What Are IIR Filters?

IIR filters, or infinite impulse response filters, are a class of digital filters in which the output depends on both present and past input samples and on past output samples. The feedback from previous outputs gives the filter an impulse response that theoretically extends to infinite length, in contrast to finite impulse response (FIR) filters, which use no feedback and produce an impulse response that is exactly zero after a finite number of samples. IIR filters are drawn from the theory of classical analog filter design: Butterworth, Chebyshev, and elliptic analog filter prototypes are mapped into the digital domain using established transformation methods, yielding digital filters that closely replicate the frequency-selective behavior of their analog counterparts.

The primary practical appeal of IIR filters is computational efficiency. For a given frequency selectivity requirement, an IIR filter typically requires far fewer multiplications and additions per output sample than an equivalent FIR design, which is a significant advantage in real-time embedded signal processing systems where computation budget is constrained.

Transfer Function and Feedback Structure

An IIR filter is characterized by a rational transfer function in the z-domain, with both a numerator polynomial (defining the zeros) and a denominator polynomial (defining the poles). The poles arise from the recursive feedback terms. In the time domain, the output y[n] is computed as a weighted sum of the M most recent input samples x[n-k] and the P most recent output samples y[n-k], governed by the difference equation coefficients a_k (feedback) and b_k (feedforward).

Poles govern the frequency-selective character of the filter: a pair of complex poles placed near a particular frequency on the unit circle in the z-plane will produce a resonant peak or sharp roll-off in the frequency response near that frequency. The University of Texas at Austin lecture notes on IIR filter design present pole-zero analysis and biquad section design, covering both cascade and parallel implementations of higher-order IIR structures.

Design from Analog Prototypes

The standard IIR design workflow begins with an analog filter prototype and applies a frequency transformation to produce a digital filter. The bilinear transform, which substitutes s = 2(1 - z^-1)/(T(1 + z^-1)) into the analog transfer function, maps the entire analog frequency axis to the digital frequency range from zero to the Nyquist rate without aliasing, at the cost of frequency-axis warping that must be compensated by pre-warping the critical frequencies before transformation.

Impulse invariance is an alternative method that preserves the analog filter's impulse response shape at the sampling instants but can introduce aliasing when the analog filter's response is not bandlimited. The NYU Digital Signal Processing course notes on IIR filter design provide a detailed derivation of both the bilinear transform and impulse invariance methods, along with worked examples for Butterworth and elliptic prototypes.

Stability and Phase Characteristics

A discrete-time IIR filter is stable in the bounded-input bounded-output (BIBO) sense if and only if all its poles lie strictly inside the unit circle in the z-plane. Because the filter coefficients are computed from floating-point or fixed-point representations, quantization of the coefficients shifts the pole locations from their ideal values, and with short word lengths (fewer than 16 bits), finite-precision effects can push poles outside the unit circle and cause the filter to become unstable. Cascaded biquad (second-order section) implementations reduce this sensitivity by keeping the denominator polynomial degree low in each section.

IIR filters generally have a non-linear phase response, meaning different frequency components are delayed by different amounts. Where linear phase is required, FIR filters are preferred, but in many audio processing, control, and biomedical measurement applications the phase non-linearity is acceptable given the computational savings. The ScienceDirect overview of IIR filter characteristics summarizes the trade-offs between IIR and FIR designs.

Applications

IIR filters have applications in a range of fields, including:

• Audio equalization and crossover networks in consumer and professional audio systems
• ECG and EEG signal conditioning in biomedical instrumentation
• Anti-aliasing and reconstruction stages in software-defined radio receivers
• Vibration analysis and mechanical resonance measurement in structural monitoring
• Speech processing and acoustic echo cancellation in telecommunications