Editing Product detector (section)

==A simple product detector==
The simplest form of product detector mixes (or heterodynes) the RF or IF signal with a locally derived carrier (the Beat Frequency Oscillator, or BFO) to produce an audio frequency copy of the original audio signal and a mixer product at twice the original RF or IF frequency.  This high-frequency component can then be filtered out, leaving the original audio frequency signal.

===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.

===Drawbacks of the simple product detector===
Although this simple detector works, it has two major drawbacks:

* The frequency of the local oscillator must be the same as the frequency of the carrier, or else the output message will fade in and out in the case of AM, or be frequency shifted in the case of SSB
* Once the frequency is matched, the phase of the carrier must be obtained, or else the demodulated message will be attenuated, but the noise will not be.

The local oscillator can be synchronized with the carrier using a [[phase-locked loop]] in a [[synchronous detector]] arrangement. For SSB, the only solution is to construct a highly stable oscillator.