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GATE 2023 EE – Question 19

Signals and Systems · Applications of Fourier Transform for continuous and discrete time signals, Laplace Transform and Z transform · 1 mark · Multiple choice

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

$$X(z)=\frac{4z}{\left(z-\frac15\right)\left(z-\frac23\right)(z-3)}\quad\text{with ROC}=R.$$

Which one of the following statements is true?

  1. Discrete-time Fourier transform of $x[n]$ converges if $R$ is $|z|>3$
  2. Discrete-time Fourier transform of $x[n]$ converges if $R$ is $\dfrac23<|z|<3$
  3. Discrete-time Fourier transform of $x[n]$ converges if $R$ is such that $x[n]$ is a left-sided sequence
  4. Discrete-time Fourier transform of $x[n]$ converges if $R$ is such that $x[n]$ is a right-sided sequence

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Correct answer: (B) Discrete-time Fourier transform of $x[n]$ converges if $R$ is $\dfrac23<|z|<3$

Explanation

The DTFT exists when the ROC contains the unit circle $|z|=1$. The poles are at $\frac15$, $\frac23$ and 3. The only ROC that includes $|z|=1$ is the annulus $\frac23<|z|<3$ (a two-sided sequence).