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 x_t and y_t and suppose we are to find a vector filter which converts them into a third series d_t. If d_t is x_t+1, this is a unit time-span prediction for filter for x_t. If d_t is a vertical seismogram and x_t and y_t 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  

$$\left[ \begin{array} {l} d_1 \\ d_2 \\ d_3 \\ d_4 \\ \\... ... \\ a_2 \\ b_2 \\ \vdots \\ a_m \\ b_m \end{array} \right]$$ (29)

If this set of equations is abbreviated  

$${\bf d \quad\approx\quad Bf}$$ (30)

then, as we have seen in an earlier chapter, the solution is of the form  

$${\bf f} \eq ({\bf B}^T{\bf B})^{-1}{\bf B}^T{\bf d}$$ (31)

We wish to inspect the matrix being inverted, call it R. For a filter with three time lags we get  

$${\bf R} \eq \sum^{}_t \; \left[ \begin{array} {l} x_t \\ ... ... \; \;y_t \; \; x_{t-1} \; \;y_{t-1} \; \;x_{t-2} \; \;y_{t-2}]$$ (32)

If we define

and likewise for r_yx(i) and r_yy(i) the matrix (32) becomes  

$${\bf R} \eq \left[ \begin{array} {l} r_{xx}(0) \quad r_{xy}... ...quad r_{yx}(0) \quad \phantom{-} r_{yy}(0) \end{array} \right]$$ (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  

$${\bf R}(\tau) \eq \left[ \begin{array} {ll} r_{xx}(\tau) &... ...\tau) & r_{yy}(\tau) \end{array} \right] \eq {\bf R}^T (- \tau)$$ (34)

then (33) in terms of the blocks defined in (34) is  

$${\bf R} \eq \left[ \begin{array} {lll} R(0) & R(-1) & R(-2)... ... R(1) & R(0) & R(-1) \\ R(2) & R(1) & R(0) \end{array} \right]$$ (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 where subject to the constraint A₀ - 1. (It can be proved that A(Z) comes out minimum-phase by examining the Levinson recursion.)