Q.A 10 MHz sinusoidal carrier wave of amplitude 10 mV is modulated by a 5 kHz sinusoidal audio signal wave of amplitude 6 mV. Find the frequency components of the resultant modulated wave and their amplitudes.

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.

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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. …