# GATE Questions & Answers of Random processes: autocorrelation and power spectral density, properties of white noise, filtering of random signals through LTI systems

## What is the Weightage of Random processes: autocorrelation and power spectral density, properties of white noise, filtering of random signals through LTI systems in GATE Exam?

Total 30 Questions have been asked from Random processes: autocorrelation and power spectral density, properties of white noise, filtering of random signals through LTI systems topic of Communications subject in previous GATE papers. Average marks 1.77.

Let () be a wide sense stationary random process with the power spectral density Sx() as shown in Figure (a), where f is in Hertz(Hz). The random process () is input to an ideal lowpass filter with the frequency response $H\left(f\right)=\left\{\begin{array}{ll}1,& \left|f\right|\le \frac{1}{2}Hz\\ 0,& \left|f\right|>\frac{1}{2}Hz\end{array}\right\$ as shown in Figure(b). The output of the lowpass filter is ().

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

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

II.                  E (X2(t)) = E (Y(t))

III.                E (Y2(t)) = 2

Select the correct option:

Consider the random process $X\left(t\right)=U+Vt$ 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________

Passengers try repeatedly to get a seat reservation in any train running between two stations until they are successful. If there is 40% chance of getting reservation in any attempt by a passenger, then the average number of attempts that passengers need to make to get a seat reserved is__________

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 12 MHz. The effective input noise temperature Te for the amplifier and the noise power Pao at the output of the preamplifier, respectively, are

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 precoded to obtain another sequence $\left\{{\beta }_{n}\right\}$, as . The sequence $\left\{{\beta }_{n}\right\}$ is used to modulate a pulse $g(t)$ to generate the baseband signal

$X\left(t\right)={\textstyle\sum_{n=-\infty}^\infty}\beta_ng\left(t-nt\right)$ where $g\left(t\right)=\left\{\begin{array}{lc}1,\;\;&0\leq t\leq T\\0,&\mathrm{otherwise}\end{array}\right.$

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

Consider a random process X(t)=3V(t)-8, where V(t) is a zero mean stationary random process with autocorrelation ${R}_{v}\left(\tau \right)=4{e}^{-5\left|\tau \right|}$. 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 R($\tau$), then the autocorrelation function R($\tau$) of the output Y(t) 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 _____

A zero mean white Gaussian noise having power spectral density $\frac{{N}_{0}}{2}$ 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

${\left\{{X}_{n}\right\}}_{n=-\infty }^{n=\infty }$ is an independent and distributed (i.i.d) random process with Xn equally likely to be +1 or -1. ${\left\{{Y}_{n}\right\}}_{n=-\infty }^{n=\infty }$ is another random process obtained as Yn = Xn+0.5Xn-1 . The autocorrelation function of ${\left\{{X}_{n}\right\}}_{n=-\infty }^{n=\infty }$ , denoted by RY[K], is

Let $X\in \left\{0,1\right\}$ and $Y\in \left\{0,1\right\}$ be two independent binary random variables. If $P(X=0)=P$ and $P(Y=0)=q$ , then $P(X+Y\geq1)$ is equal to

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 variance of the random variable X with probability density function $f\left(x\right)=\frac{1}{2}\left|x\right|{e}^{-\left|x\right|}$ is_______.

A random binary wave $y\left(t\right)$ is given by

$y\left(t\right)=\sum_\limits{n=-\infty}^\infty X_np\left(t-nT-\phi\right)$

Where $p\left(t\right)=u\left(t\right)-u\left(t-T\right),u\left(t\right)$ is the unit step function and $\phi$ is an independent random variable with uniform distribution in $\left[0,T\right]$.The sequence $\left\{{X}_{n}\right\}$ consists of independent and identically distributed binary valued random variables with $P\left\{{X}_{n}=+1\right\}=P\left\{{X}_{n}=-1\right\}=0.5$ for each n

The value of the autocorrelation ${R}_{yy}\left(\frac{3T}{4}\right)\triangleq E\left[y\left(t\right)y\left(t-\frac{3T}{4}\right)\right]$ equals_______.

Consider a discrete-time channel Y = X+ Z, where the additive noise Z is signal-dependent.
In particular, given the transmitted symbol X{−a,+a} at any instant, the noise sample Z is chosen independently from a Gaussian distribution with mean $\beta X$ and unit variance. Assume a threshold detector with zero threshold at the receiver.

When $\beta$=0, the BER was found to be Q(a)=1 × 10−8.

When $\beta$=−0.3, the BER is closest to

The power spectral density of a real process X(t) for positive frequencies is shown below. The values of E[X2(t)] and |E[X(t)]| , respectively, are

X(t) is a stationary random process with autocorrelation function ${R}_{X}\left(\tau \right)=\mathrm{exp}\left(-\pi {f}^{2}\right)$.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 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?

Consider a baseband binary PAM receiver shown below. The additive channel noise n(t) is whit with power spectral density Sk(f)=N0/2=10-20 W/Hz. The low-pass is ideal with unity gain and cutoff frequency  1MHZ Let Yk represent  the random variable y(tk)

Yk=Nk   if transmitted bit bk=0

Yk=a+N if transmitted bit bk=1

Where Nk represents the noise sample value. The noise sample has a probability density function PNk(n)=0.5αe-α|n| (This has mean zero and variance 2/α2) Assume transmitted bits to be equiprobable and thresold Z is set to a/2=10-6v

The value of the parameter α(in  V-1) is

Consider a baseband binary PAM receiver shown below. The additive channel noise n(t) is whit with power spectral density Sk(f)=N0/2=10-20 W/Hz. The low-pass is ideal with unity gain and cutoff frequency  1MHZ Let Yk represent  the random variable y(tk)

Yk=Nk   if transmitted bit bk=0

Yk=a+N if transmitted bit bk=1

Where Nk represents the noise sample value. The noise sample has a probability density function PNk(n)=0.5αe-α|n| (This has mean zero and variance 2/α2) Assume transmitted bits to be equiprobable and thresold Z is set to a/2=10-6v

The probability of bit error is

A white noise process X(t) with two-sided power spectral density 1×10-10 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 stationary random process is a sine-squared function of frequency, the shape of its autocorrelation is

Consider two independent random variables X and Y with identical distributions. The variables X and Y take values 0, 1 and 2 with probabilities $\frac{1}{2}$,$\frac{1}{4}$ and $\frac{1}{4}$ respectively. What is the conditional probability P (X + Y = 2| XY = 0)?

A discrete random variable X takes values from 1 to 5 with probabilities as shown in the table. A student calculates the mean of X as 3.5 and her teacher calculates the variance of X as 1.5. Which of the following statements is true?

 k 1 2 3 4 5 P(X=k) 0.1 0.2 0.4 0.2 0.1

The probability density function (PDF) of a random variable X is as shown below

The corresponding cumulative distribution function (CDF) has the form

Px(x) = M exp (-2|x|) + N exp (-3|x|) is the probability density function for the real random variable X, over the entire x axis. M and N are both positive real numbers. The equation relating M and N is

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

If E denotes expectation, the variance of a random variable X is given by

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