## Biased Sample Autocorrelation

The sample autocorrelation defined in (5.2) is not quite
the same as the autocorrelation function for infinitely long
discrete-time sequences defined in §2.3.6,
*viz.*,

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

Thus, can be seen as a

*Bartlett-windowed sample autocorrelation*:

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.5.1 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).

**Next Section:**

Smoothed Power Spectral Density

**Previous Section:**

Sample Power Spectral Density