1
GATE ECE 2025
MCQ (Single Correct Answer)
+1
-0.33

Consider a continuous-time, real-valued signal $f(t)$ whose Fourier transform $F(\omega)=$$\mathop f\limits_{ - \infty }^\infty $$ f(t) \exp (-j \omega t) d t$ exists.

Which one of the following statements is always TRUE?

A
$|F(\omega)| \leq \mathop f\limits_{ - \infty }^\infty|f(t)| d t$
B
$|F(\omega)|>\mathop f\limits_{ - \infty }^\infty|f(t)| d t$
C
$|F(\omega)| \leq \mathop f\limits_{ - \infty }^\infty f(t) d t$
D
$|F(\omega)| \geq \mathop f\limits_{ - \infty }^\infty f(t) d t$
2
GATE ECE 2025
MCQ (More than One Correct Answer)
+1
-0
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 (RoC) of $X(z)$ contains neither poles nor zeros
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
3
GATE ECE 2025
MCQ (Single Correct Answer)
+2
-0.67

Consider a continuous-time finite-energy signal $f(t)$ whose Fourier transform vanishes outside the frequency interval $\left[-\omega_c, \omega_c\right]$, where $\omega_c$ is in rad/sec.

The signal $f(t)$ is uniformly sampled to obtain $y(t)=f(t) p(t)$. Here

$$ p(t)=\sum_{n=-\infty}^{\infty} \delta\left(t-\tau-n T_s\right) $$

with $\delta(t)$ being the Dirac impulse, $T_s>0$, and $\tau>0$. The sampled signal $y(t)$ is passed through an ideal lowpass filter $h(t)=\omega_c T_s \frac{\sin \left(\omega_c t\right)}{\pi \omega_c t}$ with cutoff frequency $\omega_c$ and passband gain $T_s$.

The output of the filter is given by $\qquad$ .

A
$f(t)$ if $T_s<\pi / \omega_c$
B
$f(t-\tau)$ if $T_s<\pi / \omega_c$
C
$f(t-\tau)$ if $T_s<2 \pi / \omega_c$
D
$T_s f(t)$ if $T_s<2 \pi / \omega_c$
4
GATE ECE 2025
MCQ (More than One Correct Answer)
+2
-0

Let $f(t)$ be a periodic signal with fundamental period $T_0>0$. Consider the signal $y(t)=f(\alpha t)$, where $\alpha>1$.

The Fourier series expansions of $f(t)$ and $y(t)$ are given by

$$ f(t)=\sum\limits_{k = - \infty }^\infty c_k e^{j \frac{2 \pi}{T_0} k T} \text { and } y(t)=\sum\limits_{k = - \infty }^\infty d_k e^{j \frac{2 \pi}{T_0} \alpha k T} . $$

Which of the following statements is/are TRUE?

A
$c_k=d_k$ for all $k$
B
$y(t)$ is periodic with a fundamental period $\alpha T_0$
C
$c_k=d_k / \alpha$ for all $k$
D
$y(t)$ is periodic with a fundamental period $T_0 / \alpha$
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