PEF-based interpolation

A prediction-error filter (PEF) can be estimated by minimizing the following residual,  

$$\bold 0 \quad \approx \quad \bold r = \left[ \begin{array} {... ... {c} d_2 \\ d_3 \\ d_4 \\ d_5 \\ d_6 \end{array} \right] ,$$ (1)

where f_i are unknown filter values and d_i are known data values. In this case the filter has two free coefficients and there are seven data points. In practice, the PEF is multi-dimensional and contains many more coefficients. Once the PEF has been estimated it can be used in a second least-squares problem

$$\begin{eqnarray} \bold{S(m - d)} = \bold{0} \nonumber\\ \bf F m \approx 0 ,\end{eqnarray}$$ (2)

where $\bf{S}$ is a selector matrix which is 1 where data is present and where it is not, $\bf{F}$ represents convolution with the PEF and $\bf{m}$ is the desired model. The first line in equation 2 is a hard constraint while the second line is not.

In order for this method to work, the data used as input to equation 1 needs to be similar to the desired output model in equation 2. The slightly different approximations used in t-x and f-x PEFs are discussed next.