Estimation

An good estimator for the correlation function⁵ for deterministic signals could be:

$$\hat{r}_{xx}[n] = \frac{1}{L}\sum_{k=0}^{L-\vert n\vert+1} x[k]^* x[\vert n\vert+k]$$

We define bias of an estimator $\hat{X}$ of a process or random variable $X$ as:

$$\hat{\Theta}_{\hat{X}} = \vert E[\hat{X}]-X\vert$$

To check how good it is:

$$E\left\lbrace \hat{r}_{xx}[n] \right\rbrace = \frac{1}{L}\sum... ... n\vert+k] \right\rbrace = \frac{L-\vert n\vert}{L} r_{xx}[n]$$

Which shows that the estimator of the correlation is biased,If we take the limit as the number of samples $L$ goes to infinity:

$$\lim_{L\to\infty} E\left\lbrace \hat{r}_{xx}[n] \right\rbra... ...\lim_{L\to\infty} \frac{L-\vert n\vert}{L} r_{xx}[n] = r_{xx}$$

So the autocorrelation estimator $\hat{r}_{xx}[n]$ is biased, but asimptotically unbiased. In fact, an unbiased estimator of the correlation would be:

$$\check{r}_{xx}[n] = \frac{1}{\frac{L-\vert n\vert}{L}} \hat{... ...n\vert} \sum_{k=0}^{L-\vert n\vert+1} x[k]^* x[\vert n\vert+k]$$

$$E\left\lbrace \check{r}_{xx}[n] \right\rbrace = r_{xx}[n]$$

But this last one would be seldom used, and we will often use the $\hat{r}_{xx}[n]$ as the estimator for the autocorrelation. $\hat{r}_{xx}[n]$ is called the biased estimator of the correlation function, $\check{r}_{xx}[n]$ is called the unbiased estimator of the correlation function.

The variance of an estimator $\hat{X}$ of a process or random variable $X$ is defined as:

$$\sigma_{\hat{X}}^2 = E\left\lbrace (\hat{X} - E\left\lbrace X \right\rbrace)^2 \right\rbrace$$