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Welch Autocorrelation Estimate

Since $ \left\vert X_m\right\vert^2\;\longleftrightarrow\;x\star x$ which is circular (or cyclic) correlation, we must use zero padding in each FFT in order to be able to compute the acyclic autocorrelation function as the inverse DFT of the Welch PSD estimate. There is no need to arrange the zero padding in zero-phase form, since all phase information is discarded when the magnitude squared operation is performed in the frequency domain.

The Welch autocorrelation estimate is biased. That is, as discussed in §6.6, it converges as $ K\to\infty$ to the true autocorrelation $ r_x(l)$ weighted by $ M-\vert l\vert$ (a Bartlett window). The bias can be removed by simply dividing it out, as in (6.15).


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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