Almost all discrete LTI systems can be represented as a rational function in the Z-domain. This rational function can be fully characterized by a gain and the roots of the polynomials, which are, of course, the poles and the zeros of that transfer function. Any real transfer function can be broken down into cascaded second order sections each with a pair of zeros and a pair of poles. The direct interpretation of these root pairs in terms of real or imaginary part or magnitude and phase or two real roots isn’t straight forward.

In this presentation we show an alternative interpretation where each root pair can be represented as the resonance frequency and the damping of a second order resonator. We’ll show how these parameters map to the Z-plane and that each point in the z-plane can be uniquely associated with a specific frequency & damping. In other words, we can answer “What is the Q of a pole” or at least a pole pair.

Finally, we’ll demonstrate how some popular filter types can be intuitively designed using this representation.