# Heterodyne detection

Heterodyne detection is a method of detecting radiation by 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

$E_\mathrm{sig} \cos(\omega_\mathrm{sig}t+\varphi)\,$

and that of the local oscillator can be represented as

$E_\mathrm{LO} \cos(\omega_\mathrm{LO}t).\,$

For simplicity, assume that the output I of the detector is proportional to the square of the amplitude:

$I\propto \left( E_\mathrm{sig}\cos(\omega_\mathrm{sig}t+\varphi) + E_\mathrm{LO}\cos(\omega_\mathrm{LO}t) \right)^2$
$=\frac{E_\mathrm{sig}^2}{2}\left( 1+\cos(2\omega_\mathrm{sig}t+2\varphi) \right)$
$+ \frac{E_\mathrm{LO}^2}{2}(1+\cos(2\omega_\mathrm{LO}t))$
$+ E_\mathrm{sig}E_\mathrm{LO} \left[ \cos((\omega_\mathrm{sig}+\omega_\mathrm{LO})t+\varphi) + \cos((\omega_\mathrm{sig}-\omega_\mathrm{LO})t+\varphi) \right]$
$=\underbrace{\frac{E_\mathrm{sig}^2+E_\mathrm{LO}^2}{2}}_{constant\;component}+\underbrace{\frac{E_\mathrm{sig}^2}{2}\cos(2\omega_\mathrm{sig}t+2\varphi) + \frac{E_\mathrm{LO}^2}{2}\cos(2\omega_\mathrm{LO}t) + E_\mathrm{sig}E_\mathrm{LO} \cos((\omega_\mathrm{sig}+\omega_\mathrm{LO})t+\varphi)}_{high\;frequency\;component}$
$+ \underbrace{E_\mathrm{sig}E_\mathrm{LO} \cos((\omega_\mathrm{sig}-\omega_\mathrm{LO})t+\varphi)}_{beat\;component}.$

The output has high frequency ($2\omega_\mathrm{sig}$ and $2\omega_\mathrm{LO}$) and constant components. In heterodyne detection, the high frequency components and usually the constant components are filtered out, leaving the intermediate (beat) frequency at $\omega_\mathrm{sig}-\omega_\mathrm{LO}$. 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 $\omega_\mathrm{LO}$ is equal to $\omega_\mathrm{sig}$ then the beat component is a recovered version of the original signal, with the amplitude equal to the product of $E_\mathrm{sig}$ and $E_\mathrm{LO}$; that is, the received signal is amplified by mixing with the local oscillator. This is the basis for a Direct conversion receiver.