x(t) = cos2(5000πt) is DSB modulated. The carrier amplitude is 10. What is the bandwidth in Hz?

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.

x(t) = cos²(5000πt) is DSB modulated. The carrier amplitude is 10. What is the bandwidth in Hz?

Understand how to calculate the bandwidth of a Double Sideband (DSB) modulated signal given x(t) = cos²(5000πt) and a carrier amplitude of 10. This solution uses key concepts from modulation theory, trigonometric identities, and signal processing to derive the bandwidth of 10 kHz. Suitable for students studying Electrical Engineering or Communication Systems.

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