The cross-correlation of a signal with itself gives its autocorrelation:
The autocorrelation function is Hermitian:
When is real, its autocorrelation is real and even (symmetric about lag zero).
The unbiased cross-correlation similarly reduces to an unbiased autocorrelation when :
The DFT of the true autocorrelation function is the (sampled) power spectral density (PSD), or power spectrum, and may be denoted
The complete (not sampled) PSD is , where the DTFT is defined in Appendix B (it's just an infinitely long DFT). The DFT of thus provides a sample-based estimate of the PSD:^{8.10}
We could call a ``sampled sample power spectral density''.
At lag zero, the autocorrelation function reduces to the average power (mean square) which we defined in §5.8:
Replacing ``correlation'' with ``covariance'' in the above definitions gives corresponding zero-mean versions. For example, we may define the sample circular cross-covariance as
where and denote the means of and , respectively. We also have that equals the sample variance of the signal :