# Amplitude Modulation Derivation

The amplitude modulation derivation is provided here. Amplitude modulation is 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’s 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.

## Derivation of Amplitude Modulation

Mathematical representation of Amplitude Modulated waves in time domain

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

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

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

### 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:

$\mu =\frac{A_{m}}{A_{c}}$

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.

$A_{max}=A_{c}+A_{m}$

$A_{min}=A_{c}-A_{m}$

$A_{max}+A_{min}=A_{c}+A_{m}+A_{c}-A_{m}=2A_{c}$
$\Rightarrow A_{c}=\frac{A_{max}+A_{min}}{2}$

$A_{max}-A_{min}=A_{c}+A_{m}-(A_{c}-A_{m})=2A_{m}$

$\Rightarrow A_{m}=\frac{A_{max}-A_{min}}{2}$

$\mu =\frac{A_{m}}{A_{c}}$

Therefore, this is the derivation of the modulation index.

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