1
GATE ECE 2024
+2
-1.33

A source transmits a symbol $s$, taken from $\\{-4, 0, 4\\}$ with equal probability, over an additive white Gaussian noise channel. The received noisy symbol $r$ is given by $r = s + w$, where the noise $w$ is zero mean with variance 4 and is independent of $s$.

Using $Q(x) = \frac{1}{\sqrt{2\pi}} \int\limits_{x}^{\infty} e^{-\frac{t^{2}}{2}} dt$, the optimum symbol error probability is _______.

A

$\frac{2}{3} Q(2)$

B

$\frac{4}{3} Q(1)$

C

$\frac{2}{3} Q(1)$

D

$\frac{4}{3} Q(2)$

2
GATE ECE 2024
Numerical
+2
-1.33

Let $X(t) = A\cos(2\pi f_0 t+\theta)$ be a random process, where amplitude $A$ and phase $\theta$ are independent of each other, and are uniformly distributed in the intervals $[-2,2]$ and $[0, 2\pi]$, respectively. $X(t)$ is fed to an 8-bit uniform mid-rise type quantizer.

Given that the autocorrelation of $X(t)$ is $R_X(\tau) = \frac{2}{3} \cos(2\pi f_0 \tau)$, the signal to quantization noise ratio (in dB, rounded off to two decimal places) at the output of the quantizer is _________.

3
GATE ECE 2017 Set 1
+2
-0.6
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 $$f$$ is in Hertz $$(Hz)$$. The random process $$X(t)$$ is input to an ideal low pass filter with the frequency response $$H\left( f \right) = \left\{ {\matrix{ {1,} & {\left| f \right| \le {1 \over 2}Hz} \cr {0,} & {\left| f \right| > {1 \over 2}Hz} \cr } } \right.$$\$

As shown in Figure (b). The output of the low pass filter is $$y(t)$$.

Let $$E$$ be the expectation operator and consider the following statements :
$$\left( {\rm I} \right)$$ $$E\left( {X\left( t \right)} \right) = E\left( {Y\left( t \right)} \right)$$
$$\left( {{\rm I}{\rm I}} \right)$$ $$\,\,\,\,\,\,\,\,E\left( {{X^2}\left( t \right)} \right) = E\left( {{Y^2}\left( t \right)} \right)$$
$$\left( {{\rm I}{\rm I}{\rm I}} \right)\,$$ $$\,\,\,\,\,\,E\left( {{Y^2}\left( t \right)} \right) = 2$$

Select the correct option:

A
only $${\rm I}$$ is true
B
only $${\rm I}$$$${\rm I}$$ and $${\rm I}$$$${\rm I}$$$${\rm I}$$ are true
C
only $${\rm I}$$ and $${\rm I}$$$${\rm I}$$ are true
D
only $${\rm I}$$ and $${\rm I}$$$${\rm I}$$$${\rm I}$$ are true
4
GATE ECE 2016 Set 3
+2
-0.6
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 $${R_x}\left( \tau \right),$$ then the autocorrelation function $${R_x}\left( \tau \right),$$ of the output $$Y(t)$$ is equal to
A
$$2{R_X}\left( \tau \right) + {R_X}\left( {\tau - {T_0}} \right) + {R_X}\left( {\tau + {T_0}} \right)$$
B
$$2{R_X}\left( \tau \right) - {R_X}\left( {\tau - {T_0}} \right) - {R_X}\left( {\tau + {T_0}} \right)$$
C
$$2{R_X}\left( \tau \right) + 2{R_X}\left( {\tau - 2{T_0}} \right)$$
D
$$2{R_X}\left( \tau \right) - 2{R_X}\left( {\tau - 2{T_0}} \right)$$
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