Sample Autocorrelation

The *sample autocorrelation* of a sequence
,
may be defined by

where is defined as

and zero for .

In matlab, the sample autocorrelation of a vector `x`
can be computed using the `xcorr` function.^{7.3}

Example:

octave:1> xcorr([1 1 1 1], 'unbiased') ans = 1 1 1 1 1 1 1The

Note that
is the average of the *lagged product*
over all available data. For white noise, this
average approaches zero for
as the number of terms in the
average increases. That is, we must have

(7.8) |

where

(7.9) |

is defined as the

The plot in the upper left corner of Fig.6.1 shows the sample
autocorrelation obtained for 32 samples of pseudorandom numbers
(synthetic random numbers). (For reasons to be discussed below, the
sample autocorrelation has been multiplied by a Bartlett (triangular)
window.) Proceeding down the column on the left, the results of
averaging many such sample autocorrelations can be seen. It is clear
that the average sample autocorrelation function is approaching an
impulse, as desired by definition for white noise.
(The right column shows the Fourier transform of each sample
autocorrelation function, which is a smoothed estimate of the
*power spectral density*, as discussed in §6.6 below.)

For stationary stochastic processes
, the sample autocorrelation
function
approaches the true autocorrelation function
in the limit as the number of observed samples
goes to
infinity, *i.e.*,

(7.10) |

The true autocorrelation function of a random process is defined in Appendix C. For our purposes here, however, the above limit can be taken as the

At lag
, the autocorrelation function of a zero-mean random
process
reduces to the *variance*:

The variance can also be called the

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