= Coherent demodulation
= Synchronous detection
{synonym}
Coherent demodulation recovers a message by multiplying the received <signal> by a reference <carrier wave> of the same <frequency> and known relative <wave phase>, then passing the result through a <low-pass filter>. For the ideal <AM> model and reference $2\cos(\omega_ct+\varphi)$,
$$
2s(t)\cos(\omega_ct+\varphi)
=Aq(t)\cos\varphi+Aq(t)\cos(2\omega_ct+\varphi).
$$
A unity-gain ideal <low-pass filter> passing $0$ and $\Omega$ but rejecting $2\omega_c$ and $2\omega_c\pm\Omega$ leaves $Aq(t)\cos\varphi$. Such a separating cutoff exists because $\Omega<2\omega_c-\Omega$. At $\varphi=0$, subtraction of $A$ recovers $Am\cos(\Omega t)$ exactly for any finite $m\geq0$, including <overmodulation>, provided the transmitter and channel preserve this linear waveform. This is the signed factor, not its <absolute value>.
A constant phase error scales the recovered message by $\cos\varphi$: a $\pi/3$ <radian> error halves it, and a $\pi/2$ error removes it. A known error with nonzero <cosine> can be compensated in gain; a frequency mismatch makes the relative <wave phase> drift and no longer gives constant attenuation. The lecture illustrates synchronous multiplication, filtering and phase error.<Continuous-Time Modulation by Alan V. Oppenheim>[Signals and Systems, Lecture 13: Continuous-Time Modulation, transparencies 13.10–13.11 on pp. 13-7–13-8 and 13.16 on p. 13-10: multiplier, low-pass separation and phase-error cosine factor]{ref}
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