1
GATE ECE 2010
+2
-0.6
X(t) is a stationary process with the power spectral density Sx(f) > 0 for all f. The process is passed through a system shown below.

Let Sy(f) be the power spectral density of Y(t). Which one of the following statements is correct?

A
Sy(f) > 0 for all f
B
Sy(f) > 0 for $$\left| f \right|$$ > 1 kH
C
Sy(f) > 0 for f = nf0, f0 = 2kHz, n any integer
D
Sy(f) > 0 for f = (2n + 1)f0, f0 = 1 kHz, n any integer
2
GATE ECE 2010
+1
-0.3
Consider the pulse shape s(t) as shown. The impulse response h(t) of the filter matched to this pulse is
A
B
C
D
3
GATE ECE 2010
+2
-0.6
Consider a base band binary PAM receiver shown below. The additive channel noise $$n(t)$$ is white with power spectral density $${S_N}\left( f \right) = {N_0}/2 = {10^{ - 20}}$$ $$W/Hz$$. The low-pass filter is ideal with unity gain and cut -off frequency $$1MHz$$. Let $${Y_k}$$ represent the random variable $$y\left( {{t_k}} \right)$$.
$${Y_k} = {N_k}$$ if transmitted bit $${b_k} = 0$$
$${Y_k} = a + {N_k}$$ if transmitted bit $${b_k} = 1$$
Where $${b_k} = 0$$ represents the noise sample value. The noise sample has a probability density function, $${P_{{N_k}}}\left( n \right)\,\,\,\,\,\,\, = 0.5\alpha {e^{ - \alpha \left| n \right|}}$$ (This has mean zero and variance $$2/{\alpha ^2}$$). Assume transmitted bits to be equiprobable and threshold $$z$$ is set to $$a/2 = {10^{ - 6}}V$$.

The value of the parameter $$\alpha$$( in V-1 ) is

A
$${10^{10}}$$
B
$${10^{7}}$$
C
$$1.414 \times {10^{ - 10}}$$
D
$$2 \times {10^{ - 20}}$$
4
GATE ECE 2010
+2
-0.6
Consider a base band binary PAM receiver shown below. The additive channel noise $$n(t)$$ is white with power spectral density $${S_N}\left( f \right) = {N_0}/2 = {10^{ - 20}}$$ $$W/Hz$$. The low-pass filter is ideal with unity gain and cut -off frequency $$1MHz$$. Let $${Y_k}$$ represent the random variable $$y\left( {{t_k}} \right)$$.
$${Y_k} = {N_k}$$ if transmitted bit $${b_k} = 0$$
$${Y_k} = a + {N_k}$$ if transmitted bit $${b_k} = 1$$
Where $${b_k} = 0$$ represents the noise sample value. The noise sample has a probability density function, $${P_{{N_k}}}\left( n \right)\,\,\,\,\,\,\, = 0.5\alpha {e^{ - \alpha \left| n \right|}}$$ (This has mean zero and variance $$2/{\alpha ^2}$$). Assume transmitted bits to be equiprobable and threshold $$z$$ is set to $$a/2 = {10^{ - 6}}V$$.

The probability of bit error is

A
$$0.5 \times {e^{ - 3.5}}$$
B
$$0.5 \times {e^{ - 5}}$$
C
$$0.5 \times {e^{ - 7}}$$
D
$$0.5 \times {e^{ - 10}}$$
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