GATE IN 2023 Question Paper (11-Feb-2023) (Shift-1)

The impulse response of an LTI system is $h(t)=\delta (t)+0.5\delta (t-4)$, where $\delta (t)$ is the continuous-time unit impulse signal. If the input signal $x(t)=\cos (\frac{7\pi }{4}t)$, the output is ______.

The Laplace transform of the continuous-time signal $x(t)=e^{-3t}u(t-5)$ is ______, where $u(t)$ denotes the continuous-time unit step signal.

In a p-i-n photodiode, a pulse of light containing $8\times 10^{12}$ incident photons at wavelength ${\lambda }_0=1.55\text{ µm}$ gives rise to an average $4\times 10^{12}$ electrons collected at the terminals of the device. The quantum efficiency of the photodiode at this wavelength is ______ %.

Let $f(z)=j\frac{1-z}{1+z}$, where z denotes a complex number and j denotes $\sqrt{-1}$. The inverse function $f^{-1}(z)$ maps the real axis to the ______.

The simplified form of the Boolean function $F(W,X,Y,Z)=\Sigma (4,5,10,11,12,13,14,15)$ with the minimum number of terms and smallest number of literals in each term is ________.