The 𝑍-transform of a discrete signal 𝑥[𝑛] is X(z) = {4z}/{ (z−\frac{1}{5}) (z−\frac{2}{3}) (z−3)} with ROC = R. Which one of the following statements is true?

Correct Answer is : Discrete-time Fourier transform of x[n] converges if R is 23<|.z|.<3

Discrete-time Fourier transform of x[n] converges if R is |𝑧| > 3

Discrete-time Fourier transform of x[n] converges if R is \frac{2}{3} < |.z|. < 3

Discrete-time Fourier transform of x[n] converges if R is such that x[n] is a left-sided sequence

Discrete-time Fourier transform of x[n] converges if R is such that x[n] is a right-sided sequence

GATE Electrical Engineering (EE) 2023 Solved Paper

Question : 19 of 65

Marks: +1, -0

The 𝑍-transform of a discrete signal 𝑥[𝑛] is

X(z) = \;{4z}/{\;(z−\;{1}/{5}) (z−\;{2}/{3}) (z−3)}

with ROC = R.
Which one of the following statements is true?

Discrete-time Fourier transform of x[n] converges if R is |𝑧| > 3
Discrete-time Fourier transform of x[n] converges if R is $\;{2}/{3} < |.z|. < 3$
Discrete-time Fourier transform of x[n] converges if R is such that x[n] is a left-sided sequence
Discrete-time Fourier transform of x[n] converges if R is such that x[n] is a right-sided sequence

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