Question

4. (2 marks) Determine (i) the Laplace transfer function, (ii) the impulse response function, and (ii) the input-output relationship (in the form of a linear constant-coefficient differential equation) for the causal LTI systems with the input-output pairs: a) x(t)-41(t) and y(t)-tu(t) + e-2tu(t). b) x() e2tu(t) andy(t)2u(t-4).

Answer 1

X(t) = 4 u (t) = x(s) = 4/s y(t) = t u(t)+ e^-2t u (t) = y(s) = 1/s^2 + 1/s + 2 Laplace Transfer function = H(s) = y(s)/x(x) = 1/s^2 + 1/s + 2/4/s y(s)/x(x) = (s + 2 + s^2/5^2(s + 2) times 3/4 H(s) = (3^2 + s + 2/4s^2 + BS) Impulse response is H(s) = s^2 + s + 2/4s(s + 2) = -0.5/5 + 2 + 0.25/5 + 0.25 Impulse response is h(t) = [-0.5^-2t + 0.25]u(t)+0.25 d(t) iii y(s)/x(x) = s^2 + s + 2/4s(s + 2) = 4 d^2y(t)/ d t + 8 d y(t)/d t = d^x/ d t + d x /d t + 2x (t)