The GATE Grind

GATE 2025 EC – Question 27

Networks, Signals and Systems · Discrete-time Signals · 1 mark · Multiple select

Let $x[n]$ be a discrete-time signal whose $z$-transform is $X(z)$.

Which of the following statements is/are TRUE?

  1. The discrete-time Fourier transform (DTFT) of $x[n]$ always exists
  2. The region of convergence (RoC) of $X(z)$ contains neither poles nor zeros
  3. The discrete-time Fourier transform (DTFT) exists if the region of convergence (RoC) contains the unit circle
  4. If $x[n]=\alpha\delta[n]$, where $\delta[n]$ is the unit impulse and $\alpha$ is a scalar, then the region of convergence (RoC) is the entire $z$-plane

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Show answer and explanation

Correct answer: (C) The discrete-time Fourier transform (DTFT) exists if the region of convergence (RoC) contains the unit circle; (D) If $x[n]=\alpha\delta[n]$, where $\delta[n]$ is the unit impulse and $\alpha$ is a scalar, then the region of convergence (RoC) is the entire $z$-plane

Explanation

(A) is false: the DTFT needs $\sum|x[n]|<\infty$, e.g. it does not exist for $u[n]$. (B) is false: the RoC never contains poles, but it can contain zeros (for an FIR signal it is the whole plane except possibly $z=0$, zeros included). (C) is true: the DTFT is $X(z)$ evaluated on $|z|=1$. (D) is true: $X(z)=\alpha$ converges for all $z$.