# Amplitude Modulation Derivation

Amplitude modulation is a modulation technique commonly used for transmission of information via a radio carrier wave. This is the earliest modulation used in radio to transmit voice. It was developed by Landell de Moura and Reginald Fessenden in the year 1900 with the experiments of a radiotelephone. It finds applications in two-way radios, computer modems in the form of QAM, VHF aircraft radio and citizens band radio. Let us learn the amplitude modulation derivation and modulation index derivation in this article.

## Derivation of Amplitude Modulation

Mathematical representation of Amplitude Modulated waves in time domain

$$\begin{array}{l}m(t)=A_{m}cos(2\pi f_{m}t)\end{array}$$
(modulating signal)

$$\begin{array}{l}c(t)=A_{c}cos(2\pi f_{c}t)\end{array}$$
(carrier signal)

$$\begin{array}{l}s(t)=\left \lfloor A_{c}+A_{m}cos(2\pi f_{m}t) \right \rfloor cos(2\pi f_{c}t)\end{array}$$
(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

Therefore, above is the derivation of Amplitude Modulation.+
To know about modulation and demodulation in detail, watch the below video.

### Modulation Index Derivation

Modulation index is also known as modulation depth is defined for the carrier wave to describe the modulated variable of carrier signal varying with respect to its unmodulated level. It is represented as follows:

$$\begin{array}{l}\mu =\frac{A_{m}}{A_{c}}\end{array}$$

Consider maximum and minimum amplitudes of the wave as Amax and Amin

Depending upon cos(2𝜋fmt) following two equations are derived with maximum and minimum amplitude of the modulated waves.

$$\begin{array}{l}A_{max}=A_{c}+A_{m}\end{array}$$

$$\begin{array}{l}A_{min}=A_{c}-A_{m}\end{array}$$

$$\begin{array}{l}A_{max}+A_{min}=A_{c}+A_{m}+A_{c}-A_{m}=2A_{c}\end{array}$$

$$\begin{array}{l}\Rightarrow A_{c}=\frac{A_{max}+A_{min}}{2}\end{array}$$

$$\begin{array}{l}A_{max}-A_{min}=A_{c}+A_{m}-(A_{c}-A_{m})=2A_{m}\end{array}$$

$$\begin{array}{l}\Rightarrow A_{m}=\frac{A_{max}-A_{min}}{2}\end{array}$$

$$\begin{array}{l}\mu =\frac{A_{m}}{A_{c}}\end{array}$$

Therefore, this is the derivation of the modulation index.

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## Frequently Asked Questions – FAQs

### What is amplitude modulation?

Amplitude modulation is a modulation technique used in electronic communication, for transmitting messages via a radio carrier wave.

### What is a modulation index?

Modulation index is defined for the carrier wave to describe the modulated variable of carrier signal varying with respect to its unmodulated level.

### How is the modulation index represented?

$$\begin{array}{l}\mu =\frac{A_{m}}{A_{c}}\end{array}$$

### Who developed the amplitude modulation technique?

In 1900, Landell de Moura and Reginald Fessenden developed amplitude modulation.

### Give the equation for amplitude modulated wave

Amplitude modulated wave is given by

$$\begin{array}{l}s(t)=\left \lfloor A_{c}+A_{m}cos(2\pi f_{m}t) \right \rfloor cos(2\pi f_{c}t)\end{array}$$

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