Time Weighting

By José M. Requena Plens · Published Jul 6, 2026, updated Jul 26, 2026

Standards: IEC 61672 ANSI S1.4Key references: Bies et al. 2017

A displayed sound level is always a time-weighted level: before anything reaches the readout, the squared, frequency-weighted pressure passes through an exponential detector whose time constant sets how quickly the level follows the signal. This page is that detector, implemented as a sample-exact recursive filter: the Fast and Slow characteristics of IEC 61672-1:2013 (clause 5.7), the legacy asymmetric Impulse ballistics, streaming state for block processing, and the toneburst verification against the standard’s Table 4.

The names are older than the mathematics. FAST and SLOW were literal descriptions of a needle movement, standardized for analog meters in ANSI S1.4-1983 and IEC 60651 and carried into IEC 61672-1 as the F and S exponential constants; IMPULSE was added in the same era for impact noise and has since been dropped from the requirements, surviving in instruments only for legacy procedures (the choosing guidance below says when each is defensible; Bies, Hansen & Howard 2017, §3.6 covers the instrument practice).

Two sibling pages complete the chain. The detector output is the level track that the percentile levels $L_{N}$ of Levels are defined on, while the integrated metrics ( $L_{e q}$ , SEL) bypass the detector entirely; and the full IEC 61672-1 instrument chain that wraps this detector (weighting, ranges, periodic tests) is the subject of Build a sound level meter.

1. The exponential detector

A sound level meter’s needle cannot follow the pressure waveform: it shows a running mean square with an exponential memory. Formally (IEC 61672-1, 3.8):

τ \frac{d y}{d t} + y = x^{2} (t) ⟺ y (t) = \frac{1}{τ} \int_{- \infty}^{t} x^{2} (ξ) e^{- (t - ξ) / τ} d ξ

a first-order low-pass on the squared signal. The time constant τ sets the trade-off: Fast (125 ms) follows speech-like fluctuations, Slow (1 s) steadies the readout for quasi-stationary noise. After a step onset the envelope reaches 63 % of its final value in one τ and ~99.97 % after 8τ; that is why level analyses discard the first instants of a recording.

The exponential-detector chain: the band pressure is squared, smoothed by a one-pole RC low-pass with time constant tau, and converted to decibels, with the Fast, Slow and Impulse time constants listed

2. The three time weightings

Fast (fast): τ = 125 ms. Standard for noise fluctuations.
Slow (slow): τ = 1000 ms. Standard for steady noise.
Impulse (impulse): Asymmetric ballistics. 35 ms rise time for rapid onset capture, 1500 ms decay for readability.

Download the animation (WebM)

Fast, Slow and Impulse time weighting responses to a noise burst

Show the code for this figure

import numpy as np
import matplotlib.pyplot as plt
from phonometry import metrology

fs = 48000
t = np.arange(int(fs * 4)) / fs
burst = np.zeros_like(t)  # 0.5 s noise burst (Pa) starting at t = 1 s
rng = np.random.default_rng(42)
burst[fs:int(1.5 * fs)] = 0.2 * rng.standard_normal(int(0.5 * fs))

p0 = 2e-5
plt.figure()
for mode in ('fast', 'slow', 'impulse'):
    envelope = metrology.time_weighting(burst, fs, mode=mode)
    plt.plot(t, 10 * np.log10(np.maximum(envelope, 1e-12) / p0**2), label=mode)
plt.xlabel('Time [s]')
plt.ylabel('Level [dB SPL]')
plt.legend()
plt.show()

import numpy as np
from phonometry import metrology

# recording: a calibrated microphone capture (Pa) — recorded through your measurement chain. Synthesized here so the guide runs standalone.
fs = 48000
recording = 0.2 * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)

# Calculate energy envelope (Mean Square)
energy_envelope = metrology.time_weighting(recording, fs, mode='fast')
# dB SPL relative to 20 μPa
spl_t = 10 * np.log10(energy_envelope / (2e-5)**2)

print(f"Steady-state Fast level: {spl_t[-1]:.1f} dB SPL")
# Steady-state Fast level: 77.0 dB SPL

The asymmetric Impulse ballistics use two constants, a fast attack and a slow decay, switching per sample on the sign of the change:

y [n] = y [n - 1] + α (x^{2} [n] - y [n - 1]), α = {1 - e^{- 1/ (f_{s} \cdot 0.035)} 1 - e^{- 1/ (f_{s} \cdot 1.5)} x^{2} [n] > y [n - 1] otherwise

Choosing F, S or I

Fast is the default of nearly every modern method: percentile levels, impulsive-event detection, community noise, the general “level vs time” plot. Its 125 ms constant is of the same order as the ear’s own loudness integration time, so an LAF trace roughly tracks what a listener notices.
Slow suits quasi-stationary sources and any procedure that needs a steady readout: it averages away the flicker of a fluctuating source at the price of missing short events (a 100 ms burst peaks about 7.5 dB lower on S than on F). Some legacy methods prescribe it outright, most famously aircraft-certification levels, which are built from Slow-weighted samples.
Impulse is legacy, and deprecated for rating. It was a 1960s attempt to make a meter needle track the perceived loudness of impacts; it does not (the 35 ms attack still misses very short impulses, and the 1.5 s hold exaggerates duration). It entered the international standards with IEC 60651 and was dropped from the requirements of its successor IEC 61672-1, whose first edition (2002) explains why: I-weighted levels are not suitable for rating impulsive sounds. It survives in meters only for continuity with older national requirements. Modern practice rates impulsiveness with $L_{A e q}$ plus an adjustment (ISO 1996-1 Table A.1, or the onset analysis of Impulsive-sound prominence) and assesses hearing-damage risk with $L_{C p e ak}$ , never with I-weighted levels.

3. `time_weighting()` / `TimeWeighting` parameters

Parameter	Type	Units	Range / default	Notes
`x`	1D or 2D array	pressure (any scale)	non-empty	Squared internally; output is a mean-square envelope
`fs`	int	Hz	> 0
`mode`	str	—	`'fast'` (default), `'slow'`, `'impulse'`	τ = 125 ms / 1 s / 35 ms attack + 1.5 s decay
`TimeWeighting(fs, mode)` (class)	—	—	—	Stateful variant for streaming: `process(x)` carries the integrator state between blocks

The output has the units of $x^{2}$ : take 10*log10(y / p0**2) for SPL or use the level functions, which do it for you.

4. Verified ballistics (IEC 61672-1 Table 4)

The Fast envelope’s response to 4 kHz tonebursts lands exactly on the standard’s reference values: the example below verifies the 200 ms Fast burst row; the CI suite covers the full Table 4, from 1 s down to 1 ms for F and 1 s down to 2 ms for S, at class 1 acceptance limits:

Fast envelope responses to 200, 50 and 10 ms tone bursts peaking exactly at the IEC 61672-1 Table 4 reference values

Show the code for this figure

import numpy as np
import matplotlib.pyplot as plt
from phonometry import metrology

fs = 48000
t = np.arange(int(fs * 2)) / fs
tone = np.sin(2 * np.pi * 4000 * t)

# Steady-state Fast reference of the continuous tone
reference = metrology.time_weighting(tone, fs, mode='fast')[int(1.5 * fs):].mean()

# 200 ms burst of the same tone (IEC 61672-1 Table 4 target: -1.0 dB)
burst = np.zeros_like(t)
burst[int(0.5 * fs):int(0.7 * fs)] = tone[int(0.5 * fs):int(0.7 * fs)]
envelope = metrology.time_weighting(burst, fs, mode='fast')
env_db = 10 * np.log10(np.maximum(envelope / reference, 1e-6))

plt.figure()
plt.plot(t, env_db, label='Fast envelope')
plt.axhline(-1.0, linestyle='--', label='IEC target −1.0 dB')
plt.xlabel('Time [s]')
plt.ylabel('Level re steady state [dB]')
plt.legend()
plt.show()

5. Initial state

By default, the exponential integrator starts from rest (y[-1] = 0). Passing initial_state=None leaves this default unspecified, while initial_state='zero' requests the same zero state explicitly. If the recorded segment begins after a steady signal is already present, you can start from the first sample energy instead:

import numpy as np
from phonometry import metrology

# recording: a calibrated microphone capture (Pa) — recorded through your measurement chain. Synthesized here so the guide runs standalone.
fs = 48000
recording = 0.2 * np.sin(2 * np.pi * 1000 * np.arange(fs) / fs)

energy_envelope = metrology.time_weighting(recording, fs, mode='fast', initial_state='first')

6. Block processing

For block processing, pass the last output value from the previous block as the next block’s initial_state instead of resetting each block:

from phonometry import metrology

state = None

# audio_blocks: consecutive frames of your calibrated recording (Pa),
#   streamed from your sound card or read from a WAV in blocks.
for block in audio_blocks:
    energy_envelope = metrology.time_weighting(block, fs, mode='fast', initial_state=state)
    state = energy_envelope[-1]

For multichannel blocks with time on the last axis, carry one state per channel: use state = energy_envelope[..., -1]. A scalar initial_state is applied to every channel, while an array must match or broadcast to the non-time shape, such as (n_channels,) for input shaped (n_channels, n_samples).

Or let the TimeWeighting class carry the state for you:

from phonometry import metrology

tw = metrology.TimeWeighting(fs, mode='fast')
# audio_blocks: consecutive frames of your calibrated recording (Pa),
#   streamed from your sound card or read from a WAV in blocks.
for block in audio_blocks:
    energy_envelope = tw.process(block)

Concatenated block outputs are exactly equal to a single continuous call (verified for all three modes, mono and multichannel). Call tw.reset() to start from rest again.

7. Performance note

The impulse mode uses an asymmetric kernel that is JIT-compiled when numba is installed (pip install phonometry[perf]). Without numba a pure-Python fallback produces identical results, just slower.

See Integrated & Statistical Levels for Leq/LN metrics built on these envelopes, and Why phonometry for the IEC 61672-1 tone-burst verification.

What this guide covers

Covered. IEC 61672-1:2013’s exponential time-weighting detector (clause 3.8) with the Fast and Slow time constants (clause 5.7), implemented as the recursive filter of time_weighting and the streaming TimeWeighting class, verified in CI against the class 1 tone-burst references of Table 4. The legacy asymmetric Impulse ballistics (35 ms attack, 1.5 s decay), carried into the international standards from the FAST/SLOW/IMPULSE characteristics of ANSI S1.4-1983.

Not covered. IEC 61672-1’s wider instrument chain (frequency weighting, level ranges, periodic tests) is not on this page; it is the subject of Build a sound level meter. ANSI S1.4-1983 is cited only for the historical origin of the F and S constants, not implemented as its own instrument specification. Modern impulsiveness rating ( $L_{A e q}$ plus an adjustment, or onset analysis) is on the Impulsive-sound prominence page, not built on the Impulse weighting here.

References

American National Standards Institute. (1983). Specification for sound level meters (ANSI S1.4-1983). The classic analog meter specification: the FAST/SLOW dynamic characteristics the F and S constants descend from, and the IMPULSE characteristic (35 ms rise, slow decay) that IEC 61672-1 no longer specifies.
Bies, D. A., Hansen, C. H., & Howard, C. Q. (2017). Engineering noise control (5th ed.). CRC Press. https://doi.org/10.1201/9781351228152Sections 3.2 and 3.6 (sound level meters and the measurement of time-varying sound: what the F/S/I readouts mean in instrument practice). ISBN 978-1-4987-2405-0.
International Electrotechnical Commission. (2013). Electroacoustics — Sound level meters — Part 1: Specifications (IEC 61672-1:2013). The exponential time-weighting detector (clause 3.8) with the F and S time constants (clause 5.7), and the 4 kHz toneburst reference responses of Table 4 (class 1 acceptance limits) the ballistics are verified against in CI.

Created and maintained byJosé Manuel Requena Plens· GitHub· PyPI· All projects