GATE Technical: GATE 2016 Electronics and Communication Engineering (Set 1) questions with solutions

Q1. Which one of the following is an eigen function of the class of all continuous-time, linear, time-invariant systems (u(t) denotes the unit-step function)?

1. (A) e^(jω₀ t)u(t)
2. (B) cos(ω₀ t)
3. (C) e^(jω₀ t)
4. (D) sin(ω₀ t)

Answer: (C) e^(jω₀ t)

The function e^(j heta) is an eigenfunction of linear time-invariant systems because it maintains its form under linear operations and time shifts, resulting in a scaled version of itself, which is a key characteristic of eigenfunctions.

Q2. A continuous-time function x(t) is periodic with period T. The function is sampled uniformly with a sampling period Tₛ. In which one of the following cases is the sampled signal periodic?

1. (A) T = √(2) Tₛ
2. (B) T = 1.2 Tₛ
3. (C) Always
4. (D) Never

Answer: (B) T = 1.2 Tₛ

A uniformly sampled periodic signal is itself periodic (in n) iff the ratio T/Ts is rational. T=sqrt(2)Ts is irrational (not periodic); T=1.2Ts=6/5 is rational, so the sampled sequence is periodic. Answer is (B), not 'Never'.

Q3. Consider the sequence x[n] = aⁿ u[n] + bⁿ u[n], where u[n] denotes the unit-step sequence and 0 < |a| < |b| < 1. The region of convergence (ROC) of the z-transform of x[n] is

1. (A) |z| > |a|
2. (B) |z| > |b|
3. (C) |z| < |a|
4. (D) |a| < |z| < |b|

Answer: (B) |z| > |b|

The correct option is |z| > |b| because the z-transform converges for values of z outside the larger pole, which in this case is determined by the term bⁿ, as |b| is greater than |a| and both are less than 1.

Q4. Consider a two-port network with the transmission matrix: T = [[A, B],[C, D]]. If the network is reciprocal, then

1. T⁻¹ = T
2. T² = T
3. Determinant(T) = 0
4. Determinant(T) = 1

Answer: Determinant(T) = 1

In a reciprocal two-port network, the determinant of the transmission matrix must equal 1, which ensures that the power transfer characteristics are maintained in both directions, reflecting the symmetry of the network.

Q5. Consider the following statements for a metal oxide semiconductor field effect transistor (MOSFET): P: As channel length reduces, OFF-state current increases. Q: As channel length reduces, output resistance increases. R: As channel length reduces, threshold voltage remains constant. S: As channel length reduces, ON current increases. Which of the above statements are INCORRECT?

1. P and Q
2. P and S
3. Q and R
4. R and S

Answer: Q and R

Statements Q and R are incorrect because as the channel length of a MOSFET decreases, the output resistance typically decreases due to increased short-channel effects, and the threshold voltage often changes rather than remaining constant.

Q6. The block diagram of a frequency synthesizer consisting of a Phase Locked Loop (PLL) and a divide-by-N counter (comprising ÷2, ÷4, ÷8, ÷16 outputs) is sketched below. The synthesizer is excited with a 5 kHz signal (input). The free-running frequency of the PLL is set to 20 kHz. Assume that the commutator switch makes contacts repeatedly in the order 1-2-3-4. The corresponding frequencies synthesized are:

1. 10 kHz, 20 kHz, 40 kHz, 80 kHz
2. 20 kHz, 40 kHz, 80 kHz, 160 kHz
3. 80 kHz, 40 kHz, 20 kHz, 10 kHz
4. 160 kHz, 80 kHz, 40 kHz, 20 kHz

Answer: 160 kHz, 80 kHz, 40 kHz, 20 kHz

The correct option is right because the divide-by-N counter outputs frequencies that are derived from the PLL's free-running frequency of 20 kHz. By dividing this frequency by 1, 2, 4, and 8, we obtain the synthesized frequencies of 20 kHz, 40 kHz, 80 kHz, and 160 kHz, respectively, in the order specified.