## 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.,

 (7.14)

where the signal is assumed to be of finite support (nonzero over a finite range of samples), and is the DTFT of . The advantage of the definition of is that there is a simple Fourier theorem associated with it. The disadvantage is that it is biased as an estimate of the statistical autocorrelation. The bias can be removed, however, since

 (7.15)

Thus, can be seen as a Bartlett-windowed sample autocorrelation:

 (7.16)

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 kernel, and smoothing is necessary anyway for statistical stability. It also down-weights the less reliable large-lag estimates, weighting each lag by the number of lagged products that were summed, which seems natural.

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