Question

4. Laplace Transform. (15 pts) Find the Laplace Transform of the following signals and sketch the corresponding pole-zero plot for each signal. In the plot, indicate the regions of convergence (ROC). Write X(s) as a single fraction in the form of DO (a) (5 pts)-(t-e*ta(t) + e-8tu(t). Show that X(s) =は,,늚. with ROC of Re(s) >-6. (b) (5 pts)-(t) = M(-t) +Au(-t). (c) (5 pts)-(t) 6(t)-a(-t). (s+6) (s+8)

Answer 1

X(t) = e^-6t u(t) + e^-8t u(t) by taking transformation of both sides L. T {x(t)} = L. S {e^-t u(t)) x(s) = 2s + 14/(s + 6) (s + 8) Re {s} > = -6 poles rightarrow X zeros rightarrow 0 we know that ROC for left sided signal is left to the left most [ole \r\n x(t) = e^5t u(t) + e^6t u(-t) by taking transform L. T { u (t)} = -1/s: s < 0 on both sides L. T {x (t) } = L. T {e^5 u (t) } + L. T {e^6t u (t) } X(s) = -1/ s - 5 + (-1)/s - 6 X(s) = -1 (s - 6) - 1 (S - 5)/ (s - 5) (s -6) X(s) = -2s + 11/(s -5) (s - 6) Re {s} > 5 pole zero plot poles, (s - 5) (s - 6) = 0, s = 5,6 zeros -2s + 11 = 0 implies s = 11/2 = 5.5 for