Audio equalisers and biquads

An equaliser’s “mid” knob turns up one band of a song and leaves the rest alone. Below, one small filter lifts the band around 1 kHz by 12 dB and then cuts it by 12 dB. Watch the poles and the zeros move apart, and then trade places.

Boost or cut one band

A stereo with a “mid” knob lets you turn up the voices in a song without touching the bass or the cymbals. Turn it down by the same amount and the voices sink back by the same amount. I want to build that knob, and it takes one small filter.

The filter is the second-order system of Transfer functions, poles & zeros (16.3), with two zeros as well as two poles:

Its top and bottom are both quadratics in $z^{-1}$, so it is called a biquad. Five numbers set it completely. By 16.3, the gain at a frequency comes from arrows to the point $e^{j\Omega}$ on the unit circle. It is $b_0$ times the product of the two arrows from the zeros, divided by the product of the two arrows from the poles.

Choosing five numbers by hand is slow. R. Bristow-Johnson’s cookbook is a set of recipes that turns three settings you can hear into those five numbers. The settings are a centre frequency $f_0$ in hertz, a gain $G_\text{dB}$ in dB, and a width $Q$.

$Q$ is the quality factor of First- and second-order systems (6.3). There, a larger $Q$ meant a sharper resonance; here it means a narrower band. First turn $f_0$ into rad/sample, as in Frequency in discrete time (12.1): $\Omega_0=2\pi f_0/f_s$.

The cookbook’s peaking recipe boosts or cuts a band around $f_0$. It uses two helpers. One is the gain factor $10^{G_\text{dB}/40}$, whose square is the amplitude ratio of $G_\text{dB}$. The other is the width term, $\sin\Omega_0/(2Q)$, written $\alpha_Q$ in this one display:

Then divide all six by $a_0$, so that the bottom starts with 1. Two things follow straight from the lines. At $G_\text{dB}=0$ the gain factor is 1, so each $b$ equals its $a$: top and bottom are the same, and the filter passes everything unchanged.

Now flip the sign of $G_\text{dB}$. The gain factor turns into its reciprocal, so the $b$ lines become the $a$ lines and the other way round. Top and bottom trade places, so $H$ becomes $1/H$, and its gain in dB changes sign at every frequency.

Where does the recipe come from? Bristow-Johnson started from an analog bell and applied the bilinear transform of IIR design by the bilinear transform (20.2), pre-warped so that the bell’s centre lands exactly on $f_0$. Analog frequency is $\omega$ (rad/s), digital frequency is $\Omega$ (rad/sample); Oppenheim and Schafer use the opposite letters. I will not repeat that algebra here.

The picture at the top of the page runs at $f_s=48$ kHz, the usual rate for video and studio audio, with $f_0=1$ kHz. That is $\Omega_0=2\pi\cdot1000/48000=0.1309$ rad/sample, only a 48th of a turn. So the poles and zeros of an audio filter crowd close to $z=1$, and the plane is zoomed in on that corner.

The gain panel’s frequency axis is logarithmic, as in Frequency response and Bode plots (8.4). An octave is a doubling of frequency, and on this axis every octave has the same width. I measure the width of a bell in octaves, between the two frequencies where it reaches half its height in dB.

Notice the start. At 0 dB the poles sit exactly on the zeros, so each pole’s arrow is as long as its zero’s arrow and the gain is 1 everywhere. As the gain rises, the poles move toward the circle. Near 1 kHz their arrows become short, the zeros’ arrows stay longer, and the gain climbs.

At −12 dB the radii swap, just as the recipe’s lines did. The zeros are now the near pair, and the bell turns into a dip of the same shape. A tone control turned up 12 dB and then down 12 dB moves between these two pictures.

Look closely and the pairs are not exactly on the ray. At +12 dB the zeros sit at 0.0930 rad and the poles at 0.1289 rad, while the ray is at 0.1309 rad. Even so, the recipe puts the top of the bell exactly on 1 kHz: the gain there is 12.00 dB.

After the clip, drag the bell up or down, or use the arrow keys, to set the gain yourself. At +6 dB the poles sit at 0.968 and the zeros at 0.937; at −6 dB the two radii swap.

Five shapes from one recipe

The cookbook has more recipes, all built from the same three settings. A shelf lifts or lowers everything on one side of a corner frequency $f_0$. A low shelf changes the lows, below the corner, and a high shelf changes the highs, above it.

Its third setting is a slope instead of a width. I use the cookbook’s slope of 1, the steepest that does not overshoot. Its lines are longer than the peaking ones, but they use the same gain factor and the same $\cos\Omega_0$ and $\sin\Omega_0$.

A notch removes one frequency completely. As in Resonators, notches and combs (17.4), it puts its zeros on the unit circle at $\pm\Omega_0$ and its poles just inside. Its recipe is the shortest:

with the same width term, and all six divided by $a_0$ again. Its bottom line is the peaking recipe’s at 0 dB, so the notch’s poles are the ones from the first instrument’s first frame, 0.955 from the origin.

The figure puts five of these shapes on the same axes as the first instrument’s gain panel.

Three bands add up

One biquad shapes one band. To shape a whole song you run several in a row, a cascade: the output of one is the input of the next. At each frequency the first multiplies the sine by its gain, the next multiplies the result by its own gain, and so on. So the gains multiply.

In How big is a signal (1.3), multiplying by 10 always added the same number of dB, because a logarithm turns a product into a sum. So in a cascade the dB curves add. You can predict the whole equaliser by adding its bands’ curves, point by point.

A parametric equaliser is such a cascade where every band has its own $f_0$, $G_\text{dB}$ and $Q$, for example a low shelf, a peak and a high shelf. A graphic equaliser fixes the centres and gives you only the gains. The usual one has 31 bands a third of an octave apart, at the frequencies of the standard ISO 266, from 20 Hz to 20 kHz.

Below, three bands run in a row. Watch the thick total form as the sum of the three thin curves.

Notice the 2.97 dB at 100 Hz. It is the shelf’s 3.00 plus the peak’s −0.03, and the high shelf adds nothing there. The peak’s tail reaches a long way down, but only a little.

After the clip, drag each band’s handle, or use Tab and the arrow keys, to move it. Try raising the peak to 0 dB. The total is then 3.00 dB at 100 Hz and 0.00 dB at 1 kHz: only the peak’s share of the sum has changed.

Press “Hear it” to play 3 s of hiss, a random noise with every frequency in it, through the three bands. Press it again to stop.

How wide is a Q?

Audio people often give a band’s width in octaves instead of $Q$. For a bell $N$ octaves wide, the analog relation is

At $N=1$ this is $\sqrt2/1=1.41$, the first instrument’s $Q$. To go the other way, write $x=2^{N/2}$. The relation becomes $Qx^2-x-Q=0$, whose positive root gives

The formula is for the analog bell. The digital bell of the first instrument measures 0.997 octave, not 1, because the bilinear transform squeezes frequencies a little. At 1 kHz out of 48 kHz the difference is too small to see.

Worked example

Let’s compute the page’s numbers ourselves.

1. A peak of +12 dB at 1 kHz, $Q=\sqrt2$, 48 kHz. First the helpers: $\Omega_0=0.1309$ rad/sample, $\cos\Omega_0=0.991445$ and $\sin\Omega_0=0.130526$. The width term is $0.130526/(2\cdot1.414214)=0.046148$, and the gain factor is $10^{12/40}=1.995262$.

The recipe gives $b_0=1+0.046148\cdot1.995262=1.092077$, $b_1=-1.982890$ and $b_2=0.907923$. On the bottom, $a_0=1+0.046148/1.995262=1.023129$ and $a_2=0.976871$. Dividing by $a_0$:

• top: 1.067390, −1.938065, 0.887398;
• bottom: 1, −1.938065, 0.954788.

Now read off the roots. A pair $re^{\pm j\theta}$ multiplies out to $1-2r\cos\theta\,z^{-1}+r^2z^{-2}$. So the pole radius is $\sqrt{0.954788}=0.9771$, and $\cos\theta=1.938065/(2\cdot0.9771)=0.99171$ gives $\theta=0.1289$ rad, or 7.38°.

For the zeros, divide the top by $b_0$ first: $r^2=0.887398/1.067390=0.831372$, so $r=0.9118$. Then $\cos\theta=1.938065/(2\cdot1.067390\cdot0.9118)=0.99567$, so $\theta=0.0930$ rad, or 5.33°.

The gain is 12.00 dB at 1 kHz, 5.99 dB at 707 Hz, one half octave below, and 2.51 dB at 500 Hz, one octave below.

2. Shelves and a notch. Take the low shelf of the second instrument, +6 dB at 100 Hz. It gives 6.00 dB at 0 Hz, 3.00 dB at its corner and 0.38 dB at 200 Hz, one octave above. The high shelf, +4 dB at 8 kHz, gives 2.00 dB at its corner and 3.998 dB at 20 kHz.

The notch at 1 kHz with $Q=\sqrt2$ has $b_0=1/(1+0.046148)=0.955888$ after the division. It is −10.85 dB at 900 Hz and −11.65 dB at 1100 Hz. On a logarithmic axis 1100 Hz is nearer to 1 kHz than 900 Hz is, so it sits deeper in the notch.

3. The equaliser’s total. Adding the three bands’ dB at eleven frequencies gives the table.

4. Q and octaves. A third-octave band of a graphic equaliser has $N=1/3$, so $Q=2^{1/6}/(2^{1/3}-1)=4.32$. Going the other way, $Q=1$ gives $N=(2/\ln2)\ln\big((1+\sqrt5)/2\big)=1.39$ octaves.

Where you’ll meet this

Every mixing desk, car stereo and phone equaliser is a cascade of biquads like these. The Web Audio API’s BiquadFilterNode, which every web browser provides, computes its peaking, shelf and notch coefficients from the cookbook formulas.

A linear-phase FIR equaliser, which delays every frequency equally, is part of Choosing FIR or IIR (20.6). Running biquads safely with few bits is Second-order sections (21.2) and Finite word-length effects (21.3). More audio processing follows in Audio effects (29.2).

Reference card