GATE EC 2023 Question Paper (05-Feb-2023) (Shift 2)

The h-parameters of a two port network are shown below. The condition for the maximum small signal voltage gain $\frac{{\mathrm{V}}_{\mathrm{out}}}{{\mathrm{V}}_{\mathrm{s}}}$ is

Consider a discrete-time periodic signal with period 𝑁 = 5. Let the discrete-time Fourier series (DTFS) representation be $\mathrm{x}[\mathrm{n}]={\mathrm{\Sigma }}_{\mathrm{k}=0}^4{\mathrm{a}}_{\mathrm{k}}{\mathrm{e}}^{\frac{jk2\pi \mathrm{n}}{\mathrm{s}}}$, where 𝑎₀ = 1, 𝑎₁ = 3𝑗, 𝑎₂ = 2𝑗, 𝑎₃ = −2𝑗 and 𝑎₄ = −3𝑗. The value of the sum ${\mathrm{\Sigma }}_{\mathrm{n}=0}^4\mathrm{x}[\mathrm{n}]\sin \frac{4\pi \mathrm{n}}{5}$ is

Let an input 𝑥[𝑛] having discrete time Fourier transform

𝑋(𝑒${}^{\text{j𝛺}}$) = 1 − 𝑒${}^{-\text{j𝛺}}$ + 2𝑒${}^{-3\text{j𝛺}}$ be passed through an LTI system. The frequency response of the LTI system is $\mathrm{H}({\mathrm{e}}^{\mathrm{j\Omega })}=1-\frac{1}{2}{\mathrm{e}}^{-j2\Omega }$. The output 𝑦[𝑛] of the system is

Let 𝑥(𝑡) = 10 cos(10.5 𝑊𝑡) be passed through an LTI system having impulse response $\mathrm{h}(\mathrm{t})=\pi {\left(\frac{\sin \mathrm{Wt}}{\pi \mathrm{t}}\right)}^2\cos 10Wt$. The output of the system is

Let x₁(t) and x₂(t) be two band-limited signals having bandwidth 𝐵 = 4𝜋 × 10³ rad/s each. In the figure below, the Nyquist sampling frequency, in rad/s, required to sample y(t), is