1
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 ______. Your input ____ 2 GATE ECE 2014 Set 3 Numerical +2 -0 Let $${X_1},\,{X_2},$$ and $${X_3}$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,\,1} \right]$$. The probability $$P\left\{ {{X_1} + {X_2} \le {X_3}} \right\}$$ is ___________ . Your input ____ 3 GATE ECE 2014 Set 2 Numerical +2 -0 The power spectral density of a real stationary random process X(t) is given by $${S_x}\left( f \right) = \left\{ {\matrix{ {{1 \over W},\left| f \right| \le W} \cr {0,\left| f \right| > W} \cr } } \right.$$$

The value of the expectation $$E\left[ {\pi X\left( t \right)X\left( {t - {1 \over {4W}}} \right)} \right]$$\$
is ---------------.

4
GATE ECE 2014 Set 1
+2
-0.6
Consider a random process $$X\left( t \right) = \sqrt 2 \sin \left( {2\pi t + \varphi } \right),$$ where the random phase $$\varphi$$ is uniformly distributed in the interval $$\left[ {0,\,\,2\pi } \right].$$ The auto - correlation $$E\left[ {X\left( {{t_1}} \right)X\left( {{t_2}} \right)} \right]$$ is
A
$$\cos \left( {2\pi \left( {{t_1} + {t_2}} \right)} \right)$$
B
$$\sin \left( {2\pi \left( {{t_1} - {t_2}} \right)} \right)$$
C
$$\sin \left( {2\pi \left( {{t_1} + {t_2}} \right)} \right)$$
D
$$\cos \left( {2\pi \left( {{t_1} - {t_2}} \right)} \right)$$
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