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 Z-transform of a discrete signal is
with ROC = R .
Which one of the following statements is true?
with ROC = R .
Which one of the following statements is true?
Correct answer
(B) Discrete-time Fourier transform of x[n] converges if R is (1)/(2) < z < 3
Solution
The Z-transform is given as .
The poles of are at and .
For the Discrete-time Fourier Transform (DTFT) of to converge, the Region of Convergence (ROC) of its Z-transform must include the unit circle ().Let's analyze the possible ROCs based on the poles:
(A) ROC is - Does not include the unit circle.
(B) ROC is - Includes the unit circle.
(C) is a left-sided sequence, ROC is - Does not include the unit circle.
(D) is a right-sided sequence, ROC is - Does not include the unit circle.Thus, the correct statement is that the Discrete-time Fourier transform of converges if R is .
The poles of are at and .
For the Discrete-time Fourier Transform (DTFT) of to converge, the Region of Convergence (ROC) of its Z-transform must include the unit circle ().Let's analyze the possible ROCs based on the poles:
1.If is a right-sided sequence, the ROC would be . This ROC does not include the unit circle.
2.If is a left-sided sequence, the ROC would be . This ROC does not include the unit circle.
3.If is a two-sided sequence, the ROC would be the annulus . This ROC includes the unit circle ().
Therefore, for the DTFT of to converge, the ROC must be .Comparing this with the given options:(A) ROC is - Does not include the unit circle.
(B) ROC is - Includes the unit circle.
(C) is a left-sided sequence, ROC is - Does not include the unit circle.
(D) is a right-sided sequence, ROC is - Does not include the unit circle.Thus, the correct statement is that the Discrete-time Fourier transform of converges if R is .
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