ECE-220 Problem 7-2 Consider the causal LTI system with system function H(s) = Y

Question

ECE-220 Problem 7-2
Consider the causal LTI system with system function H(s) = Y(s)/X(s) = 1/(s+2)^2
a) Find a differential equation relating x(t) and y(t) b) If the input is x(t) = sin(2t) for t>0, find anapproximate expression for the response y(t) that is valid forlarge t.

Consider the causal LTI system with system function H(s) = Y(s)/X(s) = 1/(s+2)^2
a) Find a differential equation relating x(t) and y(t) b) If the input is x(t) = sin(2t) for t>0, find anapproximate expression for the response y(t) that is valid forlarge t.

Explanation / Answer

Y(s) / X(s) = 1 /(s+2)2 a) Y(s) [s+2]2 = X(s) Y(s) [ s2 + 4s + 4] =X(s) s2 Y(s) + 4sY(s) + 4Y(s) =X(s) Assume the initital condictions y(0)=0 andy'(0)=0 Take the inverse Laplace transform y"(t) + 4 y'(t) + 4 y(t) =x(t) d2y(t) /dt2 + 4 dy(t)/dt + 4y(t) = x(t) b) x(t) = sin2t    X(s) = 2/(s2 +4) Y(s) = [1 / (s+2)2 ] [ 2 /(s2 + 4) ] Y(s) = 2 / (s+2)2 (s2 +4) 2 / (s+2)2 (s2 + 4) = A/ (s+2) + B / (s+2)2 + (Cs+D) / (s2 +4) A(s+2)(s2 + 4) + B(s2+4)+ (Cs+D) (s+2)2 = 2 Put s = -2 0 + B (8) + 0 = 2 B = 1/4 Put s= 0 A(8) + B(4) + D (4) = 2 8A + 4D = 2 - (1/4)(4) 8A + 4D = 1 -------------- (1) Put s = 1 A(15) + B (5) + (C+D) (9) = 2 15A + 9C + 9D = 3/4 -----------(2) Put s = -1 A(5) + B (5) + (-C+D) (1) = 2 5A - C + D = 3/4 ---------------(3) Solving (1), (2) and (3) A = -1/8 C = -7/8 D = 1/2 Y(s) = -1/8(s+2) + 1/4(s+2)2 +(-7s/8 + 1/2) (s2+4) Y(s) = -1/8(s+2) + 1/4(s+2)2 -7s/8(s2+4) + 1/2(s2+4) Taking inverse laplace transform Y(t) = - e-2t / 8 + te-2t / 4 - 7/8 cos2t + 1/4sin2t

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