GATE EE 2020 Set 1 — Question 13
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Signals & Systems → Laplace & Z Transforms → Laplace Transform & ROC
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Question
Which of the following statements is true about the two sided Laplace transform?
Correct answer
(B) It has no poles for any bounded signal that is non-zero only inside a finite time interval.
Solution
(B) is true. If is bounded, , and non-zero only for , thenand is bounded on the finite interval for every . The integral therefore converges for all finite : the ROC is the entire -plane and is an entire function, which has no poles in the finite -plane.(A) is false. Example: for all has a Fourier transform (), but diverges for every (the part needs , the part needs ), so its two-sided Laplace transform does not exist.(C) is false. Example: gives , with one finite pole and no finite zero.(D) is false. Example: is itself a (single, unshifted) one-sided exponential, and its transform has a pole at ; a weighted sum of shifted terms gives , which in general keeps that pole.Answer: (B)
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