1
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
$${S_y}\left( f \right) = {1 \over 2}{S_x}\left( {{f \over 2}} \right){e^{ - j2\pi f}}$$
2
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 ____ 3 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 ____ 4 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 ---------------.