5.3 Distortion-product OAEs and the DP-gram

The TEOAE excites the cochlea broadband. The DPOAE excites it with two narrowband primaries at frequencies $f_1$ and $f_2$ (with $f_2 > f_1$ ), and the cochlear amplifier — being a nonlinear active element — generates intermodulation distortion products. The most prominent product, by an order of magnitude, is the cubic difference tone at $2 f_1 - f_2$ , a frequency lower than either primary. This is a clinical signal: the audiologist can sweep $f_2$ across the audiometric range and measure the DPOAE amplitude at each frequency, producing a frequency-by-frequency report card on cochlear amplifier health — a DP-gram — that is the OAE’s closest analogue to the audiogram.

f₂ = 2000 Hz

L₁ = 65 dB SPL L₂ = 55 dB SPL

cochlear loss at f₂ = 0 dB HL

Two primary tones at f₁ and f₂ enter the cochlea. They overlap on the basilar membrane near the f₂ place, where the cochlear amplifier — a nonlinear active element built into the outer hair cells — produces intermodulation distortion. The strongest cubic product is at 2f₁ − f₂, a frequency LOWER than both primaries that travels back out through the middle ear and is recorded by the probe. The Scissors protocol uses L₁ = L₂ + 10, f₂/f₁ = 1.22 — chosen to match the cochlear amplifier's natural operating point. A measurable DPOAE (SNR > 6 dB above the averaged noise floor) is strong evidence that the outer hair cells are alive and working at that cochlear place. Above ~60 dB HL of cochlear loss the OAE typically disappears entirely, making DPOAEs sensitive to mild outer-hair-cell damage that may not yet show on the audiogram.

Where the distortion comes from

A linear system, by definition, transforms each frequency independently — input $A \cos \omega_1 t + B \cos \omega_2 t$ produces output $A' \cos \omega_1 t + B' \cos \omega_2 t$ at the same two frequencies. A nonlinear system mixes the frequencies and generates new ones. The cochlea’s nonlinearity is well-modelled by a third-order polynomial:

$y \;=\; \alpha x \;+\; \beta x^2 \;+\; \gamma x^3 \;+\; \ldots$

where $x$ is the basilar-membrane displacement at the place of interest. Plugging in two sinusoids $x = A_1 \cos(\omega_1 t) + A_2 \cos(\omega_2 t)$ and expanding produces, among other terms, components at $2 \omega_1 - \omega_2$ and $2 \omega_2 - \omega_1$ (cubic difference tones), $\omega_1 + \omega_2$ , $\omega_1 - \omega_2$ , $2 \omega_1$ , $2 \omega_2$ , and their amplitudes scale as $A_1^2 A_2$ , $A_1 A_2^2$ , etc.

▶ The cubic difference tone amplitude

Take the cubic term $\gamma x^3$ alone for clarity (the linear and quadratic terms produce only the input frequencies and the sum/difference frequencies, not the cubic products). Let $x = A_1 \cos \theta_1 + A_2 \cos \theta_2$ where $\theta_i = \omega_i t$ .

Expanding $x^3$ :

$x^3 = A_1^3 \cos^3 \theta_1 + 3 A_1^2 A_2 \cos^2 \theta_1 \cos \theta_2 + 3 A_1 A_2^2 \cos \theta_1 \cos^2 \theta_2 + A_2^3 \cos^3 \theta_2.$

Apply $\cos^2 \theta = \frac{1}{2}(1 + \cos 2\theta)$ to the squared terms and product-to-sum to the products:

$\cos^2 \theta_1 \cos \theta_2 = \frac{1}{2} \cos \theta_2 + \frac{1}{4}\bigl[\cos(2 \theta_1 - \theta_2) + \cos(2 \theta_1 + \theta_2)\bigr].$

Collecting the $\cos(2 \omega_1 - \omega_2)$ terms (the cubic difference tone we care about):

$y \big|_{2 \omega_1 - \omega_2} \;=\; \gamma \cdot \frac{3}{4} A_1^2 A_2 \cos(2 \omega_1 - \omega_2) t.$

So the DP amplitude scales as $A_1^2 A_2$ — quadratic in $A_1$ , linear in $A_2$ . Converting to decibels: the DP level grows by 2 dB per dB rise in $L_1$ , and 1 dB per dB rise in $L_2$ , if the cochlea were a pure cubic nonlinearity. In the real cochlea the growth is compressive (the amplifier saturates above its operating range), so the slopes are roughly 0.4–0.6 instead of 1 or 2.

The frequency dependence on $f_2$ comes from the place of generation: the DP is generated near the $f_2$ cochlear place, where the two primaries overlap most. Cochlear damage at the $f_2$ place therefore selectively reduces the DP at that $f_2$ , giving the DP-gram its frequency selectivity.

Clinical protocol

The standard protocol — sometimes called the Scissors paradigm for its converging primary levels — fixes:

$f_2 / f_1 = 1.22$ . This ratio matches the spacing of the two primary places on the basilar membrane such that they overlap maximally at the $f_2$ place.
$L_1 = 65$ dB SPL, $L_2 = 55$ dB SPL. The 10 dB offset means $L_1 > L_2$ ; this approximates the natural amplitude ratio of the two primaries at the cochlear amplifier’s operating point at the $f_2$ place (where $f_1$ , being farther basal, has lost some amplitude relative to its $f_2$ neighbour by the time the waves overlap).

The audiologist sweeps $f_2$ across the audiometric range (typically 750, 1000, 1500, 2000, 3000, 4000, 6000, 8000 Hz) and at each $f_2$ records the level of the spectral peak at $2 f_1 - f_2$ . A DPOAE is declared present when the level exceeds the local noise floor by more than 6 dB. The recorded set of (f, DPOAE level) pairs is the DP-gram.

A typical normal DP-gram sits between 0 and 15 dB SPL at most frequencies, with a slight roll-off below 1 kHz and above 6 kHz. A flat DP-gram below the noise floor indicates absent OAEs across the whole frequency range, equivalent to bilateral sensorineural loss of at least 30–40 dB HL.

DP-gram clinical use

The DP-gram is most useful in three settings:

Diagnostic confirmation in difficult-to-test populations — children, developmentally disabled adults, malingerers. The DP-gram is objective and behavioural-independent; agreement with the audiogram is generally strong, and discrepancy (DP-gram shows OAEs at frequencies where the audiogram shows loss) suggests non-organic loss or central pathology.
Ototoxicity monitoring. Patients on cisplatin chemotherapy or aminoglycoside antibiotics are at risk of dose-dependent OHC damage that begins at the highest frequencies (8 kHz and above) and progresses downward. Routine DP-gram monitoring during treatment can detect early OHC damage at higher frequencies than standard audiometric testing reaches (some clinical DPOAE systems go to 10 kHz or 12 kHz), allowing dose modification before clinically significant audiometric loss occurs.
Noise-induced damage monitoring. Similarly, the early phase of noise-induced cochlear damage involves selective OHC death at 3–6 kHz before audiometric thresholds shift. DP-grams in industrial hearing-conservation programmes can detect the earliest cochlear damage at-risk workers sustain, supporting intervention before measurable behavioural loss develops.

DPOAE I/O functions and the cochlear compression curve

Beyond the DP-gram, the audiologist can hold $f_2$ fixed and sweep $L_2$ from low to high, recording DP level at each input level. This input/output function maps the cochlear compression curve directly. A healthy cochlea shows steep growth at low levels (the active amplifier is contributing) and shallow growth at high levels (the amplifier saturates and only the passive nonlinearity contributes). A damaged cochlea shows reduced low-level growth and a higher threshold for DP emergence — the I/O curve linearises as the amplifier disappears. The I/O function is the most quantitative single measurement of cochlear amplifier health available clinically; it is also the slowest test (multi-minute per ear per frequency), so it is reserved for diagnostic depth rather than screening.

Closing the chapter

That closes Chapter 5. The arc: in 1978 a physical observation — that the ear emits sound — overturned the passive view of cochlear mechanics; the discovery of the outer-hair-cell motor and the prestin protein supplied the mechanism; and by 2026 the consequent clinical tests have moved the average age at diagnosis of congenital deafness from age 2 to age 3 months. The TEOAE is the universal newborn screen; the DPOAE is the diagnostic depth tool. Both are limited by their middle-ear-status sensitivity and their inability to characterise the degree of loss above 30–40 dB HL — limitations that motivate the next objective test in the audiologist’s toolkit.

The next chapter introduces evoked potentials — using scalp electrodes to record the brain’s electrical response to sound, with signal averaging to extract the response from the EEG floor. The ABR (auditory brainstem response) supplies what the OAE cannot: a threshold-grade measurement, retrocochlear localisation, and an objective tool that works even when the middle ear is impaired.

Next chapter: Ch 6 — Evoked potentials.