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

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