# Inverse filtering

There is an example of inverse filtering in my textbook which I don't fully understand.

Given the two filters v(t) = (1,-0.5) and w(t) = (-0.5,1). The inverse of these filters are given by:

for v: 1:(1,-0.5) = (1, 0.5, 0.25, . . .)

for w: 1:(-0.5,1) = (-2, -4, -8, . . .)

However, there is no mathematical illustration for how these inverse filters are calculated, and I don't see the intuition here. Of course I know that the inverse of a filter v is 1/v, but why these two filters yield these different series puzzles me.

If anyone can explain this to me, I would be very grateful!

Hint: try to write your inverse filter on the form $H(z) = \frac{1}{1-0.5z^-1}$, then write the corresponding difference equation and find the impulse response. What does that give you?