GATE EC 2019 Question Paper (09-Feb-2019) (Shift 1)

Consider a causal second-order system with the transfer function $G\left(s\right)=\frac{1}{1+2s+s^2}$ with a unit-step $R\left(s\right)=\frac{1}{s}$ as an input. Let C(s) be the corresponding output. The time taken by the system output c(t) to reach 94% of its steady-state value $\underset{t\to \infty }{\lim }c\left(t\right)$, rounded off to two decimal places, is

The block diagram of a system is illustrated in the figure shown, where X(s) is the input and Y(s) is the output. The transfer function $H\left(s\right)=\frac{Y\left(s\right)}{X\left(s\right)}$ is

Let the state-space representation of an LTI system be ẋ(t) = A x(t) + B u(t), y(t) = C x(t) + d u(t) where A, B, C are matrices, d is a scalar, u(t) is the input to the system, and y(t) is its output. Let B = [0 0 1]${}^T$ and d = 0. Which one of the following options for A and C will ensure that the transfer function of this LTI system is $H\left(s\right)=\frac{1}{s^3+3s^2+2s+1}=?$

A single bit, equally likely to be 0 and 1, is to be sent across an additive white Gaussian noise (AWGN) channel with power spectral density N0/2. Binary signaling, with 0 ↦ p(t) and 1 ↦ q(t), is used for the transmission, along with an optimal receiver that minimizes the bit-error probability. Let φ₁ (t), φ₂(t) form an orthonormal signal set. If we choose p(t) = φ₁ (t) and q(t) = −φ₁ (t), we would obtain a certain bit-error probability P${}_b$. If we keep p(t) = φ₁(t), but take $q\left(t\right)=\sqrt{E}$ φ₂ (t) for what value of E would we obtain the same bit-error probability P${}_b$?

The quantum efficiency (η) and responsivity (R) at a wavelength λ (in μm) in a p-i-n photodetector are related by