A 107.6 MHz carrier signal is frequency modulated by a 7kHz sine wave . The resultant FM signal has a frequency deviation of 50 kHz. Determine the modulation index and the carrier swing of the FM wave

Options

A. 7.143 and 100 kHz
B. 8.234 and 75 kHz
C. 6.289 and 150 kHz
D. 5.103 and 14 kHz

Explanation

To solve the problem, we need to determine two quantities: the modulation index (M) and the carrier swing of the frequency-modulated (FM) signal.

### Step 1: Identify the given values
- Frequency of the carrier signal (f_c) = 107.6 MHz = 107.6 × 10^6 Hz
- Frequency of the modulating signal (f_m) = 7 kHz = 7 × 10^3 Hz
- Frequency deviation (Δf) = 50 kHz = 50 × 10^3 Hz

### Step 2: Calculate the modulation index (M)
The modulation index (M) for frequency modulation is given by the formula:
\[ M = \frac{\Delta f}{f_m} \]

Substituting the known values:
\[ M = \frac{50 \times 10^3 \text{ Hz}}{7 \times 10^3 \text{ Hz}} \]

Calculating this gives:
\[ M = \frac{50}{7} \approx 7.14 \]

### Step 3: Calculate the carrier swing
The carrier swing is defined as twice the frequency deviation (Δf):
\[ \text{Carrier Swing} = 2 \times \Delta f \]

Substituting the value of Δf:
\[ \text{Carrier Swing} = 2 \times 50 \times 10^3 \text{ Hz} = 100 \times 10^3 \text{ Hz} = 100 \text{ kHz} \]

### Final Results
- Modulation Index (M) = 7.14
- Carrier Swing = 100 kHz