psychoacoustics.fluctuation_strength

Fluctuation strength after Fastl & Zwicker / Osses et al.

Fluctuation strength F (unit: vacil) rates the slow loudness fluctuations of a sound. It has a band-pass dependence on modulation frequency peaking near 4 Hz, and its fixed point is defined (Fastl & Zwicker, Psychoacoustics: Facts and Models, Chapter 10) as 1 vacil for a 1 kHz tone at 60 dB, 100 % amplitude-modulated at 4 Hz. There is no ISO standard for fluctuation strength.

Two routes are provided:

• fluctuation_strength_am_noise — the closed form (Fastl & Zwicker Eq. 10.2) for a sinusoidally amplitude-modulated broadband noise, exact in the modulation factor, level and modulation frequency.
• fluctuation_strength — the signal model of Osses, García & Kohlrausch (2016), which estimates F from an arbitrary calibrated pressure signal. It sums specific contributions over 47 auditory filters, F = C_FS·Σ (m*_i)^p_m·|k_{i-2}·k_i|^p_k·g(z_i)^p_g (Osses 2016 Eq. 1), where m* is a generalised modulation depth, k a cross-covariance between neighbouring bands and g(z) a frequency weighting.

The closed form and the 1-vacil calibration are exact. The signal model has no normative oracle; it is implemented clean-room from the Osses 2016 paper and cross-checked against the literature fluctuation-strength values of Fastl & Zwicker (Osses 2016 Table 1) and the open SQAT reference implementation. Like the ECMA-418-2 roughness, the model reproduces the reference trends and the 1-vacil anchor within a documented tolerance rather than to the last digit.

Front-end deviations from Osses 2016 (besides the re-fitted H(fmod) corners documented at _HP_LO): the paper analyses 2 s frames with 90 % overlap and 50 ms raised-cosine gating and screens components against the absolute hearing threshold; this implementation uses 50 % overlap, a Hann analysis window and relative floors (component magnitude > max/10^4, excitation level > max - 60 dB), with k = 1 at the filter-bank edges. Measured consequences: a steady 1 kHz tone returns ~0.09 vacil instead of ~0 (analysis-window envelope leakage through the 2 Hz high-pass) and steady broadband noise returns ~0.4 vacil (partly physical level fluctuation of the noise itself).

ECMA-418-2:2025 (4th ed.) introduced a normative hearing-model fluctuation strength (its Clause 9, block size 65536); this module implements the Osses/Fastl & Zwicker model, not that method.

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fluctuation_strength

fluctuation_strength(
    signal_in: NDArray[np.float64],
    fs: float,
) -> FluctuationStrengthResult

Fluctuation strength of a calibrated signal (Osses 2016 model).

Estimates F in vacil from an arbitrary calibrated sound-pressure signal by the model of Osses, García & Kohlrausch (2016): an outer/middle-ear transmission, a 47-band excitation-pattern filter bank, the generalised modulation depth m* from the band-pass-filtered envelope, the cross-covariance k between neighbouring bands and the weighted sum of Eq. (1). The signal is analysed in 2 s frames; the overall value is the median across frames. The Osses 2016 model is defined for the free field only (its a0 transmission is the free-field characteristic); there is no diffuse-field variant, so this function takes no field argument.

Note

No ISO standard defines fluctuation strength and no numeric oracle anchors this signal model. It is implemented clean-room from the Osses 2016 paper and reproduces the literature reference values (Osses 2016 Table 1) and the 1-vacil calibration point within a documented tolerance — the reference method itself only agrees qualitatively for FM tones. For an exact figure on AM broadband noise use fluctuation_strength_am_noise.

Calibration. The constant C_FS (Eq. 1) is not the paper’s literal 0.2490 — that was fitted to the paper’s own 4096-tap FIR front-end. Here it is derived once from the 1-vacil reference stimulus (1 kHz, 60 dB, m = 1, 4 Hz) run through this implementation’s front-end, so the reference returns exactly 1.00 vacil by construction (C_FS ≈ 0.28; see _c_fs).

Achieved tolerance (against Osses 2016 Table 1 literature values): the reference tone is 1.00 vacil exactly; the AM-tone 70 dB sweep (fmod = 1, 2, 4, 8, 16, 32 Hz) has Pearson correlation ≈ 0.98 with the literature, peaks at 4 Hz and stays within a factor ≈ 2.1 at every point; the AM-tone carrier sweep at fmod = 4 Hz reproduces the Fastl & Zwicker Fig. 10.5 trend (low-mid plateau, roll-off at 8 kHz); the AM broadband-noise 60 dB sweep shows the correct band-pass shape (maximum at 4 Hz, monotone tails) but overshoots the absolute pass-band level by up to ~3x (the excitation front-end spreads the modulated energy across bands). FM-tone accuracy is not pursued — the reference method itself overestimates it above 4 Hz.

Parameters

Name	Description
signal_in	Calibrated sound-pressure signal (1-D), in Pa.
fs	Sample rate, in Hz (resampled to the model rate if needed).

Returns: A FluctuationStrengthResult.

Raises

Exception	When
ValueError	If the signal is invalid or fs is not positive.

fluctuation_strength_am_noise

fluctuation_strength_am_noise(
    level_db: float,
    modulation_factor: float,
    mod_frequency: float,
) -> float

Fluctuation strength of AM broadband noise (Fastl & Zwicker Eq. 10.2).

F = 5.8·(1.25·m − 0.25)·[0.05·(L/dB) − 1] / [(fmod/5Hz)² + (4Hz/fmod) + 1.5] vacil, the closed form for a sinusoidally amplitude-modulated broadband noise of level L, modulation factor m and modulation frequency fmod. Exact; the result is clamped at 0 (the formula goes negative below ~20 dB or m < 0.2 where the sensation vanishes).

Parameters

Name	Description
level_db	Broadband-noise level L, in dB.
modulation_factor	Modulation factor m (0..1).
mod_frequency	Modulation frequency fmod, in Hz.

Returns: Fluctuation strength, in vacil (>= 0).

Raises

Exception	When
ValueError	If modulation_factor is outside [0, 1] or mod_frequency is not positive/finite, or the level is not finite.

FluctuationStrengthResult

FluctuationStrengthResult(
    fluctuation_strength: float,
    specific: NDArray[np.float64],
    bark_axis: NDArray[np.float64],
    time_dependent: NDArray[np.float64],
)

Fluctuation strength of a signal (Osses 2016 model).

Attributes

Name	Description
fluctuation_strength	Overall fluctuation strength F, in vacil (the median over the analysis frames).
specific	Specific fluctuation strength f(z) per auditory filter, in vacil/Bark (averaged over frames), shape (47,).
bark_axis	Centre critical-band rate z_i of each filter, in Bark, shape (47,).
time_dependent	Fluctuation strength per analysis frame, in vacil.

FluctuationStrengthResult.plot()

FluctuationStrengthResult.plot(
    ax: Axes | None = None,
    *,
    language: str = 'en',
    **kwargs: Any,
) -> Axes

Plot the specific fluctuation strength against critical-band rate.