GATE EC 2025 Set 1 — Question 27

Let's analyze each statement:(A) The discrete-time Fourier transform (DTFT) of $x[n]$ always exists.
This statement is FALSE. The DTFT of a discrete-time signal $x[n]$ exists if and only if the signal is absolutely summable, i.e., $\sum_{n=-\infty}^{\infty} |x[n]| < \infty$. For example, the unit step signal $x[n] = u[n]$ is not absolutely summable, and thus its DTFT does not exist in the classical sense (it exists as a generalized function involving impulse functions).(B) The region of convergence (RoC) of $X(z)$ contains neither poles nor zeros.
This statement is FALSE. The RoC of a z-transform $X(z)$ is defined as the set of all $z$ values for which $X(z)$ converges. By definition, poles are values of $z$ for which $X(z)$ becomes infinite, so the RoC can never contain poles. However, zeros are values of $z$ for which $X(z) = 0$. If $X(z)$ is finite at a zero, then that zero is part of the RoC. For example, for $X(z) = z$, the zero is at $z=0$, and the RoC is the entire z-plane except $z=\infty$, which includes $z=0$.(C) The discrete-time Fourier transform (DTFT) exists if the region of convergence (RoC) contains the unit circle.
This statement is TRUE. The DTFT $X(e^{j\omega})$ is obtained by evaluating the z-transform $X(z)$ on the unit circle in the z-plane (i.e., for $|z|=1$). If the RoC of $X(z)$ includes the unit circle, it means that $X(z)$ converges for all $z$ such that $|z|=1$, and therefore the DTFT exists.(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.
This statement is TRUE. The z-transform of $x[n] = \alpha\delta[n]$ is given by:
$X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n} = \sum_{n=-\infty}^{\infty} \alpha\delta[n]z^{-n}$
Since $\delta[n]$ is non-zero only at $n=0$, we have:
$X(z) = \alpha\delta[0]z^{-0} = \alpha \cdot 1 \cdot 1 = \alpha$
The z-transform $X(z) = \alpha$ is a constant and converges for all finite values of $z$. Therefore, its RoC is the entire z-plane (excluding $z=\infty$ if $\alpha \neq 0$, but typically for finite signals, the RoC is considered the entire z-plane). For a finite-duration signal like an impulse, the RoC is indeed the entire z-plane.Thus, statements (C) and (D) are TRUE.The final answer is $\boxed{\text{C, D}}$