Derivation of Complex Isolated IIR Pole
Donald C. Daniel · 2011
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: yk = (1 − ǫ)yk−1 + ǫxk (1) It is not obvious from the right side of the equation that the gain is unity, because of the yk−1 term. The steady state is defined by the condition where xk = xk−1 for many iterations. The filter does not need to operate in the steady state, but the steady state is useful for discovering the properties of