We have two definitions for the autocovariance of finite samples $y\left(t\right)$and(adsbygoogle = window.adsbygoogle || []).push({});

it is given as

[tex]

\begin{equation}

\hat{r}\left(k\right)=\frac{1}{N-k}\sum_{t=K+1}^{N}y\left(t\right)y^{*}\left(t-k\right),\qquad0\le k\le N-1\end{equation}[/tex]

and

[tex]

\begin{equation}

\tilde{r}\left(k\right)=\frac{1}{N}\sum_{t=K+1}^{N}y\left(t\right)y^{*}\left(t-k\right),\qquad0\le k\le N-1\end{equation}

[/tex]

In addition we know that the autocovariance sequence for infinite

samples is

[tex]

\begin{equation}

r\left(k\right)=E\left\{ y\left(t\right)y^{*}\left(t-k\right)\right\} \end{equation}

[/tex]

where [tex]E\left\{ \cdot\right\}[/tex]is the expectation operator which

averages over the ensemble of realizations. Now I have been told that

[tex]\begin{equation}

E\left\{ \tilde{r}\left(k\right)\right\} =r\left(k\right)\end{equation}

[/tex]

and

[tex]

\begin{equation}

E\left\{ \hat{r}\left(k\right)\right\} =\frac{N-\left|k\right|}{N}r\left(k\right)\end{equation}

[/tex]

but they would be the other way round, can we proof it?

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# Autocovariance sequence (ACS)

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