Binary Golay code - Revision history

130.226.142.243: /* Mathematical definition */ punctuation

2014-09-19T09:46:29Z

Mathematical definition: punctuation

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	A '~~''product detector''' is a type of [[demodulator]] used~~ for ~~[[amplitude modulation\|AM]]~~ and [~~[Single-sideband modulation\|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. ~~A product detector is a [[frequency mixer]]~~.

	~~Product detectors can be designed to accept either [[Intermediate frequency\|IF]] or [[Radio frequency\|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\|direct-conversion receiver]].

	~~==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 carrier must be obtained, or ~~else the demodulated message will be attenuated, but the noise will not be.~~

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

	~~==Another example==~~

	~~There are~~ many ~~other kinds of product detectors as well. If one has access to [[digital signal processing]] equipment, one may experiment with~~ a ~~wide range of interesting techniques~~. ~~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.~~

	~~==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 filter~~]~~]ed~~. ~~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. These copies~~ are ~~then combined~~ to ~~produce the original message~~.

	~~==Advantages and disadvantages==~~
	~~The product demodulator has some advantages over an envelope detector for AM signal reception.~~
	* The product demodulator can decode [[overmodulation\|overmodulated]] AM and AM with suppressed carrier.
	* A signal demodulated with a product detector will have a higher [[signal to noise ratio]] than the same signal demodulated with an envelope detector.
	~~On the other hand, the envelope detector is a simple and relatively inexpensive circuit, and it can provide higher fidelity, since there is no possibility of mistuning the local oscillator.~~

	~~A product detector (or equivalent)~~ is ~~needed to demodulate SSB signals.~~

	~~{{DEFAULTSORT:Product Detector}}~~
	[[~~Category~~:~~Frequency mixers~~]]
	[[~~Category~~:~~Communication circuits]]~~
	~~[[Category:Demodulation]]~~

	~~[[de:Amplitudenmodulation#Koh~~.C3.~~A4rente_Demodulation~~]]

en>Ninly: /* Practical applications of Golay codes */

2013-11-18T22:46:40Z

Practical applications of Golay codes

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en>R.e.b.: Copyedit (major)

2012-07-06T16:22:02Z

Copyedit (major)

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