How my headphones measure my ears

What the chime is

The first patent, US 10,937,410 (Rule, filed April 2020), is where the chime is specified. It is a "relatively short confirmation sound", under a second and typically a tenth to half a second long, played automatically when the headphone detects that it has been put on. Its spectrum is ten or more tones between about 45 Hz and 16 kHz, built as harmonics of a fundamental at 46.875 Hz, with the tones above 1 kHz spaced a quarter octave or closer. Its worked example lists 24 of them: seven below 1 kHz, from 93.75 to 843.75 Hz, and seventeen from 1031 Hz up to 16969 Hz.

46.875 is not an arbitrary number. It is 48,000 divided by 1,024: at a 48 kHz sample rate, a 1,024 point FFT has bins exactly 46.875 Hz apart, and every tone in the patent sits on the centre of one of them. A periodic signal whose tones land exactly on bin centres can be read with one FFT per 21.3 ms period, with no spectral leakage between tones, and a quarter second chime gives about a dozen periods to average. So the chime is built to be read as a transfer function, tone by tone, not timed as an echo. That reading is my inference from the numbers; the patent does not put it that way.

What gets measured is G_sd, the response from the driver (d) to the feedback microphone (s) inside the cup or the nozzle. That microphone is already there: it is the error sensor of the feedback noise cancelling. The measurement costs no hardware at all.

What my ear does to a headphone

A headphone is designed on reference heads, but on a real head the driver is loading a different acoustic system every time it is put on. Two effects dominate, and the lab has a slider for each.

The first is the leak. A sealed cup or tip makes a pressure chamber: below a few hundred hertz the driver simply compresses the trapped air, and the pressure is large and flat. Any way out of that chamber, an arm of my glasses under the cushion, hair, a tip one size too small, is an acoustic mass in parallel with the trapped air, and it bleeds the bass away. Press "Broken seal" and the over-ear microphone reads 10 dB down at 100 Hz and 16 dB down at 30 Hz. In an earbud, where the trapped volume is a fraction of a cubic centimetre, a gap of 1.5 mm² costs 15 dB at 100 Hz.

The second is the canal. The ear canal is a tube roughly 25 mm long and 7.5 mm across, closed at the eardrum, and a tube closed at one end resonates at a quarter wavelength, c / 4L, which is about 3.4 kHz at 25 mm. At the eardrum that resonance is a peak; at the open end it is a pressure minimum. Set "Colour the air at" to about 3.4 kHz and look down the canal: pale at the entrance, strong at the eardrum. An earbud sits inside the canal and its microphone looks straight down it, so it sees the whole signature, shifted by wherever the tip lands. An over-ear microphone sits in a cup of about 120 cm³ of air and hears the canal only faintly.

That asymmetry turns out to decide what CustomTune can do for each kind of product.

Retuning the noise cancelling

Feedback noise cancelling is a control loop. The microphone hears the residual noise, a filter K_fb shapes it, and the driver plays it back inverted. The number that matters is the loop gain, L = G_sd K_fb. Where it is large, noise at the microphone is divided by about |1 + L|. Where it falls through 0 dB, the crossover, the loop stops helping, and if the phase has turned too far round by then it starts adding noise instead, or rings, or howls.

A fixed K_fb is designed against the average G_sd. The patent's observation is that on in-ear products the ear-to-ear variation in G_sd is largest "at and near the feedback loop gain crossover frequency", exactly where the loop is most fragile, so no single filter can be both aggressive and safe for everyone. Its answer is to take this ear's deviation from nominal at each chime tone and change the filter so the loop gain returns to its target, adjusting at least three second-order sections across at least 200 Hz to 5 kHz. The adjustment is linearised: a precomputed "influence matrix" records how each filter parameter moves the response at each tone, and its pseudo-inverse turns the measured deviation into parameter changes "with a single matrix multiplication", in under a second. In log terms, the correction subtracts the deviation.

Δθ = −(∂K_fb / ∂θ)⁺ ΔG_sd, so that G_sd K_fb ≈ L_target

The targets it states are an average feedback insertion gain that keeps reducing noise up to about 1.5 kHz or more, "a nearly 2 kHz loop magnitude crossover" with about 45 degrees of phase at crossover, and a total insertion gain, with the noise cancelling on, below −30 dB between about 1 and 2 kHz.

The lab does the same with five peaking filters. Choose "Short canal, earbuds": a short canal lifts G_sd above about 1 kHz, by more than 10 dB near 5 kHz, and with the fixed filter the loop gain does not fall through 0 dB until about 5 kHz, by which point the phase has long gone round. In hardware that is a whistle. After the chime the correction pulls that region down, and the loop is back to a 1.8 kHz crossover with about 43 degrees of margin. Choose "Loose tip": the leak takes bass out of G_sd, so it takes gain out of the loop exactly where the cancelling does most of its work. The correction lifts the lowest filters by about 8 dB and the feedback quieting between 50 and 500 Hz goes from about −10 to −13 dB. The limit shows too: a correction clamped to a sensible range cannot put back everything a big leak removes, which is why a fit test is still worth doing.

The patent notes that for on-ear and around-ear fits there is "relatively more focus" on low frequencies "due to leakage", and for in-ear fits on the high frequencies. The lab reproduces that split without being told to: on the over-ear, the canal slider barely moves the loop, and the glasses arm is the only thing that does.

One aside I did not expect: the patent proposes the fitted filter parameters, left and right combined, as a biometric "earprint" that could identify the wearer, optionally combined with the formant structure of their voice.

Retuning the music

The EQ is a different problem, because the thing to be corrected is not at the microphone. I hear the pressure at my eardrum, and the microphone is a couple of centimetres away from it, on the other side of the canal's resonance. Equalising what the microphone hears would be exactly wrong around 3 kHz.

The second patent, US 12,028,675 (Nielsen, priority December 2021), handles this with population data. In the lab, Bose measured headphones on many people and many fittings with a microphone deep in the canal standing in for the eardrum, and reduced that data to two complex constants per frequency, α and β, fitted by least squares through the model

G_ear ≈ G_sd / (α G_sd + β)

On the wearer, the headphone measures G_sd, the formula predicts what the eardrum is getting, and the EQ is the difference from what the product was tuned for. Rearranged, G_sd / G_ear = α G_sd + β, which is linear, so the lab fit is the same kind of pseudo-inverse as the noise-cancelling fit. The patent also measures the path a second time with the feedback controller running, because with feedback cancelling on, the loop suppresses the music at low frequencies along with the noise, and the EQ has to put that back. The same machinery produces the filter for aware mode.

I did the same with a synthetic population of 160 ears drawn from the model. It works where the microphone can see a difference. On the earbud, a loose tip that leaves the eardrum about 8 dB RMS wrong comes out under 2 dB, and a short or long canal comes out about half as wrong. On the over-ear, the broken seal is corrected almost completely, 3.4 dB down to about 0.2. But a 19 mm or a 32 mm canal behind an over-ear cup stays 4 to 5 dB off, because the cup microphone cannot see the canal and so the regression, correctly, returns the average ear.

What mine is actually doing

Bose's patent marking list gives US 10,937,410 and 11,600,256, the noise-cancelling customisation, against the QC Earbuds II and both generations of QC Ultra Earbuds, and not against either generation of the QC Ultra Headphones. US 12,028,675, the EQ, is listed against the headphones. Bose markets CustomTune on the second generation headphones as adapting the sound to the shape of my ears, and a reviewer describes a calibration stimulus that plays every time they power on (Headphones.com, n.d.).

Putting that together with the physics, my reading of what happens when I put mine on: the chime measures G_sd in each cup; that measurement mostly reports the seal and the air space my head and ears leave in the cup; and the EQ moves the music towards where the population data says my eardrum should be. Whether the noise-cancelling filter is also refitted on the headphones, I cannot tell from anything public. What the chime cannot do through a cup of air is measure my canal, so above a couple of kilohertz my headphones are tuned for an average ear. That is a limit of the physics rather than of Bose, and it is why earbuds, where the microphone looks down the canal, were where the technique started.

What I could not settle

Whether the over-ear headphones refit the noise-cancelling filter at all. The patents that describe it are not on their marking list, and Bose's descriptions of the headphone feature talk about sound, not cancelling.

What the product chime actually contains. The stand-in here is built from the patent's tone grid, not from a recording. Recording the chime from inside the cup at 48 kHz and taking a 1,024 point FFT would show at once whether its energy sits on the 46.875 Hz grid.

The implementation details: how many filter sections, how far they are allowed to move, whether results are averaged over several fittings (the patent allows it), and what the population behind α and β looks like. Reviews also disagree on how many microphones the headphones have, and Bose's own page does not say.