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 is $\_\_\_\_$ .

(rounded off to two decimal places)

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 $z(t) = \sum\limits_{n} b[n]x(t-nT)$ where $b[n]$ are the symbols and $x(t)$ is the overall system response to a single symbol. The received signal is sampled at $t = mT$. The Fourier transform of $x(t)$ is $X(f)$. The Nyquist condition that $X(f)$ must satisfy for zero intersymbol interference at the receiver is ______.

Let H(X) denote the entropy of a discrete random variable X taking K possible distinct real values. Which of the following statements is/are necessarily true?