List of top Communication Systems Questions asked in GATE Electronics and Communication Engineering

X and Y are Bernoulli random variables taking values in \( \{0,1\} \). The joint probability mass function of the random variables is given by:
P(X = 0, Y = 0) = 0.06, \( \quad P(X = 0, Y = 1) = 0.14 \), \( \quad P(X = 1, Y = 0) = 0.24 \), \( \quad P(X = 1, Y = 1) = 0.56 \).
The mutual information \( I(X; Y) \) is (rounded off to two decimal places).

A positive-edge-triggered sequential circuit is shown below. There are no timing violations in the circuit. Input \( P_0 \) is set to logic ‘0’ and \( P_1 \) is set to logic ‘1’ at all times. The timing diagram of the inputs \( SEL \) and \( S \) are also shown below. The sequence of output \( Y \) from time \( T_0 \) to \( T_3 \) is _________. 

The identical MOSFETs \( M_1 \) and \( M_2 \) in the circuit given below are ideal and biased in the saturation region. \( M_1 \) and \( M_2 \) have a transconductance \( g_m \) of 5 mS. The input signals (in Volts) are: \[ V_1 = 2.5 + 0.01 \sin \omega t, \quad V_2 = 2.5 - 0.01 \sin \omega t. \] The output signal \( V_3 \) (in Volts) is _________. 


 

Consider a real-valued random process \[ f(t) = \sum_{n=1}^{N} a_n p(t - nT), \] where \( T > 0 \) and \( N \) is a positive integer. Here, \( p(t) = 1 \) for \( t \in [0, 0.5T] \) and 0 otherwise. The coefficients \( a_n \) are pairwise independent, zero-mean unit-variance random variables. Read the following statements about the random process and choose the correct option.
  • (i) The mean of the process \( f(t) \) is independent of time \( t \).
  • (ii) The autocorrelation function \( E[f(t)f(t + \tau)] \) is independent of time \( t \) for all \( \tau \).
(Here, \( E[.] \) is the expectation operation.)

All the components in the bandpass filter given below are ideal. The lower -3 dB frequency of the filter is 1 MHz. The upper -3 dB frequency (in MHz, rounded off to the nearest integer) is _________. 


 

The generator matrix of a \( (6,3) \) binary linear block code is given by 

The minimum Hamming distance \( d_{{min}} \) between codewords equals (answer in integer).

The Nyquist plot of a system is given in the figure below. Let \( \omega_P, \omega_Q, \omega_R, \) and \( \omega_S \) be the positive frequencies at the points \( P, Q, R, \) and \( S \), respectively. Which one of the following statements is TRUE?

Consider an additive white Gaussian noise (AWGN) channel with bandwidth \( W \) and noise power spectral density \( \frac{N_0}{2} \). Let \( P_{{av}} \) denote the average transmit power constraint. Which one of the following plots illustrates the dependence of the channel capacity \( C \) on the bandwidth \( W \) (keeping \( P_{{av}} \) and \( N_0 \) fixed)?
The random variable \( X \) takes values in \( \{-1, 0, 1\} \) with probabilities \[ P(X = -1) = P(X = 1) = \alpha \quad {and} \quad P(X = 0) = 1 - 2\alpha, \quad 0<\alpha<\frac{1}{2}. \] Let \( g(\alpha) \) denote the entropy of \( X \) (in bits), parameterized by \( \alpha \). Which of the following statements is/are TRUE?
A 10-bit analog-to-digital converter (ADC) has a sampling frequency of 1 MHz and a full scale voltage of 3.3 V. For an input sinusoidal signal with frequency 500 kHz, the maximum SNR (in dB, rounded off to two decimal places) and the data rate (in Mbps) at the output of the ADC are _________, respectively.
A source transmits symbol \( S \) that takes values uniformly at random from the set \( \{-2, 0, 2\} \). The receiver obtains \( Y = S + N \), where \( N \) is a zero-mean Gaussian random variable independent of \( S \). The receiver uses the maximum likelihood decoder to estimate the transmitted symbol \( S \). Suppose the probability of symbol estimation error \( P_e \) is expressed as follows: \[ P_e = \alpha P(N>1), \] where \( P(N>1) \) denotes the probability that \( N \) exceeds 1. What is the value of \( \alpha \)?