The audio signal voltage is given by `V_(m) = 2 sin 12 pi xx 10^(3)t`. The band width and LSB if carrier wave has a frequency `3.14 xx10^(6) "rad"//s`.

Options

A. 12 KHz, 494 KHz
B. 6 KHz, 313 KHz
C. 6 KHz, 494 KHz
D. 18 KHz, 494 KHz

Answer

To solve the problem, we need to find the bandwidth and the lower side band (LSB) frequency based on the given audio signal voltage and the carrier wave frequency. Let's break down the solution step by step.

### Step 1: Identify the frequency of the audio signal
The audio signal voltage is given by:
\[ V_m = 2 \sin(12 \pi \times 10^3 t) \]

From this equation, we can identify the frequency of the audio signal, \( f_m \). The general form of the sine function is:
\[ V_m = A \sin(2 \pi f_m t) \]

Comparing the two equations, we see that:
\[ 2 \pi f_m = 12 \pi \times 10^3 \]

To find \( f_m \), we can simplify:
\[ f_m = \frac{12 \pi \times 10^3}{2 \pi} = 6 \times 10^3 \text{ Hz} = 6 \text{ kHz} \]

### Step 2: Calculate the bandwidth
The bandwidth \( B \) of a communication system is given by:
\[ B = 2 f_m \]

Substituting the value of \( f_m \):
\[ B = 2 \times 6 \text{ kHz} = 12 \text{ kHz} \]

### Step 3: Convert the carrier frequency from radians per second to Hertz
The carrier wave frequency is given as:
\[ f_c = 3.14 \times 10^6 \text{ rad/s} \]

To convert this to Hertz, we use the formula:
\[ f_c = \frac{\omega_c}{2\pi} \]
where \( \omega_c \) is the angular frequency in radians per second.

Calculating \( f_c \):
\[ f_c = \frac{3.14 \times 10^6}{2 \pi} \]

Calculating \( 2 \pi \):
\[ 2 \pi \approx 6.28 \]

Now substituting:
\[ f_c = \frac{3.14 \times 10^6}{6.28} \approx 500 \text{ kHz} \]

### Step 4: Calculate the lower side band (LSB) frequency
The lower side band frequency (LSB) is calculated using the formula:
\[ \text{LSB} = f_c - f_m \]

Substituting the values:
\[ \text{LSB} = 500 \text{ kHz} - 6 \text{ kHz} = 494 \text{ kHz} \]

### Final Answer
Thus, the bandwidth and lower side band frequency are:
- Bandwidth = 12 kHz
- Lower Side Band (LSB) = 494 kHz