7.4 Feedback cancellation and frequency lowering

Two algorithmic problems remain that didn’t fit naturally into the previous lessons but are central to modern hearing-aid function:

Acoustic feedback — the howling oscillation that occurs when the hearing aid’s output leaks back to its input. This is what limited gain in pre-1999 hearing aids and what still limits gain in open-fit aids without adaptive feedback cancellation.
Dead regions in the cochlea — frequency regions where the outer-hair-cell damage is so severe that amplification at that frequency produces no perceptual benefit. The clinical response is frequency lowering, the strategy of shifting unusable high-frequency information to lower frequencies where the patient can hear it.

Both problems have produced clean algorithmic solutions whose mathematical structure is worth understanding.

The closed-loop feedback problem

A hearing aid is a closed-loop system. The microphone picks up sound, the DSP applies gain $G$ , the receiver outputs amplified sound — and some fraction $H$ of that output leaks back through the vent in the eartip, around the eartip, or through the body of the patient, into the microphone. The block diagram:

   x(t) (target sound from environment)
       ↓
   [+]──────────────→ G(f) ────→ y(t) (output)
    ↑                              │
    │                              ↓
    └── H(f) (feedback path) ←─────┘

The transfer function from environmental input $x$ to output $y$ is

$\frac{y(f)}{x(f)} = \frac{G(f)}{1 - G(f) H(f)}.$

The system is stable (does not oscillate) if $|G(f) H(f)| < 1$ for all $f$ — the Nyquist stability criterion. The system oscillates when $|G(f) H(f)| = 1$ and the phase $\angle G(f) H(f) = 0$ — the closed-loop gain reaches unity with constructive phase, and a small perturbation grows exponentially. The frequency at which oscillation first occurs is whatever frequency in the audible band satisfies these two conditions soonest as $G$ is raised.

For a closed-fit ITE or a CIC hearing aid, $|H(f)|$ is small — typically −40 dB across the audible band — so $G$ can be raised to about 40 dB before oscillation. For an open-fit RIC with a large vent or a slim-tube BTE, $|H(f)|$ can be as high as −10 dB at high frequencies — limiting unaided gain to roughly 10 dB, which is essentially useless for the patient.

Adaptive feedback cancellation (AFC) is the algorithm that broke this limit. Its principle:

▶ Adaptive feedback cancellation by adaptive filtering

Add a digital filter $\hat H(f, t)$ inside the device that estimates the external feedback path $H(f)$ . Subtract the filter’s prediction of the feedback signal from the microphone input before further processing:

   x(t) ────→ [+] ────→ G(f) ────→ y(t) ──┐
              ↑                            │
              │                            ↓
              │  [+] ←─── H(f) ────────────┤
              │   │      (true path)      │
              │   ↓                       │
              └── − ←─── Ĥ(f, t) ──────────┘
                          (estimate)

If $\hat H = H$ exactly, the subtraction perfectly cancels the feedback and the system becomes effectively open-loop — gain can be as high as we like.

The estimate is built up adaptively. At each time step, the algorithm:

Computes the predicted feedback: $\hat h(t) = \hat H(t) * y(t)$ (convolution of the filter estimate with recent output).
Subtracts from the microphone input: $\tilde m(t) = m(t) - \hat h(t)$ .
Adjusts the filter estimate to minimise the mean-square error using the LMS update rule:

$\hat H(t+1) = \hat H(t) + \mu \cdot \tilde m(t) \cdot y(t-\tau).$

The update rule moves $\hat H$ slightly toward whatever filter would best predict the residual error. Over thousands of update steps, $\hat H$ converges to the true $H$ .

The step size $\mu$ trades convergence speed for stability: large $\mu$ tracks changes (e.g., the patient cupping their ear) quickly but is noisier; small $\mu$ is smoother but slow to adapt.

The catch: when the input signal $x(t)$ contains a sustained tone at frequency $f_0$ , the algorithm tends to misidentify the tone as feedback and partially cancel it — known as entrainment. Modern AFC mitigates this by adding a small probe signal (an inaudible noise) that is uncorrelated with typical input signals, providing the AFC algorithm with a clean reference to learn from. Other variants use frequency-shifting (intentionally shifting the input frequencies by a few Hz so that feedback at the output is at a slightly different frequency than the input, breaking the closed-loop constructive-phase condition).

Modern AFC provides 15–25 dB of additional usable gain over the unaided feedback limit. An open-fit RIC with AFC can apply 35–45 dB of high-frequency gain without howling — sufficient for most mild-to-moderate sensorineural losses while preserving the natural-sound advantage of an open coupling. AFC is, alongside multichannel WDRC, one of the two algorithmic innovations that defined the modern hearing-aid era.

Dead regions and frequency lowering

The audiogram measures the lowest level at which a tone is detected. Detection is a relatively coarse measurement of cochlear health: a patient may be able to detect a 4000 Hz tone at, say, 70 dB HL while having essentially zero functional information transmission at that frequency — because the only surviving auditory nerve fibres in that cochlear region are responding to off-frequency mechanical excitation. The fibres at 4000 Hz are dead; the residual response comes from the lower-frequency apical region whose tuning is broad enough to pick up the 4000 Hz tone when it’s presented at 70 dB.

Brian Moore’s TEN test (Threshold Equalising Noise, 2000) detects these dead regions by presenting a tone in a band of TEN noise designed to elevate the threshold of any normal cochlear region to a known level. If the patient’s threshold-in-noise is no higher than the normal threshold, the cochlear region at that frequency is functioning; if it’s elevated by 10+ dB, the region is dead.

Amplification at a dead region produces no benefit. A 60 dB HL audiometric threshold at 4000 Hz, if produced by a dead region, gives no audibility to amplification at 4000 Hz — the amplified tone is being detected by off-frequency listening, not by 4000 Hz cochlear filtering. Amplifying louder just produces more loudness without more information.

The response: frequency lowering. Shift the spectral content at the dead frequencies down to lower frequencies where the cochlea can receive it.

Frequency compression vs frequency transposition

Two strategies in current use:

Frequency compression (Phonak SoundRecover, Phonak SoundRecover2) gradually compresses the high-frequency region of the spectrum. Frequencies above a chosen “compression start” point are mapped to a narrower range below the start point. A patient with a dead region above 3 kHz can have high-frequency consonants (e.g., the $/s/$ at ~6 kHz, the $/f/$ and $/sh/$ in the 4–6 kHz region) compressed into the 2–3 kHz region where the cochlea is more functional.
Frequency transposition (Widex Audibility Extender, Starkey Spectral iQ) copies a high-frequency band intact down into a lower frequency band. A $/s/$ at 6 kHz becomes a $/s/$ -like sound at 3 kHz, present in addition to whatever’s already there. The transposed signal is mixed with the (compressed or attenuated) original.

Both strategies improve consonant identification for severe-loss patients in the appropriate frequency range. The clinical evidence supports both, with frequency compression slightly favoured for steady-state acoustic events and transposition slightly favoured for brief consonants. The clinical practice is to enable frequency lowering when a TEN test confirms a dead region, and to disable it when the audiogram shows good high-frequency residual hearing (where the artefact of remapping consonants would degrade rather than improve intelligibility).

What frequency lowering cannot do

Frequency lowering is a workaround for cochlear damage, not a cure. It cannot:

Restore lost frequency resolution — the lower-frequency region the high-frequency content is mapped into has its own (broader-than-normal) cochlear filters, so the remapped content arrives with degraded resolution.
Create new perceptual categories — the patient must learn that the remapped sound represents a high-frequency phoneme. Adult patients take weeks to months to adapt; children adapt faster, especially when fit pre-lingually.
Preserve all acoustic detail — frequency compression removes spectral information (compresses 6 kHz of input into 1 kHz of output); transposition adds a duplicate but at a phonetically ambiguous frequency. Either way, some perceptual fidelity is lost in exchange for audibility of phonemes that would otherwise be inaudible.

The decision to enable frequency lowering is patient-by-patient: it almost always helps when there’s a clear dead region; it almost always hurts when the high-frequency residual is intact and producing useful information. The TEN test is the standard tool for making this decision.

Closing the chapter

That closes Chapter 7. The arc: the modern hearing aid is a real-time DSP pipeline whose sixteen stages implement four key algorithmic responses to four key cochlear-damage consequences. WDRC handles the narrowed dynamic range of cochlear loss by compressing 80 dB of acoustic input into the patient’s residual 30–50 dB. Directional microphones and noise reduction handle the SNR sensitivity of the impaired cochlea by spatially and temporally suppressing background noise. Adaptive feedback cancellation handles the closed-loop instability introduced by open-fit coupling. Frequency lowering handles cochlear dead regions that resist amplification.

Each algorithm has its limits, and each contributes a few dB of benefit; collectively they produce a device that, in 2026, restores meaningful function to a billion-plus people worldwide with treatable hearing loss. But fitting one of these devices well — applying the prescribed gain at each audiometric frequency, in the actual occluded ear canal of the patient, accounting for the patient’s individual ear-canal acoustics — requires a verification step the audiogram alone cannot supply. That step is the subject of the next chapter.

Next chapter: Ch 8 — Real-ear measurement and verification.