$\text{Consider a discrete memoryless source with an alphabet of four source symbols.}s\left(t\right)\text{is a multi-level}$

$(-1,0,+1,+2)\text{signal representing a long sequence of random symbols from the above source which is}$

$\text{generating}10{}^4\text{symbols per second.}\text{Which of the following options is the correct value of equivalent Nyquist}$

$\text{bandwidth of}s\left(t\right)\text{?}$

The correct option is 5 kHz.

Step-by-Step Explanation:

1. Understand the Given Parameters:
We are given a discrete memoryless source that generates symbols from an alphabet of four source symbols: $(-1,0,+1,+2)$.
The source generates symbols at a rate of:
$R_s={10}^4\text{symbols/second}$

2. Determine the Symbol Duration:
The symbol duration $T_s$ is the reciprocal of the symbol generation rate:
$T_s=\frac{1}{R_s}=\frac{1}{{10}^4}={10}^{-4}\text{seconds}$

3. Calculate the Nyquist Bandwidth:
According to the Nyquist signaling criterion, the minimum bandwidth (or equivalent Nyquist bandwidth) required to transmit symbols at a rate of $R_s$ symbols per second without intersymbol interference (ISI) is given by:
$B=\frac{R_s}{2}$

4. Substitute the Values:
Using the symbol rate $R_s={10}^4\text{symbols/s}$:
$B=\frac{{10}^4}{2}=5000\text{Hz}=5\text{kHz}$

Thus, the equivalent Nyquist bandwidth of the multi-level signal $s\left(t\right)$ is 5 kHz.