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4. Two-level correlator

As a simple example, consider the case of a pure one-bit correlator. Here the signal is quantized to tex2html_wrap_inline1073 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 tex2html_wrap_inline1075 reduces to tex2html_wrap_inline1077. Hence the expected correlator output is


equation390
Expanding the integrand by Taylor series and integrating term by term gives


equation398

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 tex2html_wrap_inline1081, and the correlator efficiency tex2html_wrap_inline1083 0.40. The complex correlator architecture increases this by tex2html_wrap_inline977 to give 0.57. This compares with 0.64 for a one-bit correlator without digital fringe rotation (Cooper 1970).

4.1. Two-level correlator with three-level fringe rotation

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
equation412
where tex2html_wrap_inline1089 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 tex2html_wrap_inline1091 A graph of this conversion function is shown in Fig. 3 (click here). Using (12) the correlator efficiency is given by
equation438
This has a maximum for (tex2html_wrap_inline1093) where the efficiency has a value of 0.427, which is increased by the complex correlator architecture to 0.60. In practice, (tex2html_wrap_inline1095) is used as this is an exact multiple of 1 / 16 of the full fringe period phase of tex2html_wrap_inline1057. This allows the fringe rotator phase to be represented as a four-bit digital quantity. There is virtually no loss in efficiency by using (tex2html_wrap_inline1095) instead of (tex2html_wrap_inline1093).

  figure445
Figure 3: Conversion relationship for two- (tex2html_wrap_inline1095), three- (tex2html_wrap_inline1107) and four-level (tex2html_wrap_inline1109) correlators with digital fringe rotation


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