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6 Find the bilateral Laplace transform of the signals f(t) = e–at u(t) and f(t) = –e–at u(–t) and specify their regions of convergence. Solution Using the basic definition of the transform, we obtain a. F2 (s) = ∫ ∞ e − at u (t )e − st dt = −∞ ∞ ∫e − ( s + a )t 0 dt = 1 s+a and its region of convergence is Re {s} > –a For the second signal b. F2 (s) = ∫ ∞ −∞ − e − at u ( −t )e − st dt = − ∫ 0 e − ( s + a )t dt = −∞ 1 s+a and its region of convergence is Re {s} < –a Clearly, the knowledge of the region of convergence is necessary to find the time function unambiguously.

Solution Using the basic definition of the transform, we obtain a. F2 (s) = ∫ ∞ e − at u (t )e − st dt = −∞ ∞ ∫e − ( s + a )t 0 dt = 1 s+a and its region of convergence is Re {s} > –a For the second signal b. F2 (s) = ∫ ∞ −∞ − e − at u ( −t )e − st dt = − ∫ 0 e − ( s + a )t dt = −∞ 1 s+a and its region of convergence is Re {s} < –a Clearly, the knowledge of the region of convergence is necessary to find the time function unambiguously. 5. 7 For t > 0, we close the contour to the left, we obtain f (t ) = 3e st 1 = e −2 t (s − 4)(s + 1) s=−1 2 t>0 For t < 0, the contour closes to the right, and now f (t ) = e 4t 3e st 3e st 3 + = − e −t + (s − 4)(s + 2) s=−1 (s + 1)(s + 2) s= 4 5 10 t<0 These examples confirm that we must know the region of convergence to find the inverse transform.

6 Find the bilateral Laplace transform of the signals f(t) = e–at u(t) and f(t) = –e–at u(–t) and specify their regions of convergence. Solution Using the basic definition of the transform, we obtain a. F2 (s) = ∫ ∞ e − at u (t )e − st dt = −∞ ∞ ∫e − ( s + a )t 0 dt = 1 s+a and its region of convergence is Re {s} > –a For the second signal b. F2 (s) = ∫ ∞ −∞ − e − at u ( −t )e − st dt = − ∫ 0 e − ( s + a )t dt = −∞ 1 s+a and its region of convergence is Re {s} < –a Clearly, the knowledge of the region of convergence is necessary to find the time function unambiguously.

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The Handbook of Formulas and Tables for Signal Processing. “Laplace Transforms” by Poularikas A. D.


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