  Recursive filters are treated as linear time-invariant (LTI) systems but they
are not: uninitialised, they have an infinite number of outputs for any given
input, while if initialised, they are not time-invariant. This short tutorial
article explains how and why they can be treated as LTI systems, thereby
allowing tools such as Fourier analysis to be applied. It also explains the
origin of the z-transform, why the region of convergence is important, and why
the z-transform fails to find an infinite number of solutions.
