Q.Describe the principle of amplitude modulation. In particular, explain how the envelope of the modulated carrier wave is related to the original message signal.

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 $m(t)$:

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

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

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

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

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

…

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 $A_m=0$

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

…

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 $A_m=A_c$

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

…

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.

…

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.

…

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).

…

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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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):

1. First (optional) show the modulating/message signal — a low-frequency sinusoid.
2. Show the carrier — a high-frequency sinusoid of constant amplitude.
3. 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. …