Resonators, notches and combs

A notch removes one frequency and leaves the rest nearly alone: here 50 Hz mains hum in an ECG sampled at 500 Hz. Below, two poles slide toward the two zeros that make the notch. Watch the bracket on the gain curve as they come closer.

Poles behind the zeros narrow the notch

Transfer functions, poles and zeros (16.3) turned the pole-zero picture into a design method: put poles where the gain should rise and zeros where it should fall. Its resonator is the first case. Two poles $re^{\pm j\theta}$, close to the unit circle at the wanted angle $\theta$, make a peak near $\Omega=\theta$. Its width is about $2(1-r)$ rad/sample, so a pole radius nearer to 1 gives a narrower peak.

The second case is the notch: a filter that removes one frequency and leaves the rest nearly alone. Put zeros on the circle at $e^{\pm j\theta}$. At $\Omega=\theta$ the arrow from a zero to the test point has length 0, so the gain is 0 there.

I will use one running example. An ECG is often recorded at $f_s=500$ Hz, and mains hum sits at 50 Hz. From Sampling and aliasing (10.1), $\theta=2\pi f/f_s$, so here $\theta=2\pi\cdot50/500=0.2\pi$.

Zeros alone cut a wide band around 50 Hz. Poles just inside the circle at the same angle put the band back: away from $\theta$ each pole’s arrow is nearly as long as its zero’s, and they cancel. The result is

with $K$ chosen so that the gain is 1 at $\Omega=0$.

The picture at the top of the page draws this filter. When the clip ends, drag either pole along its line, or use the arrow keys, to set $r$ yourself. Try $r=0.95$: the notch is 8.1 Hz wide.

Here are the numbers behind the picture. Each width is the distance between the two $-3$ dB crossings, found numerically; the rule column is $(1-r)f_s/\pi$ in hertz, which is $2(1-r)$ rad/sample converted with $f=\Omega f_s/2\pi$.

The rule is a good guide once $r$ is close to 1. At $r=0.8$ it is already 8 percent too large, and the gain rises to 1.115 beside the notch, a little above 1. The gain at 50 Hz is exactly 0 for every $r$, because the zeros sit on the circle.

To design a notch from a width, invert the rule: $r=1-\Delta\Omega/2$, where $\Delta\Omega$ is the 3 dB width in rad/sample, the gap between the two frequencies where the gain is $1/\sqrt2$ times its peak (see How big is a signal, 1.3, for 3 dB). If you have a width $\Delta f$ in hertz, then $\Delta\Omega=2\pi\,\Delta f/f_s$. For 60 Hz mains, use $\theta=2\pi\cdot60/f_s$ in the same formula.

The gain at $\Omega=0$ is 1 when

This is $H(1)=1$ solved for $K$: put $z=1$ in the formula above and divide.

A DC blocker is the same idea at $\Omega=0$, where $\theta=0$. It has a zero at $z=1$ and a pole just inside it:

At $f_s=500$ Hz its $-3$ dB point is at 0.397 Hz, and the rule $(1-0.995)f_s/2\pi$ gives 0.398 Hz. It removes the slow baseline wander of an ECG: the gain is 0.931 at 1 Hz and 1.002 at 50 Hz.

A delay in a loop grows teeth

A delay in a loop gives a different shape, a whole set of peaks. Take an echo, as in Difference equations (6.1):

Each echo is 0.8 of the last, and they are $D$ samples apart. The poles solve $z^D=0.8$. That gives $D$ of them, all at radius $0.8^{1/D}$, one every $2\pi/D$ round the circle. A comb is a filter whose gain has teeth like this: a repeating pattern of peaks along frequency.

Below, the delay grows from 1 sample to 8. Watch the teeth: how many there are, and how tall.

When the clip ends, set $D$ yourself with the slider. Try 3: the poles are 120° apart and the peaks 2666.7 Hz apart. Watch which numbers move. The teeth’s height and depth never change; only their spacing does.

The reason is in the formula. A peak is where $e^{-j\Omega D}=1$, so $\lvert H\rvert=1/(1-0.8)=5$. A dip is where $e^{-j\Omega D}=-1$, so $\lvert H\rvert=1/(1+0.8)=0.556$. Neither depends on $D$. The peaks sit at $\Omega=2\pi m/D$, which is every $f_s/D$ in hertz: at $f_s=8$ kHz, 8000, 4000, 2000 and 1000 Hz for $D=1$, 2, 4 and 8.

The impulse response is the echo train: 1 at $n=0$, 0.8 at $n=D$, 0.64 at $2D$ and 0.512 at $3D$.

Move the delay to the input side and you get the other kind. A feed-forward comb $y[n]=x[n]+a\,x[n-D]$ has $D$ zeros instead of poles, at radius $a^{1/D}$, and its gain has notches between the peaks. For $a=0.8$ and $D=8$ the zeros are at radius 0.9725, and the gain swings between 1.8 and 0.2. Sweep $D$ slowly from 1 to 10 ms and the notches slide along the frequency axis. This is flanging.

A feed-forward comb also helps with hum. The comb $y[n]=x[n]-x[n-10]$ at 500 Hz has zeros where $z^{10}=1$: at 0, 50, 100, 150, 200 and 250 Hz. It removes the hum and every harmonic, and it also removes the ECG’s own slow part at 0 Hz. In practice you put poles behind its zeros, as in the notch above.

Notch out the mains hum

Mains hum is the 50 or 60 Hz interference that the power supply puts onto a recording, and it is why an ECG needs a notch. Tune the notch of the first instrument onto it and the hum goes, while the heartbeat barely changes. Think of tuning a radio until the whistle drops out. Below, the notch slides from 30 Hz onto the hum; watch the solid trace settle onto the dashed one.

When the clip ends, drag across the gain strip, or use the arrow keys, to tune the notch yourself. At 45 Hz, 0.234 mV of hum is left.

The hum left is the hum’s amplitude, 0.3 mV, times the notch’s gain at 50 Hz, $\lvert H(e^{j0.2\pi})\rvert$. For $r=0.95$ it is 0.299 mV with the notch at 30 Hz, 0.293 at 35, 0.280 at 40, 0.234 at 45 and 0.133 at 48. At 50 Hz it is 0. Past the hum it is 0.233 at 55 Hz, 0.279 at 60 and 0.295 at 70.

The heartbeat’s cost is small. The clean R peak is 0.996 mV. After the 50 Hz notch it is 0.988 mV at $r=0.95$, 0.976 at $r=0.9$ and 0.993 at $r=0.98$. Over 1 to 3 s the output differs from the clean trace by an RMS of 0.0085 mV at 50 Hz, against 0.199 mV when the notch sits at 40 Hz.

Real hum has harmonics, and 150 Hz is common, so use one notch per harmonic, or a comb with poles. In a 60 Hz country, retune to 60 Hz. A notch is IIR, so its phase bends near the hum; Linear-phase systems (17.3) explains why that matters. For an ECG this is accepted because the notch is narrow, and offline you can run the filter forwards and then backwards (filtfilt, covered in a later filtering chapter) to cancel the phase.

Where you’ll meet this

Biquad equalisers come in Audio equalisers and biquads (20.5), and chains of them in Second-order sections (21.2). Audio effects (29.2) builds echoes, flanging and plucked strings from the combs here. Hum that drifts in frequency needs an adaptive canceller: Adaptive filters: LMS (26.3).

Reference card