# Mathematics of the DFTExample Applications of the DFTCorrelation AnalysisAutocorrelation

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### 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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.