A carrier wave of amplitude A is used to modulate (amplitude) the signal.Modulated signal amplitude swings from 0.1 A to 1.9A,calculate modulation index.`

Answer

To calculate the modulation index for the given modulated signal, we will follow these steps:

### Step 1: Understand the given information
The amplitude of the modulated signal swings from 0.1A to 1.9A. This means:
- Minimum amplitude (A_min) = 0.1A
- Maximum amplitude (A_max) = 1.9A

### Step 2: Calculate the carrier amplitude (A_C)
The carrier amplitude (A_C) can be determined using the formula for the maximum and minimum amplitudes of the modulated signal:
- A_max = A_C + A_M
- A_min = A_C - A_M

Where:
- A_M is the amplitude of the modulating signal.

From the given values:
1. A_max = 1.9A
2. A_min = 0.1A

### Step 3: Set up the equations
From the maximum and minimum amplitudes, we can set up the following equations:
1. \( A_C + A_M = 1.9 \) (Equation 1)
2. \( A_C - A_M = 0.1 \) (Equation 2)

### Step 4: Solve the equations
To find A_C and A_M, we can add Equation 1 and Equation 2:
\[ (A_C + A_M) + (A_C - A_M) = 1.9 + 0.1 \]
This simplifies to:
\[ 2A_C = 2.0 \]
Thus, we find:
\[ A_C = 1.0A \]

Now, substitute A_C back into either equation to find A_M. Using Equation 1:
\[ 1.0 + A_M = 1.9 \]
\[ A_M = 1.9 - 1.0 = 0.9A \]

### Step 5: Calculate the modulation index (m)
The modulation index (m) is defined as:
\[ m = \frac{A_M}{A_C} \]
Substituting the values we found:
\[ m = \frac{0.9}{1.0} = 0.9 \]

### Final Answer
The modulation index is \( m = 0.9 \).

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