7.3 sideband separation

The sideband separation by mixer rejection is difficult for low IF frequencies, and currently works only at 3mm. The image rejection varies with frequency. There are other methods that cancel the signal in the unwanted side band by a larger factor. They are based on the fact that the LO1 phase $\varphi_\mathrm{\scriptscriptstyle LO1}$ appears with a different sign on the USB and LSB signals.

7.3.1 Fringe rate method

One might choose to drop the phase rotation on the second LO and let the fringes drift at their natural fringe rates. These rates are opposed in sign for the USB and LSB, and they might be separated electronically. However the natural fringe rate sometimes goes to zero (when the angular distance between source and baseline direction is minimum or maximum), and at least in these cases the method would fail.

It would be more practical to offset the LO1 and LO2 phase rates $\dot{\ensuremath{\varphi_\mathrm{\scriptscriptstyle LO1}}}$ and $\dot{\ensuremath{\varphi_\mathrm{\scriptscriptstyle LO2}}}$ from their nominal values by the same amount $\omega_\mathrm{\scriptscriptstyle OFF}$. If the offsets have the same sign, they will compensate for the USB and offset the fringe rate by $2 \ensuremath{\omega_\mathrm{\scriptscriptstyle OFF}}$ in the LSB. If $\omega_\mathrm{\scriptscriptstyle OFF}$ is large enough, the LSB signal will be cancelled. Note that offsetting $\dot{\ensuremath{\varphi_\mathrm{\scriptscriptstyle LO1}}}$ by a fixed amount is equivalent to offsetting the LO1 frequency.

This is a simple method to reject the unwanted sideband. Note that the associated noise is not rejected.

7.3.2 Phase switching method

Assume a variable phase offset $\psi_1$ is added to the LO1 phase command appropriate for compensating the geometrical delay variation:

$$\ensuremath{\varphi_\mathrm{\scriptscriptstyle LO1}}= - \ensurema... ...\scriptscriptstyle LO1}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}}+ \psi_1$$ (7.14)

$\psi_1$ will be subtracted to the phase of the USB signal, and added to that of the LSB signal. If $\psi_1$ is switched between 0 and $\pi/2$, the relative phase of the USB and LSB will be switched between 0 and $\pi$, and the signals may be separated by synchronous demodulation:

$\psi_1$	Signal
0	$\ensuremath{V_\mathrm{\scriptscriptstyle 1}} = \ensuremath{A_\mathrm{\scriptsc... ...scriptscriptstyle L}} e^{-i\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}}}$
$\pi/2$	$\ensuremath{V_\mathrm{\scriptscriptstyle 2}} = \ensuremath{A_\mathrm{\scriptsc... ...criptstyle L}} e^{i(-\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}}+\pi/2)}$

Then one may compute the visibilities in each sideband:

$$\ensuremath{A_\mathrm{\scriptscriptstyle U}}e^{i\ensuremath{\varphi_\mathrm{\scriptscriptstyle U}}} = (\ensuremath{V_\mathrm{\scriptscriptstyle 1}}+i\ensuremath{V_\mathrm{\scriptscriptstyle 2}})/2$$

$${\rm and   } \ensuremath{A_\mathrm{\scriptscriptstyle L}}e^{-i\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}}} = (\ensuremath{V_\mathrm{\scriptscriptstyle 1}}-i\ensuremath{V_\mathrm{\scriptscriptstyle 2}})/2$$ (7.15)

We have assumed here that we have a complex correlator (sine + cosine), or equivalently a spectral correlator measuring positive and negative delays (see Chapter 6).

One may also switch the phase by $\pi$, in which case the sign of all the correlated voltages is reversed. This has the advantage of suppressing any offsets in the system. Actually both switching cycles are combined in a 4-phase cycle:

$\psi_1$	Signal
0	$\ensuremath{V_\mathrm{\scriptscriptstyle 1}} = \ensuremath{A_\mathrm{\scriptsc... ...scriptscriptstyle L}} e^{-i\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}}}$
$\pi/2$	$\ensuremath{V_\mathrm{\scriptscriptstyle 2}} = \ensuremath{A_\mathrm{\scriptsc... ...criptstyle L}} e^{i(-\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}}+\pi/2)}$
$\pi$	$\ensuremath{V_\mathrm{\scriptscriptstyle 3}} = -\ensuremath{V_\mathrm{\scriptscriptstyle 1}}$
$3\pi/2$	$\ensuremath{V_\mathrm{\scriptscriptstyle 4}} = -\ensuremath{V_\mathrm{\scriptscriptstyle 2}}$

$$\ensuremath{A_\mathrm{\scriptscriptstyle U}}e^{i\ensuremath{\varphi_\mathrm{\scriptscriptstyle U}}} = (\ensuremath{V_\mathrm{\scriptscriptstyle 1}}+i\ensuremath{V_\mat... ...\mathrm{\scriptscriptstyle 3}}-i\ensuremath{V_\mathrm{\scriptscriptstyle 4}})/4$$

$${\rm and   } \ensuremath{A_\mathrm{\scriptscriptstyle L}}e^{-i\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}}} = (\ensuremath{V_\mathrm{\scriptscriptstyle 1}}-i\ensuremath{V_\mat... ...\mathrm{\scriptscriptstyle 3}}+i\ensuremath{V_\mathrm{\scriptscriptstyle 4}})/4$$ (7.16)

In a $N$ antenna system one needs to switch the relative phases of all antenna pairs. This could be done by applying the above square-wave switching on antenna 2, then on antenna 3 at twice the switching frequency, and so on. In practice the switching waveforms are orthogonal Walsh functions.