Question

2. Derive the time domain representations of the following Laplace transform expressions based on the given ROCs ROC: Refs) > 0, (b) x(s)= 2 , ROC : Re(s) <-1, (c) x(s) = , ROC : 0 < Re(s) < 1, Hint: Try not to use the inverse Laplace transform formula. Expand each expression into partial fractions and determine time domain representations based on the chart and given ROCs

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Answer 1

X(s) = 1/s: R{5} > 0 we know u(t) rightarrow 1/s hence x(t) = u(t) for given RoC X(s) = 2/s^3 - s = 2/s (s^2 - 1) = 2/s(s + 1)(s - 1) by partial fractions x(s) 2 /s(s + 1)(s - 1) = A/s + B/s + 1 + c/s - 1 2 = A(s + 1)(s - 1) + Bs(s - 1) + c (s) (s + 1) put s = 0 to get 'A' 2 = A(+ 1) (-1) + 0 + 0 Rightarrow A = -2 put s = -1 to get 'B' 2 = 0 + B(-1)(- 1 - 1) + 0 Rightarrow b = 1 put s = 1 to get 'c' 2 = 0 + 0 + c(1) (1 + 1) Rightarrow c = 1 x(s) = -2 /s + 1/s + 1 + 1/s - 1