Autocovariance sequence (ACS)

[thedean515]

We have two definitions for the autocovariance of finite samples $y\left(t\right)$and
it is given as
$$\begin{equation} \hat{r}\left(k\right)=\frac{1}{N-k}\sum_{t=K+1}^{N}y\left(t\right)y^{*}\left(t-k\right),\qquad0\le k\le N-1\end{equation}$$
and
$$\begin{equation} \tilde{r}\left(k\right)=\frac{1}{N}\sum_{t=K+1}^{N}y\left(t\right)y^{*}\left(t-k\right),\qquad0\le k\le N-1\end{equation}$$

In addition we know that the autocovariance sequence for infinite
samples is
$$\begin{equation} r\left(k\right)=E\left\{ y\left(t\right)y^{*}\left(t-k\right)\right\} \end{equation}$$
where $$E\left\{ \cdot\right\}$$is the expectation operator which
averages over the ensemble of realizations. Now I have been told that
$$\begin{equation} E\left\{ \tilde{r}\left(k\right)\right\} =r\left(k\right)\end{equation}$$
and
$$\begin{equation} E\left\{ \hat{r}\left(k\right)\right\} =\frac{N-\left|k\right|}{N}r\left(k\right)\end{equation}$$
but they would be the other way round, can we proof it?

 

[thedean515]

someone can help me

 

[oswaler]

I would first verify the given definitions for the autocovariance sequence for finite and infinite samples. This can be done by plugging in values for $k$ and $N$ and comparing the results to the definition given in equation (3). Assuming the definitions are correct, I would then proceed to prove the given equations (4) and (5).

To prove equation (4), we can use the definition of expectation and substitute in the given definition for $\tilde{r}(k)$:

\begin{align*}
E\left\{ \tilde{r}\left(k\right)\right\} &=E\left\{\frac{1}{N}\sum_{t=K+1}^{N}y\left(t\right)y^{*}\left(t-k\right)\right\} \\
&=\frac{1}{N}E\left\{\sum_{t=K+1}^{N}y\left(t\right)y^{*}\left(t-k\right)\right\} \\
&=\frac{1}{N}\sum_{t=K+1}^{N}E\left\{y\left(t\right)y^{*}\left(t-k\right)\right\} \\
&=\frac{1}{N}\sum_{t=K+1}^{N}r\left(k\right) \\
&=\frac{N-K}{N}r\left(k\right) \\
&=r\left(k\right)
\end{align*}

To prove equation (5), we can again use the definition of expectation and substitute in the given definition for $\hat{r}(k)$:

\begin{align*}
E\left\{ \hat{r}\left(k\right)\right\} &=E\left\{\frac{1}{N-k}\sum_{t=K+1}^{N}y\left(t\right)y^{*}\left(t-k\right)\right\} \\
&=\frac{1}{N-k}E\left\{\sum_{t=K+1}^{N}y\left(t\right)y^{*}\left(t-k\right)\right\} \\
&=\frac{1}{N-k}\sum_{t=K+1}^{N}E\left\{y\left(t\right)y^{*}\left(t