This document explains the mathematical formulas and concepts behind common audio metrics used in signal processing and monitoring.
The Peak Amplitude is the absolute maximum instantaneous value reached by the audio waveform within a given time window. It represents the highest excursion of the signal from the zero line.
Given a discrete audio signal represented as a sequence of samples
-
$x[n]$ : The amplitude of the audio sample at time index$n$ . -
$|x[n]$ : The absolute value (magnitude) of the sample. -
$\max_{0 \le n < N}$ : Finds the maximum value within the window of$N$ samples.
Indicates the absolute highest level reached by the signal. It's crucial for identifying potential clipping (distortion caused when the signal exceeds the maximum representable level), but it doesn't give a good sense of the perceived loudness or average power.
RMS measures the effective level or "power" of the audio signal over a time window. It's a type of average that gives more weight to higher amplitude values and is closely related to the perceived loudness of steady sounds.
For the same discrete signal
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$(x[n])^2$ : Square each individual sample's amplitude. This makes all values positive and emphasizes larger amplitudes. -
$\sum_{n=0}^{N-1}$ : Sum the squared values over the window. -
$\frac{1}{N}$ : Calculate the mean (average) of the squared values (this is the Mean Square). -
$\sqrt{\dots}$ : Take the square root of the mean square to return the value to the original units of amplitude (this is the Root Mean Square).
Provides a measure of the signal's average power or effective amplitude. It correlates much better with perceived loudness for continuous sounds than peak amplitude does. It's the foundation for calculating decibel levels.
dBFS measures the amplitude of a digital audio signal relative to the maximum possible level that the digital system can represent (Full Scale). 0 dBFS is the maximum possible level; all other values are negative.
Based on the RMS value calculated above:
However, in digital audio processing where the signal
-
$\text{RMS}$ : The Root Mean Square value of the signal (assuming values are normalized to$[-1, 1]$ ). -
$\log_{10}$ : The base-10 logarithm. -
$20 \times$ : The factor used because decibels for amplitude (like voltage or digital samples) are related to power ($P \propto V^2$ ), and$\log(V^2) = 2 \log(V)$ .
(Note: In practice, a small epsilon is often added inside the logarithm, log10(RMS + epsilon), to prevent log10(0) which is negative infinity during perfect silence.)
To quantify the level of a digital audio signal relative to its maximum possible value, primarily for preventing digital clipping and for mixing/mastering. It does not represent real-world loudness without calibration.
dBSPL measures the actual sound pressure in the real world relative to a standard reference pressure (the threshold of human hearing,
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$\text{dBFS}_{\text{measured}}$ : The dBFS value calculated from the microphone's digital output. -
$\text{Sensitivity}_{\text{dBFS}}$ : The microphone's specified sensitivity (e.g., -18 dBFS for the UMIK-1, -12 dBFS for the UMIK-2). This is the dBFS level the microphone outputs when exposed to the reference sound pressure. -
$\text{Reference}_{\text{dBSPL}}$ : The standard sound pressure level used for calibration (typically 94 dBSPL, corresponding to 1 Pascal).
For accurate dBSPL across the full frequency spectrum, the raw microphone signal is first corrected with a FIR (Finite Impulse Response) filter designed from the manufacturer's calibration file:
- Filter Design:
scipy.signal.firwin2computes coefficients that are the exact inverse of the microphone's frequency response. This is done once at startup; the result is cached as a.npyfile. - Real-Time Filtering: Every audio chunk is convolved with the filter before metrics are computed:
- dBSPL from Calibrated Signal:
To quantify the actual, physical loudness of a sound in the environment. It is the standard metric for noise measurements, acoustics, and hearing safety. Full FIR calibration adds frequency accuracy across the spectrum; gain-only calibration is accurate for broadband levels only.
Spectral Flux measures the rate of change in the frequency spectrum of an audio signal between consecutive short time frames. It quantifies how much the "timbre" or frequency content of the sound is changing.
- Divide the signal into short, overlapping frames (e.g., 20ms).
- For each frame
$t$ , compute its magnitude spectrum$S_t(k)$ , typically using the Short-Time Fourier Transform (STFT).$k$ represents the frequency bin index. - Normalize the spectrum (optional but common, e.g., unit norm). Let the normalized spectrum be
$\hat{S}_t(k)$ . - Calculate the spectral flux
$F_t$ between frame$t$ and the previous frame$t-1$ :
(Variations exist, sometimes using absolute difference or other distance metrics)
-
$\hat{S}_t(k)$ : Normalized spectral magnitude of frequency bin$k$ at frame$t$ . -
$\hat{S}_{t-1}(k)$ : Normalized spectral magnitude of frequency bin$k$ at the previous frame$t-1$ . -
$\sum_{k}$ : Sum the squared differences across all frequency bins$k$ .
Excellent for onset detection (finding the start of new sound events). Steady sounds (like hums or wind) have low spectral flux, while the beginning of a note, a bark, or speech has high spectral flux. It helps distinguish dynamic events from constant background noise.
LUFS (Loudness Units Full Scale) is a standardized measure (ITU-R BS.1770) designed to quantify the perceived loudness of audio, taking human hearing characteristics into account. For accurate results, LUFS should be calculated on the calibrated audio signal.
The full LUFS calculation is complex, involving multiple stages:
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K-Weighting Filter: The calibrated audio signal (
$x_{\text{cal}}[n]$ ) is passed through a specific frequency-weighting filter. This filter has two stages:- A high-shelf filter to boost higher frequencies (mimicking the head's acoustic effects).
- A high-pass filter to roll off very low frequencies (below ~30-40 Hz), as humans are less sensitive to them.
Let the K-weighted signal be
$x_K[n]$ .
-
Mean Square Calculation: Calculate the mean square of the K-weighted signal over specific time windows (e.g., 400ms for Momentary, 3s for Short-Term).
$$Z_i = \frac{1}{T \times \text{SR}} \sum_{n} (x_{K,i}[n])^2$$ -
$Z_i$ : Mean square value for window$i$ . -
$T$ : Window duration in seconds. - SR: Sample Rate.
-
$x_{K,i}[n]$ : Samples of the K-weighted signal within window$i$ .
-
- Channel Summation: For multi-channel audio, the mean square values are weighted and summed across channels (e.g., surrounds contribute less than fronts).
-
Logarithmic Conversion: The final loudness value is calculated logarithmically:
$$\text{Loudness} (\text{LUFS}) = -0.691 + 10 \times \log_{10} (Z_{\text{total}})$$ -
$Z_{\text{total}}$ : The summed mean square value. -
$-0.691$ : A specific offset defined in the standard.
-
To provide a consistent, perceptually relevant measure of loudness, crucial for broadcast audio, streaming services, and accurately quantifying the subjective impact of noise. It correlates much better with human perception than simple dB levels.
dBSPL(A) — also written dB(A) — is the standard metric for environmental noise regulation. It combines two corrections:
- A-weighting filter (IEC 61672): a frequency-shaping filter that attenuates low (< 1 kHz) and very high (> 6 kHz) frequencies to match the sensitivity of the human ear at moderate sound pressure levels.
- Microphone calibration: converts the digital signal level to an absolute acoustic level in Pascals.
where
The A-weighting filter is an 8-pole IIR designed from the IEC 61672 analog prototype (poles at 20.6, 107.7, 737.9, and 12194 Hz), converted to digital via the bilinear transform and normalised to 0 dB at 1 kHz.
| Frequency | A-weighting gain (IEC 61672) |
|---|---|
| 31.5 Hz | −39.4 dB |
| 100 Hz | −19.1 dB |
| 1 000 Hz | 0.0 dB (reference) |
| 4 000 Hz | +1.0 dB |
| 10 000 Hz | −2.5 dB |
The primary metric for noise regulations worldwide: OSHA (USA), WHO Environmental Noise Guidelines, EU Directive 2002/49/EC, and ABNT NBR 10151 (Brazil). When the source says "85 dB(A)" or "L_Aeq < 55 dB", they mean dBSPL(A).
Important: dBSPL(A) requires a calibrated microphone. Without calibration, the A-weighting filter can be applied but the result is relative to digital full scale — not comparable to regulatory thresholds.
L_Aeq,T is the steady-state dBSPL(A) level that would deliver the same total acoustic energy as the actual time-varying signal over a measurement period T. It is the single most important metric in environmental noise law.
where
This is energy averaging, not arithmetic averaging. A single loud event contributes far more to L_Aeq than the same duration of quiet — which is exactly the physical and perceptual reality being captured.
Two measurements: 90 dB(A) and 80 dB(A).
- Arithmetic mean: (90 + 80) / 2 = 85.0 dB(A) — incorrect.
- Energy average:
$10 \times \log_{10}\left(\frac{10^9 + 10^8}{2}\right)$ = 87.4 dB(A) — correct.
The 90 dB(A) event carries 10× the acoustic power of the 80 dB(A) event, so it dominates the average.
| Standard | Period T | Context |
|---|---|---|
| ABNT NBR 10151 | ≥ 10 minutes | Environmental noise disputes |
| ISO 1996 | Site-specific | Acoustic impact assessments |
| OSHA | 8 hours | Occupational dose (L_Aeq,8h) |
| EU Directive 2002/49/EC | Day / Evening / Night | L_den (weighted 24 h average) |
The legally required metric for noise impact assessments, planning permits, occupational health compliance, and environmental monitoring. Collect dBSPL_A() samples over T, then call AudioMetrics.L_Aeq() to obtain L_Aeq,T.
L_A90 is the A-weighted sound pressure level exceeded 90 % of the time during the measurement period T. Statistically, it is the 10th percentile of the dBSPL(A) sample distribution — the level that only the quietest 10 % of the signal falls below. It characterises the residual ambient noise in the absence of the disturbing source.
where
L_A90 and L_Aeq,T are always reported together in environmental assessments:
- L_Aeq,T — the total acoustic energy dose (dominated by loud events).
- L_A90 — the pre-existing background floor (what the environment sounds like without the source under evaluation).
The difference L_Aeq,T − L_A90 is a key indicator of how much the disturbing source stands out above the ambient noise level.
| Standard | Role of L_A90 |
|---|---|
| ISO 1996 | Characterises the residual noise in impact assessments |
| ABNT NBR 10151 | Background reference for environmental noise disputes |
| BS 4142 (UK) | Determines significance of industrial / commercial noise complaints |
| Court proceedings | Establishes the pre-existing ambient level before the disputed source |
To establish what the acoustic environment sounds like without the disturbing source — the baseline against which the source's impact is measured. Required alongside L_Aeq,T for any formal environmental noise assessment or legal dispute. Collect dBSPL_A() samples over T, then call AudioMetrics.L_A90() to obtain L_A90.