Question

A \(1000 \mathrm{KHz}\) carrier is modulated with \(800 \mathrm{~Hz}\) audio signals. What are the frequencies of first pair side band? (A) \(1000 \mathrm{KHz}\) and \(800 \mathrm{~Hz}\) (B) \(800 \mathrm{~Hz}\) and \(999.2 \mathrm{KHz}\) (C) \(1000.8 \mathrm{KHz}\) and \(999.2 \mathrm{KHz}\) (D) \(1000.8 \mathrm{KHz}\) and \(1000 \mathrm{KHz}\)

Short answer

The short answer is: \(1000.8 \mathrm{KHz}\) and \(999.2 \mathrm{KHz}\).

Step-by-step solution

Identify the carrier frequency and modulating frequency

We are given a carrier signal with a frequency of 1000 KHz and a modulating audio signal with a frequency of 800 Hz. Let's represent these frequencies as 𝑓𝑐 (carrier frequency) and 𝑓𝑚 (modulating frequency).

Convert frequencies to the same unit

Before performing any calculations, let's convert the modulating frequency to the same unit as the carrier frequency: 800 Hz = 0.8 KHz Now, 𝑓𝑐 = 1000 KHz and 𝑓𝑚 = 0.8 KHz.

Calculate the sideband frequencies

Using the formulas for the upper sideband (USB) and the lower sideband (LSB) frequencies resulting from the modulation process, we can now calculate the frequencies of the first pair of sidebands: Upper Sideband Frequency (USB) = 𝑓𝑐 + 𝑓𝑚 Lower Sideband Frequency (LSB) = 𝑓𝑐 - 𝑓𝑚

Plug in the values and compute the sideband frequencies

Let's plug in the values for 𝑓𝑐 and 𝑓𝑚 and compute the USB and LSB frequencies: USB = 1000 KHz + 0.8 KHz = 1000.8 KHz LSB = 1000 KHz - 0.8 KHz = 999.2 KHz

Determine the correct answer

Now that we have calculated the frequencies of the first pair of sidebands, we can determine the correct option: (A) 1000 KHz and 800 Hz (B) 800 Hz and 999.2 KHz (C) 1000.8 KHz and 999.2 KHz (D) 1000.8 KHz and 1000 KHz The correct answer is (C) 1000.8 KHz and 999.2 KHz.

Key concepts

Carrier Frequency

In the context of Amplitude Modulation (AM), the "carrier frequency" is a crucial element. It refers to the unmodulated frequency of the carrier signal. Think of it as the main signal that is "carrying" the information. The carrier frequency determines the main operating frequency of the transmitters and receivers involved in the communication. The carrier frequency is represented by the symbol \( f_c \). It is typically a much higher frequency compared to the modulating frequency. This allows the effective transmission of signals over long distances. In our exercise, the carrier frequency is 1000 KHz.

Sideband Frequencies

Sideband frequencies are the result of the modulation process. When the carrier frequency is modulated with an input frequency, it creates additional frequencies called sidebands. These sidebands appear on both sides of the carrier frequency. They consist of the upper and lower sidebands, which contain the real informational data of the signal. The presence of these sidebands allows AM radio to transmit audio signals over large distances.

• Upper Sideband (USB): The sum of the carrier and modulating frequencies.
• Lower Sideband (LSB): The difference between the carrier and modulating frequencies.

Modulating Frequency

The "modulating frequency" or "audio frequency" in amplitude modulation is the frequency of the signal that contains the actual information, such as voice or music. In this exercise, it is 800 Hz. The modulating frequency works by varying the amplitude of the carrier frequency, which is a much higher frequency. By converting these audio signals into radio frequencies, they can travel over long distances and be picked up by receivers. Converting the modulating frequency from Hz to KHz helps maintain a uniform unit of measurement. Here, 800 Hz is converted to 0.8 KHz for calculations.

Upper Sideband

The upper sideband (USB) is one of the sideband frequencies resulting from the process of modulation. It is found by adding the carrier frequency \( f_c \) and the modulating frequency \( f_m \). The formula is: \[USB = f_c + f_m\]This frequency lies above the carrier frequency, hence the name "upper sideband." In the context of our problem, the USB is calculated as follows:\[ 1000 ext{ KHz} + 0.8 ext{ KHz} = 1000.8 ext{ KHz} \] Thus, the upper sideband is at a frequency of 1000.8 KHz, indicating the upper limit of the modulated signal range.

Lower Sideband

The lower sideband (LSB) is the frequency below the carrier frequency that emerges in the modulation process. You can calculate it by subtracting the modulating frequency from the carrier frequency using the equation: \[LSB = f_c - f_m\]The lower sideband is located below the carrier frequency, hence referred to as the "lower sideband." In our specific example, it is calculated as: \[ 1000 ext{ KHz} - 0.8 ext{ KHz} = 999.2 ext{ KHz} \]This LSB value represents the other frequency in the first pair of sideband frequencies, indicating the lower boundary of the modulated signal range.