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Autocovariance sequence (ACS)

  1. Jul 14, 2008 #1
    We have two definitions for the autocovariance of finite samples $y\left(t\right)$and
    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?
     
  2. jcsd
  3. Jul 23, 2008 #2
    someone can help me
     
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