GATE ECE 2025 | Question: 17

Question

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

Which of the following statements is/are TRUE?

• A. The discrete-time Fourier transform (DTFT) of $x[n]$ always exists
• B. The region of convergence $(\mathrm{RoC})$ of $X(z)$ contains neither poles nor zeros
• C. The discrete-time Fourier transform (DTFT) exists if the region of convergence $(\mathrm{RoC})$ 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 $(\mathrm{RoC})$ is the entire $Z$-plane