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.

This guide was created with the help of AI, based on the presentation's transcript. Its goal is to give you useful context and background so you can get the most out of the session.

What this presentation is about and why it matters

This talk presents a compact, visual way to read and design discrete-time filters by labeling every point in the Z‑plane with two physically meaningful numbers: a resonance frequency and a damping. Instead of only working with real/imaginary parts or magnitude/phase of poles and zeros, the speaker maps each conjugate root pair to a center frequency and a damping (related to Q). The result is a set of circles in the Z‑plane: one family for constant frequency and another for constant damping. Overlaying these circles turns the Z‑plane into a map you can use to eyeball or design filters—peaks, notches, shelves, all‑pass stages, and higher‑order responses—more intuitively.

Why this matters: engineers and DSP practitioners often move between pole/zero diagrams and auditory or spectral goals (center frequency, bandwidth, peaking gain). This representation makes that mapping direct. It helps when designing biquads, composing higher‑order filters from 2nd‑order sections, understanding the effect of bilinear warping, and diagnosing why a pole/zero layout produces a certain audible behavior.

Who will benefit the most from this presentation

• Audio DSP engineers who design biquads, equalizers, and crossovers and want a faster way to interpret pole/zero plots.
• Students learning digital filter design who need geometric intuition about poles, zeros, Q, and frequency warping.
• Anyone who inspects Z‑plane plots for debugging or filter verification and wants a rule set to translate locations into filter behavior (peaks, notches, shelves, phase shapers).

What you need to know

To get the most out of the talk you should be comfortable with these core ideas:

• Complex roots and conjugate symmetry — Real filters have complex poles/zeros in conjugate pairs; these pairs form second‑order sections.
• Pole/zero basics — Poles near the unit circle produce resonant peaks; zeros near the unit circle produce notches or attenuation at corresponding frequencies.
• Second‑order resonator parameterization — A conjugate root pair can be expressed by a resonance frequency $\omega_n$ and damping $\psi$ (sometimes written $\zeta$). The speaker uses the convenient relation between damping and quality factor: $\psi = 1/(2Q)$, so $Q=1/(2\psi)$.
• Simple formulas used — In the S‑plane the resonance frequency can be written as $\omega_n = \sqrt{r_1 r_2}$ (product of roots) and damping is tied to the sum of roots. To move between digital ordinary frequency $f$ and normalized analog frequency used by the bilinear transform one common mapping is $\Omega = \tan(\pi f / f_s)$ (this is the normalized frequency used inside the bilinear mapping). The talk outlines the forward and inverse recipes: ordinary $f$ → normalized S‑plane → place S‑roots by $(\omega_n,\psi)$ → bilinear map → Z‑roots, and the inverse.
• Bilinear transform and frequency warping — The bilinear (Tustin) map warps high frequencies and maps $s=\infty$ to $z=-1$ (Nyquist). Because it is a Möbius transform it maps circles/lines to circles/lines: constant‑frequency and constant‑damping loci in the S‑plane become circles in the Z‑plane.
• Minimum‑phase vs non‑minimum‑phase — The method is described first for minimum‑phase (roots inside the unit circle). Zeros outside the circle can be treated via reflection and an all‑pass factor; equivalently the method allows negative damping or negative mapped frequency signs to represent roots outside the unit circle.

Glossary

• Z‑plane — Complex plane for discrete‑time systems; the unit circle corresponds to sinusoidal frequencies.
• Pole — A root of the denominator of a transfer function; nearby poles create resonant peaks and determine stability (must lie inside the unit circle for causal, stable digital filters).
• Zero — A root of the numerator; zeros attenuate or notch frequencies when near the unit circle.
• Second‑order section (biquad) — A pair of conjugate poles and/or zeros implemented as a 2nd‑order transfer function; the basic building block used in modular filter design.
• Resonance frequency ($\omega_n$) — The center frequency of a pole/zero pair; in this talk it is one of the two parameters used to label a Z‑plane point.
• Damping ($\psi$ or $\zeta$) — A dimensionless measure of how sharp or broad a resonance is; related to Q by $\psi=1/(2Q)$ and used as the second Z‑plane coordinate.
• Quality factor (Q) — Inverse measure of relative bandwidth; higher Q means narrower peaks or notches. Here Q is tied directly to the damping parameter.
• Bilinear transform (Tustin) — A conformal map from the S‑plane to the Z‑plane used to convert analog designs to digital, with frequency warping at high frequencies.
• Frequency warping — Nonlinear mapping of frequency introduced by the bilinear transform; low frequencies map more linearly than high frequencies.
• Minimum‑phase — Systems whose poles and zeros lie inside the unit circle; magnitude response fully determines an equivalent minimum‑phase system up to an all‑pass.