In a Double Sideband (DSB) modulation system, the bandwidth is directly related to the frequency of the modulating signal, which is the frequency of the carrier wave used.

Given that $x(t) = \cos^2(5000\pi t)$, we start by simplifying this expression using the trigonometric identity:

$\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$

So,

$x(t) = \frac{1 + \cos(2 \cdot 5000\pi t)}{2} = \frac{1 + \cos(10000\pi t)}{2}$

The term $\cos(10000\pi t)$ represents a cosine function with a frequency of $5000 \, \text{Hz}$, since $10000\pi = 2\pi \cdot 5000$.

Bandwidth Calculation:

In a DSB modulated system, the bandwidth is twice the highest frequency of the modulating signal. Here, the modulating signal has a frequency of $5000 \, \text{Hz}$.

Thus, the bandwidth $B$ is: $B = 2 \times 5000 = 10,000 \, \text{Hz}$

Therefore, the bandwidth of the signal is 10 kHz (10,000 Hz).

Would you like further details or clarifications on this?
Here are 5 related questions for further exploration:

1. How does modulation affect the bandwidth of a signal?
2. What is the difference between DSB and SSB modulation in terms of bandwidth?
3. How is the bandwidth affected if the carrier frequency increases?
4. Why is the bandwidth in DSB modulation twice the modulating signal frequency?
5. How does the amplitude of the carrier affect the modulated signal?

Tip: In DSB modulation, the carrier does not add to the bandwidth but the sidebands, which carry the information, do.