Source: https://jmrplens.github.io/phonometry/reference/api/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`](/phonometry/reference/api/psychoacoustics/fluctuation-strength/#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`](/phonometry/reference/api/psychoacoustics/fluctuation-strength/#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. > Auto-generated from the source docstrings by `scripts/generate_api_docs.py` (`make api-docs`). Do not edit by hand. ## fluctuation_strength ```python 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`](/phonometry/reference/api/psychoacoustics/fluctuation-strength/#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`](/phonometry/reference/api/psychoacoustics/fluctuation-strength/#fluctuationstrengthresult). **Raises** | Exception | When | | :--- | :--- | | ValueError | If the signal is invalid or `fs` is not positive. | ## fluctuation_strength_am_noise ```python 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 ```python 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() ```python FluctuationStrengthResult.plot( ax: Axes | None = None, *, language: str = 'en', **kwargs: Any, ) -> Axes ``` Plot the specific fluctuation strength against critical-band rate.