1
GATE ECE 2023
+2
-0.67

A random variable X, distributed normally as N(0, 1), undergoes the transformation Y = h(X), given in the figure. The form of the probability density function of Y is

(In the options given below, a, b, c are non-zero constants and g(y) is piece-wise continuous function)

A
$$a\delta (y - 1) + b\delta (y + 1) + g(y)$$
B
$$a\delta (y + 1) + b\delta (y) + c\delta (y - 1) + g(y)$$
C
$$a\delta (y + 2) + b\delta (y) + c\delta (y - 2) + g(y)$$
D
$$a\delta (y + 2) + b\delta (y - 2) + g(y)$$
2
GATE ECE 2023
Numerical
+2
-0.67

Let X(t) be a white Gaussian noise with power spectral density $$\frac{1}{2}$$W/Hz. If X(t) is input to an LTI system with impulse response $$e^{-t}u(t)$$. The average power of the system output is ____________ W (rounded off to two decimal places).

3
GATE ECE 2022
Numerical
+2
-0

Consider a real valued source whose samples are independent and identically distributed random variables with the probability density function, f(x), as shown in the figure.

Consider a 1 bit quantizer that maps positive samples to value $$\alpha$$ and others to value $$\beta$$. If $$\alpha$$* and $$\beta$$* are the respective choices for $$\alpha$$ and $$\beta$$ that minimize the mean square quantization error, then ($$\alpha$$* $$-$$ $$\beta$$*) = ___________ (rounded off to two decimal places).

4
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
EXAM MAP
Medical
NEET