Find the lower and upper side band frequencies if a signal wave of frequency 12kHz is modulated with a carrier wave of 2.51 MHz

Options

A. 2512 kHz and 2508 kHz
B. 2522 kHz and 2488 kHz
C. 2522 kHz and 2498 kHz
D. 2512 kHz and 2488 kHz

Explanation

To find the lower and upper sideband frequencies when a signal wave is modulated with a carrier wave, we can use the following formulas:

1. **Upper Sideband Frequency (USB)**:
\[ f_{USB} = f_C + f_M \]

2. **Lower Sideband Frequency (LSB)**:
\[ f_{LSB} = f_C - f_M \]

Where:
- \( f_C \) is the frequency of the carrier wave.
- \( f_M \) is the frequency of the modulating signal.

### Step-by-Step Solution

**Step 1: Identify the frequencies.**
- The frequency of the modulating signal \( f_M = 12 \text{ kHz} = 12 \times 10^3 \text{ Hz} \).
- The frequency of the carrier wave \( f_C = 2.51 \text{ MHz} = 2.51 \times 10^6 \text{ Hz} \).

**Step 2: Calculate the Upper Sideband Frequency (USB).**
Using the formula for USB:
\[ f_{USB} = f_C + f_M \]
Substituting the values:
\[ f_{USB} = (2.51 \times 10^6) + (12 \times 10^3) \]
\[ f_{USB} = 2.51 \times 10^6 + 0.012 \times 10^6 = 2.522 \times 10^6 \text{ Hz} \]
Thus, the upper sideband frequency is:
\[ f_{USB} = 2.522 \text{ MHz} \]

**Step 3: Calculate the Lower Sideband Frequency (LSB).**
Using the formula for LSB:
\[ f_{LSB} = f_C - f_M \]
Substituting the values:
\[ f_{LSB} = (2.51 \times 10^6) - (12 \times 10^3) \]
\[ f_{LSB} = 2.51 \times 10^6 - 0.012 \times 10^6 = 2.498 \times 10^6 \text{ Hz} \]
Thus, the lower sideband frequency is:
\[ f_{LSB} = 2.498 \text{ MHz} \]

### Final Answer
- Upper Sideband Frequency (USB): **2.522 MHz**
- Lower Sideband Frequency (LSB): **2.498 MHz**