2.4 Fringe Stopping and Complex Correlator

With the Earth rotation, the cosine term of Eq.2.22 modulates the correlator output quasi-sinusoidaly with a natural fringe rate of

 

$$\nu_{LO} d\tau_g/dt \simeq \Omega_{earth} {\rm b} \nu_{LO} /c$$ (2.27)

which is of order of 10 Hz for b= 300 m baselines and $\nu_{LO} = 100$ GHz, or 2'' angular resolution (since the fringe rate only depends on the effective angular resolution).

The fringe rate is somewhat too high for simple digital sampling of the visibility. An exception is VLBI (because there is no other choice), although the resolutions are < 1mas. The usual technique is to modulate the phase of the local oscillator $\Phi_{LO}$ such that $\Phi_{LO}(t) = 2\pi\nu_{LO}\tau_g(t)$at any given time. Then

$$r_r = A_o \vert V\vert {\rm cos}(\pm 2 \pi \nu_{IF}\Delta\tau - \Phi_V)$$ (2.28)

(with the + sign for USB conversion, and the - sign for LSB conversion), is a slowly varying output, which would be constant for a point source at the reference position (or delay tracking center). This process is called Fringe Stopping. After fringe stopping, we can no longer measure the amplitude |V| and the phase $\Phi_V$ separately. A second correlator, with one signal phase shifted by $\pi/2$ becomes necessary. Its output is

$$r_i = A_o \vert V\vert {\rm sin}(\pm 2 \pi \nu_{IF}\Delta\tau - \Phi_V)$$ (2.29)

With both correlators, we measure directly the real r_r and imaginary r_i parts of the complex visibility r. The device is thus called a ``complex'' correlator.

Note:

A delay tracking error $\Delta\tau$ appears as a phase slope as a function of frequency, with

$$\Phi(\nu_{IF}) = \pm 2 \pi \nu_{IF} \Delta\tau$$ (2.30)