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DESIGN OF MULTICHANNEL FILTERS

Multichannel filters are frequently useful. For example, with a vector-prediction filter one might wish to predict a time series, using its past and the past of a group of other series. With a matrix-prediction filter one could predict a group of series, using the past of the whole group. If the series are related, the group prediction should be better than self-prediction of individual channels. For definiteness, let us take two time series xt and yt and suppose we are to find a vector filter which converts them into a third series dt. If dt is xt+1, this is a unit time-span prediction for filter for xt. If dt is a vertical seismogram and xt and yt are horizontals, then the two-channel filter might be called an extrapolation filter. The set of equations which we wish to solve by least squares takes the form  
 \begin{displaymath}
\left[
\begin{array}
{l}
 d_1 \\  d_2 \\  d_3 \\  d_4 \\  \\...
 ... \\  a_2 \\  b_2 \\  \vdots \\  a_m \\  b_m \end{array} \right]\end{displaymath} (29)

If this set of equations is abbreviated  
 \begin{displaymath}
{\bf d \quad\approx\quad Bf}\end{displaymath} (30)

then, as we have seen in an earlier chapter, the solution is of the form  
 \begin{displaymath}
{\bf f} \eq ({\bf B}^T{\bf B})^{-1}{\bf B}^T{\bf d}\end{displaymath} (31)

We wish to inspect the matrix being inverted, call it R. For a filter with three time lags we get  
 \begin{displaymath}
{\bf R} \eq \sum^{}_t 
\; \left[
\begin{array}
{l}
 x_t \\  ...
 ... \; \;y_t \; \; x_{t-1} \; \;y_{t-1} \; \;x_{t-2} \; \;y_{t-2}]\end{displaymath} (32)

If we define
\begin{eqnarraystar}
r_{xx}(i) \eq \sum^{}_t \, x_t x_{t+i} \\ *
 r_{xy}(i) \eq \sum^{}_t \, x_t y_{t+i}\end{eqnarraystar}

and likewise for ryx(i) and ryy(i) the matrix (32) becomes  
 \begin{displaymath}
{\bf R}
\eq \left[
\begin{array}
{l}
 r_{xx}(0) \quad r_{xy}...
 ...quad
 r_{yx}(0) \quad \phantom{-} r_{yy}(0) \end{array} \right]\end{displaymath} (33)

We may take the 6 x 6 matrix of (33) and partition it into a 3 x 3 matrix of 2 x 2 submatrices. If we define the submatrix blocks as  
 \begin{displaymath}
{\bf R}(\tau) 
\eq \left[
\begin{array}
{ll}
 r_{xx}(\tau) &...
 ...\tau) & r_{yy}(\tau) \end{array} \right]
\eq {\bf R}^T (- \tau)\end{displaymath} (34)

then (33) in terms of the blocks defined in (34) is  
 \begin{displaymath}
{\bf R}
\eq \left[
\begin{array}
{lll}
 R(0) & R(-1) & R(-2)...
 ... R(1) & R(0) & R(-1) \\  R(2) & R(1) & R(0) \end{array} \right]\end{displaymath} (35)

The matrix in (35) is called block Toeplitz or multichannel Toeplitz. As with the ordinary Toeplitz matrix there is a trick method of solution. It will be taken up in the next section.

The reader should note that the matrix R does not depend on the desired output d. This results in a computational saving when there is more than one possible output. An example would be when it is desired to predict several different series or distances into the future on a given series.

EXERCISES:

  1. In the exercises of Chapter 2, we determined B(Z) and A(Z) such that some given power series C(Z) was expressed as C(Z) = B(Z)/A(Z). Write normal equations (do not solve them) for doing this in an approximate way by minimizing
    \begin{eqnarraystar}
\min \; (A,B) \eq \sum^{}_t \, (B_t - \sum^{}_t \, C_{t- \tau}A_\tau)^2 \nonumber\end{eqnarraystar}
    where
    \begin{eqnarraystar}
A \eq (A_0, A_1, A_2) \quad B \eq (B_0, B_1, B_2) \nonumber\end{eqnarraystar}
    subject to the constraint A0 - 1. (It can be proved that A(Z) comes out minimum-phase by examining the Levinson recursion.)

previous up next print clean
Next: LEVINSON RECURSION Up: Waveform applications of least Previous: ADAPTIVE FILTERS
Stanford Exploration Project
10/30/1997