Coherent demodulation

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 $2\cos ({\omega }_{c}t+\phi )$,A unity-gain ideal low-pass filter passing $0$ and $\Omega$ but rejecting $2{\omega }_c$ and $2{\omega }_c\pm \Omega$ leaves $Aq(t)\cos \phi$. Such a separating cutoff exists because $\Omega <2{\omega }_c-\Omega$. At $\phi =0$, subtraction of $A$ recovers $Am\cos (\Omega t)$ exactly for any finite $m\ge 0$, 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 $\cos \phi$: 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.^[ref]