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I am getting position and velocity measurements out of a GPS sensor and I want to filter these data, so I can have a better, less noisy, estimation of the true measurements. I thought of doing this using a Wiener Filter.

Doing a little research about how Wiener filters work, I found that we have to calculate the Wiener Filter coefficients in order to minimize the average square distance between filter output and the desired signal.

The equation to calculate these coefficients are:

$$ \textbf{w} = \textbf{R}^{-1}_{yy} \textbf{r}_{yx} $$

And then to find the noise-free estimate :

$$ \hat{\textbf{x}} = \textbf{Y} \textbf{w} $$

where $\textbf{w}$ is the coefficients vector, $\textbf{R}_{yy}$ is the autocorrelation matrix for the output sensor measurement (input signal in the filter) and $\textbf{r}_{yx}$ is the cross-correlation vector for the output measurement $y$ and the desired (true) measurement $x$ (the input signal $y$ in the filter and the desired signal $x$).

So my question is how should I know what the desired signal is exactly? Or is there a better Filter to use to filter such measurements?

Other examples like this Where does the "Desired Response" come from in a Wiener filter? assume that the noise source is known. But how should I know this?

Note: I want to pre-filter these measurements in order to feed them in the better way possible in a Kalman Filter afterwards.

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    $\begingroup$ Usually, I'd just go straight to a Kalman filter. There are several questions and answers on this site that show how to set it up. See this one for example. $\endgroup$
    – Peter K.
    Commented Feb 28 at 14:56
  • $\begingroup$ @PeterK. I have already implemented the Kalman but I just want to feed it with as less noisy data as possible. $\endgroup$ Commented Feb 28 at 15:13
  • $\begingroup$ Usually, you're better off rejigging the signal model (and hence the KF) rather than cascading other systems before the KF. e.g. tweaking the $Q$ and $R$ matrices (process noise and measurement noise covariances). $\endgroup$
    – Peter K.
    Commented Feb 28 at 16:42
  • $\begingroup$ To use a Wiener filter you need to know what the statistics of the signal you’re looking for are. It’s quite difficult to use otherwise. There are techniques, but the errors those techniques introduce will likely propagate through your Kalman filter. $\endgroup$
    – Baddioes
    Commented Feb 28 at 20:48
  • $\begingroup$ You are asking your question incorrectly. You need to edit your question to lead with the fact that you're proposing to pre-filter data going into your Kalman filter. The reason you need to do this is because you don't want to do that, and if you ask the question in that form then you'll get an explanation of why (short answer: it won't help, and at best it'll just add complication and increase the necessary processing power). $\endgroup$
    – TimWescott
    Commented Mar 2 at 21:44

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