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its s tep c omponents, as shown in Fig. P 2.4l9. I f get) is t he u nit s tep response of an
LTIC s ystem, show t hat t he ( zerostate) response yet) of t he s ystem t o a n i nput f (t)
c an b e expressed as 2 168 T imeDomain A nalysis o f C ontinuousTime S ystems 169 P roblems f (t) A A 5 4 t o t ( b) (a) t Fig. P 2.419 e '
2 0 o t 3 t C oulomb's law, t he e lectric field E (r) a t a d istance r from a charge q is given by
Ot E (r) =  q 47rfr2
(d) (c) 2 .421 D etermine H (s), t he t ransfer function o f a n i deal t ime delay of T seconds. F ind y our
answer by two methods; using Eq. (2.48) a nd using Eq. (2.49).
2 .51 Using t he classical m ethod, solve 1 f 2+l (D2 e' o t o if t he i nitial conditions a re y(O+) t ( a) u (t)
( f) (e) + 7D + 12) y(t) 2 .52 ( b) e tu(t) I o o 2 o t F ig. P 2.416 y(t) = I: (D2 + 6D + 25) y(t) E (x) = f (x) ( a) e  3 'u(t) 2 .55 * h(x) 1 w here h(x) = 47rfX2 H int: T he charge over a n i nterval L r l ocated a t r = n Lr is f (nLr)Lr. Also by = 2, + 3 )f(t) a nd i f t he i nput f (t) = u (t). y(t) = (D + l )f(t) ( b) e tu(t). Using t he classical m ethod, solve + 2 D)y(t) = if t he i nitial conditions a re y(O+) * g(t) 2 .420 A line c harge is located along t he x axis w ith a charge density f (x). Show t hat t he
electric field E (x) p roduced by t his line charge a t a p oint x is given by y(O+) + 4D + 4) (D2 Hint: F rom F igure P2.419, t he s haded s tep c omponent o f t he i nput is given by
( Lf)u(t  nLr) ' " [ i(r)Lr]u(t  nLr). A dd t he s ystem responses t o all such
c omponents. = 0, = (D if t he i nitial conditions are y(O+) = ~, y(O+) = 5, a nd if t he i nput f (t) is:
2 .54 j (r)g(t  r) d r = i (t) f (t) is: Using t he classical m ethod, solve t (h) (g) = 1, a nd i f t he i nput Using t he classical m ethod, solve if t he i nitial conditions a re y(O+)
2 .53 y(O+) + 2 )f(t) ( c) e  2'u(t). (D2 e' = 0, = (D = 2, y(O+) (D + l )f(t) = 1, a nd i f t he i nput is f (t) = u(t). R epeat P rob. 2.51 if t he i nput f (t) = e  3'u(t)
2 .61 Explain, w ith reasons, w hether t he L TIC s ystems described by t he following equations
are asymptotically stable, marginally stable, o r u nstable.
( a)(D2 + 8D + 12)y(t) = (D  l)f(t)
( b)D(D2 + 3D + 2 )y(t) = (D + 5 )f(t) (c)D2(D2 + 2)y(t) = f (t)
( d)(D + 1)(D2  6D + 5)y(t) = (3D + l )f(t) 1 70
: .62 2 T imeDomain A nalysis o f C ontinuousTime S ystems R epeat Prob. 2.61 if
( a)(D + 1 )(D2 + 2D + 5)2y(t) = f it)
( b)(D + 1 )(D 2 + 9 )y(t) = (2D + 9 )f(t) ( c)(D + 1)(D2 + 9)2y(t) = (2D + 9 )f(t)
( d)(D2 + 1 )(D2 + 4)(D2 + 9)y(t) = 3 Df(t)
a certain LTIC system, the impulse response h it) = u(t).
D etermine t he c haracteristic root(s) of this system.
Is this s ystem asymptotically or marginally stable, or is it unstable?
Is this s ystem BIBO stable?
W hat c an t his system be used for? : .63 For
( a)
( b)
(c)
( d) 1 .64 In Sec. 2.6 w e d emonstrated t hat for a n LTIC system, Condition (2.65) is sufficient
for BIBO s tability. Show t hat t his is also a neces...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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