Consider the signals received by two antennae observing the same source.
There will, in general, be a different velocity component in the direction
of the source at each antenna so that the correlated components in the two
signals will be shifted in frequency, with the frequency difference being
equal to the fringe rate. If the instantaneous correlation between the two
signals were to be measured with a hypothetical perfect analog correlator
it would be found to vary sinusoidally with time between 
and
, where 
is the correlation that would be measured for zero
frequency offset (Fig. 2 (click here)). This is an example of a cyclic cross
correlation used to describe the general class of cyclo-stationary random
processes (Gardner 1989).
Assuming that the joint distribution of the signal voltages S1 and S2 can be modelled by the
bivariate normal distribution function, 
, where
is 
, for a particular value of correlation , the probability that the
quantized signals are in any particular state at a
particular time can be calculated. For the AT LBA correlator four-level quantization is used,
leading to 16 possible states for the combination of the two signals. Let this probability be denoted
by 
, which is the probability that S1 is in state i and
S2 is in state j.
Here, n is the
correlator weighting given to samples in the outer level. The quantization scheme is illustrated in (1).
The parameter v is the quantization threshold and the signals are assumed to be normalised to unit
variance.
For zero mean signals of unit variance, with identical quantization
threshold, v, the probabilities, Pij, can be expressed in terms
of the integral of the bivariate normal distribution function, 
, as
If one of the sample streams is now multiplied in a digital fringe rotator, the probability that the signals
are in a certain state is altered depending on the characteristic of the multiplier and the phase of the
fringe rotating function. The probability that the signals are in a particular state after fringe rotation,
denoted by 
, and using the multiplier characteristic in Table 1, is given by
where FF is the state of the fringe rotation function.
| Fringe Function | |||||
| Weight | -n | -1 | 1 | n | |
| n | -n | -1 | 1 | n | |
| Signal | 1 | -1 | -1 | 1 | 1 |
| -1 | 1 | 1 | -1 | -1 | |
| -n | n | 1 | -1 | -n | |
Consider the situation illustrated in Fig. 2 (click here) where, for simplicity, the fringe rotation function is exactly in phase with the instantaneous correlation of the two signals. In general this would not be so but in practice the real and imaginary arms of the complex correlator would measure the in-phase and quadrature components of the correlation.
Figure 2: Instantaneous correlation and associated fringe rotation function
At a particular phase 
the probability that the two signals are in
state 
is given by 
,
where 
is the instantaneous correlation at that
phase, and the true correlation. Now consider the
sequence of sample values at , 
, 
, ... , 
.
The instantaneous correlation and state of
the fringe rotator are the same for each sample in this sequence and so, since the original signals are
assumed to be jointly stationary processes, and Nyquist sampling is assumed, the sequence forms a
stationary time series. Hence the expected value of the product of the samples in this sequence can be
calculated as
Repeating this for all the sample periods within 
of phase and averaging the result yields the mean
correlator output. As the number of samples becomes large, this averaged sum approaches an integral
and the final result for the mean correlator output is
or, by symmetry,
For this result to be correct it is strictly necessary that many fringe
periods are contained within an integration and that there be an integral
number of fringe periods within the integration. The first condition is
satisfied except when the fringe rate is very low; the second condition is
also only a problem when the fringe rate is low since, as the number of
fringe periods in an integration becomes large, the error introduced by the
non-integral number of periods becomes negligible. Dividing 
by R(1) yields the digital correlation coefficient.