Signal Processing and Linear Systems-B.P.Lathi copy

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Unformatted text preview: u (t + 5) - + 2(t - [e-(t-5) 5)e-(t-5) + 2] u (t _ 5) (a) C ausal ( b) Noncausal (c) Noncausa1 (d) Noncausal. (D + 3)Y1(t) = J (t), (D + 3)Y2(t) = D J(t) 1 .8-3 qo(t) = D~aqi(t), ( D + a )h(t) = t qi(t) C hapter 2 2 .2-1 Ziemer, R.E., W.H. Tranter, a nd D.R. Fannin, Signals and Systems, Macmillan, New York, 1983. = 6 4c2/7 1 .7-2 O nly (a) a nd (f) are time-invariant. Soliman, S., and M. S rinath, Continuous and Discrete Signal and Systems, Prentice-Hall, Englewood Cliffs, N.J., 1990. Taylor, S. J ., Principles o f Signals and Systems, McGraw-Hill, New York, 1994. P el M t) = (4 t +l)u(t+l)-6tu(t)+3u(t)+(2t-4)u(t-2), h (t) = t 2u(t)-(t2-2t+8)u(t-2)-(2t-8)u(t_4) 1 .4-2 1 .7-7 = 256/7, 2 .3-2 + 5>' + 6, >.2 + 5>' + 6 = 0, - 2, - 3, a nd e - 2t , e - 3t oCt) + ( e- 2t + e - 3t )u(t) 2 .3-4 (2 + 3 t)e- 3t u(t) (a) >.2 (b) 5 e-2t _ 3 e-3t ( b) t e-atu(t) (c) ! t 2u(t) 2 .4-6 (a) (1 - cost)u(t) ( b) sin t u(t) + ! e- 2t )u(t) ( b) ( e- 2t - e- 3t )u(t) (c) [(2 _ t )e-2t _ 2 e-3tju(t) 2 .4-5 (a) t u(t) 2 .4-8 (a) (~- ~e-3t 2 .4-10 ( a) v h[0.555 - e-2t cos(3t + I 23.7°)ju(t) ( b) 4 [e- t _ ~e-2t cos(3t + 7 l.560)ju(t) 2 .4-11 (a) ( e- t - e- 2t )u(t) ( b) e6 (e- t - e- 2t )u(t) (c) e - 6 [e-(t-3) _ e - 2(t-3)ju(t _ 3) 2 .4-13 t an- 1 t (d) (1 - e-t)u(t) - [1 - e-(t-1)ju(t - 1) +~ 2 .4-15 1 - cost for 0 ::; t::; 2.,.., c ost - 1 for 2.,..::; t::; 4 "",0 otherwise. 2 .4-21 H (s) = e -sT. 2 .5-1 (a) i e-3t - !e- 4t + ~ ( b) ! e- 3t _ ~e-4t + ~e-t (c) e -3t _ e -4t 2t 2 .5-3 ( a) ( ¥+¥t)e- _2e- 3t ( b) ( £+1ft)e- 2t 2 .5-4 £ -t e - 2t +!t 2 .5-5 ( 2-t)e-3t_2e-4t 2 .6-2 (a) Asympt. stable 2 .6-3 (a) 0 (b) marginally s table (b) marginally s table (c) unstable (c) B IBO u nstable 2 .7-2 W idth 0.6 ms, m aximum r ate 1667 pulses/sec. 2 .7-1 2 .7-3 (d) marginally stable. (a) No (a) 1 0-4 (b) Yes (b) 104 Hz (c) 104 pulses/sec. C hapter 3 3 .1-2 (a) c = 0.5 (b) E e 3 .4-1 (a) J (t) = 3 .4-3 = 1 /12 3 .2-1 C n, i + ~ 2::=1 (-sn c osnm = Cn3 = 0, Cn2 = - 1, C n, = ,,/2/.,.. = .,../2, ao = 0, bn = 0 a nd an = n~ sin(n.,../2) (c) Wo = 1, ao = 0.5, an = 0 = .,../3, ao = 0.5, bn = 0 a nd a n = ~(cos(';"') - cose;"') (a) Wo = 2, ao = 0.504, a n = 0.504( 1+{6n2 ) a nd bn = - 0.504( l+~~n2) (a) WO = .,../4, ao = 0, an = ~(n2'" s in n" - 1) a nd bn = ~ s in "','" 2 (a) Wo a nd bn = _ n1", (f) Wo 3 .4-4 3 .4-7 3 .4-9 All except (c) a nd (e) are periodic. Periods (a) 2.,.. (b) 2.,.. (d) 2 (f) I407r (g) 8.,../3 (h) 2.,.. (i) .,... 3 .4-11 J (t) ' " ! xo(t) - tX1 (t) - *X3(t) - ftx7(t) 3 .4-12 J (t) = 839 -~t + t ( ~t3 - ~t) + ... 840 3 .5-1 Answers t o Selected problems ( b) wo = 1 /5, D n = nl" s in(mr/5) (d) wo = 2, Do = 0, D n = --f,;(n~ sin(n2") - cos(n2") (e) 3 .5-5 WQ 6 .3-9 = 21r/3, Do = 1 /6, a nd D n = 2 3,,2 [e-ihn/3(~ + 1) - 1] N=3 3 .5-6 N = 6 3 .5-7 N = 2 3 .6-1 yet) = 4.1-6 4.3-2 ( a) 4 _2e-jw_2e-j2w 2W w2 - 4.3-3 ( a) ~ sin 2(wTJ ( a) ~ { sinc 2 4 .3-7 ( a) ~sinc(t)cos4t D ( b) liW2-~;w ["(W;lO)] ( (a) =;tsinc [wo(t _ to)] 3 (I_j2w_e...
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