Question

A signal wave of frequency \(12 \mathrm{KHz}\) is modulated with a carrier wave of frequency \(2.51 \mathrm{MHz}\). The upper and lower side band frequency are respectively. (A) \(2512 \mathrm{~Hz}\) and \(2508 \mathrm{KHz}\) (B) \(2522 \mathrm{KHz}\) and \(2488 \mathrm{KHz}\) (C) \(2502 \mathrm{KHz}\) and \(2498 \mathrm{KHz}\) (D) \(2522 \mathrm{KHz}\) and \(2498 \mathrm{KHz}\)

Short answer

The upper and lower sideband frequencies are 2522 KHz and 2498 KHz, respectively. So the correct answer is (D).

Step-by-step solution

Convert frequencies to the same unit

Convert the signal frequency to MHz and the carrier frequency to KHz, or both to Hz. In this case, we will convert both to KHz. Signal frequency = 12 KHz (already in KHz) Carrier frequency = 2.51 MHz = 2510 KHz

Calculate the upper sideband frequency

Add the signal frequency to the carrier frequency to calculate the upper sideband frequency. Upper sideband frequency = Carrier frequency + Signal frequency = 2510 KHz + 12 KHz = 2522 KHz

Calculate the lower sideband frequency

Subtract the signal frequency from the carrier frequency to calculate the lower sideband frequency. Lower sideband frequency = Carrier frequency - Signal frequency = 2510 KHz - 12 KHz = 2498 KHz

Identify the correct answer

Now that we have calculated the upper and lower sideband frequencies, we can compare our results to the given options and determine the correct answer. The upper sideband frequency is 2522 KHz, and the lower sideband is 2498 KHz. Comparing these values to the given options, we see that the correct answer is: (D) 2522 KHz and 2498 KHz

Key concepts

Upper Sideband Frequency

In amplitude modulation (AM), the upper sideband frequency is a fundamental concept. It is formed by adding the signal frequency to the carrier frequency. This phenomenon occurs when the signal you want to transmit modulates the carrier wave.
The upper sideband (USB) frequency is important because it carries a portion of the information from the original signal. By knowing how to calculate the USB, you'll understand how information is transmitted in a modulated wave.
For instance, if you start with a signal frequency of 12 KHz and a carrier frequency of 2.51 MHz, you convert them to the same unit, usually KHz. Thus, the process involves:

• Converting 2.51 MHz to 2510 KHz
• Adding 12 KHz to 2510 KHz

Eventually, you arrive at an upper sideband frequency of 2522 KHz.

Lower Sideband Frequency

The lower sideband frequency (LSB) plays a crucial role in the spectrum of a modulated signal. It refers to the frequency generated when the signal frequency is subtracted from the carrier frequency. This frequency also carries part of the transmitted information.
Similar to the upper sideband, obtaining the lower sideband is essential for understanding how information is separated and transmitted across different frequencies. Let's illustrate this with numbers:

• Take a carrier wave frequency of 2.51 MHz, or 2510 KHz.
• Subtract the signal frequency, 12 KHz, from 2510 KHz.

Through this calculation, you find the lower sideband frequency to be 2498 KHz. This calculation shows how a modulated signal divides into different frequency components.

Carrier Wave

The carrier wave is the cornerstone of the amplitude modulation process. It is a high-frequency signal that, unlike the signal wave, does not carry information by itself but serves to "carry" the information signal into the desired frequency spectrum. The carrier frequency is a fixed frequency around which the entire modulation process revolves.
In our example, the carrier frequency is 2.51 MHz. This frequency lies well above the typical human hearing range, allowing the modulated signal to travel long distances.
In AM, the carrier wave frequency determines the central point of modulation, influencing the positions of both upper and lower sidebands in the frequency spectrum. By manipulating this central frequency, various frequencies can coexist without interfering, which is essential in radio broadcasting.

Signal Frequency

Signal frequency is the original frequency of the information-carrying wave before it undergoes modulation. This can be any form of wave, such as voice, music, or data, that you wish to transmit. In our scenario, the signal frequency is 12 KHz.
While the signal itself cannot be sent directly over large distances, it modulates the high-frequency carrier wave, making long-distance transmission feasible. Through AM, the signal frequency is superimposed onto the carrier wave, creating new frequencies in the form of sidebands.
The modulation process thus enables these lower frequency signals to hitch a ride on higher frequency waves. This fundamental principle of AM allows for effective transmission of information over vast distances, such as radio or television broadcasting.