First order Wiener–Hopf filter design Consider a random process with auto-correlation function:
  $$r_{\rm dd} [k] = \beta^{\lvert k \rvert}\quad\text{where}\quad 0 &lt; \beta &lt; 1. 
$$
  Suppose also that the observation is:
  $$
x[n] = d[n] + v[n]
$$
  where $v[n]$ is uncorrelated white noise with variance $\sigma^2$.
  
  Design a first order Wiener–Hopf filter to reduce the noise in $x[n]$ of the form
  
  $$W(z) = w(0) + w(1)z^{-1}$$
I refer to the notation from the Wikipedia article.

Your received signal is $x[n]$, and its Autocorrelation is given by
$R_x[n]=R_d[n]+R_v[n]$ when noise and signal are uncorrelated. Hence, 

$$
R_x[n]=\beta^{|k|}+\sigma^2\delta[n]
$$

The cross-correlation between the received signal and the signal of interest is

$$R_{xd}[n]=E[d[n'](d[n'+n]+v[n'+n])]=R_d[n]$$

under the assumption again that signal and noise are uncorrelated.

Now, the Wiener-Hopf equation gets you 

$$
\begin{pmatrix}R_x[0] &amp; R_x[1]\\ R_x[1] &amp;R_x[0]\end{pmatrix}\begin{pmatrix}w[0]\\w[1]\end{pmatrix}=\begin{pmatrix}R_{xd}[0]\\R_{xd}[1]\end{pmatrix}
$$

Filling in the variables, we get

$$
\begin{pmatrix}1+\sigma^2 &amp; \beta\\ \beta &amp;1+\sigma^2\end{pmatrix}\begin{pmatrix}w[0]\\w[1]\end{pmatrix}=\begin{pmatrix}1\\\beta\end{pmatrix}
$$

Now, you just need to solve for $w[0],w[1]$ to get your filter coefficients.