Autocovariance sequence (ACS)

1. Jul 14, 2008

thedean515

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

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

2. Jul 23, 2008

thedean515

someone can help me