GATE EC Digital Communications questions

Consider a discrete memoryless source with an alphabet of four source symbols. $s(t)$ is a multi-level $(-1,0,+1,+2)$ signal representing a long sequence of random symbols from the above source which is generating $10^{4}$ symbols per second. Which of the following options is the correct value of equivalent Nyquist bandwidth of $s(t)$?

A wireless digital transmission scheme is using $16$-QAM over an additive white Gaussian noise channel and a maximum-likelihood receiver. Consider the information bit rate from source to be $4 \times 10^{6}$ bits per second. The minimum transmission bandwidth (in MHz) of the modulated signal necessary for optimum recovery of information at the receiver …

A QPSK modulated signal from an additive white Gaussian noise (AWGN) channel is received with an $E_{b} / N_{o}=8.4 \mathrm{~dB}$ at the input of a coherent QPSK demodulator. A maximum-likelihood reception method is used in the demodulator. Assume the complimentary error function …

The average bit error rate at the input of a $(7,4,1)$ Hamming decoder is $0.10$. The probability that the decoder will fail to decode a received word correctly is $\_\_\_\_\_$ (rounded off to two decimal places)

The generator matrix of a $(6,3)$ binary linear block code is given by $$G=\left[\begin{array}{llllll} 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 0 \end{array}\right]$$ The minimum Hamming distance $d_{\text{min }}$ between codewords equals _________ (answer in …

A digital communication system transmits through a noiseless bandlimited channel $[-W W]$. The received signal $z(t)$ at the output of the receiving filter is given by $z(t)=\sum_{n} b[n] x(t-n T)$ where $b[n]$ are the symbols and $x(t)$ is the overall system response to a single symbol. The received signal is sampled at $t=m T$. The Fourier …

​​​​​A source transmits a symbol $s$, taken from $\{-4,0,4\}$ with equal probability, over an additive white Gaussian noise channel. The received noisy symbol $r$ is given by $r=s+w$, where the noise $w$ is zero mean with variance 4 and is independent of $s$. Using $Q(x)=\frac{1}{\sqrt{2 \pi}} \int_{x}^{\infty} e^{\frac{-t^{2}}{2}} d t$, the …

​​​​​The information bit sequence $\{111010101\}$ is to be transmitted by encoding with Cyclic Redundancy Check $4$ (CRC-$4$) code, for which the generator polynomial is $C(x)=x^{4}+x+1$. The encoded sequence of bits is $\_\_\_\_\_\_\_$.

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 …

A symbol stream contains alternate $\text{QPSK}$ and $\text{16-QAM}$ symbols. If symbols from this stream are transmitted at the rate of $1$ mega-symbols per second, the raw (uncoded) data rate is ____________ mega-bits per second (rounded off to one decimal place).

Consider communication over a memoryless binary symmetric channel using a $(7, 4)$ Hamming code. Each transmitted bit is received correctly with probability $(1 – \in),$ and flipped with probability $\in.$ For each codeword transmission, the receiver performs minimum Hamming distance decoding, and correctly decodes the message bits if and only if the …

Consider a channel over which either symbol $x_{A}$ or symbol $x_{B}$ is transmitted. Let the output of the channel $Y$ be the input to a maximum likelihood $\text{(ML)}$ detector at the receiver. The conditional probability density functions for $Y$ given $x_{A}$ and $x_{B}$ are: $$f_{Y|x_{A}}(y) = e^{-(y+1)}u(y+1),$$ …

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 $\alpha^{\ast}$ and $\beta^{\ast}$ are the respective choices for …

A $4\:kHz$ sinusoidal message signal having amplitude $4\:V$ is fed to a delta modulator ($DM$) operating at a sampling rate of $32\:\text{kHz}$. The minimum step size required to avoid slope overload noise in the $DM$ (rounded off to two decimal places) is _____ $V$.

A digital transmission system uses a ($7,4$) systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code ($\text{m}_{i}$ ; $\text{c}_{i}$), where $\text{c}_{i}$ is the codeword corresponding to the $i^{th}$ message $\text{m}_{i}$, are known to be $(1100; 0101 100)$, …

A message signal having peak-to-peak value of $2\:V$, root mean square value of $0.1\:V$ and bandwidth of $\text{5 kHz}$ is sampled and fed to a pulse code modulation ($\text{PCM}$) system that uses a uniform quantizer. The $\text{PCM}$ output is transmitted over a channel that can support a maximum transmission rate of $\text{50 kbps}$. Assuming …

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 …

A digital communication system transmits a block of $N$ bits. The probability of error in decoding a bit is $\alpha$. The error event of each bit is independent of the error events of the other bits. The received block is declared erroneous if at least one of its bits is decoded wrongly. The probability that the received block erroneous is

In a digital communication system, a symbol $S$ randomly chosen from the set $\left \{ s_{1},s_{2},s_{3},s_{4} \right \}$ 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 …

A linear Hamming code is used to map $4-bit$ message to $7-bit$ codewords. The encoder mapping is linear. If the message $0001$ is mapped to the codeword $0000111$, and the message $0011$ is mapped to the codeword $1100110$, then the message $0010$ is mapped to