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

# a u t 1 1 2 4 12 a first order allpass filter

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Unformatted text preview: s of its s tep c omponents, as shown in Fig. P 2.4-l9. I f get) is t he u nit s tep response of an LTIC s ystem, show t hat t he ( zero-state) response yet) of t he s ystem t o a n i nput f (t) c an b e expressed as 2 168 T ime-Domain A nalysis o f C ontinuous-Time S ystems 169 P roblems f (t) A A -5 -4 t- o t- ( b) (a) t- Fig. P 2.4-19 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 .4-21 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 .5-1 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 .5-2 ( b) e -tu(t) -I o o -2 o t- F ig. P 2.4-16 y(t) = I: (D2 + 6D + 25) y(t) E (x) = f (x) ( a) e - 3 'u(t) 2 .5-5 * 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 .4-20 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 P2.4-19, t he s haded s tep c omponent o f t he i nput is given by ( Lf)u(t - nLr) ' &quot; [ 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 .5-4 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 .5-3 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. 2.5-1 if t he i nput f (t) = e - 3'u(t) 2 .6-1 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 : .6-2 2 T ime-Domain A nalysis o f C ontinuous-Time S ystems R epeat Prob. 2.6-1 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? : .6-3 For ( a) ( b) (c) ( d) 1 .6-4 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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