Random Signals and Noise · Communications · GATE ECE

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...

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

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 ...

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...

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

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...

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

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...

The amplitude spectrum of a Gaussian pulse is

The ACF of a rectangular pulse of duration T is

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

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

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

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...

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...

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...

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...

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...

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

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

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)...

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...

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...

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

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

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...

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

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 ...

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. ...

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. ...

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

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...

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...

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...

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

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

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 '...

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...

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...

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...

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

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...

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

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...

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...

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

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...