88.114.149.155: /* Formal definition */

2013-06-21T18:42:08Z

Formal definition

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			{{Unreferenced\|date=December 2009}}
			'''Heterodyne detection''' is a method of detecting radiation by [[frequency mixer\|non-linear mixing]] with radiation of a reference frequency. It is commonly used in [[telecommunications]] and [[astronomy]] for detecting and analysing signals.

			The radiation in question is most commonly either radio waves (see [[superheterodyne receiver]]) or light (see [[Optical heterodyne detection]] or [[interferometry]]). The reference radiation is known as the [[local oscillator]]. The signal and the local oscillator are superimposed at a mixer. The mixer, which is commonly a (photo-)[[diode]], has a non-linear response to the amplitude, that is, at least part of the output is proportional to the square of the input.

			The received signal can be represented as

			:<math>E_\mathrm{sig} \cos(\omega_\mathrm{sig}t+\varphi)\,</math>

			and that of the local oscillator can be represented as

			:<math>E_\mathrm{LO} \cos(\omega_\mathrm{LO}t).\,</math>

			For simplicity, assume that the output ''I'' of the detector is proportional to the square of the amplitude:
			:<math>I\propto \left( E_\mathrm{sig}\cos(\omega_\mathrm{sig}t+\varphi) + E_\mathrm{LO}\cos(\omega_\mathrm{LO}t) \right)^2</math>

			:<math> =\frac{E_\mathrm{sig}^2}{2}\left( 1+\cos(2\omega_\mathrm{sig}t+2\varphi) \right)</math>

			::<math> + \frac{E_\mathrm{LO}^2}{2}(1+\cos(2\omega_\mathrm{LO}t)) </math>

			::<math> + E_\mathrm{sig}E_\mathrm{LO} \left[
			\cos((\omega_\mathrm{sig}+\omega_\mathrm{LO})t+\varphi)
			+ \cos((\omega_\mathrm{sig}-\omega_\mathrm{LO})t+\varphi)
			\right]
			</math>
			:<math> =\underbrace{\frac{E_\mathrm{sig}^2+E_\mathrm{LO}^2}{2}}_{constant\;component}+\underbrace{\frac{E_\mathrm{sig}^2}{2}\cos(2\omega_\mathrm{sig}t+2\varphi) + \frac{E_\mathrm{LO}^2}{2}\cos(2\omega_\mathrm{LO}t) + E_\mathrm{sig}E_\mathrm{LO} \cos((\omega_\mathrm{sig}+\omega_\mathrm{LO})t+\varphi)}_{high\;frequency\;component}</math>

			::<math> + \underbrace{E_\mathrm{sig}E_\mathrm{LO} \cos((\omega_\mathrm{sig}-\omega_\mathrm{LO})t+\varphi)}_{beat\;component}.
			</math>

			The output has high frequency (<math>2\omega_\mathrm{sig}</math>, <math>2\omega_\mathrm{LO}</math> and <math>\omega_\mathrm{sig}+\omega_\mathrm{LO}</math>) and constant components. In heterodyne detection, the high frequency components and usually the constant components are filtered out, leaving the intermediate (beat) frequency at <math>\omega_\mathrm{sig}-\omega_\mathrm{LO}</math>. The amplitude of this last component is proportional to the amplitude of the signal radiation. With appropriate [[signal analysis]] the phase of the signal can be recovered as well.

			If <math>\omega_\mathrm{LO}</math> is equal to <math>\omega_\mathrm{sig} </math> then the beat component is a recovered version of the original signal, with the amplitude equal to the product of <math> E_\mathrm{sig} </math> and <math>E_\mathrm{LO} </math>; that is, the received signal is amplified by mixing with the local oscillator. This is the basis for a [[Direct conversion receiver]].

			==See also==
			*[[Heterodyne]]
			*[[Homodyne detection]]
			*[[Optical heterodyne detection]]

			{{DEFAULTSORT:Heterodyne Detection}}
			[[Category:Waves]]
			[[Category:Metrology]]

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2009-09-12T21:21:06Z

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88.114.149.155: /* Formal definition */

en>Erik9bot: remove Category:Articles lacking sources (Erik9bot)