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The "Product Detector"

This device is just another mixer - used for demodulating a signal in a receiver. The term "product" refers to the multiplication of the two input signals - with sum and difference outputs.
In the product detector the product signal ie, the dual sideband or single side band signal is again multiplied with the local carrier .
NOTE
In case of suppressed carrier systems (dont confuse carrier fully suppressed)
there is a need to get the carrier at the receiver at same frequency as that of the transmitter and the phase must be same.Hence we need highly stable oscillators free from drift.Such a process called synchronization must be done at the receiver ,which makes the receiver complex.For these reasons a pilot carrier system is preferred ,where the carrier is generated by the LO referring to the pilot carrier.
Receiver blockdiagram is available
here

In the case of AM signals a synchronous demodulator is preferrable because it offers more stability to the s/n ratio.In the synchronous demodulator we multiply the signal received by the
locally generated carrier hence the signal can be recovered
At the modulator side ,
Sin(Wct)Cos(Wmt)+Cos(Wct)Sin(Wmt)=Sin((Wc-Wm)t) {the modulated signal}
or the side band corresponding to upper frequencies.
where the Cos(Wmt) is obtained from Sin(Wmt) or vice versa thro Hilbert's Transform.
At the demodulator side,
Sin(Wct)Cos(Wmt)+Cos(Wct)Sin(Wmt)=Sin((Wc-Wm)t)
is multiplied by Sin(Wct)
so that a (Sin(Wct))square term appears which can be expressed interms of double angle
by the trig relation sothat the signal Cos(Wmt) can be filters at the output by using alow pass filter

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