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 $Pe$ is expressed as follows: $$Pe = \alpha P(N > 1),$$ where $P(N > 1)$ denotes the probability that $N$ exceeds 1. What is the value of $\alpha$ ?

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

Consider a frequency-modulated (FM) signal $f(t) = Ac \cos(2\pi fc t + 3 \sin(2\pi f1 t) + 4 \sin(6\pi f1 t))$, where $Ac$ and $fc$ are, respectively, the amplitude and frequency (in Hz) of the carrier waveform. The frequency $f1$ is in Hz, and assume that $fc > 100 f1$. The peak frequency deviation of the FM signal in Hz is .

The random variable $X$ takes values in $\{-1, 0, 1\}$ with probabilities $P(X = -1) = P(X = 1) = \alpha$ and $P(X = 0) = 1 - 2\alpha$, where $0 < \alpha < 1/2$. Let $g(\alpha)$ denote the entropy of $X$ (in bits), parameterized by $\alpha$. Which of the following statements is/are TRUE?

Consider a message signal $m(t)$ which is bandlimited to $-W, W$, where $W$ is in Hz. Consider the following two modulation schemes for the message signal: - Double sideband-suppressed carrier (DSB-SC): $f{DSB}(t) = Ac m(t) \cos(2\pi fc t)$ - Amplitude modulation (AM): $f{AM}(t) = Ac (1 + \mu m(t)) \cos(2\pi fc t)$ Here, $Ac$ and $fc$ are the amplitude and frequency (in Hz) of the carrier, respectively. In the case of AM, $\mu$ denotes the modulation index. Consider the following statements: (i) An envelope detector can be used for demodulation in the DSB-SC scheme if $m(t) > 0$ for all $t$. (ii) An envelope detector can be used for demodulation in the AM scheme only if $m(t) > 0$ for all $t$. Which of the following options is/are correct?

Consider an additive white Gaussian noise (AWGN) channel with bandwidth $W$ and noise power spectral density $\frac{N0}{2}$. Let $P{av}$ denote the average transmit power constraint. Which one of the following plots illustrates the dependence of the channel capacity $C$ on the bandwidth $W$ (keeping $P{av}$ and $N0$ fixed)?

An amplitude modulator has output (in Volts) $s(t) = A \cos(400\pi t) + B \cos(360\pi t) + B \cos(440\pi t)$ The carrier power normalized to $1 \Omega$ resistance is $50$ Watts. The ratio of the total sideband power to the total power is $1/9$. The value of $B$ (in Volts, rounded off to two decimal places) is .

A source transmits symbols from an alphabet of size $16$. The value of maximum achievable entropy (in bits) is .

Let $X(t) = A\cos(2\pi f0 t + \theta)$ 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 $RX(\tau) = \frac{2}{3}\cos(2\pi f0 \tau)$, the signal to quantization noise ratio (in 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 $\frac{N0}{2}$, when applied to a first-order RC low pass filter produces an output $n(t)$. At a particular time $t = tk$, the variance of the random variable $n(tk)$ is .