# Product detector

A product detector is a type of demodulator used for AM and SSB signals. Rather than converting the envelope of the signal into the decoded waveform like an envelope detector, the product detector takes the product of the modulated signal and a local oscillator, hence the name.

Product detectors can be designed to accept either IF or RF frequency inputs. A product detector which accepts an IF signal would be used as a demodulator block in a superheterodyne receiver, and a detector designed for RF can be combined with an RF amplifier and a low-pass filter into a direct-conversion receiver.

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## A simple product detector

The simplest form of product detector multiplies an incoming signal by its carrier to produce a copy of the original message and another AM signal at twice the original carrier frequency. This high-frequency component can then be filtered out leaving the original signal.

### Mathematical model of the simple product detector

If m(t) is the original message, the AM signal can be shown by

[itex]x(t)=(C+m(t))\cos(\omega t).[itex]

Multiplying the AM signal x(t) by an oscillator at the same frequency as and in phase with the carrier yields

[itex]y(t)=(C+m(t))\cos(\omega t)\cos(\omega t),[itex]

which can be re-written as

[itex]y(t)=(C+m(t))(1/2 + 1/2\cos(2\omega t)).[itex]

After filtering out the high-frequency component based around cos(2ωt) 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 carrier must be obtained, or else the demodulated message will be attenuated, but the noise will not be.

Frequency of an AM carrier can accurately determined with a phase-locked loop, but for SSB, the only solution is to construct a highly stable oscillator.

## A more sophisticated product detector

A more sophisticated product detector can be constructed in a way much like a single-sideband modulator. Two copies of the modulated input signals are created. The first copy is mixed with a local oscillator and low-pass filtered. The second copy is mixed with a 90° phase-shifted copy of the oscillator and the output of this mixer is also 90° phase-shifted and then low-pass filtered. Both of these copies are then combined to produce the original message.

The product demodulator has several advantages over an envelope detector.

• The product demodulator can decode overmodulated AM, AM with suppressed carrier, and SSB in addition to regular AM
• A signal demodulated with a product detector will have a higher signal to noise ratio than the same signal demodulated with an envelope detector.

The only real disadvantage of a product demodulator is the increased complexity.

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