# Amplitude Modulation

Amplitude modulation is a process by which the wave signal is transmitted by modulating the amplitude of the signal. It is often called as AM and is commonly used in transmitting a piece of information through a radio carrier wave. Amplitude modulation is mostly used in the form of electronic communication.

Currently, this technique is used in many areas of communication such as in portable two-way radios; citizens band radio, VHF aircraft radio and in modems for computers. Amplitude modulation is also used to mention the mediumwave AM radio broadcasting.

## What is Amplitude Modulation?

Amplitude modulation or just AM is one of the earliest modulation methods that is used in transmitting information over the radio. This technique was devised in the 20th century at a time when  Landell de Moura and Reginald Fessenden were conducting experiments using a radiotelephone in the 1900s. After successful attempts, the modulation technique was established and used in electronic communication.

In general, amplitude modulation definition is given as a type of modulation where the amplitude of the carrier wave is varied in some proportion with respect to the modulating data or the signal.

As for the mechanism, when amplitude modulation is used there is a variation in the amplitude of the carrier. Here, the voltage or the power level of the information signal changes the amplitude of the carrier. In AM, the carrier does not vary in amplitude. However, the modulating data is in the form of signal components consisting of frequencies either higher or lower than that of the carrier. The signal components are known as sidebands and the sideband power is responsible for the variations in the overall amplitude of the signal.

The AM technique is totally different from frequency modulation and phase modulation where the frequency of the carrier signal is varied in the first case and in the second one the phase is varied respectively.

## Types of Amplitude Modulation

There are three main types of amplitude modulation. They are;

• Double sideband-suppressed carrier modulation (DSB-SC).
• Single Sideband Modulation (SSB).
• Vestigial Sideband Modulation (VSB).

## Communication Systems and Modulation

We are studying modulation under communication systems. They are used to transmit and receive the message (information) from one place to another place in the form of electronic signals and they carried out in two different ways.

(i) Analog signal transmission

(ii) Digital signal transmission

So we can represent an analogue electronic signal (information) as follows;

$\left.\begin{matrix} m(t)={{A}_{m}}\cos ({{\omega }_{m}}t+\theta ) \\ (or) \\ m(t)={{A}_{m}}\sin ({{\omega }_{m}}t+\theta ) \end{matrix}\right\}…………….1$.

We can represent the analogue electronic signal either as sine (or) cosine wave. Every wave will have an amplitude and phase.

Where m(t) – modulating signal (input signal) (or) Baseband signal.

Am 🡪 Amplitude of the modulating signal (ωmt + Ɵ) 🡪 phase of the signal phase contains both frequency (ωmt) ad angle (Ɵ) term.

### What is Modulation?

Basically, it is a process in a communication system. For communication, we need some fundamental elements. One is the high-frequency carrier wave and the other is the information which has to be transmitted (modulating signal) (or) input signal. These are essential for communication which is done using a device from one place to another. All in all, we need the help of the communication system.

An electronic communication system converts our message (information) into an electronic signal and the electronic signal carried out by carrier waves to the destination.

Message (information)

(or)

Modulating signal

Superposition of modulating signal onto a carrier wave is known as modulation.

Modulation is defined as,

Varying any one of the fundamental parameters of a carrier wave in accordance with the modulating signal. A carrier wave can be represented as a sine (or) cosine.

C(t) = Ac sin (ωct + Ɵ)

### Amplitude Phase

If we vary the amplitude of the carrier wave in accordance with the modulating signal (input signal) it is known as Amplitude Modulation.

Similarly, it can be frequency modulation and phase modulation also in other words modulation is the phenomenon of “Superimposition of modulating signal (input signal) into the carrier wave”.

## Why Do We Need Modulation?

Practically speaking, modulation is required for;

• High range transmission
• Quality of transmission
• To avoid the overlapping of signals.

### High Range Transmission: (Effective Length of Antenna)

For effective communication, the length of the antenna should be $\frac{\lambda }{4}$times of the modulating signal.

Hmin = $\frac{\lambda }{4}$

λ – wavelength of the modulating signal (or) transmitting signal H> $\frac{\lambda }{4}$

(e – x) if i ned to transmit signal of frequency of f = 20 kHz as we know c = 8 λ

3 × 108 = 20 × 103 (λ)

$\frac{3\times {{10}^{8}}}{2\times {{10}^{4}}}=\lambda$ $\lambda =\frac{3}{2}\times {{10}^{4}}$

So ${{H}_{\min }}\simeq \frac{\lambda }{4}=\frac{3}{8}\times {{10}^{4}}=0.375\times {{10}^{4}}m$

Hmin = 3750 m

Hmin = 3750 m is practically impossible, for that we can transmit our modulating signal onto a carrier wave of frequency 1MHz, what we did? We raised our transmission frequency from 20kHz to 1mHz.

Now let us what is the Hmin needed for good transmission?

c = fλ

3 × 108 = 1×106 (λ) If we increase transmitting frequency, wavelengths

$\frac{3\times {{10}^{8-6}}}{1}=\lambda$ will decreases so, Hmin also decreases.

3 × 102 = λ $\Rightarrow f\uparrow and \; \lambda \downarrow \to H\min \downarrow$

λ = 300 m

Hmin = ${}^{\lambda }/{}_{4}=\frac{300}{4}=75m$

Which is practically possible, so to transmit a low-frequency signal we need modulation to increase the transmission frequency.

### Quality of Transmission: (Power of Transmission by Antenna)

Since, from Q-factor, we know sharpness (or) quality is maximum when power is maximum

Sharpness (or) quality α power

Power radiated by a linear antenna is

$P\alpha \frac{1}{{{\lambda }^{2}}}\Rightarrow Power=\frac{\ell }{{{\lambda }^{2}}}$

Where $\ell \to$ length of the antenna

$\lambda \to$ wavelength of the transmitting signal

$\left.\begin{matrix} for\,small(\lambda ) \\ (or) \\ High\,frequency \end{matrix}\right\} \to Transmission\,power\,is\,high\to quality\,of\,transmission\,high$.

$\left.\begin{matrix} for\,high(\lambda ) \\ (or) \\ small\,frequency \end{matrix}\right\} \to Transmission\,power\,is\,small\to quality\,of\,transmission\,low$

### Avoiding the Overlapping of Signals

Two different transmitting stations transmits signal of same frequency they will get mixed up (or) over lap one on other to avoid this we need to modulate this signals by different carriers waves.

When we talk about amplitude modulation it is a technique that is used to vary the amplitude of the high-frequency carrier wave in accordance with the amplitude of the modulating signal. But the frequency of the carrier wave remains constant. Now let us see, what are carrier wave and modulating signal.

## Common Terms

Carrier Wave (High frequency)

Amplitude and frequency of carrier wave remains constant generally it will high frequency generally it will be sine (or) cosine wave of electronic signal it can be represented as

C(t) = Ac sin wct ……………. 1

Modulating Signal

Modulating signal is nothing input signal (electronic signal), which has to be transmitted it is also sine (or) cosine wave it can be represented as

m(t) = Am sin wmt

Where

Ac and Am 🡪 Amplitude of the carrier wave and the modulating signal.

Sin wct 🡪 phase of the carrier wave

Sin wmt 🡪 phase of the modulating signal

## Expression for Amplitude Modulated Wave

We have carrier wave and modulating signal,

$\left.\begin{matrix} m(t)={{A}_{m}}\,\sin \,{{\omega }_{m}}t\,and \\ \\ c(t)={{A}_{c}}\sin {{\omega }_{c}}t \end{matrix}\right\} \to 1$.

m(t) 🡪 modulating signal

c(t) 🡪 carrier wave.

Am and Ac 🡪 are Amplitude of modulating signal and carrier wave respectively in Amplitude modulation. We are superimposing modulating signal into carrier wave and also varying the amplitude of the carrier wave in accordance with the amplitude of the modulating signal and the amplitude-modulated wave Cm(t) will be

Cm(t) = (Ac + Am sin ωmt) sin ωct ……………….. 2

This is the general form of amplitude modulated wave Cm(t) 🡪 is the amplitude-modulated wave.

Where,

A = Ac + Am sin ωmt → is the amplitude of the modulated wave

Sin wct → phase of modulated wave

${{C}_{m}}(t)={{A}_{c}}\left( 1+\frac{{{A}_{m}}}{{{A}_{c}}}\sin \,{{\omega }_{m}}t \right)\sin {{\omega }_{c}}t$

= ${{A}_{c}}\sin {{\omega }_{c}}t+\frac{{{A}_{m}}}{{{A}_{c}}}{{A}_{c}}\sin {{\omega }_{m}}t\,\sin {{\omega }_{c}}t$

Where,

$\frac{{{A}_{m}}}{{{A}_{c}}}=\mu =\bmod ulation\,index$

Cm (t) = Ac sin ωct + Acμ sin ωmt

We can rewrite the above equation as

$\sin A\sin B=\frac{1}{2}\left[ \cos (A-B)-\cos (A+B) \right]$ $={{A}_{c}}\sin {{\omega }_{c}}t+{{A}_{c}}\mu \frac{1}{2}\left[ \cos ({{\omega }_{c}}-{{\omega }_{m}})-\cos \left( {{\omega }_{c}}+{{\omega }_{m}} \right) \right]$ ${{c}_{m}}(t)={{A}_{c}}\sin {{\omega }_{c}}t+\frac{{{A}_{c}}\mu }{2}\cos ({{\omega }_{c}}-{{\omega }_{m}})-\frac{{{A}_{c}}\mu }{2}\cos \left( {{\omega }_{c}}+{{\omega }_{m}} \right)……\,3$

From equation 3 we can see Amplitude modulated wave is sum of three sine (or) cosine waves.

### Frequencies of Amplitude Modulated Wave

There are three frequencies in amplitude modulated wave f1, f2 and f3 corresponding to ωc, ωc + ωm and ωc – ωm respectively.

ω1 = ωc → it is corresponding f1 = fc

ω2 = ωc + ωm → it is corresponding f2 = fc + fm

ω3 = ωc – ωm → it is corresponding f3 = fc – fm

where fc → carrier wave frequency

fc + fm → upper side band frequency

fc – fm → lower side band frequency

fm → modulating signal frequency

in general fc > > fm

Band width: (W)

W = upper side band frequency – lower side band frequency (fc – fm)

w = fc + fm – fc + fm = 2 fm

w = 2fm = twice the frequency of modulating signal

Modulation Index

Is the ratio of Amplitude of modulating signal to amplitude of the carrier wave.

$\mu =\frac{{{A}_{m}}}{{{A}_{c}}}=\frac{Amplitude\,of\bmod ulating\,signal}{Amplitude\,of\,carrier\,wave}$

Waveform representation of Amplitude modulated wave:

1. Carrier wave →

2. Modulating signal →

3. Super position of the carrier wave and →

modulating signal

4. Amplitude modulated wave →

Summary:

Carrier wave, c(t) = Ac sin wct

Modulating single m(t) = Am sin wmt

Amplitude modulate wave (m(t) = (Ac + Am sin ωmt) sin ωct

$\bmod ulatin\,index(\mu )=\frac{{{A}_{m}}}{{{A}_{c}}}=\frac{Amplitude\,of\bmod ulcting\sin gal}{Amplitude\,carrier\,wave}$

Frequencies of modulated wave → fc, fc + fm, fc – fm

Bandwidth (w) = fc + fm – (fc – fm) = 2fm

 Advantages Disadvantages Amplitude Modulation is easier to implement. When it comes to power usage it is not efficient. Demodulation can be done using few components and a circuit. It requires a very high bandwidth that is equivalent to that of the highest audio frequency. The receiver used for AM is very cheap. Noise interference is highly noticeable.

## Solved Problems

Ex 1

Carrier wave of frequency f = 1mHz with pack voltage of 20V used to modulated a signal of frequency 1kHz with pack voltage of 10v. Find out the following

(i) μ?

(ii) Frequencies of modulated wave?

(iii) Bandwidth

Solution:

(i) $\mu =\frac{{{A}_{m}}}{{{A}_{c}}}=\frac{10v}{20v}=\frac{1}{2}=0.5$

(ii) frequencies of modulated wave

f → fc, fc + fm and fc – fm

fc = 1mHz, fm = 1kHz

fc + fm = 1×106 + 1×103 = 1001 ×103 = 1001 kHz

fc – fm = 1×106 – 1×103 = 999 × 103 = 999 kHz

(iii) Band width: (W)

(W) = upper side band frequency – lower side band frequency

= fc + fm – (fc – fm)

= 2fm = 1001 kHz – 999 kHz = 2 kHz

Ex: 2

y = 10 cos (1800 πt) + 20 cos 2000 πt + 10 cos 220 πt. Find the modulation index (μ) of the given wave.

Solution:

As we know the expression for amplitude modulated wave.

Cm(t) = (Ac + Am cos ωmt) cos ωct ……………… 1

So, we have to bring the given wave equation into known form

y = 10 [cos(1800 πt) + cos (2200πt)] + 20 cos 2000 πt we can rewrite the above equation as

$\cos A+\cos B=2\cos \left( \frac{A+B}{2} \right)\cos \left( \frac{A-B}{2} \right)$

cos(2000 πt) + cos (1800 πt) = 2 cos 2000 πt cos 200 πt

$y=10\left[ 2\cos (2000\pi t)\cos (200\pi t) \right]+20\cos 2000\pi t$ $y=\cos 2000\pi t\left[ 20+20\cos (200\pi t) \right]……2$

Compare equation 1 and 2

Ac = 20

Am = 20

Then modulation index (μ)

$\mu =\frac{Am}{Ac}=\frac{20}{20}=1$

We can also find the frequencies of the modulated wave and B and width

(ii) Frequencies off the modulated wave:

We know frequencies are fc, fc + fm and fc – fm from the modulated wave expression, we can get it.

$y=\cos 2000\pi t[20+20\cos (2000\pi t)]\,…………\,3$

y = (Ac + Am cos ωmt) cos ωct ……………….. 4

Comparing the equation 3 and 4

Cos ωmt = cos (200 πt)

ωm = 200 π

2 πfm = 200 π

fm = 100 Hz

Similarly,

cos ωct = cos 2000 πt

ωc = 2000 π

2πfc = 2000 π

fc = 1000 Hz

fc, fc + fm and fc – fm respectively 1000 Hz, 1100 Hz and 900 Hz.

(iii) Bandwidth (W)

W = fc + fm – (fc – fm) = 2fm

W = 200 Hz

## NCERT Questions on Amplitude Modulation

1. Why carrier waves are of higher frequency compared to modulating signal?

(i) High-frequency carrier wave, effectively reduces the size of antenna which increases transmission range.

(ii) Converts wideband signal into a narrowband signal which can easily be recovered at the receiving end.

2. Define modulation index?

The modulation index is defined as the ratio of the amplitude of the modulating signal to the amplitude of the carrier wave (μ)

$\mu =\frac{{{A}_{m}}}{{{A}_{c}}}$

3. What happens if μ > 1?

As we know the range of modulation index (μ) should 0 < μ < 1 if μ > 1 it is said to be over modulated and distortion will take place in the modulated signal.

4. Why we need modulation?