Poles, Zeros & Frequency Response | DSP-101

The Pole-Zero Representation

Every rational transfer function H(z) can be written as a ratio of two polynomials in z. Factoring both numerator and denominator reveals two special sets of points in the complex plane: zeros — values of z where H(z) = 0 — and poles — values of z where H(z) → ∞. Together, these points completely characterize the transfer function up to a scalar gain.

Factored Transfer Function

H(z) = g\,\frac{\prod_{k=1}^{M}(z - z_k)}{\prod_{k=1}^{N}(z - p_k)}

H(z) expressed in factored form: z_k are zeros, p_k are poles, and g is the overall gain. The pole-zero plot places each z_k (circle ○) and p_k (cross ×) on the complex z-plane.

The pole-zero plot is the most compact description of a filter. A glance at it reveals stability (are the poles inside the unit circle?), filter type (where are the zeros relative to the unit circle?), and frequency response shape — all without computing a single number. Learning to read and design pole-zero plots is one of the most powerful skills in DSP.

From H(z) to Frequency Response

The frequency response H(e^jω) is obtained by evaluating H(z) on the unit circle: substitute z = e^jω for ω ∈ [−π, π]. This makes the unit circle the "window" through which we view the filter's spectral behavior.

Frequency Response from Pole-Zero Form

H(e^{j\omega}) = g\,\frac{\prod_{k=1}^{M}|e^{j\omega} - z_k|}{\prod_{k=1}^{N}|e^{j\omega} - p_k|}\,e^{j\angle H}

Each factor (e^{jω} − z_k) is the complex vector from zero z_k to the point e^{jω} on the unit circle. Similarly (e^{jω} − p_k) is the vector from pole p_k to e^{jω}. The magnitude response is the product of zero distances divided by the product of pole distances.

This geometric picture is the key insight: evaluating the frequency response at angle ω amounts to measuring the distance from each pole and zero to the point e^jω on the unit circle. Poles that are close to the unit circle make the denominator product small, boosting the magnitude. Zeros that are on the unit circle make the numerator product zero, creating a perfect null.

The Geometric Interpretation

Imagine "sweeping" a point around the unit circle as ω increases from −π to π. At each angle, compute the distance to every pole and zero. The magnitude response |H(e^jω)| equals the product of distances to zeros divided by the product of distances to poles.

Effect

Pole Near Unit Circle

At the angle nearest the pole, the denominator distance becomes very small → magnitude shoots up. Creates a resonance peak in the frequency response.

Effect

Zero On Unit Circle

At the angle of the zero, the numerator distance is exactly zero → |H| = 0. Creates a perfect null (infinite attenuation) at that frequency.

Effect

Pole Far From Unit Circle

The denominator distance barely changes as ω sweeps → flat, weak influence on frequency response. Distant poles matter little.

Effect

Zero Far From Unit Circle

The numerator distance changes slowly → moderate, broad attenuation rather than a sharp null. Distant zeros spread their influence.

This geometric view transforms filter design from an abstract polynomial problem into an intuitive layout task. Want a notch at ω = π/2? Place a zero at z = j. Want a sharp resonance at ω = π/3? Place a pole close to e^jπ/3 inside the unit circle.

The Boundary: Marginal Stability

Everything above assumed a stable filter — one whose poles lie strictly inside the unit circle, so its impulse response decays and any bounded input produces a bounded output. Pushing a pole toward the circle sharpens the resonance, which raises the obvious question: what happens on the circle itself? That boundary has a name. A pole whose radius is exactly r = 1 makes the system marginally stable — the dividing line between the stable interior and the unstable exterior.

At r = 1 the impulse response neither decays nor grows; it settles into a sustained, constant-amplitude oscillation. A single real pole at z = 1 accumulates its input forever — the discrete-time integrator — while a conjugate pole pair at e^±jω₀ rings at ω₀ and never dies out, which is precisely how a digital oscillator is built. As a filter this is unusable: it is not BIBO-stable, because an input at the pole's own frequency drives the output to infinity, and a repeated pole on the circle grows without bound even from an impulse. Marginal stability is a feature when you want an oscillator and a failure when you want a filter.

This is why the 60 Hz notch below places its poles at r slightly less than 1, never on the circle. Zeros may sit anywhere — a zero exactly on the circle is a harmless perfect null — but a pole on the circle turns the filter into an oscillator. As r → 1 the notch becomes infinitely narrow precisely because the design is approaching that boundary: the price of a sharper notch is a filter that rings longer and tolerates less numerical error.

Phase Response and Group Delay

The pole-zero plot also determines the phase response ∠H(e^jω). The phase at frequency ω is the sum of angles from each zero minus the sum of angles from each pole to the point e^jω on the unit circle.

Phase as Sum of Angles

\angle H(e^{j\omega}) = \angle g + \sum_{k=1}^{M}\angle(e^{j\omega}-z_k) - \sum_{k=1}^{N}\angle(e^{j\omega}-p_k)

The total phase is the angle of the gain g plus the sum of angles from zeros minus the sum of angles from poles, where each angle is measured from the corresponding root to the evaluation point e^{jω}.

The group delay τ(ω) = −d∠H/dω measures how much different frequency components are delayed. A linear phase response — where all frequencies are delayed by the same amount — is achieved by symmetric (FIR) designs. IIR filters with poles generally have non-linear phase.

Real Filters: Conjugate Symmetry

If the filter coefficients h[n] are real-valued, then H(z) has a conjugate symmetry property: whenever p is a pole, so is p*; whenever z₀ is a zero, so is z₀*. This means poles and zeros of real filters always appear in conjugate pairs (unless they lie on the real axis).

Conjugate Symmetry Rule

For a real filter: poles and zeros off the real axis always come in conjugate pairs (p, p*). This ensures H(e^jω) has even magnitude and odd phase — a real-valued impulse response in the time domain.

Real h[n] ⟺ Poles & zeros appear as conjugate pairs

This constraint halves the design freedom: specifying one complex pole at re^jθ automatically places its conjugate at re^−jθ. The combined pair contributes a second-order section with real coefficients, which is the standard building block for IIR filter implementation.

Classic Filter Types from the Pole-Zero Plot

Filter classification by frequency band — lowpass, highpass, bandpass, bandstop — can be read directly from the pole-zero plot by observing where the zeros (nulls) and poles (boosts) are placed relative to the unit circle.

Type

Lowpass

Zeros near ω = π (z = −1) suppress high frequencies. Poles near ω = 0 (z = +1) reinforce low frequencies.

Type

Highpass

Zeros near ω = 0 (z = +1) null out DC. Poles near ω = π (z = −1) reinforce high frequencies.

Type

Bandpass

Conjugate pole pair near the passband angle. Zeros at z = ±1 (DC and Nyquist) suppress both low and high extremes.

Type

Notch (Bandstop)

Conjugate zero pair on the unit circle at the notch angle. A nearby conjugate pole pair restores gain outside the notch band.

A Simple Design Example

Suppose we want a digital notch filter to remove a 60 Hz interference from a signal sampled at 8 kHz. The normalized frequency is ω₀ = 2π×60/8000 ≈ 0.0471 rad/sample — that is 0.0150π, so the notch sits 1.5% of the way from DC to f_s/2. (Watch the convention: 0.0471 is the angle in radians per sample; the factor of π is already inside 2π×60/8000 and must not be written again.) We place a conjugate zero pair on the unit circle at ±ω₀, then add a conjugate pole pair at radius r < 1 at the same angles to restore gain at nearby frequencies.

Notch Filter Transfer Function

H(z) = \frac{(z - e^{j\omega_0})(z - e^{-j\omega_0})}{(z - r e^{j\omega_0})(z - r e^{-j\omega_0})} = \frac{z^2 - 2\cos\omega_0\,z + 1}{z^2 - 2r\cos\omega_0\,z + r^2}

The notch filter has zeros exactly on the unit circle at ±ω₀ (guaranteeing |H| = 0 there) and poles at radius r slightly inside the unit circle at the same angles, controlling the notch bandwidth. As r → 1, the notch becomes infinitely narrow.

This example illustrates the power of the pole-zero design approach: a real-world engineering requirement (remove 60 Hz) maps directly to a geometric placement problem (put zeros on the unit circle at the right angle). No optimization algorithm required — just geometry.

The next lesson, Inverse Z-Transform Methods, develops the systematic techniques — partial fractions, long division, and contour integration — for recovering time-domain sequences from H(z).

Key Takeaways

Poles are values of z where H(z) → ∞; zeros are values where H(z) = 0. Together they characterize a rational filter up to a gain constant.
The frequency response H(e^jω) is the ratio of distances from zeros to e^jω over distances from poles to e^jω on the unit circle.
Poles close to the unit circle create magnitude peaks (resonances); zeros on the unit circle create perfect nulls.
A pole exactly on the unit circle (r = 1) is marginally stable: it oscillates forever instead of decaying — the basis of a digital oscillator, but never a stable filter. Keep poles strictly inside.
The phase response equals the sum of zero angles minus the sum of pole angles; non-linear phase arises from asymmetric pole-zero placement.
Real-coefficient filters have conjugate-symmetric poles and zeros; complex roots always appear in conjugate pairs.
Lowpass, highpass, bandpass, and notch filters each have characteristic pole-zero topologies on the z-plane.
Notch filter design: place zeros on the unit circle at the interference frequency and poles just inside at the same angles to control bandwidth.