One of the principle differences between a VLBI correlator and a correlator as used on a connected-element array is that with the former fringe rotation is usually done digitally after the antenna signals have been sampled and recorded. This is because it is difficult to achieve a sufficiently low residual fringe rate for antennae spaced by many hundreds of kilometres by fringe rotating at the antennae. Since some precorrelation fringe rotation will usually be necessary to remove this residual fringe rate, it is simplest to do all the fringe rotation at this stage. This has the added advantage of allowing other sources within the primary beam of the array to be observed by simply changing the phase tracking centre to the new source.
A digital correlator operating on sampled signals estimates the correlation coefficient of the two original analog signals. It is essential to know the relationship between this digital estimate, based on sampled and quantized values of the input signals, and the true continuous correlation coefficient, so that the measured digital correlation can be corrected to what it would have been without quantization. As a result of the coarse one- or two-bit quantization used in radio astronomy this relationship is nonlinear. For a conventional correlator where fringe rotation is done by altering the phase of the local oscillator signal before sampling, a simple relationship can be derived based on the assumption that the signals entering the samplers have Gaussian statistics (Cooper 1970; Bowers & Klinger 1974). When one of the inputs is digitally fringe rotated before correlation this approach is no longer possible as the sequence of sample values coming from the digital fringe rotator forms a non-stationary time series, making conventional ergodic time series analysis invalid. In addition, the digital multiplier may have a biased characteristic. For example, in the case of a four-level correlator, as used on the Australia Telescope Long Baseline Array (AT LBA) correlator, it multiplies two four-level signals resulting in six possible output levels which must be reduced back to a four-level representation. Hence some multiplication products are biased relative to others, introducing further distortion.
The method generally used to quantify these effects is to introduce a loss
factor, 
, the fringe rotation loss (Thompson et al.
1986). This describes the loss in signal-to-noise ratio, at small
values of correlation, relative to a correlator without digital fringe
rotation. To calculate the conversion relationship from measured digital
correlation to the true correlation for the full range of correlation
coefficient generally requires a lengthy computer simulation of the
correlator although a constant scaling correction can be applied for small
values of the correlation coefficient (Thompson et al.
1986). In the present work a method is presented which allows the
conversion relationship to be calculated directly and permits an analytical
expression for the signal-to-noise ratio performance of the correlator to be
derived.
The VLBI correlator considered in this work is shown in Fig. 1 (click here).
This is a type of complex correlator. The sampled IF from one of the
antennae is digitally multiplied by a digital approximation to a sine
function to reduce the fringe rate to near zero. The input data streams are
then correlated and the measured digital correlation at each lag corrected
for distortion due to quantization. The correlation function is then Fourier
transformed to give the cross power spectrum. Doing the same sequence of
operations but with a digital sine function that is in quadrature to the
original digital sine function results in a cross power spectrum with a 
phase difference compared to the first spectrum but with independent
noise. Averaging the two spectra after correcting for the phase
difference gives a spectrum with a signal to noise ratio 
greater
than either of the individual spectra. The digital multiplier also produces
an unwanted image response, but the fringes resulting from this image band
generally have a high enough frequency that, with all but very low fringe
rates, they are reduced to a negligible level by the time averaging of the
correlator. In addition, the illustrated correlator configuration forms a single-sideband correlator. Any
residual response to the image band differs in phase by in the real and imaginary arms of the correlator so that, while the final phase shift and addition reinforces for the signal
band, it cancels for the image band, removing the image response from
the final spectrum.
Figure 1: The VLBI correlator architecture considered in this work