Source: https://jmrplens.github.io/phonometry/signals/levels/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 5.8), 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/signals/levels/levels/) are defined on, while the integrated metrics ($L_\mathrm{eq}$, 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/signals/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 $\tau$ 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 $\tau$ and ~99.97 % after $8\tau$; that is why level analyses discard the first instants of a recording. ## 2. The three time weightings * **Fast (`fast`):** $\tau = 125\ \text{ms}$. Standard for noise fluctuations. * **Slow (`slow`):** $\tau = 1000\ \text{ms}$. 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 filters 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 = filters.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 filters # 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 = filters.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: $$ y[n] = y[n-1] + \alpha \ (x^2[n] - y[n-1]), \qquad \alpha = \begin{cases}1 - e^{-1/(f_\mathrm{s} \cdot 0.035)} & x^2[n] > y[n-1]\\[2pt] 1 - e^{-1/(f_\mathrm{s} \cdot 1.5)} & \text{otherwise}\end{cases} $$ The asymmetry has a consequence worth stating: unlike F and S, Impulse is not a filter. Its coefficient depends on the signal, so an I-weighted level is not additive — the I-weighted level of two sources running together cannot be derived from the two measured separately, while their $L_\mathrm{eq}$ values can. ### 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 $L_\mathrm{AF}$ 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 7.6 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 $L_\mathrm{Aeq}$ plus an adjustment (ISO 1996-1 Table A.1, or the onset analysis of [Impulsive-sound prominence](https://jmrplens.github.io/phonometry/environment/assessment/impulsive-sound/)) and assesses hearing-damage risk with $L_\mathrm{Cpeak}$, 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'` | $\tau$ = 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 filters 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 = filters.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 = filters.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`); `initial_state=None` and `initial_state='zero'` are the same thing. Passing `'first'` seeds the integrator with the square of the very first sample, $x[0]^2$ — an unbiased but extremely noisy estimate of the mean square (one degree of freedom), and for a tone it depends entirely on the phase at the cut: a sine starting at a zero crossing makes `'first'` bit-identical to `'zero'`. A float resumes a state you saved yourself (which is what block processing does in section 6), and an array broadcastable to the input shape without the time axis does the same per channel. If the recorded segment is a continuation of a signal already running, seed the integrator with something robust — the mean square of the first half second is a far better estimate than a single sample. If the record is the *start* of the event, do not seed at all: let it start from rest and discard the first $5\tau$ (0.6 s Fast, 5 s Slow), which is exactly what `ln_levels` does internally before computing percentiles. ```python import numpy as np from phonometry import filters # 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) # Robust seed: the mean square of the first half second, not one sample. energy_envelope = filters.time_weighting( recording, fs, mode='fast', initial_state=float(np.mean(recording[:fs // 2] ** 2))) ``` ## 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 filters 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 = filters.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 filters tw = filters.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/signals/levels/levels/) for $L_\mathrm{eq}$/$L_N$ metrics built on these envelopes, and [Why phonometry](https://jmrplens.github.io/phonometry/start/why-phonometry/) for the IEC 61672-1 tone-burst verification. ## See also - [Levels](https://jmrplens.github.io/phonometry/signals/levels/levels/): the percentile levels defined on the detector's output, and the integrated metrics that bypass it. - [Build a sound level meter](https://jmrplens.github.io/phonometry/signals/sound-level-meter/): the complete IEC 61672-1 instrument chain around this detector. - [Frequency Weighting](https://jmrplens.github.io/phonometry/signals/levels/weighting/): the A/C/Z filters applied before the detector. - [Block Processing](https://jmrplens.github.io/phonometry/signals/filters/block-processing/): streaming the detector over frames without state discontinuities. - [Impulsive-sound prominence](https://jmrplens.github.io/phonometry/environment/assessment/impulsive-sound/): the onset-based rating of impulses that the international standards moved to. - [Spanish Noise Regulation](https://jmrplens.github.io/phonometry/environment/assessment/spanish-noise-regulation/): a rating in force that still requires the I weighting, through the $L_\mathrm{AIeq} - L_\mathrm{Aeq}$ impulsive correction $K_\mathrm{i}$. - [Calibration and dBFS](https://jmrplens.github.io/phonometry/signals/metrology/calibration/): the IEC 61672-3 periodic tests, in which these ballistics are spot-checked against the class limits on a real instrument. - API reference: [`filters.weighting`](https://jmrplens.github.io/phonometry/reference/api/filters/weighting/). - Theory: [Time Integration](https://jmrplens.github.io/phonometry/reference/theory/signal-analysis/#time-integration): the first-order equation the exponential detectors solve, and what integrating it over a fixed block instead would change. ## 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 design-goal time constants (clause 5.8.1), and the 4 kHz toneburst reference responses of Table 4 (class 1 acceptance limits) used to verify the ballistics in CI.