Using the definition of the Laplace transform, find the Laplace transform of eac

Question

Using the definition of the Laplace transform, find the Laplace transform of each of the following signals, and specify the corresponding regions of convergence: x(t) - S(t + 1) + S (r) + e-2(t + 3) u(r + 1) x(t) = e-21 u (t) + e-4t u(t) Determine the inverse Laplace transform of each of the following functions using partial fraction expansion method X(s) = s + 2/s2 + 7s + 12 X(s) = 2s2 - 9s - 35/s2 + 4s +2 Design a continuous time system to generate output y(t) = sin(wt) u(t) when the input is a unit step function Specify the system using a differential equation

Explanation / Answer

1) a) using laplace transform, x(t) = delta(t+1) + delta(t) + e^(-6) * e^(-2t) u(t+1) X(s) = e^(-1)s + 1 + e^(-6) (1/s+2)e^(-s) b) X(s) = (1/(s + 2) ) + ( 1/(s+4) ) 2) a) ( (s+2) )/( s^2 + 7s + 12) = (s+2)/( (s+3)(s+4) ) = A/s+3 + B/s+4 (s+2) = A(s+4) + B(s+3) --------->(1) put s = -3 -1 = A(1) , A = -1 put s = -4 -2 = B(-1), B = 2 = (-1)/(s+3) + 2/s+4 inverse laplace, x(t) = -e^(-3t) + (2) e^(-4t)

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