Estimation Basics and Periodogram

Quite similar to what was done in TMA4245 - Statistics.

If an estimator is consistent, this means that it approaches the true value as $N\to \infty$

Properties of a good estimator

Unbiased - $m_{\hat{\Theta }}=E(\hat{\Theta })=\Theta$ (Can also have a weaker requirement where it approaches $\Theta$ as $N->\infty$) Small variance - ${\sigma }_{\hat{\Theta }}^2\ll$. Decreasing variance - as $N$ gets smaller, the variance should get smaller.

Crude estimate of PDS

When using the estimator

$${\hat{\gamma }}_{xx}[l]=\{\begin{array}{ll}\frac{1}{N}{\sum }_{n=0}^{N-|l|-1}x[n]x[n+|l|],& |l|<N\\ 0& \text{ce}otherwise\end{array}$$

Where $W_B$ is the Bartlett Window