0
$\begingroup$

Consider a random process with auto-correlation function: $$r_{\rm dd} [k] = \beta^{\lvert k \rvert}\quad\text{where}\quad 0 < \beta < 1. $$ 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

$$W(z) = w(0) + w(1)z^{-1}$$

$\endgroup$
  • $\begingroup$ Is this homework? Can you show your attempt at solving this? $\endgroup$ – Atul Ingle Jan 3 '17 at 16:43
  • $\begingroup$ the w=R^(-1) * p , which R is the correlation matrix of X[n] and p is the cross-correlation vector between d[n] and X[n]. i should compute the inverse of R and p to solve the function to get the W. is that right? it is my opinion. $\endgroup$ – Haoming Li Jan 3 '17 at 22:27
  • $\begingroup$ @HaomingLi , your question should have the homework tag. $\endgroup$ – Gilles Jan 3 '17 at 22:49
  • $\begingroup$ -1 for removing the homework tag. $\endgroup$ – msm Jan 3 '17 at 22:59
  • $\begingroup$ Sorry about that, i am fresh here and i update it now, my solution above is correct or not plz guide me 0.0 $\endgroup$ – Haoming Li Jan 3 '17 at 23:20
1
$\begingroup$

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

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

Filling in the variables, we get

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

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.