Discrete Correlation and the Power Spectral Density

The cross correlation of two signals and is given by

(8.5.7) |

where denotes the number of samples by which is delayed, is the maximum delay (). By definition is a random variable. The expectation value of converges to when . The autocorrelation of the time series is also obtained using a similar equation as Eq. 8.5.8 by replacing 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

(8.5.9) |

(8.5.10) |

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,
is

(8.5.11) |

(8.5.12) |

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.