2.1 Basic principle

The antenna produces a Voltage proportional to the linear superposition of the incident electric field pattern. For a simple monochromatic case:

$$U(t) = E {\rm cos} (2 \pi \nu t + \Phi)$$ (2.1)

In the receiver, a mixer superimposes the field generated by a local oscillator to the antenna output.

$$U_{LO}(t) = Q {\rm cos} (2 \pi \nu_{LO} t + \Phi_{LO})$$ (2.2)

The mixer is a non-linear element (such as a diode) whose output is

$$I(t) = a_0 + a_1 (U(t)+U_{LO}(t)) + a_2 (U(t)+U_{LO}(t))^2 + a_3 (U(t)+U_{LO}(t))^3 + ...$$ (2.3)

The second order (quadratic) term of Eq.2.3 can be expressed as

$$I(t) = ...$$

$$+ a_2 E^2 {\rm cos}^2(2\pi\nu t + \Phi)$$

$$+ 2 a_2 EQ {\rm cos}(2 \pi\nu t + \Phi) {\rm cos}(2 \pi\nu_{LO} t + \Phi_{LO})$$

$$+ a_2 Q^2 {\rm cos}^2(2 \pi\nu_{LO} t + \Phi_{LO})$$

$$+ ...$$ (2.4)

Developping the product of the two cosine functions, we obtain

$$I(t) = ...$$

$$+ a_2 EQ {\rm cos}(2 \pi(\nu+\nu_{LO}) t + \Phi + \Phi_{LO})$$

$$+ a_2 EQ {\rm cos}(2 \pi(\nu-\nu_{LO}) t + \Phi - \Phi_{LO})$$

$$+ ...$$ (2.5)

There are obviously other terms in $2 \nu_{LO}$, $2 \nu$, $3 \nu_{LO} \pm \nu$, etc...in the above equation, as well as terms at very different frequencies like $\nu$, $3\nu$, etc...

By inserting a filter at the output of the mixer, we can select only the term such that

$$\nu_{IF} - \Delta \nu /2 \leq \vert\nu -\nu_{LO}\vert \leq \nu_{IF} + \Delta \nu /2$$ (2.6)

where $\nu_{IF}$, the so-called Intermediate Frequency, is a frequency which is significantly different from than the original signal frequency $\nu$ (which is often called the Radio Frequency $\nu_{RF}$).

Hence, after mixing and filtering, the output of the receiver is

$$I(t) \propto EQ {\rm cos}(2 \pi(\nu-\nu_{LO}) t + \Phi - \Phi_{LO})$$ (2.7)

$$or$$

$$I(t) \propto EQ {\rm cos}(2 \pi(\nu_{LO}-\nu) t - \Phi + \Phi_{LO})$$ (2.8)

i.e.

• changed in frequency: $\nu \rightarrow \nu-\nu_{LO}$ or $\nu \rightarrow \nu_{LO}-\nu$
• proportional to the original electric field of the incident wave: $\propto E$
• with a phase relation with this electric field: $\Phi \rightarrow \Phi-\Phi_{LO}$ or $\Phi \rightarrow \Phi_{LO}-\Phi$
• proportional to the local oscillator voltage: $\propto Q$

The frequency change, usually towards a lower frequency, allows to select $\nu_{IF}$ such that amplifiers and transport elements are easily available for further processing. The mixer described above accepts simultaneously frequencies which are (see Fig.2.1)

• higher than the local oscillator frequency.
  This is called Upper Side Band (USB) reception
• lower than the local oscillator frequency.
  This is called Lower Side Band (LSB) reception

Figure 2.1: Relation between the IF, RF and local oscillator frequencies in an heterodyne system

and cannot a priori distinguish between them. This is called Double Side Band (DSB) reception. Some receivers are actually insensitive to one of the frequency range, either because a filter has been placed at the receiver input, or because their response is very strongly frequency dependent. Such receivers are called Single Side Band (SSB) receivers.

An important property of the receiving system expressed by Eq.2.8 is that the sign of the phase is changed for LSB conversion. This property can be easily retrieved recognizing that the Frequency $\nu$ is the time derivative of the Phase $\Phi$. Assume the phase varies linearly with time:

$$\Phi(t) = 2 \pi n t$$

$$n = \frac{1}{2 \pi} \frac{d\Phi}{dt}$$ (2.9)

In this case, the signal

$$I(t) \propto \cos (2 \pi \nu t + \Phi(t))$$

$$\propto \cos (2 \pi(\nu+n) t )$$ (2.10)

is just another monochromatic signal with slightly shifted frequency.