next up previous print clean
Next: Normal equations Up: MULTIVARIATE LEAST SQUARES Previous: MULTIVARIATE LEAST SQUARES

Inverse filter example

Let us take up a simple example of time-series analysis. Given the input, say $(\cdots ,0, 0, 2, 1,0,0,\cdots )$, to some filter, say $\bold f = (f_0, f_1)$, then the output is necessarily $\bold c = (2f_0, \, f_0 + 2f_1, \, f_1)$. To design an inverse filter, we would wish to have boldc come out as close as possible to (1, 0, 0). So the statement of wishes (17) is  
 \begin{displaymath}
\left[ 
\begin{array}
{c}
 1 \  
 0 \  
 0 \end{array} \ri...
 ...\; \left[ 
\begin{array}
{c}
 f_0 \  
 f_1 \end{array} \right]\end{displaymath} (19)
The method of solution is to premultiply by the matrix $\bold B'$, getting
\begin{displaymath}
\left[ 
\begin{array}
{ccc}
 2 & 1 & 0 \  0 & 2 & 1 \end{ar...
 ... 
\left[ 
\begin{array}
{ccc}
 f_0 \  f_1 \end{array} \right] \end{displaymath} (20)
Thus,
\begin{displaymath}
\left[ 
\begin{array}
{c}
 2 \  0 \end{array} \right] 
\eq
...
 ...
\; \left[ 
\begin{array}
{c}
 f_0 \  f_1 \end{array} \right] \end{displaymath} (21)
and the inverse filter comes out to be
\begin{displaymath}
\left[ 
\begin{array}
{c}
 f_0 \  f_1 \end{array} \right] 
...
 ...rray}
{r}
 {10 \over 21} \  -{4 \over 21} \end{array} \right] \end{displaymath} (22)
Inserting this value of (f0,f1) back into (19) yields the actual output $({20 \over 21}, +{2 \over 21}, -{4 \over 21})$,which is not a bad approximation to (1, 0, 0).


next up previous print clean
Next: Normal equations Up: MULTIVARIATE LEAST SQUARES Previous: MULTIVARIATE LEAST SQUARES
Stanford Exploration Project
10/21/1998