Also of interest is the signal-to-noise ratio of the correlator measurement relative to an ideal correlator
processing the same number of samples. This is known as the correlator efficiency. Most important
is the efficiency when the correlation is small as this is the region in which the correlator generally
operates. For two input sample streams, *x*_{i} and *y*_{i}, the correlator output is given by

where *N* is the number of samples processed. The variance of the correlator measurement is

where is the expectation.

In the limit as the correlation approaches zero the second term approaches zero and the two processes become independent. Under these conditions, and with Nyquist sampling so that successive samples are uncorrelated, the expression for the variance reduces to

Hence the signal-to-noise ratio of the correlator measurement is

For the ideal case with no quantization this becomes

Thus the efficiency of the correlator is

This result applies to both the real and imaginary outputs of the complex correlator and so the final correlator efficiency is greater than (12).

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