Autocorrelation estimators

In reality we don’t have access to the real autocorrelation, so we have to estimate it

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Estimator 1:

$${\hat{\gamma }}_{xx}[l]=\frac{1}{N}\sum _{n=0}^{N-|l|-1}x[n]x[n+|l|]$$

Variance of estimator goes to zero for a fixed $l$ and $N\to \infty$

$$E\{{\hat{\gamma }}_{xx}[l]\}=\left(1-\frac{|l|}{N}\right){\gamma }_{xx}[l]$$

Disadvantages:

• Biased for $l\ne 0$

Estimator 2:

$${\hat{\gamma }}_{xx}[l]=\frac{1}{N-|l|}\sum _{n=0}^{N-|l|-1}x[n]x[n+|l|]$$

This estimator is unbiased and the variance goes towards zero as $N\to \infty$.

Disadvantages:

• Variance increases as $|l|\to N$