5.1 A causal signal x(t) having a Laplace transform with poles in the open-left s-plane (i.e., not including the jN axis) has a Fourier transform that can be found from its Laplace transform. Consider the following signals: x1(t) = e-2"u (t), x2(t) =r(t), x3(t) = x1(t)x2(t). 354 CHAPTER 5 FREQUENCY ANALYSIS: THE FOURIER TRANSFORM (a) Determine the Laplace transform of the above signals indicating their corresponding region of convergence. (b) Determine for which of these signals you can find its Fourier transform from its Laplace transform. Explain. (c) Give the Fourier transform of the signals that can be obtained from their Laplace transform. Answers: (a) X2(s) =1/s², o > 0; (b) x1(t) and x3(t).

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5.10 PROBLEMS
5.10.1 BASIC PROBLEMS
5.1 A causal signal x(t) having a Laplace transform with poles in the open-left s-plane (i.e., not
including the j2 axis) has a Fourier transform that can be found from its Laplace transform.
Consider the following signals:
x1(t) = e-2'u (t), x2(t)=r(t), x3(t) = x1(t)x2(t).
354
CHAPTER 5 FREQUENCY ANALYSIS: THE FOURIER TRANSFORM
(a) Determine the Laplace transform of the above signals indicating their corresponding region
of convergence.
(b) Determine for which of these signals you can find its Fourier transform from its Laplace
transform. Explain.
(c) Give the Fourier transform of the signals that can be obtained from their Laplace transform.
Answers: (a) X2(s) = 1/s², o > 0; (b) x1(t) and x3(t).
Transcribed Image Text:5.10 PROBLEMS 5.10.1 BASIC PROBLEMS 5.1 A causal signal x(t) having a Laplace transform with poles in the open-left s-plane (i.e., not including the j2 axis) has a Fourier transform that can be found from its Laplace transform. Consider the following signals: x1(t) = e-2'u (t), x2(t)=r(t), x3(t) = x1(t)x2(t). 354 CHAPTER 5 FREQUENCY ANALYSIS: THE FOURIER TRANSFORM (a) Determine the Laplace transform of the above signals indicating their corresponding region of convergence. (b) Determine for which of these signals you can find its Fourier transform from its Laplace transform. Explain. (c) Give the Fourier transform of the signals that can be obtained from their Laplace transform. Answers: (a) X2(s) = 1/s², o > 0; (b) x1(t) and x3(t).
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