Derivation of Amplitude Modulation

Amplitude Modulation (AM) is a fundamental technique in communication systems, widely used for transmitting information over long distances. At the JEE level, understanding its mathematical derivation is key to grasping the concepts of modulation index, bandwidth, and power distribution. This guide provides a clear, step-by-step derivation of the AM wave equation, its spectral analysis, and the power required for transmission.

1.0What 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 that 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.

2.0Basics of Communication System

Need for Modulation

• The baseband signal (voice, audio, or data) has a low frequency and cannot travel long distances effectively.
• Direct transmission would cause interference, signal attenuation, and limited range.
• To overcome these issues, the signal is modulated with a high-frequency carrier wave.

3.0Types of Modulation

Modulation techniques are broadly classified into:

1. Amplitude Modulation (AM)
2. Frequency Modulation (FM)
3. Phase Modulation (PM)

Among these, AM is the simplest and most widely studied.

4.0What is Amplitude Modulation (AM)?

Amplitude modulation is defined as:

“A modulation process in which the amplitude of a high-frequency carrier wave is varied in accordance with the instantaneous amplitude of the low-frequency modulating signal, while the frequency and phase of the carrier remain constant.”

Thus, the amplitude of the carrier contains the information from the baseband signal.

• The first figure represents the modulating (message) signal.
• The second figure shows the carrier wave, a high-frequency signal with no information.
• The third figure depicts the modulated wave (resultant signal).
• The positive and negative peaks of the carrier form an imaginary line called the envelope.
• The envelope reproduces the exact shape of the message signal.

5.0Derivation of the AM Wave Equation

The core idea of AM is that the amplitude of the carrier wave, Ac​, is made to vary linearly with the instantaneous value of the message signal, m(t).

modulating signal 

m(t) = Aₘ cos(2π fₘ t)

carrier signal

$c(t)=A_c\cos (2\pi f_{c}t)$

Aₘ and $A_c$ represent the amplitudes of the modulating and carrier signals, respectively.

fₘ and $f_c$ represent the frequencies of the modulating and carrier signals, respectively.

Representation of Amplitude Modulated wave

$s(t)=\left[A_c+A_m\cos (2\pi f_{m}t)\right]\cos (2\pi f_{c}t)$

6.0The Modulated Carrier Wave

The new, modulated carrier wave, $c_{AM}(t)$, will have a time-varying amplitude. Let's define this new amplitude as A(t). This amplitude is the original carrier amplitude, Ac​, plus a term proportional to the message signal, m(t).

$A(t)=A_c+k_{a}m(t)$

Here, $k_a$ is the amplitude sensitivity of the modulator, which determines how much the carrier amplitude changes for a given change in the message signal.

Substituting the expression for the message signal m(t):

$A(t)=A_c+k_a{A}_{m}cos({\omega }_{m}t)$

Final AM Wave Equation

The AM wave itself is a cosine wave with this time-varying amplitude A(t) and the carrier frequency ${\omega }_c$: $c_{A}M(t)=A(t)cos({\omega }_{c}t)$

Substitute the expression for A(t): $c_{A}M(t)=[A_c+k_a{A}_{m}cos({\omega }_{m}t)]cos({\omega }_{c}t)$

7.0Modulation Index

The Modulation Index (or Modulation Depth) represents the extent of modulation a carrier wave undergoes, indicating how much the carrier’s amplitude varies with the message signal.

$$\begin{gathered} s(t)=A_c[1+(A_m/A_c)cos(2\pi f_{m}t)]cos(2\pi f_{c}t) \\ \Rightarrow s(t)=A_c[1+\mu cos(2\pi f_{m}t)]cos(2\pi f_{c}t) \end{gathered}$$

μ is Modulation index and it is equal to the ratio of $A_{m}and{A}_c.$

$\mu =A_m/A_c$

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($2\pi f_{m}t$) is 1.

$\Rightarrow A_{max}=A_c+A_m$

Minimum amplitude of the modulated wave, when cos(2π fₘ t) is −1

$$\begin{gathered} \Rightarrow A_{m}in=A_c-A_m \\ {A}_{m}ax+A_{m}in=A_c+A_m+A_c-A_m=2A_c \\ \Rightarrow A_c=(A_{m}ax+A_{m}in)/2 \\ {A}_{m}ax-A_{m}in=A_c+A_m-(A_c-A_m)=2A_m \\ \Rightarrow A_m=(A_{m}ax-A_{m}in)/2 \\ {A}_m/A_c=[(A_{m}ax-A_{m}in)/2]/[(A_{m}ax+A_{m}in)/2] \\ \Rightarrow \mu =(A_{m}ax-A_{m}in)/(A_{m}ax+A_{m}in) \end{gathered}$$

• The Modulation Index or Modulation Depth is expressed as a percentage, called the Percentage of Modulation.
• It is calculated as: Percentage of Modulation = Modulation Index × 100
• For perfect modulation, the modulation index = 1 (i.e., 100% modulation).
• If the modulation index < 1 (e.g., 0.5), it is called under-modulation, and the resulting wave is an under-modulated wave.

If the modulation index exceeds 1 (e.g., 1.5), the signal becomes over-modulated, resulting in an over-modulated wave When the modulation index exceeds 1, the carrier undergoes a 180° phase reversal, creating extra sidebands and distortion. This over-modulated wave causes interference that cannot be removed.

When the modulation index exceeds 1, the carrier undergoes a 180° phase reversal, creating extra sidebands and distortion. This over-modulated wave causes interference that cannot be removed.

8.0Bandwidth of AM Wave

Bandwidth (BW) is the difference between the highest and lowest frequencies of a signal, expressed as:

$$\begin{gathered} BW=f_{m}ax-f_{m}in \\ s(t)=A_c[1+\mu cos(2\pi f_{m}t)]cos(2\pi f_{c}t) \\ \Rightarrow s(t)=A_{c}cos(2\pi f_{c}t)+A_c\mu cos(2\pi f_{c}t)cos(2\pi f_{m}t) \\ s(t)=A_{c}cos(2\pi f_{c}t)+(A_c\mu /2)cos[2\pi (f_c+f_m)t]+(A_c\mu /2)cos[2\pi (f_c-f_m)t] \end{gathered}$$

Hence, the amplitude modulated wave has three frequencies. Those are carrier frequency $f_c,$ upper sideband frequency $f_c+f_m$ and lower sideband frequency $f_c-f_m$

$$\begin{gathered} f_{m}ax=f_c+f_{m}and{f}_{m}in=f_c-f_m \\ BW=(f_c+f_m)-(f_c-f_m) \\ \Rightarrow BW=2f_m \end{gathered}$$

The bandwidth of an AM wave is twice the modulating signal’s frequency.

9.0Power Calculations of AM Wave

Equation of amplitude modulated wave,

$s(t)=A_{c}cos(2\pi f_{c}t)+(A_c\mu /2)cos[2\pi (f_c+f_m)t]+(A_c\mu /2)cos[2\pi (f_c-f_m)t]$

The total power of an AM wave is the sum of the powers of the carrier, upper sideband, and lower sideband.

$$\begin{gathered} P_t=P_c+P_{U}SB+P_{L}SB \\ P=v_{r}m{s}^2/R=(v_m/\surd 2{)}^2/R \end{gathered}$$

• $v_{rms}$ is the RMS value of a cosine signal.
• $v_m$ is the peak value of a cosine signal.

Carrier power: $P_c=(A_c/\surd 2{)}^2/R=A_c^2/(2R)$

Upper Sideband Power: $P_{USB}=(A_c\mu /(2\surd 2){)}^2/R=A_c^2{\mu }^2/(8R)$

Lower Sideband Power: $P_{LSB}=A_c^2{\mu }^2/(8R)$

Power of AM wave

$$\begin{gathered} P_t=A_c^2/(2R)+A_c^2{\mu }^2/(8R)+A_c^2{\mu }^2/(8R) \\ \Rightarrow P_t=(A_c^2/2R)(1+{\mu }^2/4+{\mu }^2/4) \\ \Rightarrow P_t=(A_c^2/2R)(1+{\mu }^2/2) \\ \Rightarrow P_t=P_c(1+{\mu }^2/2) \end{gathered}$$

For modulation index μ = 1 (perfect modulation):

• AM wave power = 1.5 × carrier power
• Transmission power required is 1.5 times the carrier power

10.0Example Problems for Derivation of Amplitude Modulation

Example 1: Calculating Modulation Index and Frequencies

An AM wave is given by the equation: s(t) = 20 [1 + 0.6 cos(2π × 10³ t)] cos(2π × 10⁵ t)

Find the carrier frequency, modulating frequency, modulation index, and the frequencies of the upper and lower sidebands.

Solution: We compare the given equation with the standard AM wave equation: $s(t)=A_c[1+\mu cos(2\pi f_{m}t)]cos(2\pi f_{c}t)$

• Carrier Amplitude (Ac​): By direct comparison, the amplitude of the carrier wave is Ac​=20 V.
• Modulation Index (μ): The coefficient of the cosine term in the envelope is the modulation index, so μ=0.6.
• Modulating Frequency ($f_m)$: From the term $2\pi f_{m}t=2\pi \times 10^{3}t$ we find $f_m=10^3$ Hz or 1 kHz.
• Carrier Frequency (fc​): From the term $2\pi f_{c}t=2\pi \times 10^{5}t,$ we find $f_c=10^{5}Hz$ or 100 kHz.
• Upper Sideband (USB) Frequency: The frequency of the USB is the sum of the carrier and modulating frequencies.  $f_{USB}=f_c+f_m=100kHz+1kHz=101kHz$
• Lower Sideband (LSB) Frequency: The frequency of the LSB is the difference between the carrier and modulating frequencies. $f_{LSB}=f_c-f_m=100kHz-1kHz=99kHz$

Example 2: Power Calculation in AM

A 400 W carrier wave is modulated to a depth of 75%. Calculate the total power transmitted and the power in each sideband.

Solution: Given: Carrier Power, Pc​=400 W Modulation Index, μ=75%=0.75

Total Power (Pₜ): The total power is given by the formula: $P_t=P_c(1+{\mu }^2/2)$

Pₜ = 400 (1 + (0.75)² / 2)
= 400 (1 + 0.5625 / 2)
= 400 (1 + 0.28125)
= 400 (1.28125)
= 512.5 W

Power in each Sideband: The total sideband power (Psb​) is the difference between the total power and the carrier power:  $P_{sb}=P_t-P_c=512.5W-400W=112.5W$

Since the power is equally distributed between the upper and lower sidebands: $P_{USB}=P_{LSB}=P_{sb}/2=112.5/2=56.25W$

Alternatively, using the formula: 

$P_{USB}=P_{LSB}=({\mu }^2/4)P_c=(0.75{)}^2/4\times 400=0.5625/4\times 400=56.25W$

11.0Applications of Amplitude Modulation

1. Radio Broadcasting: AM is widely used in medium-wave (MW) and short-wave (SW) broadcasting.
2. Two-Way Communication: Used in aircraft and military communication systems.
3. Amplitude Modulation in TV Transmission: Picture transmission in analog TV used AM for video signals.
4. Signal Transmission in Early Telephony: AM was used for long-distance voice communication.
5. Radar and Aeronautics: Some radar systems and aircraft communications rely on AM principles.