5.3.2 Phase switching method

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

$$\ensuremath{\varphi_\mathrm{\scriptscriptstyle 1}} = - \ensur... ...le 1}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}} + \psi_1$$ (5.14)

$\psi_1$ 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	$\ensuremath{V_\mathrm{\scriptscriptstyle 1}} = \ensuremath{A_\mathrm{\scriptscr... ...criptscriptstyle L}} e^{-i\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}} }$
$\pi/2$	$\ensuremath{V_\mathrm{\scriptscriptstyle 2}} = \ensuremath{A_\mathrm{\scriptscr... ...riptstyle L}} e^{i(-\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}} +\pi/2)}$

Then one may compute the visibilities in each side band:

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

$${\rm and~~~} \ensuremath{A_\mathrm{\scriptscriptstyle L}} e^{-i\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}} } (\ensuremath{V_\mathrm{\scriptscriptstyle 1}} -i\ensuremath{V_\mathrm{\scriptscriptstyle 2}} )/2$$ = (5.15)

We have assumed here that we have a complex correlator (sine + cosine), or equivalently a spectral correlator measuring positive and negative delays (see Chapter 4).

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	$\ensuremath{V_\mathrm{\scriptscriptstyle 1}} = \ensuremath{A_\mathrm{\scriptscr... ...criptscriptstyle L}} e^{-i\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}} }$
$\pi/2$	$\ensuremath{V_\mathrm{\scriptscriptstyle 2}} = \ensuremath{A_\mathrm{\scriptscr... ...riptstyle L}} e^{i(-\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}} +\pi/2)}$
$\pi$	$\ensuremath{V_\mathrm{\scriptscriptstyle 3}} = -\ensuremath{V_\mathrm{\scriptscriptstyle 1}}$
$3\pi/2$	$\ensuremath{V_\mathrm{\scriptscriptstyle 4}} = -\ensuremath{V_\mathrm{\scriptscriptstyle 2}}$

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

$${\rm and~~~} \ensuremath{A_\mathrm{\scriptscriptstyle L}} e^{-i\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}} } (\ensuremath{V_\mathrm{\scriptscriptstyle 1}} -i\ensuremath{V_\ma... ...athrm{\scriptscriptstyle 3}} +i\ensuremath{V_\mathrm{\scriptscriptstyle 4}} )/4$$ = (5.16)

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.