For the case of a two-bit, four-level correlator the expected correlator output cannot be solved for explicitly. Instead the integral (6) is solved numerically. Using the multiplier table give in Table 1, and an outer sample weighting of four, the conversion function shown in Fig. 3 (click here) is found.

By using a first-order in approximation to the integral of the bivariate normal distribution function,

and evaluating (6) and (12), an exact expression for the correlator efficiency can be derived. The
efficiency in this case is a function of three variables: *v* - the threshold for samples to be considered in
the outer level; *n* - the relative weighting for samples in the outer levels and - the
phase at which the
fringe rotation function jumps from the lower to the higher level. The result is

This function has a maximum at (*v* = 0.922, *n* = 3.84, = 0.544) where the efficiency is 0.602.
Again, the complex correlator architecture increases this to 0.851. This compares to 0.88 for a four-level
correlator without digital fringe rotation (Cooper 1970). In
the AT LBA correlator (*v* = 0.94, *n* = 4.0, )
is used for simplicity in digital implementation and compatability reasons. There is only a slight drop in
efficiency to 0.846.

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