GATE EC 2025 Set 1 Q28 — Continuous-time Signals

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 $\mathrm{rad}$ / $\mathrm{sec}$.

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 $\_\_\_\_\_\_\_$