Question

Digital Signal Processing Homework #4 1. Find the solution of the differential equation: y+4y+3y = x+2x for x(t)-e'u(t) and initial conditions y(0) 0, (0) 1 What is the transfer function of a LTI system that is describable by the equation above? 2. Find the transfer functions of the LTI systems A and B in the configuration shown below when you are given that v v-z and y-x

Answer 1

y^dotdot + 4y^dot + 3y = x^dot + 2x taking Laplace transform s^2 y(s) - s y(0) - y^'(0) + 4[s y(s) - y(0)] + 3 y(s) = s x(s) - x(0) + 2 x(s) s^2 y(s) - 1 + 4s y(s) + 3 y(s) = s x(s) + 2 x(s) y(s) [s^2 + 4s + 3] = (s + 2) x(s) + 1 y(s) = s + 2/s^2 + 4s + 3 x(s) + 1/s^2 + 4s + 3 y(s) = s + 2/(s + 3)(s + 1) x(s) + 1/(s + 1)(s + 3) if x(t) = e^-t u(t) x(s) = 1/s + 1 So y(s) = (s + 2)/(s + 1)^2(s + 3) + 1/(s + 1)(s + 3) y(s) = y_1(s) + y_2(s) y_1(s) = s + 2/(s + 1)^'(s + 3), y_2(s) = 1/(s + 1)(s + 3) \r\n y_1(s) = A/s + 1 + B/(s + 1)^2 + C/(s + 3) B