Introduction to IIR Filters and the Importance of Zeros

Infinite Impulse Response (IIR) filters are a cornerstone of modern signal processing, used in everything from digital audio equalizers to radar systems. Unlike their Finite Impulse Response (FIR) counterparts, IIR filters rely on both feedback (poles) and feedforward (zeros) to achieve sharp frequency selectivity with low computational cost. While poles dominate the filter’s stability and gain, zeros are equally critical—they define how the filter attenuates specific frequencies and shape the overall magnitude and phase response. This article provides an in-depth look at the role of zeros in IIR filter design, moving beyond basic definitions to explore practical placement strategies, interactions with poles, and real-world applications.

For readers new to the topic, a solid foundation in IIR filter theory is recommended. We will build on that knowledge to demystify zero placement and its impact on frequency response shaping.

What Are Filter Zeros in IIR Systems?

A digital IIR filter is typically described by its transfer function H(z), expressed as a ratio of polynomials in z⁻¹:

H(z) = (b0 + b1 z⁻¹ + … + bM z⁻ᴹ) / (1 + a1 z⁻¹ + … + aN z⁻ᴺ)

The numerator polynomial defines the zeros of the filter: the values of z (points in the complex plane) for which H(z) = 0. If the denominator polynomial is nonzero at those points, the output becomes zero when the input is a complex sinusoid at the corresponding frequency. Zeros are therefore the frequencies where the filter completely blocks or strongly attenuates the input signal.

Formally, a zero is a root of the numerator equation: B(z) = 0. For a real-coefficient filter, zeros occur in complex-conjugate pairs. Their locations are visualized on a pole-zero plot, where zeros are marked with circles and poles with crosses. This graphical representation is essential for understanding frequency response behavior at a glance.

Zeros and the Complex Plane

The unit circle in the z-plane corresponds to discrete-time frequencies from 0 to the Nyquist frequency. A zero’s angular position (angle θ) determines the normalized frequency f = θ / (2π) in cycles per sample. For example, a zero at θ = π/2 (90°) corresponds to a frequency of f = 0.25 (quarter of the sampling rate). The zero’s radial distance from the origin determines the width and depth of the attenuation notch: zeros exactly on the unit circle produce perfect nulls, while zeros inside or outside yield finite attenuation.

Understanding this geometry is the first step to mastering IIR filter design.

The Impact of Zeros on Frequency Response

The magnitude response of an IIR filter at a given frequency is proportional to the product of distances from all zeros to the point on the unit circle representing that frequency, divided by the product of distances from all poles. This geometric interpretation makes it clear why zeros placed near the unit circle create deep attenuation valleys. The closer a zero is to the unit circle at a certain angle, the lower the magnitude response at the corresponding frequency.

Zeros on the Unit Circle: Notch Filters

Placing a zero exactly on the unit circle (|z| = 1) creates an infinite notch—the filter output becomes zero at that frequency. This is the basis for digital notch filters used to eliminate power-line hum (50/60 Hz) or narrowband interference. For instance, a second-order IIR notch filter with poles close to the zeros can achieve very narrow stopbands while minimally affecting adjacent frequencies. Practical implementations often use a pair of complex-conjugate zeros on the unit circle at the desired notch frequency.

Zeros Inside the Unit Circle: Broad Attenuation

Zeros placed inside the unit circle (|z| < 1) produce finite, broadened notches. They still attenuate frequencies near their angle but do not nullify them completely. This is useful when a filter must suppress a range of frequencies without total cancellation—for example, in audio applications where removing all energy at a resonance frequency would sound unnatural. The radial distance controls the “sharpness” of the attenuation: zeros at radius 0.9 create deeper and narrower dips than zeros at radius 0.5.

Zeros Outside the Unit Circle and Nonminimum-Phase Systems

Zeros can also lie outside the unit circle (|z| > 1). Such zeros are generally avoided in causal, stable filters because they produce nonminimum-phase responses—though they are sometimes necessary for linear-phase approximations or for analyzing the forward-backward filtering (filtfilt) technique. In communication systems, nonminimum-phase zeros can model transmission channels that introduce delay dispersion. However, for most IIR designs, zeros are kept inside or on the unit circle to maintain stability and a manageable phase response.

Phase Response Contribution

Zeros also influence the phase response. Each zero contributes a phase shift that increases with frequency, and the combined effect can be visualized using the group delay. For a zero on the unit circle at angle θ, the phase abruptly jumps by π radians at that frequency. In contrast, zeros inside the circle produce more gradual phase changes. Understanding this is critical when designing filters for applications sensitive to phase distortion, such as audio crossovers or ECG processing.

Zero Placement Strategies for Common Filter Types

Engineers use zero placement to achieve specific filter characteristics. Below are three common scenarios where zeros play a starring role.

1. Notch Filters

A notch filter is designed to reject a single frequency (or very narrow band). The classic approach places a pair of complex-conjugate zeros directly on the unit circle at the target frequency and then pairs them with poles positioned slightly inside the unit circle at the same angle. The pole radius controls the notch bandwidth: poles closer to the unit circle yield a narrower notch. This technique is widely used in MATLAB’s iirnotch function.

Example: 60 Hz Hum Filter

For a sampling rate of 1 kHz, a 60 Hz hum corresponds to a normalized frequency of 0.06 cycles/sample (angle ≈ 0.3768 rad). Zeros are placed at z = e^{±j0.3768}, poles at z = 0.95 e^{±j0.3768}. The result is a deep notch at 60 Hz with approximately 40 dB of attenuation, while frequencies 50 Hz away are unaffected to within 0.5 dB.

2. Comb Filters

Comb filters require multiple evenly spaced notches or peaks. They are built by cascading multiple notch filters or by using a single delay-based structure. For example, a feedforward comb filter with transfer function H(z) = 1 – α z⁻ᴰ places zeros at the Dth roots of α on the unit circle. When α = 1, the zeros lie exactly on the unit circle at D equally spaced frequencies, creating a harmonic series of notches. This is used to remove periodic interference, such as engine noise in audio recordings.

3. Bandstop Filters

A bandstop (band-reject) filter suppresses a continuous range of frequencies. While a notch filter targets one frequency, a bandstop design uses multiple zeros spread over the stopband. For example, a fourth-order IIR bandstop filter may have two pairs of complex zeros on the unit circle at the stopband edges and two pairs of poles to shape the transition bands. The zeros ensure deep attenuation across the band, while the poles maintain a flat passband response.

Analog Devices provides an excellent tutorial on using pole-zero placement for custom bandstop filters in digital audio.

Interaction Between Zeros and Poles

In IIR filters, zeros and poles work together. Poles amplify signals near their frequencies (for example, creating a resonance peak), while zeros attenuate. The overall frequency response is the quotient of the zero and pole contributions. Well-designed filters often place poles near the unit circle to create sharp transitions, and zeros near or on the unit circle to cancel out unwanted frequencies or to balance the response.

Pole-Zero Cancellation

Sometimes a pole and a zero are placed at nearly the same location in the z-plane. This technique—called pole-zero cancellation—can simplify implementation or remove unwanted dynamics. If the pole and zero exactly coincide, they cancel each other, reducing the filter order. However, in fixed-point arithmetic or with coefficient quantization, exact cancellation is impossible; closely placed pole-zero pairs can cause instability or poor performance. Careful analysis using pole-zero plots is essential.

Stability Considerations

While zeros do not affect stability (that is governed solely by pole locations inside the unit circle), they do affect the output magnitude. A zero very close to a pole can create numerical issues in the filter’s internal state variables. For instance, if a zero at radius 0.99 is near a pole at radius 0.98, the filter may exhibit large transient oscillations before the steady-state notching takes effect. Practical designs maintain sufficient separation—typically a radial difference of at least 0.1.

Advanced Topics: Minimum-Phase and Allpass Filters

Zeros also determine whether an IIR filter is minimum-phase. A filter is minimum-phase if all its zeros lie inside the unit circle. Such filters have the minimum group delay for a given magnitude response, making them ideal for applications where latency must be minimized. Conversely, filters with zeros outside the unit circle are nonminimum-phase and have increased group delay (and possibly phase distortion).

Allpass filters are a special case where the poles and zeros are reciprocal pairs (zero = 1 / pole*). They pass all frequencies with constant magnitude but modify the phase. Allpass sections are used to correct group delay distortions introduced by other IIR stages, enabling linear-phase-like behavior in a feedback filter.

Practical Design Workflow

Modern digital filter design is typically done using software tools that allow interactive pole-zero editing. Here is a typical workflow for an engineer using DSP System Toolbox or an open‑source alternative:

  1. Define filter specifications: passband and stopband frequencies, ripple, attenuation, and phase constraints.
  2. Choose a prototype (e.g., Butterworth, Chebyshev, Elliptic) that meets the magnitude requirements with minimal order.
  3. Map the analog prototype to the digital domain using bilinear transform or impulse invariance; this yields initial pole-zero locations.
  4. Manually adjust zero placements to eliminate residual interference or to meet notch depth requirements. For example, moving zeros slightly off the unit circle trades notch depth for lower sensitivity to coefficient quantization.
  5. Simulate the frequency response and verify stability. Check pole-zero plots for near-cancellations that could cause long transients.
  6. Quantize coefficients and test under fixed-point arithmetic (if applicable). Iterate on zero radii until performance meets specifications.

For a textbook reference with detailed design equations, see Digital Signal Processing: Principles, Algorithms, and Applications by Proakis and Manolakis (Chapter 11).

Conclusion

Filter zeros are far more than mathematical trivia—they are the primary mechanism by which an IIR filter attenuates unwanted frequencies. From creating deep, narrow notches on the unit circle to shaping broad stopbands with interior zeros, their placement directly determines a filter’s frequency response, phase characteristics, and practical performance. By carefully pairing zeros with poles, engineers can build filters that meet demanding specifications in audio, communications, biomedical, and control systems. Mastery of zero placement—visualized through pole-zero plots and verified through simulation—remains one of the most valuable skills in the digital signal processing toolkit.