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M. H. Hayes calls in his book "Statistical digital signal processing" autocorrelation sequences $r[k]$.

For optimal filters the desired autocorrelation has the index d -> $r_d[k]$. However, often there is also an $r_{dx}[k]$. Are they the same? Or is there an assumption where you can interchange them?

Because in some cases it seems like he does just that. No clue why the index changed in some formulas.

This is how d and x are connected:

process

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All these notation should be considered within the context of the development.

So $r_d[k]$ may refer to the auto-correlations sequence of the desired signal whereas the $r_{dx}[k]$ is the cross correlation between the input $x[n]$ and desired signal $d[n]$. So they are two different things.

The latter is used in Wiener optimal filtering to solve the Wiener-Hopf equations (aka The Normal Equations) but I cannot recall where the former is used, can you cite the formula, chapter etc. that makes use of it?

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  • $\begingroup$ There are several occasions. One I just have in mind is converted to power spectral density in question 7.8 p.383. The solution for H is $\frac{P_{dx}(e^{jw})}{P_x(e^{jw})}$ which becomes $\frac{P_d(e^{jw})}{P_d(e^{jw})+P_{\omega}(e^{jw})}$. Is it because it is assumed that d and the noise are not correlated? $\endgroup$ – Mr.Sh4nnon Feb 7 at 11:14
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    $\begingroup$ As you have stated, such cases arise because of particular assumptions or relations between $x[n]$, $d[n]$ ad noise on them. $\endgroup$ – Fat32 Feb 7 at 12:00

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