Marks 1
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 $$.
The value of K shou...
For a narrow band noise with Gaussian quadrature and inphase components, the probability density function of its enveolope will be
Marks 2
Let $$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\,.$$ Then the variance of - kx (where k is a positive constant ) is
Marks 4
White Gaussian noise with zero mean and double - sided power spectral density $$\eta /2$$ is the input $$x(t)$$ to a linear system with impulse respon...