Random Signals And Noise Practice Questions

Consider a real-valued random process $\mathrm{f}(\mathrm{t})=\sum_{\mathrm{n}=1}^{\mathrm{N}} \mathrm{a}_{\mathrm{n}} \mathrm{p}(\mathrm{t}-\mathrm{nT})$ , where $\mathrm{T}>0$ and N is a positive integer. Here, $p(t)=1$ for $t \in[0,0.5 T]$ and 0 otherwise. The coefficients $\mathrm{a}_{\mathrm{n}}$ 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 $f(t)$ is independent of time t . (ii) The autocorrelation function $\mathrm{E}[\mathrm{f}(\mathrm{t}) \mathrm{f}(\mathrm{t}+\tau)]$ is independent of time t for all $\tau$ . (Here, $\mathrm{E}[\cdot]$ is the expectation operation.)

A source transmits symbol $S$ that takes values uniformly at random from the set $\{-2,0,2\}$ . The receiver obtains $Y=S+N$ , where $N$ is a zero-mean Gaussian random variable independent of $S$ . The receiver uses the maximum likelihood decoder to estimate the transmitted symbol $S$ . Suppose the probability of symbol estimation error $\mathrm{P}_{\mathrm{e}}$ is expressed as follows: $\mathrm{P}_{\mathrm{e}}=\alpha \mathrm{P}(\mathrm{N}>1)$ , where $\mathrm{P}(\mathrm{N}>1)$ denotes the probability that $N$ exceeds 1. What is the value of $\alpha$ ?

Let $X(t)=A \cos \left(2 \pi f_{0} t+\theta\right)$ be a random process, where amplitude $A$ and phase $\theta$ are independent of each other, and are uniformly distributed in the intervals $[-2,2]$ and $[0,2 \pi]$ , respectively. $X(t)$ is fed to an 8-bit uniform mid-rise type quantizer. Given that the autocorrelation of $X(t)$ is $R_{x}(\tau)=\frac{2}{3} \cos \left(2 \pi f_{0} \tau\right)$ , the signal to quantization noise ratio (in $\mathrm{dB}$ , rounded off to two decimal places) at the output of the quantizer is ______

A White Gaussian noise $w(t)$ with zero mean and power spectral density $\frac{N_{0}}{2}$ , when applied to a first-order $R C$ low pass filter produces an output $n(t)$ . At a particular time $t=t_{k}$ , the variance of the random variable $n\left(t_{k}\right)$ is _____

For a real signal, which of the following is/are valid power spectral density/densities?

Consider an FM broadcast that employs the pre-emphasis filter with frequency response $H_{pe}(\omega )=1+\frac{j\omega }{\omega _0}$ where $\omega _0=10^4$ rad/sec. For the network shown in the figure to act as a corresponding de-emphasis filter, the appropriate pair(s) of ( $R,C$ ) values is/are ________.

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).

The frequency response $H(f)$ of a linear time-invariant system has magnitude as shown in the figure. Statement I: The system is necessarily a pure delay system for inputs which are bandlimited to $-\alpha \leq f\leq \alpha$ . Statement II: For any wide-sense stationary input process with power spectral density $S_X(f)$ , the output power spectral density $S_Y(f)$ obeys $S_Y(f)=S_X(f)$ for $-\alpha \leq f\leq \alpha$ . Which one of the following combinations is true?

The autocorrelation function $R_{X}\left ( \tau \right )$ of a wide-sense stationary random process $X(t)$ is shown in the figure. The average power of $X(t)$ is ________________

Consider a polar non-return to zero $(\text{NRZ})$ waveform, using $+2\:V$ and $-2\:V$ for representing binary '1' and '0' respectively, is transmitted in the presence of additive zero-mean white Gaussian noise with variance $0.4\:V^{2}$ . If the a priori probability of transmission of a binary '1' is 0.4, the optimum threshold voltage for a maximum a posteriori $(\text{MAP})$ receiver (rounded off to two decimal places) is ______ V.

Two continuous random variables X and Y are related as $Y=2X+3$ Let $\sigma ^{2}_{X}$ and $\sigma ^{2}_{Y}$ denote the variances of X and Y, respectively. The variances are related as

In a digital communication system, a symbol S randomly chosen from the set ( $s_1, s_2, s_3, s_4$ ) is transmitted. It is given that $s_1=-3, s_2=-1, s_3=+1$ and $s_4=+2$ . The received symbol is Y = S + W. W is a zero-mean unit-variance Gaussian random variable and is independent of S. $P_i$ is the conditional probability of symbol error for the maximum likelihood (ML) decoding when the transmitted symbol $S=s_i$ . The index i for which the conditional symbol error probability $P_i$ is the highest is ______.

A binary random variable X takes the value +2 or -2. The probability $P(X=+2)= \alpha$ . The value of $\alpha$ (rounded off to one decimal place), for which the entropy of X is maximum, is __________

X is a random variable with uniform probability density function in the interval [-2,10]. For Y = 2X-6, the conditional probability $P(Y \leq 7|X \geq 5)$ (rounded off to three decimal places) is _____.

The random variable $Y=\int_{-\infty }^{\infty }W(t)\phi (t) dt$ where

and W(t) is a real white Gaussian noise process with two-sided power spectral density $S_W(f) = 3$ W/Hz, for all f. The variance of Y is _________.

Let a random process Y(t) be described as Y(t)=h(t)*X(t)+Z(t), where X(t) is a white noise process with power spectral density $S_X(f)=5$ W/Hz. The filter h(t) has a magnitude response given by |H(f)|=0.5 for $-5\leq f\leq 5$ , 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).

Consider a white Gaussian noise process N(t) with two-sided power spectral density $S_{N}(f)=0.5 W/Hz$ as input to a filter with impulse response $0.5e^{\frac{-t^{2}}{2}}$ (where t is in seconds) resulting in output Y(t). The power in Y(t) in watts is

Consider the random process X(t)=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 ____________

Let X(t) be a wide sense stationary random process with the power spectral density $S_{X}(f)$ 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 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. $E(X^{2}(t))=E(Y^{2}(t))$ III. $E(Y^{2}(t))=2$ Select the correct option: