Editing Product detector (section)

===Mathematical model of the simple product 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 ''x''(''t'') by an oscillator at the same frequency as and in phase with the carrier yields 
:<math>\,y(t)=(C+m(t))\cos(\omega t)\cos(\omega t),</math>
which can be re-written as
:<math>\,y(t)=(C+m(t))\left(\tfrac{1}{2} + \tfrac{1}{2}\cos(2\omega t)\right).</math>

After filtering out the high-frequency component based around cos(2ω''t'') and the DC component ''C'', the original message will be recovered.