Consider a real-valued random process

where  and  is a positive integer. Here,  for  and 0 otherwise. The coefficients  are pairwise independent, zero-mean unit variance random variables.

Read the following statements about the random process and choose the correct option.

(i) The mean of the process  is independent of time .

(ii) The autocorrelation function  is independent of time  for all .

(Here,  is the expectation operation.)

The random variable  takes values in  with probabilities  and , where . Let  denote the entropy of  (in bits), parameterized by . Which of the following statements is/are TRUE?

A white Gaussian noise  with zero mean and power spectral density , when applied to a first-order RC low pass filter produces an output . At a particular time , the variance of the random variable  is _________.

Suppose  and  are independent and identically distributed random variables that are distributed uniformly in the interval . The probability that  is _________.

Let  be a random process, where amplitude  and phase  are independent of each other, and are uniformly distributed in the intervals  and , respectively.  is fed to an 8 -bit uniform mid-rise type quantizer. Given that the autocorrelation of  is , the signal to quantization noise ratio (in , rounded off to two decimal places) at the output of the quantizer is_________. 

A random variable , distributed normally as , undergoes the transformation , given in the figure. The form of the probability density function of  is (In the options given below,  are non-zero constants and  is piecewise continuous function)

Let H(X) denote the entropy of a discrete random variable X taking K possible distinct real values. Which of the following statements is/are necessarily true?         

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

Consider a 1 bit quantizer that maps positive samples to value  and others to value . If  and  are the respective choices for  and  that minimize the mean square quantization error, then  __________(rounded off to two decimal places).

The autocorrelation function  of a wide-sense stationary random process  is shown in the figure.

The average power of X(t) is __________.

The random variable

where

and  is a real white Gaussian noise process with two-sided power spectral density  , for all . The variance of  is _______.  

A binary random variable  takes the value +2 or -2. The probability . The value of  (rounded off to one decimal place), for which the entropy of  is maximum, is ________.

Let Z be an exponential random variable with mean 1. That is the cumulative distribution function of Z is given by

Then , rounded off to two decimal places, is equal to ________

If X and Y are random variables such that and , then

______

A single bit, equally likely to be 0 and 1, is to be send across an additive white Gaussian noise (AWGN) channel with power spectral density . Binary signaling, with 0p(t) and 1q(t), is used for the transmission along with an optimal receiver that minimizes the bit-error probability.

Let , from an orthogonal signal set

If we choose and,we would obtain a certain bit error probability.

If we keep , but take, for what value of E would we obtain the same bit-error probability?

 Let a random process Y(t) be described as  where X(t) is a white noise process with power spectral density . The filter  has a magnitude response given by  for , and zero elsewhere, Z(t)  is a stationary random process,  uncorrelated with X(t) , with power spectral density as shown in the figure. The power in Y(t), in watts, is equal to _______________ W (rounded off to two decimal places).

A random variable X takes values  and  with probabilities 0.2 and 0.8, respectively. It is transmitted across a channel which adds noise N, so that the random variable at the channel output is . The noise N is independent of X, and is uniformly distributed over the interval. The receiver makes a decision.

Where the threshold  is chosen so as to minimize the probability of error . The minimum probability of error, rounded off to 1 decimal place, is ____________.

A binary source generates symbols which are transmitted over a noisy channel. The probability of transmitting X = 1 is 0.5. Input to the threshold detector is . The probability density function of the noise N is shown below.

If the detection threshold is zero, then the probability of error (correct to two decimal places) is ______  

Let  and be independent normal random variables with zero mean and unit variance. The probability thatis the smallest among the four is _________

Consider a white Gaussian noise process N(t) with two-sided power spectral density  as input to a filter with impulse response (where t is in seconds) resulting in output Y(t). The power Y(t) in watts is         

Let X(t) be a wide sense stationary random process with the power spectral density  as shown in Figure (a), where f is in Hertz (Hz). The random process X(t) is input to an ideal lowpass filter with the frequency response  as shown in
Figure (b). The output of the lowpass filter is Y(t).

Let E be the expectation operator and consider the following statements:

I. E(X(t)) = E(Y(t))

II.  

III.

Select the correct option:

Consider the random process

where U is a zero-mean Gaussian random variable and V is a random variable uniformly distributed between 0 and 2. Assume that U and V are statistically independent. The mean value of the random process at t = 2 is _____________         

The second moment of a Poisson-distributed random variable is 2. The mean of the random variable is ___________        

An analog pulse s(t) is transmitted over an additive white Gaussian noise (AWGN) channel. The received signal is r(t) = S(t) + n(t), where n(t) is additive white Gaussian noise with power spectral density . The received signal is passed through a filter with impulse response h(t). Let and  denote the energies of the pulse s(t) and the filter h(t), respectively. When the signal-to-noise ratio (SNR) is maximized at the output of the filter , which of the following holds?

An antenna pointing in a certain direction has a noise temperature of 50 K. The ambient temperature is 290 K. The antenna is connected to a pre-amplifier that has a noise figure of 2 dB and an available gain of 40 dB over an effective bandwidth of 12MHz. The effective input noise temperature  for the amplifier and the noise power at the output of the preamplifier, respectively, are

An information source generates a binary sequence . 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 , a. The sequence
is used to modulate a pulse g(t) to generate the baseband signal

 , where .

If there is a null at  in the power spectral density of X(t), then k is ________.

Consider a random process, where V(t) is a zero mean stationary random process with autocorrelation .  The power in X(t) is ________.

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  , then the autocorrelation function  of the output Y(t) is equal to

A voice-grade AWGN (additive white Gaussian noise) telephone channel has a bandwidth of 4.0 kHz and two-sided noise power spectral density  watt per Hz. If information at the rate of 52 kbps is to be transmitted over this channel with arbitrarily small bit error rate, then the minimum bit-energy  (in mJ/bit) necessary is __________        

A zero mean white Gaussian noise having power spectral density   is passed through an LTI filter whose impulse response h(t) is shown in the figure. The variance of the filtered noise at t = 4 is                

 is an independent and identically distributed (i.i.d.) random process with equally likely to be +1 or −1.  is another random process obtained as . The autocorrelation function of , denoted by , is

Let  and  be two independent binary random variables. If and , then is equal to

Let the random variable X represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of X is _______.

Let  and  be independent and identically distributed random variables with the uniform distribution on [0,1].The probability
P{} is _________

Consider a random process , where the random phase φ is uniformly distributed in the interval [0,2π]. The auto-correlation  is

The capacity of a band-limited additive white Gaussian noise (AWGN) channel is given by  bits per second (bps), where W is the channel bandwidth, P is the average power received and   is the one-sided power spectral density of the AWGN.

For a fixed  , the channel capacity (in kbps) with infinite bandwidth  is approximately

The power spectral density of a real stationary random process X(t) is given by

 .

The value of the expectation  is _____________

Let X(t) be a wide sense stationary (WSS) random process with power spectral density . If Y(t) is the process defined as Y(t)=X(2t-1), the power spectral density  is

Let , , and  be independent and identically distributed random variables with the uniform distribution on [0,1]. The probability  is _________.

A real band-limited random process X(t) has two-sided power spectral density

Where f is the frequency expressed in Hz. The signal X(t) modulates a carrier and the resultant signal is passed through an ideal band-pass filter of
unity gain with centre frequency of 8 kHz and band-width of 2 kHz. The output power (in Watts) is _______.

If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier or future calls, the probability distribution function of the total number of calls in a fixed time interval will be

Let X be a zero mean unit variance Gaussian random variable.   is equal to _____.

Consider a communication scheme where the binary valued signal X satisfies  and . The received signal Y = X + Z, where Z is a Gaussian random variable with zero mean and variance . The received signal Y is fed to the threshold detector. The output of the threshold detector X is:          

.

To achieve a minimum probability of error , the threshold τ should be

The variance of the random variable X with probability density functionis _______.        

A random binary wave y(t) is given by

Where, u(t) is the unit step function and ϕ is an independent random variable with uniform distribution in [0,T]. The sequence  consists of independent and identically distributed binary valued random variables with

for each n.

The value of the autocorrelation equals ________.        

A fair die with faces {1, 2, 3, 4, 5, 6} is thrown repeatedly till ‘3’ is observed for the first time. Let X denote the number of times the die is thrown. The expected value of X is ________.

The power spectral density of a real process X(t) for positive frequencies is shown below. The values of and  , respectively, are

Let U and V be two independent and identically distributed random variables such that. The entropy in bits is

X(t) is a stationary random process with autocorrelation function . This process is passed through the system shown below. The power spectral density of the output process Y(t) is        

X(t) is a stationary process with the power spectral density  for all f. The process is passed through a system shown below.

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

A white noise process X(t) with two-sided power spectral density  W/Hz is input to a filter whose magnitude squared response is shown below.

The power of the output process Y(t) is given by

If the power spectral density of a stationary random process is a sinc-squared function of frequency, the shape of its autocorrelation is

If the signal to quantization noise ratio required in uniformly quantizing the signal is 43.5 dB, the step size of the quantization is approximately

If the positive values of the signal are uniformly quantized with a step size of 0.05 V, and the negative values are uniformly quantized with a step size of 0.1 V, the resulting signal to quantization noise ratio is approximately

Noise with double-sided power spectral density of K over all frequencies is passed through a RC low pass filter with 3 dB cut-off frequency of . The noise power at the filter output is

If R(τ) is the autocorrelation function of a real, wide-sense stationary random process, then which of the following is NOT true?

If S(f) is the power spectral density of a real, wide-sense stationary random process, then which of the following is ALWAYS true?

A uniformly distributed random variable X with probability density function         

Where u (.) is the unit step function is passed through a transformation given in the figure below. The probability density function of the transformed random Variable Y would be

A zero-mean white Gaussian noise is passed through an ideal low-pass filter of bandwidth 10 kHz. The output is then uniformly sampled with sampling period msec. The samples so obtained would be

The following question refer to wide sense stationary stochastic processes

It is desired to generate a stochastic process (as voltage process) with power spectral density

By driving a Linear-Time-Invariant system by zero mean white noise (as voltage process) with power spectral density being constant equal to 1. The system which can perform the desired task could be:         

The following question refer to wide sense stationary stochastic processes

It is desired to generate a stochastic process (as voltage process) with power spectral density

By driving a Linear-Time-Invariant system by zero mean white noise (as voltage process) with power spectral density being constant equal to 1.
The parameters of the system obtained would be

Noise with uniform power spectral density of  is passed through a filter followed by an ideal low pass filter of bandwidth B Hz. The output noise power in Watts is

An output of a communication channel is a random variable with the probability density function as shown in figure. The mean square value of is        

A 1mW video signal having a bandwidth of 100 MHZ is transmitted to a receiver through a cable that has 40 dB loss. If the effective one-sided noise spectral density at the receiver is , then the signal to noise ratio at the receiver is

A random variable X with uniform density in the interval 0 to 1 is quantized as follows:

If ,                

If ,                

Where  is the quantized value of X

The root-mean square value of the quantization noise is

The noise at the input to an ideal frequency detector is white. The detector is operating above threshold. The power spectral density of the noise at the output is

Let X and Y be two statistically independent random variables uniformly distributed in the ranges
(-1, 1) and (-2, 1) respectively. Let Z = X + Y. then the probability that  is

X(t) is a random process with a constant mean value of 0 and the autocorrelation function .

Let X be the Gaussian random variable obtained by sampling the process at  and let .

The probability that  is

X(t) is a random process with a constant mean value of 0 and the autocorrelation function .

Let Y and Z be the random variables obtained by sampling X(t) at t = 2 and t = 4 respectively. Let. The variance of W is        

If the variance  of is one-tenth the variance  of a stationary zero mean discrete time signal x(n), then the normalized autocorrelation function  at k=1 is

A deterministic signal  is passed through a differentiator as shown in Fig.

(a) Determine the autocorrelation   and the power spectral density .

(b) Find the output power spectral density.

(c) Evaluate and.

The PDF of a Gaussian random variable X is given by . The probability of the event {X = 4} is

During transmission over a communication channel, bit errors occur independently with probability p. If a block of n bits is transmitted, the probability of at most one bit error is equal to

The PSD and the power of a signal g(t) are, respectively,  and . The PSD and the power of the signal ag(t) are, respectively

The Hilbert transform of  is

Zero mean white Gaussian noise with a two-sided power spectral density of  is passed through an ideal low pass filter with a cut-off frequency of 2 KHz and a pass band gain of 1, to produce the noise output n(t).        

(a) Obtain the total power in n(t).

(b) Find the autocorrelation function  of the noise n(t) as a function of .

(c) Two noise samples are taken at times and . Find the spacing  so that the product  has the most negative expected value and obtain this most negative expected value.

The power spectral density (PSD) of a noise process is given by        

The noise is passed through a unit-gain ideal band pass filter, centered at 50 MHz and having a bandwidth of 2MHz.

(a) Sketch neatly the PSD of the output noise process.

(b) Determine the output noise power.

(c) Using the band-pass representation for the output noise process, sketch the PSD of the in-phase and Quadrature noise components, and determine their respective powers.

The amplitude spectrum of a Gaussian pulse is

The ACF of a rectangular pulse of duration T is

The spectral density of a real valued random process has

The probability density function of the envelope of narrow band Gaussian noise is

White noise of two-sided spectral density  is applied to a simple R-C low pass filter whose 3 dB cut off frequency is 4 kHz. Find the mean squared value of the noise output        

A probability density function is given by . The value of K should be

The power spectral density of a deterministic signal given by  where f is frequency. The autocorrelation function of this signal in the time domain is

An amplifier a has 6dB gain and  input and output impedances. The noise figure of this amplifier as shown in the figure is (a) is 3 dB. A cascade of two such amplifiers as in the figure will have a noise figure of        

Find the mean of a function , where  is a constant and T is a random variable. The PDF of T is given by,

for  & 0 for

The autocorrelation function of an energy signal has

For a narrow band noise with Gaussian quadrature components, the probability density function of its envelope will be

Noise figure of an amplifier is always greater than 1. (True=1,False=0)

The function shown in figure can represent a probability density function for A _______

It is desired to generate a random signal x(t), with autocorrelation function, , by passing white noise n(t), with power spectral density

watt/Hz, through a LTI system. Obtain an expression for the transfer function H(f) of the system.

For a random variable x following the probability density function, p(x), shown in figure the mean and the variance are, respectively,

Two resistors  and  (in ohms) at temperatures  and  respectively, are connected in series. Their equivalent noise temperatures is ________ K.

(a) A Gaussian random variable with zero mean and variance is input to a limiter with input output characteristic given by

Determine the probability density function of the output random variable.

(b) A random process X(t) is wide sense stationary. If

Determine the auto correlation function  and power spectral density  of Y(t) in terms of those of X(t).