Editing Electronic mixer (section)

===Mathematical treatment===
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{{Clarify|reason=But at double the frequency?|date=October 2016}}. This is the basis for a [[Direct conversion receiver]].