1
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).

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2
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.

GATE ECE 2022 Communications - Random Signals and Noise Question 7 English

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).

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3
GATE ECE 2017 Set 1
MCQ (Single Correct Answer)
+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 $$y(t)$$.

GATE ECE 2017 Set 1 Communications - Random Signals and Noise Question 28 English 1 GATE ECE 2017 Set 1 Communications - Random Signals and Noise Question 28 English 2

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 2
Numerical
+2
-0
An information source generates a binary sequence $$\left\{ {{\alpha _n}} \right\}.{\alpha _n}$$ can take one of the two possible values −1 and +1 with equal probability and are statistically independent and identically distributed. This sequence is pre-coded to obtain another sequence $$\left\{ {{\beta _n}} \right\},$$ as $${\beta _n} = {\alpha _n} + k{\mkern 1mu} {\alpha _{n - 3}}$$ . The sequence $$\left\{ {{\beta _n}} \right\}$$ is used to modulate a pulse $$g(t)$$ to generate the baseband signal

$$x\left( t \right) = \sum\limits_{n = - \infty }^\infty {{\beta _n}g\left( {t - nT} \right),} $$ where $$g\left( t \right) = \left\{ {\matrix{ {1,} & {0 \le t \le T} \cr 0 & {otherwise} \cr } } \right.$$

If there is a null at $$f = {1 \over {3T}}$$ in the power spectral density of $$X(t)$$, then $$k$$ is _________.

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