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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.^[ref]

For an original single-message-frequency voltage model, chooseHere $t$ 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 $z(t)=Aq(t)e^{\mathrm{i}{\omega }_{c}t}$ isThe 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 $\Omega <{\omega }_c$.

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.

The modulation property of the Fourier transform adds shifted copies of the message spectrum. If those copies overlap, their complex amplitudes add before taking a magnitude; a frequency filter then acts on this sum. For a real even triangular message spectrum proportional to $(1-|\omega |{)}_{+}$ and carrier frequency $1/2<\Omega <1$, the translated spectra overlap over $|\omega |<1-\Omega$. A filter retaining $|\omega |<\Omega$ therefore leaves a central plateau and sloping outer sections, rather than two disjoint triangular bands. Spectral Dirac delta lines at the cutoff need an explicitly specified filter gain: multiplying a distribution by a discontinuous mask is not defined by pointwise reasoning alone.

Use the Fourier transform conventionSince $\mathcal{F}[e^{i\Omega t}]=2\pi \delta (\omega -\Omega )$ in distribution theory, the carrier has transformThe modulation property of the Fourier transform translates the message spectrum to the two carrier frequencies. ThereforeThe first term represents the unmodulated carrier wave; the other terms are the amplitude modulation sidebands.

Multiply $H$ by the exponential form of the sine and use the modulation property of the Fourier transform:Inserting part (iii) givesThe reciprocal terms are real principal-value distributions and the delta terms are imaginary. Equivalently, exponential damping $H(t)e^{-ϵt}$ gives ${\omega }_0/[(ϵ+i\omega {)}^2+{\omega }_0^2]$, whose distributional limit is the boxed expression. The shifted-difference formula remains valid at ${\omega }_0=0$, where all terms cancel.