The cross correlation of two signals and is given by
The correlation function estimated from the quantized samples
in general deviates from the measurements taken with infinite
amplitude precision. The deviation depends on the true correlation value
of the signals. The relationship between the two measurement
can be expressed as
Note that the correction function is non-linear and hence this correction should be applied before any further operation on the correlation function. If the number of bits used for quantization is large then over a large range of correlation values the correction function is approximately linear.
The power spectral density (PSD) of a stationary stochastic process is
defined to be the FT of its auto-correlation function (the Wiener-Khinchin
theorem). That is if
then the PSD,
i.e. the function is a decomposition of the variance (i.e. `power') of into different frequency components.
For sampled signals, the PSD is estimated by the Fourier transform of the discrete auto-correlation function. In case the signal is also quantized before the correlation, then one has to apply a Van Vleck correction prior to taking the DFT. Exactly as before, this estimate of the PSD is related to the true PSD via convolution with the window function.
One could also imagine trying to determine the PSD of a function in the following way. Take the DFTs of the sampled signal for several periods of length and average them together and use this as an estimate of the PSD. It can be shown that this process is exactly equivalent to taking the DFT of the discrete auto-correlation function.
The cross power spectrum of the two signals is defined as the FT of the cross correlation function and the estimator is defined in a similar manner to that of the auto-correlation case.