Question

4. the LTI system which we are given the following information: h(t) X(s) s +2 s-2 andel-)-jeuc). (a) Determine H(s) and its region of convergence (b) Determine ht) (c) If keep the same system, but change the input x(t)-e-31 , determine the output y(t)

Answer 1

x (s) = s + 2/s - 2, x (t) = 0, t > 0 and y (t) = -2/3 e^2 t u (- t) + 1/3 e^- t u (t). (a) H (s) = y (s)/x (s) y (s) = integral^infinity_- infinity y (t) e^- s t dt = integral^infinity_- infinity [-2/3 e^2 t u/- t) + 1/3 e^- t u/t)] e^- s t dt = integral^infinity_- infinity - 2/3 e^2t u (- t) e^- s t dt + 1/3 integral^infinity_- infinity e^- t u (t) e^- s t dt = -2/3 integral^0_- infinity e^- (s - 2) t dt + 1/3 integral^infinity_0 e^- (s + 1) t dt = -2/3 integral^0_- infinity e^(2 - s) t dt + 1/3 integral^infinity_0 e^- (s + 1) t dt = -2/3 [e^(2 - s)t/2 - s]^0_- infinity + 1/3 [- e^- (s + 1) t/s + 1]^infinity_0