1
GATE ECE 2015 Set 2
MCQ (Single Correct Answer)
+2
-0.6
A zero mean white Gaussian noise having power spectral density $${{{N_0}} \over 2}$$ is passed through an $$ LTI $$ filter whose impulse response $$h(t)$$ is shown in the figure. The variance of the filtered noise at $$t = 4$$ is GATE ECE 2015 Set 2 Communications - Random Signals and Noise Question 33 English
A
$${3 \over 2}{A^2}{N_0}$$
B
$${3 \over 4}{A^2}{N_0}$$
C
$${A^2}{N_0}$$
D
$${1 \over 2}{A^2}{N_0}$$
2
GATE ECE 2015 Set 2
MCQ (Single Correct Answer)
+2
-0.6
$$\mathop {\left\{ {{X_n}} \right\}}\nolimits_{n = - \infty }^{n = \infty } $$ is an independent and identically distributed (i.i.d) random process with $${X_n}$$ equally likely to be $$+1$$ or $$-1$$. $$\mathop {\left\{ {{Y_n}} \right\}}\nolimits_{n = - \infty }^{n = \infty } \,$$ is another random process obtained as $${Y_n} = {X_n} + 0.5{X_{n - 1}}.\,\,\,$$
The autocorrelation function of $$\mathop {\left\{ {{Y_n}} \right\}}\nolimits_{n = - \infty }^{n = \infty } $$, denoted by $${r_y}\left[ K \right],$$ is
A
GATE ECE 2015 Set 2 Communications - Random Signals and Noise Question 32 English Option 1
B
GATE ECE 2015 Set 2 Communications - Random Signals and Noise Question 32 English Option 2
C
GATE ECE 2015 Set 2 Communications - Random Signals and Noise Question 32 English Option 3
D
GATE ECE 2015 Set 2 Communications - Random Signals and Noise Question 32 English Option 4
3
GATE ECE 2015 Set 2
MCQ (Single Correct Answer)
+2
-0.6
Let $$X \in \left\{ {0,1} \right\}$$ and $$Y \in \left\{ {0,1} \right\}$$ be two independent binary random variables.

If $$P\left( {X\,\, = 0} \right)\,\, = p$$ and $$P\left( {Y\,\, = 0} \right)\,\, = q,$$ then $$P\left( {X + Y \ge 1} \right)$$ is equal to

A
$$pq + \left( {1 - p} \right)\left( {1 - q} \right)$$
B
$$pq$$
C
$$p\left( {1 - q} \right)$$
D
$$1 - pq$$
4
GATE ECE 2015 Set 3
Numerical
+2
-0
A random binary wave $$y(t)$$ is given by $$$y\left( t \right) = \sum\limits_{n = - \infty }^\infty {{X_n}p\left( {t - nT - \phi } \right)} $$$

where $$p(t) = u(t) - u(t - T)$$, $$u(t)$$ is the unit step function and $$\phi $$ is an independent random variable with uniform distribution in $$[0, T]$$. The sequence $$\left\{ {{X_n}} \right\}$$ consists of independent and identically distributed binary valued random variables with $$P\left\{ {{X_n} = + 1} \right\} = P\left\{ {{X_n} = - 1} \right\} = 0.5$$ for each $$n$$.

The value of the autocorrelation $${R_{yy}}\left( {{{3T} \over 4}} \right)\underline{\underline \Delta } E\left[ {y\left( t \right)y\left( {t - {{3T} \over 4}} \right)} \right]\,\,$$


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