Key Equations and Concepts in Amplitude Modulation
Amplitude modulation is considered to be a process in which the wave signals are transmitted by modulating the amplitude of the signal. The amplitude modulation is often called AM. This technique was established in the 20th century by Landell de Moura and Reginald Fessenden when they were conducting experiments using a radiotelephone. It is commonly used in the transmission of information through a radio carrier wave. It is mostly used in the form of electronic communication.
The amplitude modulation technique is used in different areas of communication. Some such areas of communication are portable two-way radios, citizens band radio, VHF aircraft radio. Also, in the modems of the computers Amplitude modulation is also used in mentioning the medium wave AM radio broadcasting.
Amplitude Modulation Equation Derivation
Here, one will get a clear knowledge of how to derive an expression for amplitude modulation.
The mathematical representation of amplitude-modulated waves in the time domain is as follows.
m(t) = Am cos (2πfmt) (modulating signal)
m(t) = Ac cos (2πfct) (carrier signal)
s(t) = [ Ac + Am cos (2πfmt) ] cos (2πfct) (equation of amplitude modulated wave)
Where,
Am: Amplitude of modulating signal
Ac: Amplitude of carrier signal
fm: Frequency of modulating signal
fc: Frequency of carrier signal
From the above information, a student can learn how to derive an expression for amplitude modulated waves.
Modulation Index Derivation
The modulation index is also considered as modulation depth. It is defined for the carrier wave in order to describe the modulated variable of the carrier signal, which is showing some variation with respect to its unmodulated level. The modulation index can be given as:
\[\mu =\frac{A_{m}}{A_{c}}\]
Consider the maximum and minimum wave amplitudes as Amax and Amin.
Considering the equation cos (2πfmt), the following two equations are derived, which are having the maximum and minimum amplitude of the modulated waves.
Amax = Ac + Am
Amax = Ac - Am
Amax + Amin = Ac + Am + Ac - Am = 2 Ac ⇒ \[A_{c}=\frac{A_{max}+A_{min}}{2}\]
Amax - Amin = Ac + Am - (Ac - Am) = 2 Am ⇒ \[A_{m}=\frac{A_{max}-A_{min}}{2}\]
\[\mu =\frac{A_{m}}{A_{c}}\]
From this above information, students can learn about the concept of modulation index derivation.
Types of Amplitude Modulation
From the above information, students have gained a clear knowledge of what amplitude modulation is and how to derive an expression for amplitude modulated wave. Now, they will have a clear knowledge of the types of amplitude modulation from the below-given information.
There are three types of amplitude modulation, and they are discussed as follows.
Double Sideband-suppressed carrier (DSB - SC) Modulation
In this type of amplitude modulation, the transmitted wave only has two sidebands, which are the upper sideband and the lower sideband. The channel bandwidth requirement here is the same as before.
Single Sideband (SSB) Modulation
In this type of amplitude modulation, the transmission wave consists of only one sideband, either it can be the upper band or the lower band. It is required for translating the spectrum of the modulating signal into a new location that is present in the frequency domain.
Vestigial Sideband (VSB) Modulation
In this type of amplitude modulation, one sideband has almost passed through the transmission wave, and just a trace of the other sideband is retained. Here, the requirement of channel bandwidth is slightly more than the message bandwidth by an amount which is the same as the vestigial sideband's width.
Advantages and Disadvantages of Amplitude Modulation
Advantages
The advantages of amplitude modulation are as follows.
Amplitude modulation is considered to be economical and is easily obtainable.
The implementation of amplitude modulation is simple and easy.
It can be demodulated easily just by using a circuit with a smaller number of components.
The receivers of amplitude modulation are cheap and inexpensive as it does not require any specialized components for continuing the process.
Disadvantages
The disadvantages of amplitude modulation are as follows.
Amplitude modulation uses a lot of energy, which in turn reduces its efficiency.
Amplitude frequency is used several times in this process for modulating the signal by a carrier signal.
The original signal quality is poor in the receiving end, thus causing various troubles.
They generate a lot of noise, which can turn out to be irritating.
This application is limited to VHF and radios only.
It is possible only in one-to-one communication, which is a major drawback.
The process by which a wave signal is transmitted by modulating the amplitude of the signal. Called AM, it is commonly used in transmitting a piece of information through a radio carrier wave. It is mostly used in the form of electronic communication.
Established in the 20th century by Landell de Moura and Reginald Fessenden when they were conducting experiments using a radiotelephone, the technique of Amplitude modulation technique is used in various areas of communication like portable two-way radios; citizens band radio, and modems for computers. It is also used in medium wave AM radio broadcasting.
Derivation of Amplitude Modulation
m(t) = Am cos (2πfmt) (modulating signal)
m(t) = Ac cos (2πfct) (carrier signal)
s(t) = [ Ac + Am cos (2πfmt) ] cos (2πfct) (equation of amplitude modulated wave)
Where,
Am: Denotes the amplitude of a modulating signal
Ac: Denotes the amplitude of a carrier signal
fm: Denotes the frequency level of the modulating signal
Lfc: Denotes the frequency level of the carrier signal.
This helps to learn how to derive an expression for the amplitude modulated waves.
FAQs on Amplitude Modulation Derivation Explained Step-by-Step
1. What is amplitude modulation (AM) and what is its fundamental purpose in communication systems?
Amplitude Modulation (AM) is a modulation technique used in electronic communication where the amplitude of a high-frequency carrier wave is varied in proportion to the instantaneous amplitude of a lower-frequency message signal (like audio). The primary purpose of AM is to encode information onto a carrier signal that can travel long distances and be easily transmitted and received, which is not possible for the original low-frequency message signal on its own.
2. How do you derive the mathematical expression for an amplitude modulated (AM) wave for the 2025-26 CBSE syllabus?
To derive the expression for an AM wave, we start with two signals:
- The modulating signal (message): m(t) = Am cos(2πfmt)
- The carrier signal: c(t) = Ac cos(2πfct)
Here, Am and fm are the amplitude and frequency of the modulating signal, while Ac and fc are for the carrier signal. The amplitude of the carrier, Ac, is varied according to m(t). The instantaneous amplitude of the AM wave is Ac + m(t). This modulated amplitude is then multiplied by the carrier wave, giving the final expression for the AM wave, s(t):
s(t) = [Ac + Am cos(2πfmt)] cos(2πfct)
3. What is the modulation index (μ) in AM, and what is its physical significance?
The modulation index (μ), also known as modulation depth, is a measure of how much the carrier wave's amplitude is varied by the message signal. It is defined as the ratio of the amplitude of the modulating signal (Am) to the amplitude of the carrier signal (Ac):
μ = Am / Ac
Its significance lies in determining the quality of the transmission:
- μ < 1 (Under-modulation): This is the ideal condition for clear signal transmission without distortion.
- μ = 1 (100% modulation): The maximum amplitude of the AM wave is double the carrier amplitude, and the minimum is zero. This is the maximum modulation without distortion.
- μ > 1 (Over-modulation): This causes severe distortion as the carrier wave is cut off during negative peaks of the modulating signal, leading to loss of information.
4. What are the three frequency components that constitute an amplitude modulated wave?
An amplitude modulated wave is composed of three distinct frequency components:
- Carrier Frequency (fc): This is the original frequency of the high-frequency carrier wave.
- Upper Sideband (USB): This is a frequency component located at fc + fm, which is the sum of the carrier and modulating signal frequencies.
- Lower Sideband (LSB): This is a frequency component located at fc - fm, which is the difference between the carrier and modulating signal frequencies.
5. Why is the bandwidth required to transmit an AM signal twice the frequency of the message signal?
The bandwidth of a signal is the range of frequencies it occupies. In amplitude modulation, the signal consists of a carrier frequency (fc), an upper sideband (fc + fm), and a lower sideband (fc - fm). The total frequency spectrum spans from the lowest frequency (fc - fm) to the highest frequency (fc + fm). Therefore, the bandwidth is the difference between the highest and lowest frequencies:
Bandwidth = (fc + fm) - (fc - fm) = 2fm
This shows that the channel must have a bandwidth that is twice the highest frequency present in the modulating message signal to transmit all the information without loss.
6. In what scenarios would Single Sideband (SSB) modulation be preferred over standard AM?
Single Sideband (SSB) modulation is preferred over standard AM in scenarios where power efficiency and bandwidth conservation are critical. In standard AM, the carrier contains no information and consumes over two-thirds of the transmitted power, and both sidebands carry the same information. SSB saves power and bandwidth by transmitting only one of the sidebands and suppressing the carrier and the other sideband. This makes it ideal for long-distance voice communications like amateur radio and military applications.
7. Besides radio broadcasting, what are some important real-world applications of amplitude modulation?
While famous for AM radio, amplitude modulation has several other important applications:
- VHF Aircraft Radio: It is used for air-to-ground and air-to-air voice communication because AM circuits are simple and reliable.
- Quadrature Amplitude Modulation (QAM): A more sophisticated form of AM, QAM is used extensively in digital communication systems like DSL internet modems, cable modems, and Wi-Fi standards to transmit large amounts of data.
- Computer Modems: Early modems used AM principles to encode digital data into audible frequencies for transmission over telephone lines.
- Citizens Band (CB) Radio: AM is widely used for short-distance, two-way personal communication.
