Pole-zero / z-plane explorer

Computed live From the course lessons Illustrative defaults

Every curve and every number on this page is computed, in this browser, by real complex arithmetic. H(e^jω) is evaluated on the actual unit circle as g·∏(z − z_k) / ∏(z − p_k); the impulse response h[n] is obtained by running the difference equation forward from n = 0, which is why it really does diverge when you drag a pole outside the circle; the group delay is a genuine derivative of the unwrapped phase. Nothing here is a plausible-looking curve that was drawn rather than calculated.

From the course lessons, not invented here: the notation (H(z) = g ∏(z−z_k)/∏(z−p_k), ω in rad/sample, a pole at r e^jθ, τ(ω) = −d∠H/dω) is m4-l3.html; the difference equation y[n] = Σb_kx[n−k] − Σa_ky[n−k] and the BIBO test Σ|h[n]| < ∞ are m3-l4.html; the exterior ROC |z| > |a| is m4-l2.html; and the worked notch (60 Hz at f_s = 8 kHz, so ω₀ = 2π×60/8000 = 0.0471 rad/sample, which is 0.0150π) is m4-l3.html:412. It is recomputed here rather than quoted — and stated in both units, because 0.0471 is the value in radians per sample and 0.0150π is the same number expressed as a fraction of π. Those two are the pair to quote; "0.0471π" would be π times too large.

What is illustrative: the starting radii and angles of each preset (r = 0.85, 0.90, 0.95; θ = 0.25π, …) are chosen so the effect is legible on a small plot — they are not measurements of any real filter. The heat map's dB compression and its ±40 dB clip are display choices. Move any of them and all four views recompute exactly.

The z-plane, and the four views of one filter

Drag a pole (×) or a zero (○) and watch the magnitude response, the phase, the group delay and the impulse response all move at once. The single idea this page exists to make visible: the frequency response is a slice through the surface |H(z)| taken along the unit circle. A pole near the circle pushes that slice up into a peak; a zero on the circle pulls it down to a true null. Poles and zeros are told apart by shape, not colour. Keyboard: tab to any marker, then arrow keys to move it — or type r and θ into the table.

Every thumbnail above is live — drawn from the same arithmetic as the full-size plots, so it moves the instant you move a root. Hover or tab to a block and it marks the plot it produces and the controls that change it. Block 3 is the one to stare at: |H(z)| is a surface over the whole plane, and the response is its height along the unit circle. Nothing else on this page is as worth understanding.

1 · Placement — where the poles and zeros go

Filter form conjugate-pair lock

Drag a pole toward the unit circle and the magnitude peak gets taller and narrower, and h[n] rings for longer. Cross the circle and h[n] blows up — the filter is unstable. Drag a zero onto the circle and the response hits exactly zero at that angle.

Turning the conjugate-pair lock off lets a complex root stand alone: the coefficients then go complex and h[n] grows an imaginary part, so no real filter can be built. Values are shown as a radius (a plain ratio — 1.00 is the unit circle) and an angle in degrees and as a fraction of π rad/sample.

2 · H(z) — the gain in front of the polynomials

Gain g × (dimensionless) normalise to unity DC gain

g moves the whole magnitude curve up or down and changes nothing else — not the peak positions, not the phase, not the stability. Normalising to unity DC gain divides g by |H(1)|, which is the usual way to make a lowpass pass a constant through unchanged. It is undefined for a filter with a null at DC, and the verdict panel says so instead of dividing by nearly zero.

3 · The slice — evaluating H on the unit circle

ω cursor 0.000π ×π rad/sample |H(z)| heat map f_s Hz

The cursor is a point walking the unit circle, and the dashed lines are the distances that produce |H| there — numerator distances from the zeros divided by denominator distances from the poles. Watch the cursor pass close to a pole and the magnitude plot rise at the same instant.

f_s only relabels the axis in Hz: the filter is defined in ω, radians per sample, and ω = π is always f_s/2. Changing f_s from 8 kHz to 48 kHz moves no pole and no curve.

1 · The z-plane — |z| = 1 is the unit circle

Poles ×, zeros ○, the unit circle, and (optionally) the |H(z)| surface underneath as a heat map. The ω cursor is the small square walking the circle.

× pole ○ zero □ ω cursor on the unit circle · faint marker = conjugate partner (moves with its handle) |z| = 1 the unit circle — the line the stability rule is about

Numeric entry — the same control, by typing

A full equivalent to dragging. Radius r and angle θ (in units of π) for every root. With the conjugate lock on, a root with 0 < θ < π stands for a pair at ±θ.

kind	r	θ / π	θ (deg)	as	remove

2 · Magnitude |H(e^jω)|

computed

3 · Phase ∠H and group delay τ(ω)

computed

4 · Impulse response h[n]

computed

What each view is telling you

Conventions, stated rather than assumed.

Frequency. ω is in radians per sample and the plots run 0 … π. ω = π is f_s/2, the Nyquist frequency; ω = 0 is DC. The Hz row on the axis is only a relabelling using the f_s you typed — nothing about the filter changes with f_s. Because the coefficients are real, |H| is even in ω, so the negative half carries no new information and is not drawn.
Decibels. Magnitude is 20 log₁₀|H|, because |H| is an amplitude ratio. The course's 10 log₁₀ rule is for power; the two differ by the square, not by an error.
Order. The verdict panel names the denominator order N (the number of poles, including any at the origin) and the number of zeros M separately. It never says just "order", and it never says "taps".
Roots at the origin. A root at z = 0 is invisible in the magnitude response, because |e^jω − 0| = 1 at every ω. It contributes only a pure delay: −ω of phase and +1 sample of group delay per root. That is why this page can add poles at the origin to make a zero-heavy design causal without touching |H| or the stability verdict — and it is the precise sense in which an FIR filter "has all its poles at the origin".
Stability is a property of H(z) and its ROC, not of the poles alone. Every verdict on this page assumes a causal, right-sided h[n], so the ROC is the exterior |z| > max|p_k| and BIBO stability means that region contains the unit circle. The same pole-zero picture read as a left-sided (anticausal) sequence has the ROC inside the smallest pole circle, and is then stable exactly when the poles are outside. The picture does not decide; the ROC does.
Conjugate pairs. Real filter coefficients require that every complex root come with its conjugate (m4-l3.html). Turn the lock off and the coefficients go complex, h[n] acquires an imaginary part, and the panel says so instead of hiding it.

Where this sits in the course. It is the lab declared at courses/dsp101/SYLLABUS.md:169-170 — "Interactive pole-zero placement on z-plane / Real-time frequency response update / Drag poles/zeros and see filter shape change / Stability indicator". Lesson 4.3 makes the geometric argument in prose: evaluating |H| at ω "amounts to measuring the distance from each pole and zero to the point e^jω". Lesson 4.4 gives the BIBO criterion and the marginal case. Lesson 3.4 gives the difference equation this page runs to get h[n]. Nothing here extends the syllabus; it draws what those three lessons already say.

The z-plane, and the four views of one filter

1 · The z-plane — |z| = 1 is the unit circle

Numeric entry — the same control, by typing

2 · Magnitude |H(ejω)|

3 · Phase ∠H and group delay τ(ω)

4 · Impulse response h[n]

What each view is telling you

The arithmetic, in full

2 · Magnitude |H(e^jω)|