Derivation of Complex Isolated IIR Pole

The complex isolated IIR pole is derived. References given derive a zero from the pole, and illustrate how they are applied to make complex time domain filters. We will start by deriving an IIR representation for a negative real pole, then generalize to the complex case. A formal derivation might derive the difference equation from a differential equation, but a heuristic derivation is more accessible, and will be presented here. We want a unity gain negative real pole, a “leaky integrator”. Let ǫ be a positive real number much smaller than one. A good guess is: