Filter estimation

When PEFs are estimated, the matrix ${\bf A}$ is unknown. If ${\bf y}$is the data vector from which we want to estimate the filters, we minimize the vector ${\bf r_y}$ as follows:  

$${\bf 0 \approx r_y = Ay}$$ (125)

which can be rewritten  

$${\bf 0 \approx r_y = Ya},$$ (126)

where ${\bf Y}$ is the matrix representation of the non-stationary convolution or combination with the input vector ${\bf y}$.The transition between equations () and () is not simple. In particular, the shape of the matrix ${\bf Y}$ is quite different if we are doing non-stationary convolution or combination. For the convolution, we have  

$$\bf{Y_{conv}}=\left( \begin{array} {c\vert c\vert c\vert c} ... ...bf{Y^2_{conv}} & \cdots \end{array} \right ), \; \mbox{where}$$ (127)

$$\bf{Y^0_{conv}} = \left( \begin{array} {cccc} y_0 & 0 & 0 &... ... & \vdots & \vdots & \vdots \end{array} \right )\mbox{etc...}$$

We see that for the convolution case, the ${\bf Y^k_{convo}}$ matrices are diagonal operators, translating the need for one filter to be applied to one input point. The size of the matrix $\bf{Y_{conv}}$ is $(n \times (m\times nf))$ where nf is the number of filter coefficients. Now, for the combination, we have  

$$\bf{Y_{comb}}=\left( \begin{array} {c\vert c\vert c\vert c} ... ...bf{Y^2_{comb}} & \cdots \end{array} \right ), \; \mbox{where}$$ (128)

$$\bf{Y^0_{comb}} = \left( \begin{array} {cccc} y_0 & 0 & 0 &... ... & \vdots & \vdots & \vdots \end{array} \right )\mbox{etc...}$$

We see that for the combination case, the ${\bf Y^k_{combo}}$ matrices are row operators, translating the need for one filter to be constant for one output point. The size of $\bf{Y_{comb}}$ is equal to the size of $\bf{Y_{conv}}$.For the vector ${\bf a}$ in equation () we have  

$$\bf{a}=\left( \begin{array} {c} \bf{a_0} \\ \hline \bf{a_... ...,k} \\ a_{2,k} \\ \vdots \\ a_{nf-1,k} \end{array} \right )$$ (129)

where nf is the number of coefficients per filter. This definition of ${\bf a}$ is independent of ${\bf Y}$.We might want to have one filter common to different input or output points instead of one filter per point. In that case, the matrix ${\bf Y}$ is obtained by adding successive ${\bf Y^k}$ matrices depending on how many points have a similar filter. Note that in the stationary case, for both the convolution and the combination case we have $\bf{a_0}=\bf{a_1}=\bf{a_2}=\cdots=\bf{a}$and

$$\bf{Ya}= \left( \bf{Y_0} + \bf{Y_1} + \bf{Y_2} + \cdots \right )\bf{a}.$$ (130)

Therefore, for the matrix ${\bf Y}$, we have to add all the ${\bf Y^k}$ matrices together. If we take advantage of the special structure of $\bf{Y_k}$ for the convolution and the combination, we obtain for the stationary case  

$$\bf{Ay}=\bf{Ya}= \left( \begin{array} {cccccc} y_0 & 0 & 0 ... ...1 \\ a_1 \\ a_2 \\ a_3 \\ \vdots \\ \end{array} \right),$$ (131)

which is the matrix formulation of the stationary convolution. With the definitions given in equations (), () and (), the fitting goal in equation () can be rewritten  

$${\bf 0 \approx r_y = Y^0_{conv}a_0 + Y^1_{conv}a_1 + Y^2_{conv}a_2 + \cdots}$$ (132)

or  

$${\bf 0 \approx r_y = Y^0_{comb}a_0 + Y^1_{comb}a_1 + Y^2_{comb}a_2 + \cdots}$$ (133)

Each vector ${\bf a_k}$ has one constrained coefficient. We can then rewrite equations () and () as follows:  

$${\bf 0 \approx r_y = y+Y^0_{conv}Ma_0 + Y^1_{conv}Ma_1 + Y^2_{conv}Ma_2 + \cdots}$$ (134)

and  

$${\bf 0 \approx r_y = y+Y^0_{comb}Ma_0 + Y^1_{comb}Ma_1 + Y^2_{comb}Ma_2 + \cdots}$$ (135)

with

$${\bf M}= \left( \begin{array} {cccc} 0 & 0 & 0 & \vdots \\... ...ts \\ \vdots & \vdots & \vdots & \vdots \end{array} \right )$$ (136)

The definition of ${\bf M}$ assumes that the first coefficient of each filter is known. Note that ${\bf M}$ is equal for both convolution and combination methods. Having defined the matrix ${\bf M}$, we can now rewrite equation () as follows:  

$$\bf{0} \approx r_y = \bf{YKa}+\bf{y}$$ (137)

where the square matrix ${\bf K}$ is

$$\bf{K}=\left( \begin{array} {c\vert c\vert c\vert c} \bf{M}... ... \hline \vdots & \vdots & \vdots & \vdots \end{array}\right).$$ (138)

The next step consists of estimating the filter coefficients in a least-squares sense.