DAF Fairbanks Curve Explorer (2D)

DAF Parameters

Feedback delay τ 180 ms

Feedback gain K 0.70

Masking noise level 0%

Target speech rate 4.0 syll/s

Live Readouts

Produced rate

4.0 syll/s

Phase error

0°

Disfluency index (period CV)

0.0%

Curve peak (this K / mask)

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How it works

Same phase-locked-oscillator model as the 3D DAF simulator, viewed two different ways in 2D: an oscilloscope of the production and delayed-feedback signals, and a live-computed Fairbanks disruption curve — disfluency plotted against delay itself, not just its current value.

dθ/dt = ω₀·[1 + K·(1−mask)·sin(θ(t−τ) − θ(t))] + ω₀·σ·ξ(t)
CV(period) = std(last 24 inter-syllable intervals) / mean(...)

Discrepancy found and fixed: the base equation above (without the ω₀σξ(t) motor-noise term) is exactly what the 3D version runs. Numerically integrating it shows it always relaxes to a zero-error phase-locked orbit for any fixed delay once transients die out — because when the delayed-feedback error reaches 0 the sin(·) coupling term vanishes and stops correcting, so CV decays to ~0 regardless of τ. That can never reproduce Fairbanks' classic non-monotonic delay-response curve, which is only observed in real speakers because human speech timing has intrinsic motor jitter. This page adds that jitter as an Euler–Maruyama noise term (verified numerically, see report) — with it restored, sweeping τ reproduces the real non-monotonic shape: low disruption near τ=0 and near a full syllable period, a peak in the clinical 150–220 ms band, matching Fairbanks (1955).

Delay τ — feedback-loop latency in milliseconds; also the x-axis of the lower curve panel.
Gain K — coupling strength between the delayed copy and the timing generator.
Masking noise — fraction of the feedback signal masked before it can act.
Target speech rate — natural syllable rate ω₀/2π (also sets the natural period marked on the curve).

Production signal Delayed feedback Disruption vs delay (Fairbanks curve) Current delay τ

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