Integrated and Statistical Levels

Environmental noise metrics computed directly from the raw (calibrated) signal.

Leq and LAeq

The equivalent continuous level integrates the squared pressure over the measurement time:

$$L_{eq} = 10\log_{10}\left(\frac{1}{T}\int_0^T \frac{p^2(t)}{p_0^2}\ dt\right) \text{ dB}, \qquad p_0 = 20\ \mu\text{Pa}$$

and $L_{Aeq}$ is the same integral after A-weighting the signal. $L_N$ is the level exceeded $N\ \%$ of the time — the $(100-N)$-th percentile of the time-weighted level distribution.

from phonometry import leq, laeq

# Equivalent continuous level of the whole recording
level = leq(signal, calibration_factor=sensitivity)

# A-weighted Leq (the standard environmental noise metric)
la = laeq(signal, fs, calibration_factor=sensitivity)

Both accept 1D signals (returning a scalar) or 2D [channels, samples] arrays (returning one level per channel), and support dbfs=True for digital full-scale analysis (calibration does not apply in dBFS mode).

Why the energy mean and not the arithmetic mean of dB values? Because sound doses add as energy: two periods at 60 dB and 80 dB do not average to 70 dB — the 80 dB half dominates and $L_{eq}$ = 77 dB. Averaging decibels directly underestimates every fluctuating noise. $L_{eq}$ is the level of the steady sound carrying the same energy as the real, fluctuating one, which is why regulations are written in terms of it.

leq() / laeq() parameters

Parameter	Type / shape	Units	Range / default	Notes
x	1D or 2D array	digital units (or Pa if calibrated)	non-empty	2D is [channels, samples]; returns one level per channel
fs	int	Hz	> 0 (laeq only)	leq needs no sample rate (pure RMS integral)
calibration_factor	float	Pa per digital unit	default 1.0	From calculate_sensitivity()
dbfs	bool	—	default False	True: 0 dBFS = full-scale RMS sine; ignores calibration

Percentile levels (LN)

ln_levels computes statistical levels from the time-weighted envelope: L10 is the level exceeded 10 % of the time (event peaks), L50 the median, L90 the background level.

from phonometry import ln_levels

stats = ln_levels(signal, fs, n=(10, 50, 90), weighting="A")
print(f"LA10={stats[10]:.1f}  LA50={stats[50]:.1f}  LA90={stats[90]:.1f} dB")

L10 tracks the event peaks, L50 the median level and L90 the background.

Options: mode selects the envelope ballistics ('fast', 'slow', 'impulse'), weighting applies A/C weighting first, and calibration_factor/dbfs behave as in leq. The integrator attack transient (~2τ) is discarded before taking percentiles.

Formally, $L_N$ is the $(100-N)$-th percentile of the distribution of the time-weighted level: the recording is first turned into a level-vs-time envelope (Fast by default), and $L_{10}$ is the envelope value exceeded 10 % of the time. That makes the ballistics choice part of the metric: an $L_{10}$ from a Slow envelope is systematically lower than from a Fast one on impulsive noise, so regulations always name the time weighting.

ln_levels() parameters

Parameter	Type / shape	Units	Range / default	Notes
x	1D or 2D array	digital units	non-empty	2D returns per-channel dicts
fs	int	Hz	> 0	Needed by the envelope detector
n	tuple of ints	%	default (10, 50, 90)	Any exceedance percentages, e.g. (1, 5, 95)
mode	str	—	'fast' (default), 'slow', 'impulse'	IEC 61672-1 ballistics of the envelope
weighting	str or None	—	'A', 'C', 'G', 'Z', None (default)	Frequency weighting before the envelope
calibration_factor / dbfs	float / bool	—	as leq	Same semantics as in leq()

Peak, event and occupational metrics

from phonometry import lc_peak, sel, sound_exposure, lex_8h

# C-weighted peak (IEC 61672-1 §5.13) - occupational action limits use this
peak = lc_peak(signal, fs, calibration_factor=sensitivity)

# Sound exposure level: single-event level normalized to 1 s (LAE)
lae = sel(event, fs, weighting="A", calibration_factor=sensitivity)

# Daily noise dose (IEC 61252): exposure in Pa²·h and LEX,8h / LEP,d
E = sound_exposure(shift_sample, fs, duration_hours=8, calibration_factor=sensitivity)
lex = lex_8h(shift_sample, fs, duration_hours=8, calibration_factor=sensitivity)

lc_peak is verified against the one-cycle/half-cycle reference responses of IEC 61672-1:2013 Table 5, sel against the Table 4 LAE toneburst column, and the dose functions against the IEC 61252 anchors (3.2 Pa²h ↔ exactly 90 dB). With duration_hours, the input is treated as a representative sample of that exposure period; without it, the input is the whole event.

SEL: comparing events of different duration

A 4 s train pass-by and a 30 s one cannot be compared by their $L_{Aeq}$ alone — the longer event delivers more energy at the same level. The sound exposure level compresses the whole event energy into exactly one second:

$$L_E = L_{eq,T} + 10\log_{10}\frac{T}{T_0}, \qquad T_0 = 1\ \text{s}$$

so events of any duration become directly comparable, and $N$ identical events sum as $+10\log_{10}N$. This is the building block of airport and railway noise models.

Noise dose: sound exposure and LEX,8h

Occupational regulations limit the daily dose, not the level. IEC 61252 expresses it as sound exposure $E$ in pascal-squared-hours — the time integral of the squared A-weighted pressure — and the equivalent normalized 8 h level:

$$E = \int_0^T p_A^2(t)\ dt \quad [\text{Pa}^2\text{h}], \qquad L_{EX,8h} = 10\log_{10}\frac{E}{8\ \text{h} \cdot p_0^2}$$

The anchor worth memorizing: 3.2 Pa²h ⇔ exactly 90 dB over 8 h (the CI suite enforces it). Half the dose is −3 dB; double duration at the same level is +3 dB.

Peak / event / dose parameters

Function	Key parameters	Returns	Standard anchor
lc_peak(x, fs, calibration_factor=1.0, dbfs=False)	dbfs=True references full-scale peak (1.0), not RMS	LCpeak [dB]	IEC 61672-1 §5.13, Table 5 tone bursts
sel(x, fs, weighting=None, ...)	weighting='A' gives LAE	SEL [dB]	IEC 61672-1 Table 4 (LAE column)
sound_exposure(x, fs, duration_hours=None, ...)	duration_hours treats x as a sample of that period	E [Pa²h]	IEC 61252
lex_8h(x, fs, duration_hours=None, ...)	same sampling semantics	LEX,8h [dB]	IEC 61252 (≡ LEP,d)

Loudness level of pure tones (ISO 226:2023)

The normal equal-loudness-level contours relate the SPL of a pure tone to its perceived loudness level in phons (the SPL of an equally loud 1 kHz tone). equal_loudness_contour(phon) evaluates ISO 226:2023 Formula (1) at the 29 preferred third-octave frequencies of Table 1, loudness_level(spl, frequency) is the exact inverse (Formula 2), and hearing_threshold() returns the threshold-of-hearing column:

from phonometry import equal_loudness_contour, loudness_level

freqs, spl = equal_loudness_contour(40.0)   # the classic 40-phon contour
phon = loudness_level(73.0, 63.0)           # 73 dB @ 63 Hz -> 40 phon

Validity per clause 4.1: 20-90 phon (80 phon above 4 kHz); the implementation is verified against the Annex B tables in CI. Note this is the loudness of pure tones - loudness of arbitrary signals (ISO 532 sones) is a different, upcoming feature.

Prominent discrete tones (ECMA-418-1)

Tonal components in machinery noise are far more annoying than their level suggests. ECMA-418-1:2024 (referenced by ECMA-74 Annex D) gives two FFT-based methods to decide whether a discrete tone is prominent: tone_to_noise_ratio() compares the tone level with the masking noise in its critical band (clause 11), and prominence_ratio() compares the critical band centred on the tone with the two contiguous bands (clause 12). Both return a structured verdict against the frequency-dependent prominence criteria:

from phonometry import tone_to_noise_ratio, prominence_ratio

tnr = tone_to_noise_ratio(x, fs)            # highest peak, or tone_freq=...
pr = prominence_ratio(x, fs, tone_freq=1000.0)
print(tnr.ratio_db, tnr.criterion_db, tnr.prominent)

The methods hinge on the critical band — the ear’s analysis bandwidth, $\Delta f_c = 25 + 75\ [1 + 1.4(f/1000)^2]^{0.69}$ Hz (162 Hz at 1 kHz): a tone is masked only by the noise inside its critical band, so both ratios compare the tone against exactly that noise, not the whole spectrum.

A TNR above $8 + 8.33\log_{10}(1000/f_t)$ dB (8 dB from 1 kHz up) classifies the tone as prominent; the PR criterion is $9 + 10\log_{10}(1000/f_t)$ dB. Low frequencies get higher thresholds because wider relative bands mask more.

Proximate secondary tones in the same critical band are combined per clause 11.6; for harmonic complexes assess each component (tone_freq=). Both methods work on Hann-windowed, RMS-averaged spectra and need no absolute calibration (the ratios are level differences).

tone_to_noise_ratio() / prominence_ratio() parameters

Parameter	Type	Units	Range / default	Notes
x	1D array	any (uncalibrated OK)	≥ fs/resolution_hz samples	Ratios are level differences: calibration cancels out
fs	int	Hz	> 0
tone_freq	float, optional	Hz	89.1–11 200; default None	None assesses the highest peak in the range of interest
resolution_hz	float	Hz	> 0; default 1.0	Tone band must stay within 15 % of the critical band (clause 11.2)

Both return a ToneAssessment(frequency, ratio_db, criterion_db, prominent).

Environmental noise: Lden, Ldn and rating levels (ISO 1996-1)

Regulatory noise assessment weights evenings and nights more heavily. lden() implements the day-evening-night level of ISO 1996-1:2016 (3.6.4: +5 dB evening, +10 dB night, default 12/4/8 h periods — adjustable, since countries define them differently), ldn() the day-night variant (3.6.5), and composite_rating_level() the general whole-day composite of clause 6.5 (Formulae 5-6) for arbitrary periods with source or character adjustments (Table A.1: e.g. +5 dB regular impulsive, +12 dB highly impulsive, +3 to +6 dB prominent tones):

from phonometry import lden, composite_rating_level

l = lden(63.2, 58.1, 51.4)                      # from LAeq per period
r = composite_rating_level([(63.2, 12, 0.0),    # day
                            (58.1, 4, 5.0),     # evening (+5)
                            (51.4, 8, 10.0)])   # night  (+10) == lden

lden() / ldn() / composite_rating_level() parameters

Function	Key parameters	Notes
lden(lday, levening, lnight, hours=(12, 4, 8))	period LAeq values [dB]; hours must sum to 24	+5 dB evening, +10 dB night (3.6.4)
ldn(lday, lnight, hours=(15, 9))		+10 dB night (3.6.5)
composite_rating_level(periods)	iterable of (level_db, hours, adjustment_db); hours positive, finite and summing to 24	General Formulae (5)-(6); adjustments per Table A.1

Combine with laeq() per time period to go from recordings to Lden, and with tone_to_noise_ratio() / prominence_ratio() to justify tonal adjustments.

Octave Spectrogram (levels over time)

Short-time fractional-octave analysis: one level per band per window, time-aligned across bands.

from phonometry import OctaveFilterBank

bank = OctaveFilterBank(fs=48000, fraction=3)
levels, freq, times = bank.spectrogram(signal, window_time=0.125, overlap=0.5)
# levels: (bands, frames) — ready for pcolormesh(times, freq, levels)

A logarithmic sweep plus two tone bursts, resolved in time and in standardized 1/3-octave bands.

• Multichannel input (channels, samples) returns (channels, bands, frames).
• times holds each window’s center in seconds.
• mode='peak' gives per-window peak-holding levels instead of RMS.
• zero_phase=True filters bands forward-backward so per-band group delay does not skew the frames (offline analysis only).

OctaveFilterBank.spectrogram() parameters

Parameter	Type	Units	Range / default	Notes
x	1D or 2D array	digital units	non-empty	2D returns (channels, bands, frames)
window_time	float	s	> 0; default 0.125	Frame length (0.125 s mirrors Fast)
overlap	float	—	0 ≤ overlap < 1; default 0.5	Fraction of window overlap (0 = none)
mode	str	—	'rms' (default) or 'peak'	Per-window detector
zero_phase	bool	—	default False	Forward-backward filtering (offline only)
calibration_factor / dbfs	—	—	constructor-only	Set on OctaveFilterBank(...), not per call

import matplotlib.pyplot as plt

fig, ax = plt.subplots()
mesh = ax.pcolormesh(times, freq, levels, shading="auto")
ax.set_yscale("log")
ax.set_xlabel("Time [s]")
ax.set_ylabel("Frequency [Hz]")
fig.colorbar(mesh, label="Level [dB]")

See Calibration and dBFS to convert digital units to physical SPL, and Time Weighting for the envelope details.