Editing Product detector (section)

==Another example==

There are many other kinds of product detectors as well, which are practical if one has access to [[digital signal processing]] equipment. For instance, it is possible to multiply the incoming signal by the carrier, times the square of another carrier 90° out of phase with it. This will produce a copy of the original message, and another AM signal at the fourth harmonic, by means of the [[trigonometric identity]]

:<math>\sin^2\theta \cos^2\theta = \frac{1 - \cos 4\theta}{8}</math>

The high-frequency component can again be filtered out, leaving the original signal.

===Mathematical model of the detector===
If ''m''(''t'') is the original message, the AM signal can be shown to be 
:<math>\,x(t)=(C+m(t))\cos(\omega t).</math>
Multiplying the AM signal by the new set of frequencies yields
:<math>\,y(t)=(C+m(t))\sin^2(\omega t)\cos^2(\omega t)</math>
::<math>=(C+m(t))\frac{1 - \cos 4\omega t}{8}</math>
::<math>=\frac{(C+m(t))}{8} - \frac{(C+m(t))\cos 4\omega t}{8}.</math>

After filtering out the component based around cos(4ω''t'') and the DC component C, the original message will be recovered.