Specific Fluctuation Strength (Hearing Model)

Specific Fluctuation Strength (Hearing Model) resolves the perception of slow amplitude modulation across the frequency axis using the hearing model defined in ECMA-418-2, Edition 4 (2025). Instead of a single overall vacil value, the analysis delivers a distribution of fluctuation strength over 53 half-Bark bands, so that the frequency region responsible for a throbbing or pulsing character can be localized directly.

This analysis complements the classic fluctuation strength model described on the Fluctuation Strength page. The classic model follows the Zwicker/Fastl formulation; the hearing-model variant follows the standardized ECMA-418-2 auditory processing chain. Both are available side by side and are selected according to the reporting requirement of the project.

Theory

Hearing-Model Front End

The input signal passes through the ECMA-418-2 auditory filter bank, which divides the audible range into 53 overlapping bands spaced half a Bark apart, covering 0.5 – 26.5 Bark. Before the filter bank, an ear transfer function corresponding to the selected sound field (free field or diffuse field) is applied. The envelope of every band output is then extracted and analysed for low-rate modulation.

Modulation Evaluation and Time Grid

Within each band, modulation components in the fluctuation-relevant range are weighted so that sensitivity peaks near 4 Hz, matching the temporal resolution limit of the auditory system. The band results are produced on a fixed 50 Hz result grid — one value every 20 ms — and a non-linear low-pass stage models the temporal integration (build-up and decay) of the ear, so that short bursts do not create unrealistically sharp peaks in the result.

Specific Value and Total Value

The band result is a density: fluctuation strength per unit of critical-band rate, expressed in vacil/Bark. Integrating the density over the critical-band-rate axis yields the total fluctuation strength in vacil:

F = ∫₀^{26.5 Bark} F′(z) dz [vacil]

The single value reported in the footer is the 10th percentile F₁₀ of the total fluctuation strength over time — the level that is exceeded during 10% of the evaluated duration. This follows the same percentile convention used elsewhere in psychoacoustics, such as the loudness value N₅; it is a statistic of the signal, not a tolerance or an uncertainty. All three variants of this analysis share this same single-value definition, so a band spectrum, a vs Time colour map and a vs RPM colour map computed from the same recording report an identical single value.

Parameters

Parameter	Options	Description
Sound Field	Automatic (default) / Free field / Diffuse field	Ear transfer function of the hearing model. In Automatic mode the sound field is determined per channel from the equalization information stored with the recording; channels without that information fall back to free field
Standard Edition	ECMA-418-2 Edition 4 (2025)	Hearing-model edition applied by this analysis
Band Resolution	53 half-Bark bands, 0.5 – 26.5 Bark (fixed)	Band layout defined by the standard; not user-adjustable
Result Rate	50 Hz (fixed)	One result frame every 20 ms; applies to the vs Time variant and to the sampling grid of the vs RPM variant
Speed Step	50 rpm (default)	vs RPM variant only. Spacing of the speed axis
Slope	Automatic (default) / Run-up / Coast-down	vs RPM variant only. Direction of the speed change to be evaluated

Output Quantities

Quantity	Unit	Description
F′(z)	vacil/Bark	Specific fluctuation strength per band — 53 values, plotted over the band centre frequency
F′(z, t)	vacil/Bark	Specific fluctuation strength as a colour map over band and time
F′(z, n)	vacil/Bark	Specific fluctuation strength as a colour map over band and speed of rotation
F₁₀	vacil	Single value: 10th percentile of the total fluctuation strength over time; identical for all three variants

The Three Variants

The hearing-model specific fluctuation strength is offered as three analyses that share one computation core and one single-value definition, but differ in how the result is presented:

Analysis	Result Form	Axes	Typical Use
Specific Fluctuation Strength (Hearing Model)	Band spectrum (one curve per channel)	x = band centre frequency, y = vacil/Bark	Stationary operating points; identifying which frequency region fluctuates
Specific Fluctuation Strength (Hearing Model) vs. Time	Colour map	x = time, y = band centre frequency, colour = vacil/Bark	Transient events, load changes, cycle-by-cycle comparison
Specific Fluctuation Strength (Hearing Model) vs. RPM	Colour map	x = speed of rotation, y = band centre frequency, colour = vacil/Bark	Run-up and coast-down; locating speed ranges with pronounced fluctuation

The vs RPM variant is intended for run-up and coast-down measurements, so the recording must contain speed information. For an overview of all psychoacoustic vs RPM analyses, see Psychoacoustics vs RPM.

Sound Field Selection

The hearing model requires an ear transfer function, and the correct choice depends on how the signal was recorded. The analysis therefore offers an Automatic setting that reads the equalization information stored with each recorded channel:

Free-field equalized channels are evaluated with the free-field transfer function.
Diffuse-field equalized channels — and independent-of-direction equalized artificial head channels — are evaluated with the diffuse-field transfer function.
Channels carrying no usable equalization information fall back to free field.

The sound field is resolved per channel. In a multi-channel measurement that mixes an artificial head with free-field microphones, the Automatic setting will not apply one common transfer function to all channels. If a report requires a uniform basis, select the sound field explicitly.

Interpretation Guidelines

Reference value: 1 vacil corresponds to the fluctuation strength of a 1 kHz tone at 60 dB SPL, 100% amplitude-modulated at 4 Hz. The band values F′(z) are densities, so they must be integrated over the Bark axis before being compared against overall vacil values.

Flat, low distribution — No frequency region stands out; the sound is temporally steady.
Single narrow peak — One band group carries the fluctuation. Convert the band centre frequency into a physical source frequency to locate the mechanism.
Two separated peaks — Typically two independent modulation mechanisms, e.g. a low-frequency combustion irregularity plus a high-frequency beating between two closely spaced tones.
Broad elevated plateau — Modulation of a broadband source such as flow or brush noise rather than of a discrete tone.

Practical Application Notes

Because the analysis is based on the standardized hearing model, results are comparable across projects and laboratories when the same edition and sound field are declared. Always state both in the report.
Low-rate modulation needs time to become visible. Recordings that are too short do not provide enough result frames for a stable percentile; allow several seconds per operating point.
The band spectrum answers "which frequency region fluctuates", the vs Time map answers "when", and the vs RPM map answers "at which speed". Diagnosing a complaint usually requires at least two of the three.
Fluctuation strength and roughness describe the same physical phenomenon — envelope modulation — at different modulation rates. If the band spectrum shows little fluctuation but the sound is still perceived as uneven, evaluate Roughness as well.
The single value is a percentile, not a maximum. Isolated single-frame outliers therefore do not dominate the reported figure, which makes it robust for pass/fail limits.

Standard References

ECMA-418-2:2025 — Psychoacoustic metrics for ITT equipment — Part 2: Models based on human perception (Edition 4)
ECMA-418-1 — Psychoacoustic metrics for ITT equipment — Part 1: Prominent discrete tones (companion part)
R. Sottek, "A Hearing Model Approach to Time-Varying Loudness," Acta Acustica united with Acustica, vol. 102, 2016
E. Zwicker & H. Fastl, Psychoacoustics: Facts and Models, 3rd ed., Springer, 2007, Chapter 10: Fluctuation Strength