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# 20/11/2018 Analog Communication Amplitude Modulation

AMPLITUDE MODULATION
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A continuous-wave goes on continuously without any intervals and it is the baseband message signal, which
contains the information. This wave has to be modulated.

According to the standard definition, “The amplitude of the carrier signal varies in accordance with the
instantaneous amplitude of the modulating signal.” Which means, the amplitude of the carrier signal containing
no information varies as per the amplitude of the signal containing information, at each instant. This can be well
explained by the following figures.

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The first figure shows the modulating wave, which is the message signal. The next one is the carrier wave, which is
a high frequency signal and contains no information. While, the last one is the resultant modulated wave.

It can be observed that the positive and negative peaks of the carrier wave, are interconnected with an imaginary
line. This line helps recreating the exact shape of the modulating signal. This imaginary line on the carrier wave is
called as Envelope. It is the same as that of the message signal.

Mathematical Expressions
Following are the mathematical expressions for these waves.

## Let the modulating signal be,

m (t) = Am cos(2πfm t)

## and the carrier signal be,

c (t) = Ac cos(2πfc t)

Where,

Am and A are the amplitude of the modulating signal and the carrier signal respectively.
c

fm and f are the frequency of the modulating signal and the carrier signal respectively.
c

## s(t) = [Ac + Am cos(2πfm t)] cos(2πfc t) Equation1

Modulation Index

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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.

## Rearrange the Equation 1 as below.

Am
s(t) = Ac [1 + ( ) cos(2πfm t)] cos(2πfc t)
Ac

## ⇒ s (t) = Ac [1 + μ cos(2πfm t)] cos(2πfc t) Equation2

Where, μ is Modulation index and it is equal to the ratio of A and A . Mathematically, we can write it as
m c

Am
μ =
Ac

Equation3

Hence, we can calculate the value of modulation index by using the above formula, when the amplitudes of the
message and carrier signals are known.

Now, let us derive one more formula for Modulation index by considering Equation 1. We can use this formula for
calculating modulation index value, when the maximum and minimum amplitudes of the modulated wave are
known.

Let A max and A min be the maximum and minimum amplitudes of the modulated wave.

We will get the maximum amplitude of the modulated wave, when cos(2πf m t) is 1.

⇒ Amax = Ac + Am Equation4

We will get the minimum amplitude of the modulated wave, when cos(2πf m t) is -1.

⇒ Amin = Ac − Am Equation5

A max +A min
⇒ Ac =
2

Equation6

A max −A min
⇒ Am =
2

Equation7

## The ratio of Equation 7 and Equation 6 will be as follows.

Am (Amax − Amin ) /2
=
Ac (Amax + Amin ) /2

A max −A min
⇒ μ = Equation8
A max +A min

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Therefore, Equation 3 and Equation 8 are the two formulas for Modulation index. The modulation index or
modulation depth is often denoted in percentage called as Percentage of Modulation. We will get the percentage
of modulation, just by multiplying the modulation index value with 100.

For a perfect modulation, the value of modulation index should be 1, which implies the percentage of modulation
should be 100%.

For instance, if this value is less than 1, i.e., the modulation index is 0.5, then the modulated output would look
like the following 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 following figure.

As the value of the modulation index increases, the carrier experiences a 180o phase reversal, which causes
additional sidebands and hence, the wave gets distorted. Such an over-modulated wave causes interference, which

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cannot be eliminated.

Bandwidth of AM Wave
Bandwidth BW is the difference between the highest and lowest frequencies of the signal. Mathematically, we
can write it as

BW = fmax − fmin

## ⇒ s (t) = Ac cos(2πfc t) + Ac μ cos(2πfc t) cos(2πfm t)

Ac μ Ac μ
⇒ s (t) = Ac cos(2πfc t) + cos[2π (fc + fm ) t] + cos[2π (fc − fm ) t]
2 2

Hence, the amplitude modulated wave has three frequencies. Those are carrier frequency f , upper sideband c

c m c m

Here,

## Substitute, f max and f

min values in bandwidth formula.

BW = fc + fm − (fc − fm )

⇒ BW = 2fm

Thus, it can be said that the bandwidth required for amplitude modulated wave is twice the frequency of the
modulating signal.

## Power Calculations of AM Wave

Consider the following equation of amplitude modulated wave.
Ac μ Ac μ
s (t) = Ac cos(2πfc t) + cos[2π (fc + fm ) t] + cos[2π (fc − fm ) t]
2 2

Power of AM wave is equal to the sum of powers of carrier, upper sideband, and lower sideband frequency
components.

P t = P c + P U SB + P LSB

– 2
2
vrms (vm /√ 2)
P = =
R 2

Where,

## vm is the peak value of cos signal.

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First, let us find the powers of the carrier, the upper and lower sideband one by one.

Carrier power
– 2
(Ac /√ 2) Ac
2

Pc = =
R 2R

## Upper sideband power

– 2 2 2
(Ac μ/2√ 2) Ac μ
P U SB = =
R 8R

Similarly, we will get the lower sideband power same as that of the upper side band power.
2 2
Ac μ
P LSB =
8R

Now, let us add these three powers in order to get the power of AM wave.
2 2 2 2 2
Ac Ac μ Ac μ
Pt = + +
2R 8R 8R

2 2 2
Ac μ μ
⇒ Pt = ( ) (1 + + )
2R 4 4

2
μ
⇒ P t = P c (1 + )
2

We can use the above formula to calculate the power of AM wave, when the carrier power and the modulation
index are known.

If the modulation index μ = 1 then the power of AM wave is equal to 1.5 times the carrier power. So, the power
required for transmitting an AM wave is 1.5 times the carrier power for a perfect modulation.

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