3. Signal-to-noise performance

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).