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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.,

$\displaystyle (v\star v)_l$ $\displaystyle \isdef$ $\displaystyle \sum_{n=-\infty}^{\infty} \overline{v(n)} v(n+l)
\protect$ (6.4)
  $\displaystyle \longleftrightarrow$ $\displaystyle \left\vert V(\omega)\right\vert^2$  

where the signal $ v(n)$ is assumed to be of finite support (nonzero over a finite range of samples), and $ V(\omega)$ is the DTFT of $ v$. The advantage of the definition of $ v\star v$ 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

$\displaystyle \hat{r}_{v,N}(l) \isdef \frac{1}{N-\vert l\vert} (v\star v)(l), \quad \vert l\vert<N. \protect$ (6.5)

Thus, $ v\star v$ can be seen as a Bartlett-windowed sample autocorrelation:

$\displaystyle (v\star v)(l) = \left\{\begin{array}{ll} (N-\left\vert l\right\ve...
...ots,\pm (N-1) \\ [5pt] 0, & \vert l\vert\geq N. \\ \end{array} \right. \protect$ (6.6)

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$ 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).

Previous: Sample Power Spectral Density
Next: Smoothed Power Spectral Density

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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 for details.


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