What are the relative amplitudes of the fourth pair of sidebands for an FM signal with a deviation ratio of 8?

To calculate the relative amplitudes of the fourth pair of sidebands for an FM signal with a deviation ratio of 8, we need to use **Bessel functions of the first kind**. The relative amplitude of the nth sideband is determined by \( J_n(\beta) \), where \( J_n \) is the Bessel function of order \( n \), and \( \beta \) is the modulation index (also called the deviation ratio).

Given:
- Deviation ratio \( \beta = 8 \)
- We are interested in the fourth pair of sidebands, so \( n = 4 \).

We need to calculate \( J_4(8) \), which is the value of the Bessel function of the first kind for \( n = 4 \) and \( \beta = 8 \).

By referring to tables or using computational tools to find Bessel function values:

- \( J_4(8) \approx 0.228 \)

Thus, the relative amplitude of the fourth pair of sidebands is approximately **0.228**.

### Key Steps:
1. Identify the modulation index \( \beta = 8 \).
2. Use the Bessel function \( J_4(8) \).
3. The amplitude of the fourth sideband is about 0.228.

Would you like more details or have any questions?

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Here are 5 related questions:
1. What is the role of Bessel functions in FM signal analysis?
2. How does the deviation ratio affect the bandwidth of an FM signal?
3. How do we interpret higher-order sidebands in frequency modulation?
4. What is the total power distribution among sidebands in FM?
5. How can we calculate the bandwidth of an FM signal using Carson’s rule?

**Tip:** In FM signals, most of the signal power is concentrated in the first few sidebands when the modulation index is relatively small.

Learn how to calculate the relative amplitudes of the fourth pair of sidebands in an FM signal with a deviation ratio of 8 using Bessel functions of the first kind. This example explains key concepts in frequency modulation and sideband amplitude analysis. Suitable for college-level students in Electrical Engineering.

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