# Random Signals and Noise · Communications · GATE ECE

Start Practice## Marks 1

GATE ECE 2024

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 ...

GATE ECE 2023

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

GATE ECE 2022

The frequency response H(f) of a linear time-invariant system has magnitude as shown in the figure.
Statement I : The system is necessarily a pure del...

GATE ECE 2017 Set 2

Consider the random process
x(t) = U + Vt.
Where U is a zero mean Gaussian random variable and V is a random variable uniformly distributed between ...

GATE ECE 2014 Set 4

If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier or future calls, th...

GATE ECE 2012

Two independent random variable X and Y are uniformly distributed in the interval [ - 1, 1]. The probability that max [X, Y] is less than 1/2 is

GATE ECE 2012

The power spectral density of a real process X(t) for positive frequencies is shown below. The value of $$E\,\left[ {{X^2}\,(t)} \right]$$ and $$E\,\l...

GATE ECE 2007

If E denotes expectation, the variance of a random variable X is given by

GATE ECE 2001

The PDF of a Gaussian random variable X is given by $${p_x}(x) = \,{1 \over {3\sqrt {2\pi } }}\,\exp \,[ - \,{(x - 4)^2}/18]$$.
The probability of th...

GATE ECE 1998

The amplitude spectrum of a Gaussian pulse is

GATE ECE 1998

The probability density function of the envelope of narrow band Gaussian noise is

GATE ECE 1998

The ACF of a rectangular pulse of duration T is

GATE ECE 1997

A probability density function is given by $$p(x) = \,K\,\,\exp \,\,( - \,{x^2}/2),\,\, - \,\infty \, < \,x < \,\infty $$.
The value of K shou...

GATE ECE 1995

For a narrow band noise with Gaussian quadrature and inphase components, the probability density function of its enveolope will be

## Marks 2

GATE ECE 2024

A source transmits a symbol $s$, taken from $\\{-4, 0, 4\\}$ with equal probability, over an additive white Gaussian noise channel. The received noisy...

GATE ECE 2024

Let $X(t) = A\cos(2\pi f_0 t+\theta)$ be a random process, where amplitude $A$ and phase $\theta$ are independent of each other, and are uniformly dis...

GATE ECE 2023

A random variable X, distributed normally as N(0, 1), undergoes the transformation Y = h(X), given in the figure. The form of the probability density ...

GATE ECE 2023

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(...

GATE ECE 2022

Consider a real valued source whose samples are independent and identically distributed random variables with the probability density function, f(x), ...

GATE ECE 2017 Set 1

Let $$X(t)$$ be a wide sense stationary random process with the power spectral density $${S_x}\left( f \right)$$ as shown in figure (a), where $$f$$ i...

GATE ECE 2016 Set 2

An information source generates a binary sequence $$\left\{ {{\alpha _n}} \right\}.{\alpha _n}$$ can take one of the two possible values −1 and +1 wit...

GATE ECE 2016 Set 2

Consider random process $$X(t) = 3V(t) - 8$$, where $$V$$ $$(t)$$ is a zero mean stationary random process with autocorrelation $${R_v}\left( \tau \r...

GATE ECE 2016 Set 3

A wide sense stationary random process $$X(t)$$ passes through the $$LTI$$ system shown in the figure. If the autocorrelation function of $$X(t)$$ is...

GATE ECE 2016 Set 1

An antenna pointing in a certain direction has a noise temperature of 50K. The ambient
temperature is 290K. The antenna is connected to a pre-amplifie...

GATE ECE 2015 Set 2

A zero mean white Gaussian noise having power spectral density $${{{N_0}} \over 2}$$ is passed through an $$ LTI $$
filter whose impulse response $$h(...

GATE ECE 2015 Set 2

$$\mathop {\left\{ {{X_n}} \right\}}\nolimits_{n = - \infty }^{n = \infty } $$ is an independent and identically distributed (i.i.d) random process w...

GATE ECE 2015 Set 2

Let $$X \in \left\{ {0,1} \right\}$$ and $$Y \in \left\{ {0,1} \right\}$$ be two independent binary random variables.
If $$P\left( {X\,\, = 0} \right)...

GATE ECE 2015 Set 3

A random binary wave $$y(t)$$ is given by
$$$y\left( t \right) = \sum\limits_{n = - \infty }^\infty {{X_n}p\left( {t - nT - \phi } \right)} $$$
whe...

GATE ECE 2014 Set 3

Let $$X(t)$$ be a wide sense stationary $$(WSS)$$ random procfess with power spectral density $${S_x}\left( f \right)$$. If $$Y(t)$$ is the process de...

GATE ECE 2014 Set 3

A real band-limited random process $$X( t )$$ has two -sided power spectral density
$$${S_x}\left( f \right) = \left\{ {\matrix{
{{{10}^{ - 6}}\lef...

GATE ECE 2014 Set 3

Let $${X_1},\,{X_2},$$ and $${X_3}$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,\,1} \ri...

GATE ECE 2014 Set 2

The power spectral density of a real stationary random process X(t) is given by
$$${S_x}\left( f \right) = \left\{ {\matrix{
{{1 \over W},\left| f...

GATE ECE 2014 Set 1

Consider a random process $$X\left( t \right) = \sqrt 2 \sin \left( {2\pi t + \varphi } \right),$$ where the random phase $$\varphi $$ is uni...

GATE ECE 2014 Set 1

Let $$Q\left( {\sqrt y } \right)$$ be the BER of a BPSK system over an AWGN channel with two - sided noise power spectral density N0/2. The parameter ...

GATE ECE 2013

Bits 1 and 0 are transmitted with equal probability. At the receiver, the pdf of the respective received signals for both bits are as shown below.
...

GATE ECE 2013

Bits 1 and 0 are transmitted with equal probability. At the receiver, the pdf of the respective received signals for both bits are as shown below.
...

GATE ECE 2011

X(t) is a stationary random process with autocorrelation function Rx$$\left( \tau \right)$$= exp$$\left( { - \pi {\tau ^2}} \right)$$. This process i...

GATE ECE 2010

X(t) is a stationary process with the power spectral density Sx(f) > 0 for all f. The process is passed through a system shown below.
Let Sy(f...

GATE ECE 2008

Noise with double-sided power spectral density of K over all frequencies is passed through a RC low pass filter with 3-dB cut-off frequency of fc. The...

GATE ECE 2006

A zero-mean white Gaussian noise is passed through an ideal low-pass filter of bandwidth 10 kHz. The output is then uniformly sampled with sampling pe...

GATE ECE 2006

The following question refer to wide sense stationary stochastic process:
It is desired to generate a stochastic process (as voltage process) wi...

GATE ECE 2006

The following question refer to wide sense stationary stochastic process:
The parameters of the system obtained in Q. 12 would be

GATE ECE 2005

An output of a communication channel is a random variable 'V' with the probability density function as shown in the figure. The mean square value of '...

GATE ECE 2005

Noise with uniform power spectral density of N0 W/Hz is passed through a filter H(ω ) = 2exp (-jωtd) followed by an ideal low pass filter of bandwidt...

GATE ECE 2004

A 1 mW video signal having a bandwidth of 100 MHz is transmitted to a receiver through a cable that has 40 dB loss. If the effective one-sided noise s...

GATE ECE 2002

If the variance $$\sigma _d^2$$ of d(n) = x(n - 1) is one-tenth the variance $$\sigma _x^2$$ of a stationary zero-mean discrete-time signal x(n), then...

GATE ECE 1992

For a random variable 'X' following the probability density function, p (x), shown in figure, the mean and the variance are, respectively.
...

GATE ECE 1991

Two resistors $$\,{R_1}$$ and $$\,{R_2}$$ (in ohms) at temperatures $${T_1}{}^ \circ K$$ and $${T_2}{}^ \circ K$$ respectively, are connected in serie...

GATE ECE 1989

Zero mean Gaussian noise of variance N is applied to a half wave rectifier. The mean squared value of the rectifier output will be:

GATE ECE 1989

A part of a communication system consists of an amplifier of effective noise temperature, $$Te = \,\,21\,\,{}^ \circ K\,$$, and a gain of 13 dB, follo...

GATE ECE 1987

White Gaussian noise is passed through a linear narrow band filter. The probability density function of the envelope of the noise at the filter outpu...

GATE ECE 1987

The variance of a random variable X is $$\sigma _x^2\,.$$ Then the variance of - kx (where k is a positive constant ) is

## Marks 4

GATE ECE 1988

White Gaussian noise with zero mean and double - sided power spectral density $$\eta /2$$ is the input $$x(t)$$ to a linear system with impulse respon...