A signal of low frequency $f_m$ is to be transmitted using a carrier wave of frequency $f_c$. Derive the expression for the amplitude modulated wave and deduce expressions for the lower and upper sidebands produced. Hence, obtain the expression for modulation index.

A signal of low frequency fm is to be transmitted using a carrier wave of frequency fc. Derive the expression

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f_m

CBSECBSE Class XII Board 2019Subjective· 3mImportance★★★★★

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Concept understanding — Amplitude Modulation

Amplitude Modulation: From Intuition to Precision

Imagine you need to send a voice signal — say, your friend's speech — across a city. The voice itself is a low-frequency wave, maybe a few hundred hertz. If you just broadcast that directly, it won't travel far; low frequencies get absorbed quickly and need enormous antennas. So you need a "vehicle" that can carry that voice over long distances. That vehicle is a high-frequency carrier wave — think of it as a clean, steady sine wave at, say, 1 MHz.

Now, how do you make the carrier carry your voice? You can't just add the two waves together — that would give a messy sum, not a proper transmission. Instead, you let the voice signal control the carrier's amplitude. When your voice is loud, the carrier's amplitude grows; when your voice is soft, the carrier's amplitude shrinks. The carrier's frequency stays fixed — only its envelope (the outer shape of the wave) follows the message.

That is the core intuition: Amplitude Modulation (AM) is the process of imprinting a low-frequency message onto a high-frequency carrier by varying the carrier's amplitude in step with the message.

The Precise Statement

Let the carrier wave be a pure sinusoid:

c(t) = A_c \cos(2\pi f_c t)

where $A_c$ is the carrier amplitude and $f_c$ is the carrier frequency (much higher than the message frequency).

Let the message signal be $m(t)$ . In standard AM, the modulated signal is:

s(t) = \left[ A_c + m(t) \right] \cos(2\pi f_c t)

The amplitude of the carrier is no longer constant — it is $A_c + m(t)$ . That sum is the instantaneous amplitude of the carrier, and it varies exactly as $m(t)$ varies.

s(t) = A_c \left[ 1 + \mu \, m_n(t) \right] \cos(2\pi f_c t)

The modulation index $\mu$ controls how much the carrier amplitude swings. If $\mu > 1$ , the envelope distorts — the message is lost. If $\mu = 0$ , there is no modulation at all — just the bare carrier.

What You Actually See

If you plot $s(t)$ against time, you see a high-frequency wave whose peaks and troughs trace out the shape of $m(t)$ . The upper envelope is $A_c + m(t)$ , the lower envelope is $-(A_c + m(t))$ . A simple AM receiver (like a diode detector) extracts that envelope and recovers the original message. …

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Amplitude modulation superimposes a low-frequency message signal onto a high-frequency carrier by varying the carrier's amplitude. The modulated wave contains the carrier frequency and two sidebands at $f_c \pm f_m$ ; the modulation index $\mu = \frac{A_m}{A_c}$ quantifies the depth of modulation.

Why amplitude modulation works

A low-frequency signal cannot be radiated efficiently by an antenna because the required antenna size is inversely proportional to frequency. We therefore "piggyback" the message signal onto a high-frequency carrier wave by letting the message control the carrier's amplitude. The carrier acts as a vehicle that can travel through space, while the message rides along as variations in its envelope.

Mathematically, we multiply the carrier by a time-varying amplitude factor that depends on the message signal. This multiplication in the time domain creates new frequency components—the sidebands—that carry the information.

Derivation of the amplitude modulated wave

Define the message and carrier signals

The message (modulating) signal is sinusoidal:

m(t) = A_m \sin(2\pi f_m t)

where $A_m$ is the message amplitude and $f_m$ is the message frequency.

The carrier wave is:

c(t) = A_c \sin(2\pi f_c t)

where $A_c$ is the carrier amplitude and $f_c \gg f_m$ is the carrier frequency.

Construct the modulated amplitude

In amplitude modulation, the instantaneous amplitude of the carrier varies linearly with the message signal. We write:

A(t) = A_c + m(t) = A_c + A_m \sin(2\pi f_m t)

This can be rewritten as:

A(t) = A_c \left[1 + \frac{A_m}{A_c} \sin(2\pi f_m t)\right]

Form the AM wave

The amplitude-modulated signal is the carrier with time-varying amplitude:

s(t) = A(t) \cdot \sin(2\pi f_c t)

s(t) = A_c \left[1 + \frac{A_m}{A_c} \sin(2\pi f_m t)\right] \sin(2\pi f_c t)

Define the modulation index $\mu = \frac{A_m}{A_c}$ , which measures the fractional change in amplitude:

s(t) = A_c [1 + \mu \sin(2\pi f_m t)] \sin(2\pi f_c t)

Expand to reveal the frequency components

Distribute the carrier:

s(t) = A_c \sin(2\pi f_c t) + \mu A_c \sin(2\pi f_m t) \sin(2\pi f_c t)

Apply the product-to-sum trigonometric identity:

\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]

With $A = 2\pi f_m t$ and $B = 2\pi f_c t$ :

\sin(2\pi f_m t) \sin(2\pi f_c t) = \frac{1}{2}[\cos 2\pi(f_c - f_m)t - \cos 2\pi(f_c + f_m)t]

Write the complete AM expression

Substituting back:

s(t) = A_c \sin(2\pi f_c t) + \frac{\mu A_c}{2} \cos 2\pi(f_c - f_m)t - \frac{\mu A_c}{2} \cos 2\pi(f_c + f_m)t

Important

The AM wave consists of three frequency components:

Carrier: frequency $f_c$ , amplitude $A_c$
Lower sideband (LSB): frequency $f_c - f_m$ , amplitude $\frac{\mu A_c}{2}$
Upper sideband (USB): frequency $f_c + f_m$ , amplitude $\frac{\mu A_c}{2}$

Sideband expressions …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2024Set ANNUAL1 markMCQ
$Q. The process in which the amplitude of the carrier wave is made proportional to the instantaneous amplitude of the signal wave is called$
(a) Amplitude modulation
(b) Demodulation
(c) Amplification
(d) Rectification.
›Reveal solutionSolution

This is exactly the definition of amplitude modulation (AM): the carrier wave's amplitude tracks the message signal's instantaneous value while its frequency and phase stay fixed.

In amplitude modulation, a high-frequency carrier wave $c(t) = A_c\sin(\omega_c t)$ has its amplitude $A_c$ made to vary in step with (proportional to) the instantaneous amplitude of a lower-frequency message/signal wave $m(t)$ :

$s(t) = [A_c + k\, m(t)]\sin(\omega_c t)$ …

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CBSE 2024Set ANNUAL1 markMCQ
$Q. A speech signal of 3kHz is used to modulate a carrier wave of frequency 1MHz, using amplitude modulation. The frequencies of the side bands will be -$
(a) 1.003 MHz and 0.997 MHz
(b) 3001 kHz and 2997 kHz
(c) 1003 kHz and 1000 kHz
(d) 1 MHz and 0.997 MHz
›Reveal solutionSolution

In amplitude modulation, sidebands appear at $f_c \pm f_m$ .

Carrier frequency $f_c = 1\text{ MHz} = 1000\text{ kHz}$ , modulating (speech) frequency $f_m = 3\text{ kHz}$ .

$f_c + f_m = 1000+3 = 1003\text{ kHz} = 1.003\text{ MHz}$

$f_c - f_m = 1000-3 = 997\text{ kHz} = 0.997\text{ MHz}$

…

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CBSE 2020Set OC1 mark
$Q. Calculate the modulation factor of an amplitude-modulated wave when the signal amplitude is zero.$
›Reveal solutionSolution

With zero message amplitude, there is nothing to modulate the carrier with, so the modulation factor is simply zero.

Formula

$\mu = \frac{A_m}{A_c}$

Substitute $A_m=0$

$\mu = \frac{0}{A_c} = 0$

…

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CBSE 2020Set OC1 mark
$Q. Calculate the modulation factor of an amplitude-modulated wave when the signal amplitude is equal to the carrier amplitude.$
›Reveal solutionSolution

When the message swing exactly matches the carrier's own amplitude, the modulation factor hits its maximum "safe" value of 1.

Formula

$\mu = \frac{A_m}{A_c}$

Substitute $A_m=A_c$

$\mu = \frac{A_c}{A_c} = 1$

…

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CBSE 2020Set OC1 mark
$Q. Calculate the modulation factor of an amplitude-modulated wave when the signal amplitude is half the carrier amplitude.$
›Reveal solutionSolution

A message amplitude that's half the carrier's gives exactly half the maximum possible modulation depth.

Formula

$\mu = \frac{A_m}{A_c}$

Substitute $A_m=A_c/2$

$\mu = \frac{A_c/2}{A_c} = 0.5$

i.e. 50% modulation.

…

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CBSE 2019Set ANNUAL1 mark
$Q. What is carrier wave?$
›Reveal solutionSolution

Low-frequency signals like voice cannot be transmitted efficiently on their own; a high-frequency carrier wave is used to "carry" them, with the message impressed onto the carrier via modulation.

Definition

A carrier wave is a high-frequency sinusoidal electromagnetic wave, $c(t)=A_c\sin(\omega_ct+\phi)$ , generated at the transmitter, onto which the (much lower frequency) message/information signal is superimposed through a process called modulation (varying $A_c$ , $\omega_c$ , or $\phi$ in step with the message). The carrier's high frequency allows practical antenna sizes and efficient long-distance radiation, which the low-frequency message signal alone could not achieve.

…

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CBSE 2019Set ANNUAL1 mark
$Q. What is modulation factor?$
›Reveal solutionSolution

The modulation factor simply measures, as a ratio, how large the message's amplitude swing is compared to the carrier's own amplitude.

Definition

In amplitude modulation, if $A_c$ is the carrier's amplitude and $A_m$ is the message (modulating) signal's amplitude, the modulation index/factor is

$\mu = \frac{A_m}{A_c}$

(often expressed as a percentage, "percentage modulation" $=\mu\times100\%$ ). It quantifies the depth of modulation; $\mu$ should not exceed 1 to avoid distortion (over-modulation).

…

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CBSE 2018Set ANNUAL1 mark
$Q. What is amplitude modulation?$
›Reveal solutionSolution

AM is a modulation technique in which the carrier wave's amplitude is made to vary in step with the message signal, so the message can "ride" on a high-frequency carrier suitable for radio transmission.

Definition

If the carrier is $c(t) = A_c\sin(\omega_c t)$ and the message signal is $m(t) = A_m\sin(\omega_m t)$ , amplitude modulation produces

$c_m(t) = (A_c + A_m\sin\omega_m t)\sin\omega_c t$

so the envelope of the transmitted wave follows the shape of the message signal, while the carrier frequency $\omega_c$ and phase stay unchanged. The ratio $\mu = A_m/A_c$ is called the modulation index.

§ Honest curriculum note

…

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CBSE 2018Set ANNUAL1 mark
$Q. Draw a diagram of amplitude modulated wave.$
›Reveal solutionSolution
An AM wave is a fast carrier whose amplitude envelope follows the slow message signal.

Description of the diagram to draw (plot amplitude vs time):

First (optional) show the modulating/message signal — a low-frequency sinusoid.

Show the carrier — a high-frequency sinusoid of constant amplitude.

The amplitude-modulated wave is the carrier whose peak amplitude is no longer constant but varies according to the message signal. Draw a fast sinusoid whose upper and lower peaks are bounded by two mirror-image envelope curves having the shape of the modulating signal. …
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Q.A signal of low frequency fmf_mfm​ is to be transmitted using a carrier wave of frequency fcf_cfc​. Derive the expression for the amplitude modulated wave and deduce expressions for the lower and upper sidebands produced. Hence, obtain the expression for modulation index.

Why amplitude modulation works

Derivation of the amplitude modulated wave

Sideband expressions …

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Q.A signal of low frequency $f_m$ is to be transmitted using a carrier wave of frequency $f_c$ . Derive the expression for the amplitude modulated wave and deduce expressions for the lower and upper sidebands produced. Hence, obtain the expression for modulation index.

Unlock everything free for 14 days