Inverse filters

Let b_t denote a filter. Then a_t is its ``inverse filter'' if the convolution of a_t with b_t is an impulse function. Filters are said to be inverse to one another if their Fourier transforms are inverse to one another. So in terms of Z-transforms, the filter A(Z) is said to be inverse to the signal of B(Z) if A(Z)B(Z)=1. What we have seen so far is that the inverse filter can be stable or unstable depending on the location of its poles. Likewise, if B(Z) is a filter, then A(Z) is a usable filter inverse to B(Z), if A(Z)B(Z)=1 and if A(Z) does not have coefficients that tend to infinity.

Another approach to inverse filters lies in the Fourier domain. There a filter inverse to b_t is the a_t made by taking the inverse Fourier transform of $1/B(Z(\omega))$.If B(Z) has its zeros outside the unit circle, then a_t will be causal; otherwise not. In the Fourier domain the only danger is dividing by a zero, which would be a pole on the unit circle. In the case of Z-transforms, zeros should not only be off the circle but also outside it. So the $\omega$-domain seems safer than the Z-domain. Why not always use the Fourier domain? The reasons we do not always inverse filter in the $\omega$-domain, along with many illustrations, are given in chapter .