4 0  4  0   1 1   cost sint u(t)  0 0   2 2  1 s 1  1 s X(s)  0  0 2 s2 2 2 s2 2 2 s2 2 0 0 0 (d)   x(t)e3t cos t u(t)    0 3      e3t cos costsin sint u(t)         3  0  3  0  1 3  e3t  cost sintu(t) 2 0 2 0   1 s3 3  1 s3 3 X(s)  0  0 2 s32 2 2 s32 2 2 s32 2 0 0 0 2.5 (a) 1 1  j  j 2 2 X(s)  s2 j s2 j 1  1   x(t)  j e(2j)t   j e(2j)t u(t)      2  2   1 1  e2t ejt  jejt  ejt  jejt u(t)   2 2  e2tcos(t)2sin(t)u(t)  5e2t cost63.4u(t) (b) 4 4 4 X(s)   s2 s1 s12 x(t)4e2t 4et 4tetu(t)4e2t t1etu(t)     (c) 4 4 4 X(s)   s2 s22 s1 x(t)4e2t 4te2t 4etu(t)4et t1e2tu(t)     2.6 (a) 3 2 1 X(s)   s3 s2 s1 x(t)3e3t 2e2t etu(t)   (b) 229 135 54 216 108 13 X(s)      s3 s32 s33 s2 s22 s1 x(t)229135t26t2e3t 216108te2t 13etu(t)   (c) 32 31 30 21 20 1 X(s)      s3 s32 s33 s2 s22 s1 x(t)3231t30t2e3t 2120te2t etu(t)   2.7 (a) y(t)a y(t)a y(t)bx(t)b x(t) initial conditions 1 2 1 0 (b) The poles are at s a  a2 4a 1 1 0 (i) For real and distinct poles, we requirea2 4a . In this case, the poles 1 0 ares a  a2 4a . 1 1 0 (ii) For real and repeated poles, we requirea2 4a . In this case, the poles 1 0 ares a ,s a . 1 1 (iii) For complex conjugate poles, we requirea2 4a . In this case, the poles are 1 0 s a  j 4a a2 1 0 1 (c) (i) a   a  1 n 0 2 a 0 In general, the poles are at s   2 1 n n (ii) For real and distinct poles, we require1. In this case, the poles ares   2 1 n n (iii) For real and repeated poles, we require1. In this case, the poles ares ,s. n n (iv) For complex conjugate poles, we require01. In this case, the poles ares   j 12 n n (v) This form is preferred because the nature of the poles is determined by a single parameter. (d) (i) b b 0 0 b 2 2 1 2 2 1 H(s) 0  n  n s2 2s2 s 2 1 s 2 1 n n n n b   21t  21t h(t) 0 e n e n u(t)   2 2 1  n b ent   21t  21t  0 e n e n u(t)   2 2 1  n b     0 entsinh  2 1 t u(t) n  2 1 n (ii) b H(s) 0 s2 n h(t)btentu(t) 0 (iii) b b 0 0 j2 12 j2 12 H(s) n  n     s  j 12 s  j 12 n n b h(t) 0 entejn 12t entejn 12tu(t) j2 12   n b ent     0 sin  12 t u(t) n  12 n (e) An overdamped system does not display any oscillations in its impulse response, whereas the underdamped system does display oscillations. Hence, the term damped refers to oscillations: if oscillations are present, the oscillations have not been damped; if there are no oscillations, the oscillations have been damped. 2.8 (a) 6 6 5 5 5 Y(s) X(s)   s6 s6 s s s6 y(t)51e6tu(t)   (b) x(t) 6 y(t) 4 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t (c) 6s 6s 5 30 V(s) X(s)  s6 s6 s s6 v(t)30e6tu(t) (d) 30 v(t) 20 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t 2.9 (a) 6 6 5 5 5/6 5/6 Y(s) X(s)    s6 s6 s2 s2 s s6 y(t)5t51e6tu(t)    6  (b) x(t) 6 y(t) 4 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t (c) 6s 6s 5 30 5 5 V(s) X(s)    s6 s6 s2 s(s6) s s6 v(t)51e6tu(t)   (d) 5 v(t) 4 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t