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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.,

$\displaystyle (v\star v)_l$ $\displaystyle \isdef$ $\displaystyle \sum_{n=-\infty}^{\infty} \overline{v(n)} v(n+l) \protect$ (7.14)
  $\displaystyle \longleftrightarrow$ $\displaystyle \left\vert V(\omega)\right\vert^2$  

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

$\displaystyle \hat{r}_{v,N}(l) \isdef \frac{1}{N-\vert l\vert} (v\star v)(l), \quad \vert l\vert<N. \protect$ (7.15)

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

$\displaystyle (v\star v)(l) = \left\{\begin{array}{ll} (N-\left\vert l\right\vert) \hat{r}_{v,N}(l), & l=0,\pm1,\pm2,\ldots,\pm (N-1) \\ [5pt] 0, & \vert l\vert\geq N. \\ \end{array} \right. \protect$ (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 $ \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.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