Biased Sample Autocorrelation
The sample autocorrelation defined in (6.6) is not quite
the same as the autocorrelation function for infinitely long
discrete-time sequences defined in §2.3.6,
viz.,
where the signal
![$ v(n)$](http://www.dsprelated.com/josimages_new/sasp2/img1029.png)
![$ V(\omega)$](http://www.dsprelated.com/josimages_new/sasp2/img1134.png)
![$ v$](http://www.dsprelated.com/josimages_new/sasp2/img1068.png)
![$ v\star v$](http://www.dsprelated.com/josimages_new/sasp2/img1135.png)
Thus,
![$ v\star v$](http://www.dsprelated.com/josimages_new/sasp2/img1135.png)
It is common in practice to retain the implicit Bartlett (triangular) weighting in the sample autocorrelation. It merely corresponds to smoothing of the power spectrum (or cross-spectrum) with the
![$ \hbox{asinc}^2$](http://www.dsprelated.com/josimages_new/sasp2/img1103.png)
The left column of Fig.6.1 in fact shows the Bartlett-biased sample autocorrelation. When the bias is removed, the autocorrelation appears noisier at higher lags (near the endpoints of the plot).
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Smoothed Power Spectral Density
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Sample Power Spectral Density