GATE Technical: GATE 2019 questions with solutions

Q1. 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 the 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(s) = 1/(s³ + 3s² + 2s + 1)?

1. A = [[0,1,0],[0,0,1],[-1,-2,-3]] and C = [1 0 0]
2. A = [[0,1,0],[0,0,1],[-3,-2,-1]] and C = [1 0 0]
3. A = [[0,1,0],[0,0,1],[-1,-2,-3]] and C = [0 0 1]
4. A = [[0,1,0],[0,0,1],[-3,-2,-1]] and C = [0 0 1]

Answer: A = [[0,1,0],[0,0,1],[-1,-2,-3]] and C = [1 0 0]

This option correctly represents the system dynamics and output configuration needed to achieve the specified transfer function. The matrix A defines the system's characteristic polynomial, which matches the desired transfer function, while the output matrix C ensures that the output is taken from the appropriate state variable.

Q2. 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(t) = √E φ2(t), for what value of E would we obtain the same bit-error probability P_b?

1. 0
2. 1
3. 2
4. 3

Answer: 3

BER depends only on the squared distance between the two signal points. Antipodal p=phi1, q=-phi1 gives distance^2 = |2 phi1|^2 = 4. For orthogonal p=phi1, q=sqrt(E) phi2 the distance^2 = 1+E. Setting 1+E=4 gives E=3, not the stored E=1.

Q3. The quantum efficiency (η) and responsivity (R) at a wavelength λ (in μm) in a p-i-n photodetector are related by

1. R = η×λ/1.24
2. R = λ/(η×1.24)
3. R = 1.24×λ/η
4. R = 1.24/(η×λ)

Answer: R = η×λ/1.24

The correct option expresses the relationship between quantum efficiency and responsivity, indicating that responsivity is directly proportional to quantum efficiency and the wavelength, while being inversely proportional to a constant factor (1.24), which accounts for the conversion between energy and wavelength.