= Amplitude modulation
{wiki}
= AM
{c}
{synonym}
Amplitude modulation encodes a message by varying the multiplying <wave amplitude> of a <carrier wave>. Conventional <AM> adds a positive constant before multiplication, allowing a simple <envelope detector> to recover the message within its limits. Multiplication translates the message's <frequency spectrum> into a band around the carrier, adapting a slow message to a faster channel. This is an application of the <modulation property of the Fourier transform>.<Continuous-Time Modulation by Alan V. Oppenheim>[Signals and Systems, Lecture 13: Continuous-Time Modulation, printed pp. 13-1–13-2: channel matching, spectrum translation, carrier injection and demodulation]{ref}
For an original single-message-<frequency> voltage model, choose
$$
s(t)=A[1+m\cos(\Omega t)]\cos(\omega_ct),\qquad
A>0,\quad 0<\Omega<\omega_c,\quad m\geq0.
$$
Here $t$ is time, $A$ is the peak voltage <amplitude> of the unmodulated <carrier wave>, $\omega_c$ is its <angular frequency>, $\Omega$ is the message <angular frequency>, and $m$ is the dimensionless <AM modulation index>. The message shape is $\cos(\Omega t)$; $m$ measures how deeply it changes the multiplying factor. Define the signed factor $q(t)=1+m\cos(\Omega t)$. The nonnegative <wave envelope> of the representation $z(t)=Aq(t)e^{\mathrm i\omega_ct}$ is
$$
E(t)=|z(t)|=A|q(t)|.
$$
The instantaneous voltage $s(t)$ crosses zero every carrier cycle, even when $E(t)$ is large. Its value at one time, its <wave envelope>, and its carrier-component <amplitude> are three different quantities. The narrow-envelope intuition additionally needs $\Omega\ll\omega_c$; the algebraic model only assumes $\Omega<\omega_c$.
\Image[/amplitude-modulation-envelope-and-phase-reversal.png]
{title=AM waveform, signed modulation factor and envelope before and after overmodulation}
{description=Original voltage model with $\omega_c/\Omega=20$ and $m=0.5$ or $1.4$. Orange curves bound the waveform by $\pm|q|$; the purple dashed curve is the signed $q$. Shading marks negative $q$, where the carrier phase reverses. These are calculated curves, not oscilloscope traces or detector measurements.}
The beauty is that two views reveal different parts of the same operation. In time, the message appears to ride on the oscillations. In the <frequency spectrum>, <amplitude modulation sidebands> show it as a pair of translated components. An <envelope detector> reads their combined shape without needing the carrier's <wave phase>; <coherent demodulation> instead uses a synchronized reference and preserves the signed factor.
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