# Modulation - Principles of Communication

## What is Amplitude Modulation?

Among the varieties of modulation techniques, the key classification is Continuous-wave Modulation and Pulse Modulation. The uninterrupted wave modulation methods are further separated into Amplitude Modulation and Angle Modulation.

A continuous-wave goes on endlessly without any intermissions and it is the baseband message signal, which comprises the information. This wave has to be modulated.

According to the standard definition, “The amplitude of the carrier signal differs in unity with the immediate amplitude of the modulating signal.” Which means, the amplitude of the carrier signal which comprises no information differs as per the amplitude of the signal, at each instant, which contains information? This can be well described by the resulting figures.

The modulating wave which is presented first is the message signal. The following one is the carrier wave, which is just a high frequency signal and comprises no information. While the last one is the subsequent modulated wave.

It can be detected that the positive and negative peaks of the carrier wave, are organized with an imagined line. This line helps re-forming the exact shape of the modulating signal. This imaginary line on the carrier wave is called as Envelope. It is the similar as the message signal.

## Mathematical Expression

Resulting are the mathematical expression for these waves.

### Time-domain Representation of the Waves

Let modulating signal be −

$m\left(t\right)={A}_{m}cos\left(2\pi {f}_{m}t\right)$ Let carrier signal be −
$c\left(t\right)={A}_{c}cos\left(2\pi {f}_{c}t\right)$ Where Am = maximum amplitude of the modulating signal
Ac = maximum amplitude of the carrier signal
The standard form of an Amplitude Modulated wave is defined as −
$S\left(t\right)={A}_{c}\left[1+{K}_{a}m\left(t\right)\right]cos\left(2\pi {f}_{c}t\right)$ $S\left(t\right)={A}_{c}\left[1+\mu cos\left(2\pi {f}_{m}t\right)\right]cos\left(2\pi {f}_{c}t\right)$ $Where,\mu ={K}_{a}{A}_{m}$

## Modulation Index

A carrier wave, after being modulated, if the modulated level is calculated, then such an attempt is called as Modulation Index or Modulation Depth. It states the level of modulation that a carrier wave undergoes.

The maximum and minimum values of the envelope of the modulated wave are represented by Amax and Amin respectively.
Let us try to develop an equation for the Modulation Index.

${A}_{max}={A}_{c}\left(1+\mu \right)$

Since, at Amax the value of cos θ is 1

${A}_{min}={A}_{c}\left(1-\mu \right)$

Since, at Amin the value of cos θ is -1

$\frac{{A}_{max}}{{A}_{min}}=\frac{1+\mu }{1-\mu }$ ${A}_{max}-\mu {A}_{max}={A}_{min}+\mu {A}_{min}$ $-\mu \left({A}_{max}+{A}_{min}\right)={A}_{min}-{A}_{max}$ $\mu =\frac{{A}_{max}-{A}_{min}}{{A}_{max}+{A}_{min}}$

Therefore, the equation for Modulation Index is attained. µ means the modulation index or modulation depth. This is often signified in percentage called as Percentage Modulation. It is the degree of modulation denoted in percentage, and is denoted by m.

For an impeccable modulation, the value of modulation index should be 1, which means the modulation depth should be 100%.

For example, if this value is less than 1, i.e., the modulation index is 0.5, and then the modulated output would look like the resulting figure. It is called as Under-modulation. Such a wave is called as an under-modulated wave.

If the value of the modulation index is greater than 1, i.e., 1.5 or so, then the wave will be an over-modulated wave. It would look like the resulting figure.

As the value of modulation index rises, the carrier experiences a 180° phase reversal, which origins additional sidebands and therefore, the wave gets distorted. Such over modulated wave roots interference, which cannot be removed.

## Bandwidth of Amplitude Modulation

The bandwidth is the alteration amongst lowest and highest frequencies of the signal.

For amplitude modulated wave, the bandwidth is given by

$BW={f}_{USB}-{f}_{LSB}$ $\left({f}_{c}+{f}_{m}\right)-\left({f}_{c}-{f}_{m}\right)$ $=2{f}_{m}=2W$

Where W is the message bandwidth

Therefore we got to know that the bandwidth necessary for the amplitude modulated wave is twice the frequency of the modulating signal.