Recursive Filters as Linear Time-Invariant Systems

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.