Source: https://jmrplens.github.io/phonometry/guides/time-weighting/ # Time Weighting and Integration 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 F, S or I" 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](https://jmrplens.github.io/phonometry/reference/api/levels/levels/) are defined on, while the integrated metrics ($L_{eq}$, 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](https://jmrplens.github.io/phonometry/guides/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): $$ \tau\ \frac{dy}{dt} + y = x^2(t) \quad\Longleftrightarrow\quad y(t) = \frac{1}{\tau} \int_{-\infty}^{t} x^2(\xi)\ e^{-(t-\xi)/\tau}\ d\xi $$ 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. ## 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. [Watch the high-resolution video (WebM)](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/images/anim_time_weighting.webm)

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```python 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() ```

```python 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: $$ y[n] = y[n-1] + \alpha \ (x^2[n] - y[n-1]), \qquad \alpha = \begin{cases}1 - e^{-1/(f_s \cdot 0.035)} & x^2[n] > y[n-1]\\[2pt] 1 - e^{-1/(f_s \cdot 1.5)} & \text{otherwise}\end{cases} $$ ### 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 $L_{Aeq}$ plus an adjustment (ISO 1996-1 Table A.1, or the onset analysis of [Impulsive-sound prominence](https://jmrplens.github.io/phonometry/reference/api/environment/impulse-prominence/)) and assesses hearing-damage risk with $L_{Cpeak}$, 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:

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```python 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: ```python 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: ```python 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: ```python 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](https://numba.pydata.org/) is installed (`pip install phonometry[perf]`). Without numba a pure-Python fallback produces identical results, just slower. See [Integrated & Statistical Levels](https://jmrplens.github.io/phonometry/reference/api/levels/levels/) for Leq/LN metrics built on these envelopes, and [Why phonometry](https://jmrplens.github.io/phonometry/reference/why-phonometry/) for the IEC 61672-1 tone-burst verification. ## References - International Electrotechnical Commission. (2013). *Electroacoustics — Sound level meters — Part 1: Specifications* (IEC 61672-1:2013). [IEC webstore](https://webstore.iec.ch/en/publication/5708). The exponential-detector definition, the F and S time constants and the Table 4 toneburst responses the ballistics are verified against in CI. - American National Standards Institute. (1983). *Specification for sound level meters* (ANSI S1.4-1983). [ANSI webstore](https://webstore.ansi.org/standards/asa/ansiasas11983). 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. [doi:10.1201/9781351228152](https://doi.org/10.1201/9781351228152). Sections 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. ## Standards IEC 61672-1:2013, *Electroacoustics — Sound level meters — Part 1: Specifications*: 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) used to verify the ballistics in CI.