Frequency Response from Poles and Zeros

Reading H(e^{jΩ}) from a pole–zero plot, zeros on the unit circle, resonators and notches, all-pass filters and minimum phase

Leiden, 1901 — the heart on a wire

Physiologists knew that the beating heart produces electrical currents, but the instruments of the day could not measure them accurately without electrodes on the heart itself. Beginning in 1901, Willem Einthoven builds a series of string galvanometers: a very thin conducting filament between very strong electromagnets, whose shadow is recorded on moving photographic paper. The first machine needs water cooling for its magnets, five people to operate it, and weighs some 270 kilograms. In 1924 Einthoven receives the Nobel Prize for the first practical system of electrocardiography (Willem Einthoven). A century later the problem has changed shape. An ECG records signals of about a millivolt — the standard paper scale is 10 mm per millivolt — from a body that also picks up the 50 or 60 Hz hum of the mains wiring around it; circuits such as the right-leg driver reduce that interference (Electrocardiography), and digital filters remove what is left. The classic tool is a notch: a pair of zeros on the unit circle, with two poles just inside.

Derivation

The frequency response as geometry

Factor a transfer function by its zeros

z_i

and poles

p_i

and set

z = e^{j\Omega}

H(e^{j\Omega}) = K\,\frac{\prod_i \left(e^{j\Omega} - z_i\right)}{\prod_i \left(e^{j\Omega} - p_i\right)} .

Each factor is a vector from a pole or zero to the point

e^{j\Omega}

on the unit circle. So

\big|H(e^{j\Omega})\big| = |K|\,\frac{\prod_i \text{distance to zero } i}{\prod_i \text{distance to pole } i}, \qquad \angle H = \sum \text{zero angles} - \sum \text{pole angles} .

Walk once round the circle and you can read the whole response from the picture: dips where the point passes close to a zero, peaks where it passes close to a pole.

💡Zeros on the circle, poles near it

A zero on the unit circle at angle $\Omega_0$ makes the response exactly zero at $\Omega_0$ : a frequency the filter cannot pass at all.
A pole just inside the circle at angle $\Omega_0$ makes a sharp peak there, sharper as its radius approaches 1: a resonator.
Put both at the same angle — zero on the circle, pole at radius $r$ just inside — and you have a notch: away from $\Omega_0$ the pole's vector and the zero's vector are almost equal and cancel, so the response is flat; at $\Omega_0$ the zero wins. The closer $r$ is to 1, the narrower the notch.

📐A 50 Hz notch

Sampled at 500 Hz, 50 Hz is

\Omega_0 = 2\pi \times 50/500

, or 36°. SciPy's iirnotch(50, 30, fs=500) puts its zeros on the unit circle at ±36° and its poles at radius 0.9896 at the same angles. The response is −0.11 dB at 45 Hz, −2.26 dB at 49 Hz and zero at 50 Hz, and the notch is

f_0/Q = 1.67

Hz wide. Applied forwards and backwards (filtfilt) to a synthetic ECG carrying 0.3 mV of hum, it leaves no measurable hum and changes the R-wave peak by 6 µV, from 0.991 to 0.985 mV.

💡All-pass filters

Place a pole at

a

inside the circle and a zero at

1/a^*

, its mirror image outside. For every point on the circle the two distances are in the fixed ratio

1/|a|

, so with the right gain

|H(e^{j\Omega})| = 1

at every frequency. Only the phase changes: the first-order all-pass

(z^{-1} - a)/(1 - az^{-1})

with

a = 0.6

delays low frequencies by 4 samples and high frequencies by a quarter of a sample. All-pass sections correct the phase of other filters without touching their magnitude.

💡Minimum phase

Reflecting a zero from

z_0

1/z_0^*

changes the magnitude response only by a constant — the all-pass argument again — but it changes the phase. Among all filters with the same magnitude, the one with every zero inside the circle has the least phase lag and packs its energy earliest: it is minimum phase. The filters

1 - 0.5z^{-1}

and

-0.5 + z^{-1}

have identical magnitudes, yet the first puts 80 % of its energy in its first sample and the second only 20 %. A minimum-phase filter also has a stable, causal inverse — the one that can be undone.

The heart of animals is the foundation of their life, the sovereign of everything within them, the sun of their microcosm, that upon which all growth depends, from which all power proceeds.

– William Harvey, dedication to King Charles I of On the Motion of the Heart and Blood (1628), quoted on Wikiquote

🕰A heart recorded at a distance

Einthoven's galvanometer greatly increased the sensitivity of earlier instruments, and with it he went on to describe the electrocardiographic features of a number of heart diseases. His Nobel lecture of December 1925 was titled "The String Galvanometer and the Measurement of the Action Currents of the Heart" (Willem Einthoven).

⚠A narrow notch rings

The narrower the notch, the closer its poles sit to the circle — and poles near the circle ring for a long time after any sudden change, at the notch frequency. A notch also removes any genuine signal at that frequency. Use the widest notch that does the job, and filter forwards and backwards (filtfilt) when the delay of a causal filter does not matter.

💡Where you will use this

Pole–zero reasoning guides every filter design in Unit 9 and every check of a design's response. Notches clean measurement signals of hum, vibration and known interference; all-pass sections equalise phase in audio and communications; minimum-phase systems are the ones whose effect a stable filter can undo.

📚Further reading

Oppenheim and Willsky, Signals and Systems, section 10.4 — geometric evaluation of the Fourier transform from the pole–zero plot.
Alan V. Oppenheim and Ronald W. Schafer, Discrete-Time Signal Processing (3rd ed., Pearson, 2010), chapter 5 — all-pass and minimum-phase systems.
All-pass filter on Wikipedia.

↓But there is a problem

A notch, a resonator, an all-pass: a few points placed by hand. But a real specification reads differently — pass 0 to 3 kHz within 0.1 dB, reject everything above 4 kHz by 80 dB, with exactly linear phase. No one places poles and zeros by hand for that. In 1972 two engineers published an algorithm that finds the best possible filter of a given length for such a specification, and it is still the one engineers reach for.