GATE EC 2026 Set 1 Q22 — Digital Communications

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 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 …