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.