https://r2.erweima.ai/mathbot/image/1724983280004l37dsgv6.jpg

For the AM waveform s(t) shown, the carrier
signal has a peak amplitude of 0.8 V. What is the modulating index? 


To determine the modulation index (\(m\)) of the AM waveform shown in the image, we can use the following relationship between the carrier amplitude and the maximum and minimum values of the modulated waveform:

\[
m = \frac{A_{\text{max}} - A_{\text{min}}}{A_{\text{max}} + A_{\text{min}}}
\]

Where:
- \(A_{\text{max}}\) is the maximum amplitude of the modulated signal.
- \(A_{\text{min}}\) is the minimum amplitude of the modulated signal.

Given in the problem:
- The carrier amplitude \(A_c = 0.8 \text{ V}\).

Looking at the AM waveform:
- \(A_{\text{max}}\) corresponds to the peak amplitude of the envelope.
- \(A_{\text{min}}\) corresponds to the trough (minimum value) of the envelope.

**Step-by-Step Process**:
1. Identify \(A_{\text{max}}\) and \(A_{\text{min}}\) from the waveform.
2. Use the formula to calculate the modulation index \(m\).

### From the waveform:
- \(A_{\text{max}} \approx 0.8 + 0.4 = 1.2 \text{ V}\) (since the peak is above the carrier by approximately 0.4 V).
- \(A_{\text{min}} \approx 0.8 - 0.4 = 0.4 \text{ V}\) (since the trough is below the carrier by approximately 0.4 V).

### Calculation of \(m\):
\[
m = \frac{1.2 - 0.4}{1.2 + 0.4} = \frac{0.8}{1.6} = 0.5
\]

Thus, the modulation index \(m\) is **0.5**.

Would you like any additional details or explanations?

### Relative Questions:
1. What is the significance of the modulation index in AM signals?
2. How does the modulation index affect the power of the AM signal?
3. What would happen if the modulation index exceeds 1?
4. Can you explain the relationship between \(A_{\text{max}}\), \(A_{\text{min}}\), and the carrier amplitude?
5. How would you measure the modulation index in a real-world AM signal?

### Tip:
To avoid over-modulation in AM signals, keep the modulation index \(m\) less than or equal to 1. Over-modulation can lead to signal distortion.

Learn how to calculate the modulation index (m) for an AM waveform given a carrier signal with a peak amplitude of 0.8V. Understand the significance of modulation index in AM signals and its calculation using the formula m = (A_max - A_min) / (A_max + A_min). Suitable for undergraduate level studies.

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