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Random Processes and Analog Communications(17)Information Theory and Digital Communications(18)

Random Processes and Analog Communications

17 Questions

GATE→2026→Communications→Random Processes and Analog Communications

Q1

MCQ[2 Marks]

Let X, N, Y and Z be random variables. The variables X and N are independent of each other. X is uniformly distributed between -1 and 1; N follows Normal distribution with zero mean and unity variance. Y and Z are defined as, $Y=X+N$ and $Z=X^{2}+N$ Which of the following pairs represents the values of correlation between X and Y and that between X and Z?

(A)

$1/3$ and 0

(B)

$1/3$ and $1/9$

(C)

$1/3$ and $1/3$

(D)

1 and 0

Explanation

GATE→2026→Communications→Random Processes and Analog Communications

Q2

MCQ[2 Marks]

Consider a real baseband signal $x(t)=e^{-2t}$ for t (in seconds) $\ge0.$ If 99% of energy of $x(t)$ lies within B Hz, then which of the following options is TRUE for the value of B?

(A)

$B>1~kHz$

(B)

$63/\pi~Hz<B<64/\pi~Hz$

(C)

126/π Hz < Β < 128/π Hz

(D)

$B<1~Hz$

Explanation

GATE→2026→Communications→Random Processes and Analog Communications

Q3

MSQ[2 Marks]

Consider a real, narrowband signal $x(t)=A(t)cos[2\pi f_{c}t+\theta(t)]$ where the maximum frequency components of $A(t)$ and $\theta(t)$ are $f_{M}$ and $f_{c}(=1000f_{M})$ respectively. Which of the following statements is/are correct for∞ $-\infty<t<\infty$?

(A)

$x(t)$ represents a PSK modulated signal for suitable choices of $A(t)$ and $\theta(t).$

(B)

$x(t)$ represents an amplitude modulated signal for suitable choices of $A(t)$ and $\theta(t)$ gegen

(C)

$x(t)$ represents a band-limited Gaussian noise process.

(D)

$x(t)$ never represents a narrowband FM signal.

Explanation

GATE→2025→Communications→Random Processes and Analog Communications

Q4

MCQ[1 Mark]

Consider a frequency-modulated (FM) signal $f(t) = A_c \cos(2\pi f_c t + 3\sin(2\pi f_1 t) + 4\sin(6\pi f_1 t))$, where $A_c$ and $f_c$ are, respectively, the amplitude and frequency (in Hz) of the carrier waveform. The frequency $f_1$ is in Hz, and assume that $f_c > 100 f_1$. The peak frequency deviation of the FM signal in Hz is

(A)

$15 f_1$

(B)

$12 f_1$

(C)

$4 f_1$

(D)

$2 f_1$

Explanation

GATE→2025→Communications→Random Processes and Analog Communications

Q5

MSQ[1 Mark]

Consider a message signal $m(t)$ which is bandlimited to $[-W, W]$, where W is in Hz. Consider the following two modulation schemes for the message signal: Double sideband-suppressed carrier (DSB-SC): $f_{DSB}(t) = A_c m(t) \cos(2\pi f_c t)$ Amplitude modulation (AM): $f_{AM}(t) = A_c (1 + \mu m(t)) \cos(2\pi f_c t)$ Here, $A_c$ and $f_c$ are the amplitude and frequency (in Hz) of the carrier, respectively. In the case of AM, $\mu$ denotes the modulation index. Consider the following statements: (i) An envelope detector can be used for demodulation in the DSB-SC scheme if $m(t) > 0$ for all t. (ii) An envelope detector can be used for demodulation in the AM scheme only if $m(t) > 0$ for all t.

(A)

(i) is TRUE

(B)

(i) is FALSE

(C)

(ii) is TRUE

(D)

(ii) is FALSE

Explanation

GATE→2025→Communications→Random Processes and Analog Communications

Q6

MCQ[2 Marks]

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.

(A)

(i) is TRUE and (ii) is FALSE

(B)

Both (i) and (ii) are TRUE

(C)

Both (i) and (ii) are FALSE

(D)

(i) is FALSE and (ii) is TRUE

Explanation

GATE→2024→Communications→Random Processes and Analog Communications

Q7

MCQ[1 Mark]

A white Gaussian noise $w(t)$ with zero mean and power spectral density $\frac{N_0}{2}$, when applied to a first-order RC low pass filter produces an output $n(t)$. At a particular time $t = t_k$, the variance of the random variable $n(t_k)$ is

(A)

$\frac{N_0}{4RC}$

(B)

$\frac{N_0}{2RC}$

(C)

$\frac{N_0}{RC}$

(D)

$\frac{2N_0}{RC}$

Explanation

GATE→2024→Communications→Random Processes and Analog Communications

Q8

NAT[1 Mark]

An amplitude modulator has output (in Volts) $s(t) = A\cos(400\pi t) + B\cos(360\pi t) + B\cos(440\pi t)$. The carrier power normalized to $1~\Omega$ resistance is 50 Watts. The ratio of the total sideband power to the total power is $1/9$. The value of B (in Volts, rounded off to two decimal places) is

Explanation

GATE→2024→Communications→Random Processes and Analog Communications

Q9

NAT[2 Marks]

Let $X(t) = A \cos(2\pi f_0 t + \theta)$ be a random process... The signal to quantization noise ratio (in dB, rounded off to two decimal places) at the output of an 8-bit uniform mid-rise quantizer is

Explanation

GATE→2023→Communications→Random Processes and Analog Communications

Q10

MCQ[1 Mark]

Let $m(t)$ be a strictly band-limited signal with bandwidth B and energy E. Assuming $\omega_0 = 10B$, the energy in the signal $m(t) \cos \omega_0 t$ is

(A)

$\frac{E}{4}$

(B)

$\frac{E}{2}$

(C)

E

(D)

2E

Explanation

GATE→2023→Communications→Random Processes and Analog Communications

Q11

MSQ[1 Mark]

For a real signal, which of the following is/are valid power spectral density/densities?

(A)

$S_x(\omega) = \frac{2}{9 + \omega^2}$

(B)

$S_x(\omega) = e^{-\omega^2} \cos^2 \omega$

(C)

Option C

(D)

Option D

Explanation

GATE→2023→Communications→Random Processes and Analog Communications

Q12

MCQ[2 Marks]

Let a frequency modulated (FM) signal $x(t) = A \cos(\omega_c t + k_f \int_{-\infty}^t m(\lambda)d\lambda)$, where $m(t)$ is a message signal of bandwidth W. It is passed through a non-linear system with output $y(t) = 2x(t) + 5(x(t))^2$. Let $B_T$ denote the FM bandwidth. The minimum value of $\omega_c$ required to recover $x(t)$ from $y(t)$ is

(A)

$B_T + W$

(B)

$\frac{3}{2} B_T$

(C)

$2B_T + W$

(D)

$\frac{5}{2} B_T$

Explanation

GATE→2023→Communications→Random Processes and Analog Communications

Q13

MCQ[2 Marks]

Let $x_1(t)$ and $x_2(t)$ be two band-limited signals having bandwidth $B = 4\pi \times 10^3$ rad/s each. In the figure below, the Nyquist sampling frequency, in rad/s required to sample $y(t)$, is

(A)

$20\pi \times 10^3$

(B)

$40\pi \times 10^3$

(C)

$8\pi \times 10^3$

(D)

$32\pi \times 10^3$

Explanation

GATE→2023→Communications→Random Processes and Analog Communications

Q14

NAT[2 Marks]

Let $X(t)$ be a white Gaussian noise with power spectral density $\frac{1}{2}$ W/Hz. If $X(t)$ is input to an LTI system with impulse response $e^{-t} u(t)$, the average power of the system output is W (rounded off to two decimal places).

Explanation

GATE→2022→Communications→Random Processes and Analog Communications

Q15

MCQ[1 Mark]

The frequency response $H(f)$ of a linear time-invariant system has magnitude as shown in the figure... Which one of the following combinations is true?

(A)

Statement I is correct, Statement II is correct

(B)

Statement I is correct, Statement II is incorrect

(C)

Statement I is incorrect, Statement II is correct

(D)

Statement I is incorrect, Statement II is incorrect

Explanation

GATE→2022→Communications→Random Processes and Analog Communications

Q16

MSQ[2 Marks]

A state transition diagram with states A, B, and C, and transition probabilities $p_1, p_2, \dots, p_7$ is shown in the figure. For this state diagram, select the statement(s) which is/are universally true.

(A)

$p_2 + p_3 = p_5 + p_6$

(B)

$p_1 + p_3 = p_4 + p_6$

(C)

$p_1 + p_4 + p_7 = 1$

(D)

$p_2 + p_5 + p_7 = 1$

Explanation

GATE→2022→Communications→Random Processes and Analog Communications

Q17

MSQ[2 Marks]

Consider an FM broadcast that employs the pre-emphasis filter response $H_{pe}(\omega) = 1 + \frac{j\omega}{\omega_0}$ where $\omega_0 = 10^4$ rad/sec. For the network shown in the figure to act as a corresponding de-emphasis filter, the appropriate pair(s) of (R, C) values is/are

(A)

$R = 1~\text{k}\Omega, C = 0.1~\mu\text{F}$

(B)

$R = 2~\text{k}\Omega, C = 1~\mu\text{F}$

(C)

$R = 1~\text{k}\Omega, C = 2~\mu\text{F}$

(D)

$R = 2~\text{k}\Omega, C = 0.5~\mu\text{F}$

Explanation

Information Theory and Digital Communications

18 Questions

GATE→2026→Communications→Information Theory and Digital Communications

Q18

MCQ[1 Mark]

Consider a discrete memoryless source with an alphabet of four source symbols. $s(t)$ is a multi-level $(-1,0,+1,+2)$ signal representing a long sequence of random symbols from the above source which is generating $10^{4}$ symbols per second. Which of the following options is the correct value of equivalent Nyquist bandwidth of $s(t)$?

(A)

10 kHz

(B)

64 kHz

(C)

5 kHz

(D)

20 kHz

Explanation

GATE→2026→Communications→Information Theory and Digital Communications

Q19

NAT[1 Mark]

A wireless digital transmission scheme is using 16-QAM over an additive white Gaussian noise channel and a maximum-likelihood receiver. Consider the information bit rate from source to be $4\times10^{6}$ bits per second. The minimum transmission bandwidth (in MHz) of the modulated signal necessary for optimum recovery of information at the receiver is ______ (rounded off to two decimal places)

Explanation

GATE→2026→Communications→Information Theory and Digital Communications

Q20

MCQ[2 Marks]

A QPSK modulated signal from an additive white Gaussian noise (AWGN) channel is received with an Eb/N = 8.4 dB at the input of a coherent QPSK demodulator. A maximum-likelihood reception method is used in the demodulator. Assume the complimentary error function $erfc(u)\cong[1/(u\sqrt{\pi})]exp(-u^{2})$. Which is the nearest bit error rate (BER) at the output of the demodulator?

(A)

$10^{-3}$

(B)

$10^{-4}$

(C)

$10^{-5}$

(D)

$10^{-6}$

Explanation

GATE→2026→Communications→Information Theory and Digital Communications

Q21

MSQ[2 Marks]

Let the relevant bandwidth (B) of a digital communication system be 1 MHz and $kT=-174~dBm/Hz$, where k is Boltzmann's constant and '7" is equivalent noise temperature of the receiver. The power (S) of signal received through an additive Gaussian channel is -80 dBm. Which of the following options is/are TRUE about Shannon capacity (C) of the channel?

(A)

$C=B$

(B)

$C=2B$

(C)

$C>3B$

(D)

$C<B$

Explanation

GATE→2026→Communications→Information Theory and Digital Communications

Q22

NAT[2 Marks]

The average bit error rate at the input of a (7, 4, 1) Hamming decoder is 0.10. The probability that the decoder will fail to decode a received word correctly is ______ (rounded off to two decimal places)

Explanation

GATE→2025→Communications→Information Theory and Digital Communications

Q23

MCQ[1 Mark]

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)?

(A)

(A)

(B)

(B)

(C)

(C)

(D)

(D)

Explanation

GATE→2025→Communications→Information Theory and Digital Communications

Q24

NAT[1 Mark]

The generator matrix of a $(6,3)$ binary linear block code is given by $G = \begin{bmatrix} 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 0 \end{bmatrix}$. The minimum Hamming distance $d_{min}$ between codewords equals ______ (answer in integer).

Explanation

GATE→2025→Communications→Information Theory and Digital Communications

Q25

MCQ[2 Marks]

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 $\alpha$?

(A)

$\frac{1}{3}$

(B)

1

(C)

$\frac{2}{3}$

(D)

$\frac{4}{3}$

Explanation

GATE→2024→Communications→Information Theory and Digital Communications

Q26

MCQ[1 Mark]

A digital communication system transmits through a noiseless bandlimited channel [-W, W]. The Nyquist condition that $X(f)$ must satisfy for zero intersymbol interference at the receiver is

(A)

Option A

(B)

Option B

(C)

Option C

(D)

Option D

Explanation

GATE→2024→Communications→Information Theory and Digital Communications

Q27

NAT[1 Mark]

A source transmits symbols from an alphabet of size 16. The value of maximum achievable entropy (in bits) is

Explanation

GATE→2024→Communications→Information Theory and Digital Communications

Q28

MCQ[2 Marks]

A source transmits a symbol s, taken from $\{-4, 0, 4\}$ with equal probability, over an additive white Gaussian noise channel. The received noisy symbol r is given by $r = s + w$ where noise w is zero mean with variance 4. Using $Q(x)$, the optimum symbol error probability is

(A)

$\frac{2}{3} Q(2)$

(B)

$\frac{4}{3} Q(1)$

(C)

$\frac{2}{3} Q(1)$

(D)

$\frac{4}{3} Q(2)$

Explanation

GATE→2024→Communications→Information Theory and Digital Communications

Q29

MCQ[2 Marks]

The information bit sequence {111010101} is to be transmitted by encoding with Cyclic Redundancy Check 4 (CRC-4) code, for which the generator polynomial is $C(x) = x^4 + x + 1$. The encoded sequence of bits is

(A)

{1110101011100}

(B)

{1110101011101}

(C)

{1110101011110}

(D)

{1110101010100}

Explanation

GATE→2023→Communications→Information Theory and Digital Communications

Q30

NAT[2 Marks]

The frequency of occurrence of 8 symbols (a-h) is shown in the table below. A symbol is chosen and it is determined by asking a series of "yes/no" questions which are assumed to be truthfully answered. The average number of questions when asked in the most efficient sequence, to determine the chosen symbol, is

Explanation

GATE→2022→Communications→Information Theory and Digital Communications

Q31

MSQ[1 Mark]

Let $H(X)$ denote the entropy of a discrete random variable X taking K possible distinct real values. Which of the following statements is/are necessarily true?

(A)

$H(X) \le \log_2 K$ bits

(B)

$H(X) \le H(2X)$

(C)

$H(X) \le H(X^2)$

(D)

$H(X) \le H(2^X)$

Explanation

GATE→2022→Communications→Information Theory and Digital Communications

Q32

NAT[1 Mark]

A symbol stream contains alternate QPSK and 16-QAM symbols. If symbols from this stream are transmitted at the rate of 1 mega-symbols per second, the raw (uncoded) data rate is _______ mega-bits per second (rounded off to two decimal places).

Explanation

GATE→2022→Communications→Information Theory and Digital Communications

Q33

NAT[2 Marks]

The transition diagram of a discrete memoryless channel with three input symbols and three output symbols is shown in the figure... The parameter $\alpha$ lies in the interval $[0.25, 1]$. The value of $\alpha$ for which the capacity of this channel is maximized, is _______ (rounded off to two decimal places).

Explanation

GATE→2022→Communications→Information Theory and Digital Communications

Q34

NAT[2 Marks]

Consider communication over a memoryless binary symmetric channel using a $(7, 4)$ Hamming code. Each transmitted bit is received correctly with probability $(1-\epsilon)$ and flipped with probability $\epsilon$. For each codeword transmission, the receiver performs minimum Hamming distance decoding, and correctly decodes the message bits if and only if the channel introduces at most one bit error. For $\epsilon = 0.1$, the probability that a transmitted codeword is decoded correctly is _______ (rounded off to two decimal places).

Explanation

GATE→2022→Communications→Information Theory and Digital Communications

Q35

NAT[2 Marks]

Consider a channel over which either symbol $x_A$ or symbol $x_B$ is transmitted. Let the output of the channel Y be the input to a maximum likelihood (ML) detector at the receiver. The conditional probability density functions for Y given $x_A$ and $x_B$ are given... The probability of symbol error for this system is _______ (rounded off to two decimal places).

Explanation