GATE EE 2023 Set 1 — Question 19
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Signals & Systems → Laplace & Z Transforms → Z Transform & ROC
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Question
The -transform of a discrete signal isWhich one of the following statements is true?
Correct answer
(B) Discrete-time Fourier transform of x[n] converges if R is (2)/(3) < z < 3
Solution
The poles ofare at , and .The discrete-time Fourier transform of is evaluated on the unit circle, . It therefore converges only if the region of convergence contains the unit circle .The pole magnitudes divide the -plane into the following possible regions of convergence:
1. — corresponds to a left-sided ; does not contain .
2. — two-sided; does not contain .
3. — two-sided; contains .
4. — corresponds to a right-sided ; does not contain .
Hence the DTFT exists only for the annular region , which rules out options (A), (C) and (D).Answer: (B) the discrete-time Fourier transform of converges if is Continue learning with Success Tracker
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