IIR Filter Pole-Zero Placement Calculator
Design a 2-pole IIR resonator filter: enter a center frequency and bandwidth, get pole location, coefficients, and a stability check.
🎯 What is IIR Filter Pole-Zero Placement?
Pole-zero placement is the process of choosing the exact location of a digital filter's poles and zeros in the z-plane so that the filter's frequency response does what you want, a resonant boost, a notch, a low-pass roll-off, or anything in between. For an IIR (Infinite Impulse Response) filter, the poles determine where the filter's transfer function goes to infinity in the complex z-plane, and their distance from the unit circle directly controls how sharp and how stable the resulting response is.
Audio engineers use two-pole resonators like this one inside parametric equalizers, wah pedals, and formant filters for speech synthesis, each pole pair creating one resonant peak at a chosen center frequency. Control engineers place poles when designing digital compensators, since pole location determines settling time and oscillation. Musical instrument physical-modeling synthesizers chain many resonators together, each one built from exactly this math, to simulate the resonant modes of strings, tubes, and membranes.
A common misconception is that any pole location is usable. It is not: a pole must sit strictly inside the unit circle (radius r < 1) for the filter to be stable, meaning its impulse response eventually decays to zero rather than growing without bound. Placing a pole exactly on the unit circle produces a filter that oscillates forever; placing it outside produces an output that blows up exponentially.
This calculator takes the two numbers that matter most in practice, the resonant frequency and the -3 dB bandwidth around it, and converts them into the underlying pole radius, pole angle, and the a1/a2 coefficients needed to implement the filter as a two-line difference equation.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Speech Formant Resonator
fs = 8,000 Hz, f0 = 1,000 Hz, BW = 50 Hz
Example 2 — Narrow Musical Note Resonator
fs = 44,100 Hz, f0 = 440 Hz (A4), BW = 10 Hz
Example 3 — Quarter-Nyquist Resonator
fs = 48,000 Hz, f0 = 12,000 Hz, BW = 200 Hz
❓ Frequently Asked Questions
🔗 Related Calculators
What does pole-zero placement mean in an IIR filter?
Pole-zero placement is choosing where a filter's poles (and zeros) sit in the z-plane to shape its frequency response. For a simple 2-pole resonator, placing a conjugate pole pair near the unit circle at a chosen angle creates a sharp resonant peak at the corresponding frequency.
How do you calculate pole radius from bandwidth?
Pole radius is r = e^(-pi * BW / fs), where BW is the desired -3 dB bandwidth in Hz and fs is the sampling rate. A narrower bandwidth (smaller BW) pushes r closer to 1, producing a sharper resonance.
How do you calculate pole angle from frequency?
Pole angle is theta = 2*pi*f0/fs radians, where f0 is the target resonant frequency and fs is the sampling rate. This maps the continuous frequency range 0 to fs/2 onto the angle range 0 to pi radians around the upper half of the unit circle.
Why do poles come in conjugate pairs?
A real-valued (not complex) filter's transfer function must have real coefficients, which forces any complex poles to appear in conjugate pairs (a + jb and a - jb). A single pole pair at angle theta and its mirror at -theta together produce a real, physically realizable resonant filter.
What makes an IIR filter unstable?
An IIR filter is unstable if any pole lies on or outside the unit circle (|z| >= 1) in the z-plane. Because IIR filters use feedback, an unstable pole causes the output to grow without bound instead of settling, unlike FIR filters, which have no poles and are always stable.
How are a1 and a2 used in the filter's difference equation?
For the two-pole resonator H(z) = 1 / (1 - 2r*cos(theta)*z^-1 + r^2*z^-2), the difference equation is y[n] = x[n] - a1*y[n-1] - a2*y[n-2], with a1 = -2*r*cos(theta) and a2 = r^2. These two coefficients are all that is needed to implement the resonator in code or hardware.
What is the relationship between pole radius and ringing time?
The decay time constant is tau = 1 / (pi * BW) seconds, independent of the sampling rate. A pole closer to the unit circle (smaller BW, larger r) rings down more slowly; a pole further inside (larger BW, smaller r) decays quickly but produces a broader, less selective resonance.
Can this pole placement create a notch filter instead of a resonator?
This calculator places poles only, which boosts a frequency band (a resonator/peaking response). A notch (band-reject) filter additionally places a zero pair on the unit circle at the same angle theta, canceling that frequency exactly while the nearby poles shape how quickly the response recovers on either side.
Why is the pole angle capped below the Nyquist frequency?
The pole angle theta = 2*pi*f0/fs only has meaning for f0 between 0 and fs/2 (the Nyquist frequency), which maps to theta between 0 and pi. Requesting a resonant frequency at or above fs/2 would fold back (alias) onto a lower frequency, so this calculator rejects it as invalid input.
How does this relate to Butterworth and Chebyshev filter design?
Butterworth and Chebyshev filters are built by placing several pole pairs (and for Chebyshev, a specific radius pattern) around the z-plane rather than a single pair. This single-resonator calculator shows the same underlying pole-radius/pole-angle math on a scale simple enough to compute by hand.
What sampling rate should I use for a musical resonator?
Use whatever sampling rate your audio system runs at, commonly 44,100 Hz or 48,000 Hz. The pole angle and radius formulas automatically scale to any sampling rate, so the same center frequency and bandwidth in Hz produce the equivalent resonance regardless of fs.