As a simple example, consider the case of a pure one-bit correlator. Here the signal is quantized to according to whether the signal is greater or less than zero. In this case the fringe rotation function is a simple square wave of unit amplitude, and reduces to . Hence the expected correlator output is

Expanding the integrand by Taylor series and integrating term by term gives

The digital correlation coefficient is obtained by dividing this by the normalizing value *R*(1) = 0.5. This
expression allows the digital correlation coefficient to be calculated to any desired accuracy although
only a few terms are required for small correlations. To convert from measured digital correlation back
to true correlation, the above series can be inverted or, as is more usual, a lookup table or interpolating
function can be used.

In the pure one-bit case , and the correlator efficiency 0.40. The complex correlator architecture increases this by to give 0.57. This compares with 0.64 for a one-bit correlator without digital fringe rotation (Cooper 1970).

In practice, most two-level correlators use a three-level fringe rotation
function, where the fringe rotation function is zero for a portion of the
cycle to give a better approximation to a sine function. The fringe rotation
function so obtained is as shown in Fig. 2 (click here), but with *n* equal to
unity and the intermediate levels equal to zero. The three-level fringe
rotation is implemented by blanking the correlator when the fringe rotation
function is zero.

Using (6) the expected correlator output is

where is the phase at which the fringe rotation function jumps from zero to the higher level.
The digital correlation coefficient is obtained by dividing by the normalizing value
A graph of this conversion function
is shown in Fig. 3 (click here).
Using (12) the correlator efficiency is given by

This has a maximum for () where the efficiency has a value of 0.427, which is increased by the
complex correlator architecture to 0.60. In practice, () is used as this is an exact multiple
of 1 / 16 of the full fringe period phase of . This allows the fringe rotator phase to be represented as a
four-bit digital quantity. There is virtually no loss in efficiency by using () instead of
().

**Figure 3:** Conversion relationship for two- (), three-
() and four-level () correlators with digital fringe rotation

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