IIR Filter
Canonical Definition
An infinite impulse response filter is a discrete-time filter whose current output depends on current or past input samples and one or more past output states. Feedback permits long or theoretically unending decay with few coefficients, while stability, precision, saturation, phase, and state behavior depend on the implemented transfer function and numeric architecture.
It does not automatically mean analog, minimum phase, stable, low latency, or acoustically correct.
Basic Form
y[n] = b0x[n] + b1x[n-1] + ... - a1y[n-1] - a2y[n-2] - ...The b terms are feedforward coefficients. The a terms feed previous output states back into the calculation. A common audio implementation is the second-order section or biquad.
Why IIR Is Common
- useful crossover, shelf, notch, parametric-EQ, and all-pass shapes with few coefficients
- low memory use and usually low instruction count
- direct real-time tuning through familiar frequency, gain, Q, and slope controls
- efficient implementation on processors such as SigmaDSP
Efficiency does not remove the need to inspect the resulting magnitude, phase, group delay, internal level, and acoustic response.
Poles, Zeros, And Stability
Poles and zeros describe the transfer function. Pole locations control resonance and decay. A causal discrete IIR must keep its poles inside the unit circle for stability. Coefficient quantization can move the implemented poles away from the mathematical design, especially for high-Q and very-low-frequency sections.
- record coefficient format and precision mode
- prefer appropriate double-precision sections where the platform recommends them
- check cascaded-section order and internal headroom
- slew or mute during large coefficient changes
Phase And Minimum Phase
Many standard stable causal IIR EQ and crossover designs are minimum phase, but IIR as a category is not automatically minimum phase. Added delay, all-pass sections, parallel paths, and the installed acoustic system can create excess-phase behavior.
A named electrical alignment also does not guarantee the same acoustic alignment after driver response, enclosure, mounting, spacing, polarity, and delay are included.
Verification Record
- sample rate, filter type, order, frequency, gain, Q, coefficients, and precision
- block order and every stage that can add gain
- electrical magnitude, phase, group delay, and peak gain
- internal meters and physical analog output
- preset load, bypass, reset, and coefficient-update behavior
- installed driver magnitude, phase, distortion, and acoustic sum