1
GATE ECE 2015 Set 2
+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
B
C
D
2
GATE ECE 2015 Set 2
+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
$$1 - pq$$
3
GATE ECE 2014 Set 3
+2
-0.6
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 defined as $$Y(t) = X(2t - 1)$$, the power spectral density $${S_y}\left( f \right)$$ is .
A
$${S_y}\left( f \right) = {1 \over 2}{S_x}\left( {{f \over 2}} \right){e^{ - j\pi f}}$$
B
$${S_y}\left( f \right) = {1 \over 2}{S_x}\left( {{f \over 2}} \right){e^{ - j\pi f/2}}$$
C
$${S_y}\left( f \right) = {1 \over 2}{S_x}\left( {{f \over 2}} \right)$$
D
$${S_y}\left( f \right) = {1 \over 2}{S_x}\left( {{f \over 2}} \right){e^{ - j2\pi f}}$$
4
GATE ECE 2014 Set 3
Numerical
+2
-0
A real band-limited random process $$X( t )$$ has two -sided power spectral density $${S_x}\left( f \right) = \left\{ {\matrix{ {{{10}^{ - 6}}\left( {3000 - \left| f \right|} \right)Watts/Hz} & {for\left| f \right| \le 3kHz} \cr 0 & {otherwise} \cr } } \right.$$\$

Where f is the frequency expressed in $$Hz$$. The signal $$X( t )$$ modulates a carrier cos $$16000$$ $$\pi t$$ and the resultant signal is passed through an ideal band-pass filter of unity gain with centre frequency of $$8kHz$$ and band-width of $$2kHz$$. The output power (in Watts) is ______.