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  

$$\left[ \begin{array} {c} 1 \ 0 \ 0 \end{array} \ri... ...\; \left[ \begin{array} {c} f_0 \ f_1 \end{array} \right]$$ (19)

The method of solution is to premultiply by the matrix $\bold B'$, getting

$$\left[ \begin{array} {ccc} 2 & 1 & 0 \ 0 & 2 & 1 \end{ar... ... \left[ \begin{array} {ccc} f_0 \ f_1 \end{array} \right]$$ (20)

Thus,

$$\left[ \begin{array} {c} 2 \ 0 \end{array} \right] \eq ... ... \; \left[ \begin{array} {c} f_0 \ f_1 \end{array} \right]$$ (21)

and the inverse filter comes out to be

$$\left[ \begin{array} {c} f_0 \ f_1 \end{array} \right] ... ...rray} {r} {10 \over 21} \ -{4 \over 21} \end{array} \right]$$ (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).