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QUESTION: 1

Consider a low-pass random process with a white-noise power spectral density as shown in fig.

Que: The auto correlation function R_{x}(τ) is

Solution:

QUESTION: 2

Consider a low-pass random process with a white-noise power spectral density as shown in fig.

Que: The power P_{X} is

Solution:

QUESTION: 3

If X(t) is a stationary process having a mean value E[X(t)] = 3 and autocorrelation function

The variance of random variable Y =

Solution:

QUESTION: 4

A random process is defined by X(t) = Acos(πt) where A is a gaussian random variable with zero mean and variance σ_{π}^{2}. The density function of X(0)

Solution:

QUESTION: 5

The two-level semi-random binary process is defined by X(t) = A or -A

where (n-1)T < t < nt and the levels A and -A occur with equal probability. T is a positive constant and

Que: The mean value E[X(t)] is

Solution:

QUESTION: 6

The two-level semi-random binary process is defined by X(t) = A or -A

where (n-1)T < t < nt and the levels A and -A occur with equal probability. T is a positive constant and

Que: The auto correlation R_{xx} = (t_{1} = 0.5T, t_{2} = 0.7 T) will be

Solution:

QUESTION: 7

A random process consists of three samples function X(t, s_{1} ) = 2, X(t, s_{2} ) = 2cos t_{1} and X(t, s_{3} ) = 3sint - each occurring with equal probability. The process is

Solution:

The mean value is time dependent so X (t) is not stationary in any sense.

QUESTION: 8

The auto correlation function of a stationary ergodic random process is shown in fig.

Que: The mean value E[X(t)] is

Solution:

We know that for ergodic with no periodic component

QUESTION: 9

The auto correlation function of a stationary ergodic random process is shown in fig.

Que: The E[X^{2}(t)] is

Solution:

QUESTION: 10

The auto correlation function of a stationary ergodic random process is shown in fig.

Que: The variance σ_{x}^{2 } is

Solution:

QUESTION: 11

A stationary zero mean random process X(t) is ergodic has average power of 24 W and has no periodic component. The valid auto correlation function is

Solution:

For (A) : It has a periodic component.

For (B) ; It is not even in τ, total power is also incorrect.

For (C) It depends on t not even in τ and average power is ∞

QUESTION: 12

A stationary random process X(t) is applied to the input of a system for which If E[X(t)] = 2, the mean value of the system's response Y(t) is

Solution:

QUESTION: 13

A random process X(t) is applied to a network with impulse response where a > 0 is a constant. The cross correlation of X(t) with the output Y(t) is known to have the same form

Que: The auto correlation of Y(t) is

Solution:

QUESTION: 14

A random process X(t) is applied to a network with impulse response where a > 0 is a constant. The cross correlation of X(t) with the output Y(t) is known to have the same form

Que: The average power in Y(t) is

Solution:

QUESTION: 15

A random noise X(t) having a power spectrum is applied to a differentiator that has a transfer function H(ω) = j(ω). The output is applied to a network for which

Que : The average power in X(t) is

Solution:

QUESTION: 16

A random noise X(t) having a power spectrum is applied to a differentiator that has a transfer function H(ω) = j(ω). The output is applied to a network for which

Que : The power spectrum of Y(t) is

Solution:

QUESTION: 17

White noise with power density N_{0} /2 is applied to a low pass network for which |H(0)| = 2. It has a noise bandwidth of 2 MHz. If the average output noise power is 0.1 W in a 1 - Ω( resistor, the value of N_{0 }is

Solution:

QUESTION: 18

An ideal filter with a mid-band power gain of 8 and bandwidth of 4 rad/s has noise X(t) at its input with power spectrum The noise power at the network's output is (F(2) = 0.9773)

Solution:

QUESTION: 19

White noise with power density N_{0} /2 = 6 μW/Hz is applied to an ideal filter of gain 1 and bandwidth W rad/s. If the output's average noise power is 15 watts, the bandwidth W is

Solution:

QUESTION: 20

A system have the transfer function where W is a real positive constant. The noise bandwidth of the system is

Solution:

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