### Autocorrelation

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 :

 (8.2)

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 :

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Unbiased Cross-Correlation