Source: https://jmrplens.github.io/phonometry/guides/electroacoustics/ # Electroacoustics: distortion and frequency response (IEC 60268-3 / Bendat & Piersol) Two staples of audio-equipment characterisation, from a captured signal: how much an amplifier or transducer **distorts** a test tone, and what **frequency response** an input/output measurement reveals. This page covers the IEC 60268-3 distortion set: total and nth-order harmonic distortion, THD+N and SINAD through the AES17 measurement bandwidth, the per-order modulation and difference-frequency intermodulation, dynamic intermodulation (DIM) and the ITU-R 468 weighted THD, and the Bendat & Piersol frequency-response estimators `H1`/`H2` with the ordinary coherence `γ²`. Every quantity has an exact analytic oracle, so the numbers are verifiable rather than tuned. The closing sections cover the sensitivity conventions of IEC 60268-4 (microphones) and IEC 60268-5 (loudspeakers), where most cross-datasheet comparisons go wrong, the baffled-piston radiation model behind loudspeaker directivity, and the IEC 60268-5 loudspeaker and IEC 60268-4 microphone rated-characteristics reports that render the standards' data sheets from a measured response. ## 1. Harmonic distortion (IEC 60268-3 14.12.2–5) A non-linear device fed a pure sine at `f₁` returns the fundamental plus harmonics at `2f₁, 3f₁, …`. The **total harmonic distortion** combines the harmonic amplitudes `aₙ`, either relative to the fundamental (`kind='F'`) or to the total RMS (`kind='R'`): $$ \mathrm{THD}_F = \frac{\sqrt{\sum_{n\ge 2} a_n^2}}{a_1}, \qquad \mathrm{THD}_R = \frac{\sqrt{\sum_{n\ge 2} a_n^2}}{\sqrt{\sum_{n\ge 1} a_n^2}}, \qquad d_n = \frac{a_n}{\sqrt{\sum_{k\ge 1} a_k^2}}. $$ `dₙ` is the nth-order harmonic distortion (the nth harmonic relative to the total). The tones should fall on FFT bins: use coherent sampling (an integer number of periods) or a low-leakage window so the amplitudes are read without spectral leakage. ```python from phonometry import electroacoustics thd_f = electroacoustics.thd(signal, fs, 1000.0, kind="F") # relative to fundamental thd_r = electroacoustics.thd(signal, fs, 1000.0, kind="R") # relative to total RMS d2 = electroacoustics.harmonic_distortion(signal, fs, 1000.0, 2) # 2nd-order harmonic ``` `harmonic_analysis` bundles the fundamental, the harmonic amplitudes and the THD (both conventions), THD+N and SINAD into one plottable result: ```python from phonometry import electroacoustics res = electroacoustics.harmonic_analysis(signal, fs, 1000.0) print(res.thd_f, res.thd_r, res.thd_plus_noise, res.sinad_db) res.plot() # annotated harmonic spectrum (needs matplotlib) ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import electroacoustics fs = 48000 n = fs # 1 s -> 1 Hz bins; harmonics land on bins t = np.arange(n) / fs f0 = 1000.0 amps = {1: 1.0, 2: 0.02, 3: 0.012, 4: 0.006, 5: 0.003} sig = sum(a * np.sin(2 * np.pi * k * f0 * t) for k, a in amps.items()) sig = sig + np.random.default_rng(2026).standard_normal(n) * 1.2e-2 res = electroacoustics.harmonic_analysis(sig, fs, f0, n_harmonics=len(amps)) res.plot() plt.show() ```

## 2. THD+N and SINAD (AES17-2015 6.3) Where THD counts only the harmonics, **THD+N** compares *everything but the fundamental* (harmonics **and** noise) with the total signal. AES17 removes the fundamental with a standard notch filter (`1.2 ≤ Q ≤ 3`, validated on the *applied* zero-phase response per clause 5.2.8) and takes the ratio of the residual to the total RMS; **SINAD** is its reciprocal in dB: $$ \mathrm{THD{+}N} = \frac{V_\text{residual}}{V_\text{total}}, \qquad \mathrm{SINAD} = -20\lg(\mathrm{THD{+}N})\ \mathrm{dB}. $$ Both voltages are measured through the AES17 measurement bandwidth (a 20 Hz high-pass plus the standard low-pass at 20 kHz, clauses 5.2.5 and 6.3.1) so a DC offset or ultrasonic noise at a high sample rate does not count as "noise". The band edge is configurable, and `bandwidth=None` disables the chain and measures the full Nyquist band. ```python from phonometry import electroacoustics ratio = electroacoustics.thd_plus_noise(signal, fs, 1000.0) # ratio (0..) db = electroacoustics.thd_plus_noise(signal, fs, 1000.0, as_db=True) # 20·lg(ratio) dB sinad_db = electroacoustics.sinad(signal, fs, 1000.0) # = -db wide = electroacoustics.thd_plus_noise(signal, fs, 1000.0, bandwidth=None) # full Nyquist ``` Because THD+N counts the in-band noise floor, it is at or above the harmonic-only THD; `SINAD` is the corresponding signal-to-noise-and-distortion headroom in dB (a quantity derived from the AES17 THD+N; AES17 itself does not define SINAD). The notch discards a start/stop transient internally, so the measurement wants a steady, sufficiently long capture. ### 2.1 Weighted THD (IEC 60268-3 14.12.11) `weighted_thd` frequency-weights the notched residual before taking the ratio, so the perceptual emphasis of the distortion products is accounted for. The default weighting is the network the clause requires: the IEC 60268-1 Appendix A curve, i.e. ITU-R BS.468-4, which peaks at +12.2 dB near 6.3 kHz (exposed as `itu_r_468_weighting`); `'A'` and `'C'` remain as labelled options. Per the clause, the weighted measurement is valid for fundamental frequencies between 31.5 Hz and 400 Hz: ```python from phonometry import electroacoustics print(electroacoustics.weighted_thd(signal, fs, 100.0)) # ITU-R 468 network print(electroacoustics.weighted_thd(signal, fs, 100.0, weighting="A")) # A-weighted variant print(electroacoustics.itu_r_468_weighting([6300.0])) # [+12.2] dB ``` ### 2.2 Dynamic range and idle channel noise (AES17-2015 6.4) The two catalogue figures of any converter share the AES17 **CCIR-RMS** weighting (5.2.7): the ITU-R BS.468-4 curve shifted by a flat `-5.63` dB so it is unity at 2 kHz. **Dynamic range** (6.4.1) drives the device with a 997 Hz sine 60 dB below full scale, removes the fundamental with the standard notch (5.2.8) and weights the residual noise-plus-distortion, then reports the ratio of the full-scale sine to that weighted residual (also known as the signal-to-noise ratio). **Idle channel noise** (6.4.2) is the same weighted level measured with the device driven by digital zero, reported relative to full scale: ```python from phonometry import electroacoustics # Dynamic range: capture the output of the -60 dBFS 997 Hz test, scaled so 1.0 # is digital full scale. dr = electroacoustics.dynamic_range(output, fs, 997.0) # dB CCIR-RMS # Idle channel noise: capture the output with a digital-zero input. idle = electroacoustics.idle_channel_noise(idle_output, fs) # dBFS CCIR-RMS ``` Both reuse the notch and the ITU-R 468 curve of the THD+N chain, and both are measured through the AES17 band (a 20 Hz high-pass plus the standard low-pass at `bandwidth`). Because the CCIR-RMS filter reads `-5.63` dB at 1 kHz, a 1 kHz tone measures its own dBFS minus 5.63 dB, which pins the weighting exactly. ## 3. Intermodulation distortion (IEC 60268-3 14.12.7–10) When two tones pass through a non-linearity they beat against each other, producing sum and difference products. IEC 60268-3 standardises three tests, each with its own per-order definition: - **Modulation distortion** (14.12.7): a large low tone `f_low` and a small high tone `f_high` (preferably 4:1). The per-order values are *arithmetic* sums of the sideband amplitudes relative to the `f_high` output: `d_m,2 = (a_{f₂+f₁} + a_{f₂−f₁})/a_{f₂}` and `d_m,3 = (a_{f₂+2f₁} + a_{f₂−2f₁})/a_{f₂}`. The result also carries the combined-RMS `smpte` value that SMPTE-type analyzers report (not an IEC quantity). - **Difference-frequency distortion** (14.12.8): two equal high tones `f₁ < f₂`, referenced to `U_{2,ref} = 2·U_{2,f₂}` (the sum of both tone amplitudes): `d_d,2 = a_{f₂−f₁}/(a_{f₁}+a_{f₂})` and the arithmetic `d_d,3 = (a_{2f₂−f₁} + a_{2f₁−f₂})/(a_{f₁}+a_{f₂})`. - **Total difference-frequency distortion** (14.12.10): a specific two-tone test (`f₁ = 2f₀`, `f₂ = 3f₀ − δ`; the standard tones 8 kHz and 11.95 kHz are the defaults) where only the two in-band products at `f₀ ∓ δ` count: `d_TDFD = √(a²_{f₂−f₁} + a²_{2f₁−f₂}) / (a_{f₁} + a_{f₂})`. - **Dynamic intermodulation** (DIM, 14.12.9): a 15 kHz sine plus a low-pass-filtered 3.15 kHz square wave (1:4 peak-to-peak). The DIM is the RMS of the intermodulation products `|k·f_square ± f_sine|` that fall below `f_sine` (IEC 60268-3 Table 2), relative to the 15 kHz sine amplitude (the 14.12.9.1 definition; the 14.12.9.2 f) print of the denominator is an editorial defect, see [ERRATA](https://jmrplens.github.io/phonometry/reference/errata/)). ```python from phonometry import electroacoustics md = electroacoustics.modulation_distortion(signal, fs, 60.0, 7000.0) print(md.d2, md.d3, md.smpte) # 14.12.7 md.plot() # carrier and modulation sidebands, d2/d3 annotated (needs matplotlib) dfd2 = electroacoustics.difference_frequency_distortion(signal, fs, 13e3, 14e3, order=2) tdfd = electroacoustics.total_difference_frequency_distortion(signal, fs) # 8/11.95 kHz dim = electroacoustics.dynamic_intermodulation_distortion(signal, fs) # DIM (15k/3.15k) ``` The `ModulationDistortionResult` is plottable: `.plot()` draws the output amplitude at the carrier `f₂` (the 0 dB reference) and the four modulation sidebands at `f₂ ± f₁` and `f₂ ± 2f₁`, the modulation counterpart of the harmonic spectrum of section 1. The asymmetric ratio between the `n = 2` and `n = 3` pairs reads the balance between the quadratic and cubic terms of the non-linearity at a glance:

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import electroacoustics fs = 48000 t = np.arange(fs) / fs # 1 s -> tones land on FFT bins x = np.sin(2 * np.pi * 60.0 * t) + 0.25 * np.sin(2 * np.pi * 7000.0 * t) signal = x + 0.04 * x**2 + 0.012 * x**3 # weakly non-linear device output md = electroacoustics.modulation_distortion(signal, fs, 60.0, 7000.0) md.plot() plt.show() ```

## 4. Frequency response and coherence (Bendat & Piersol) Given an input `x` and the output `y` of a device, the **frequency response** `H(f)` is estimated from the Welch-averaged cross- and auto-spectra. Bendat & Piersol (*Random Data*, 4th ed.) give two estimators, differing in which channel carries the noise, plus the **ordinary coherence** `γ²`, the fraction of the output power linearly explained by the input: $$ H_1 = \frac{G_{xy}}{G_{xx}}, \qquad H_2 = \frac{G_{yy}}{G_{yx}}, \qquad \gamma^2 = \frac{|G_{xy}|^2}{G_{xx}\,G_{yy}} \in [0, 1]. $$ `H1` is unbiased when the noise is on the output, `H2` when it is on the input; for a noiseless linear path both recover the true response and `γ² = 1`. Additive output noise biases `H2` upward and pulls the coherence down to `SNR/(1+SNR)`. ```python from phonometry import electroacoustics res = electroacoustics.transfer_function(x, y, fs, estimator="H1") print(res.magnitude_db, res.phase, res.coherence) res.plot() # Bode magnitude/phase + coherence (needs matplotlib) freqs, gamma2 = electroacoustics.coherence(x, y, fs) ``` Coherence needs averaging over several Welch segments to be meaningful; a single segment gives `γ² ≡ 1` by construction.

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```python import matplotlib.pyplot as plt import numpy as np from scipy import signal as sp from phonometry import electroacoustics fs = 48000 n = 400000 rng = np.random.default_rng(7) x = rng.standard_normal(n) b, a = sp.butter(2, [400.0, 4000.0], btype="band", fs=fs) # device under test y = sp.lfilter(b, a, x) y = y + rng.standard_normal(n) * np.sqrt(np.mean(y ** 2)) * 0.05 # output noise res = electroacoustics.transfer_function(x, y, fs, estimator="H1") res.plot() plt.show() ```

## 5. Sensitivity conventions (IEC 60268-4 / IEC 60268-5) Distortion and response figures only compare across devices when the *sensitivity* conventions behind them match, and two conventions trip people constantly: the microphone reference level and the loudspeaker power-and-distance normalization. **Microphones (IEC 60268-4 clause 11).** The sensitivity `M` is the output voltage per unit sound pressure, quoted in mV/Pa; the sensitivity *level* is `LM = 20 lg(M / 1 V/Pa)` dB, which is negative for every real microphone (a 50 mV/Pa studio condenser sits at −26 dB re 1 V/Pa). The classic pitfall is the reference: −26 dB re 1 V/Pa and +34 dB re 1 mV/Pa are the *same* microphone, 60 dB apart on paper, so a sensitivity level without its reference is meaningless. The standard also distinguishes free-field, diffuse-field and pressure sensitivity; for the same capsule they diverge at high frequency, so the stated type matters as much as the number. **Loudspeakers (IEC 60268-5 clause 20.3).** The characteristic sensitivity is the sound pressure produced at 1 m on the reference axis, in the free field, referred to an input of 1 W into the rated impedance; expressed as a level re 20 µPa (clause 20.4) it is the familiar "dB @ 1 W/1 m". Two normalizations hide in that sentence: - *The electrical one.* The test voltage is `Up = √(R · 1 W)`, numerically `√R`: 2.83 V into a rated 8 Ω. A datasheet quoting "dB @ 2.83 V/1 m" for a 4 Ω loudspeaker is feeding it 2 W, which flatters the figure by 3 dB against a true 1 W/1 m rating; check the rated impedance before comparing. - *The geometric one.* 1 m is a **reference** distance, not necessarily the measurement distance. The standard has you measure in the far field (at 0.5 m or an integer number of metres, clause 7.1) and refer the result back with the inverse-distance law, `Lp(1 m) = Lp(r) + 20 lg(r / 1 m)`. That scaling only holds where the level actually falls 6 dB per doubling of distance, hence the free field of the diagram; at 1 m from a large multi-way cabinet the near field may not have ended yet, and the quoted "1 m" figure is then a referred quantity, not what a microphone placed at 1 m would read. ## 6. Radiating piston: radiation impedance and directivity The rigid circular **piston in an infinite baffle** is the canonical radiator behind a loudspeaker cone, the open end of a duct and the radiation efficiency of any finite vibrating surface (Beranek & Mellow §4.19, §13.7). Its mechanical radiation impedance is `Z_r = rho c S (R1 + j X1)` with `S = pi a^2` and the dimensionless resistance and reactance functions (Eqs. (13.117), (13.118)) ``` R1(x) = 1 - 2 J1(x) / x , X1(x) = 2 H1(x) / x , x = 2ka, ``` with `J1` the Bessel function and `H1` the Struve function of order one. At low frequency `R1 -> (ka)^2 / 2` and the reactance is mass-like with the radiation mass `M_r = 8 rho a^3 / 3` (Eq. (4.151)); at high frequency `R1 -> 1`, `X1 -> 0` and the piston radiates as into an infinite tube. The far field follows the directivity `D(theta) = 2 J1(ka sin theta) / (ka sin theta)`, whose first null is at `ka sin theta = 3.8317`. ```python import numpy as np from phonometry import radiating_piston res = radiating_piston(radius=0.1, frequencies=np.geomspace(20, 20000, 200), angles=np.linspace(0.0, np.pi / 2, 91)) print(round(res.radiation_mass, 4)) # 8 rho a^3 / 3, kg print(round(float(res.directivity_index[0]), 2)) # 3.01 dB half-space limit res.plot() # R1 and X1 vs ka ``` *The `.plot()` of the result is the classic Beranek & Mellow impedance figure computed for this piston: below `ka ≈ 1` the load is almost purely mass-like (`X1` dominates and `R1 ∝ (ka)²`, the regime where a loudspeaker's output is stiffness- and mass-limited), and above it the resistance settles on `ρcS` with interference ripples while the reactance dies away.*

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import radiating_piston res = radiating_piston(radius=0.075, frequencies=np.geomspace(20, 20000, 400)) res.plot() # normalized R1 and X1 against ka plt.show() ```

`radiating_piston` returns a `RadiatingPistonResult` with the normalized `resistance`/`reactance`, the mechanical `radiation_resistance`/`radiation_reactance`, the `radiation_mass`, the `directivity_index`, the far-field `directivity` pattern (when `angles` are given) and `.plot()`. The building blocks `piston_resistance`, `piston_reactance` and `piston_directivity` are also callable directly. The piston is the companion radiator of the [industrial noise-control silencers](https://jmrplens.github.io/phonometry/guides/noise-control/). The far-field **directivity pattern** is a plottable result in its own right: `piston_directivity_pattern(ka)` samples `D(theta)` at one or more `ka` values and returns a `PistonDirectivity` (the polar angle grid, the linear `directivity` and its dB form `directivity_db`, and the `ka` values) whose `.plot()` draws the classic polar beam pattern. Several `ka` are shown as one family, so the main lobe narrowing and the side lobes appearing read at a glance: ```python from phonometry import piston_directivity_pattern pattern = piston_directivity_pattern([3.0, 8.0, 16.0]) print(pattern.directivity_db.shape) # (3, 361): one row per ka pattern.plot() # polar beam pattern in dB (needs matplotlib) ```

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```python import matplotlib.pyplot as plt from phonometry import piston_directivity_pattern piston_directivity_pattern([3.0, 8.0, 16.0]).plot() plt.show() ```

## 7. Loudspeaker characteristics report (IEC 60268-5) The rated characteristics IEC 60268-5 defines around a measured on-axis response gather into a single **loudspeaker characteristics** result that renders the standard's rated-characteristics data sheet. Two of the numbers are computed from the response rather than merely repeated: - **Characteristic sensitivity level** (20.3/20.4). The on-axis level averaged over a stated band, referred to 1 W into the rated impedance `R` at 1 m: `LM = L_mean + 20 lg(d/d0) + 20 lg(Up/U)` with `Up = sqrt(R · 1 W) = sqrt(R)`. The default drive is that `sqrt(R)` (2.83 V into 8 Ω), so with a 1 m response the sensitivity level is the band mean. - **Effective frequency range** (21.2). The band over which the response stays within 10 dB of the level averaged over the one-octave band in the region of maximum sensitivity; troughs narrower than 1/9 octave are neglected. ```python import numpy as np from phonometry import loudspeaker_characteristics, radiating_piston, ReportMetadata freqs = np.geomspace(30, 24000, 320) spl = 87.0 + 1.2 * np.sin(2 * np.log2(freqs / 900.0)) spl -= 10 * np.log10(1 + (50.0 / freqs) ** 6) # low-frequency roll-off spl -= 10 * np.log10(1 + (freqs / 16000.0) ** 7) # high-frequency roll-off result = loudspeaker_characteristics( freqs, spl, rated_impedance=8.0, sensitivity_band=(200.0, 4000.0), impedance=(np.geomspace(20, 20000, 260), 6.6 + 24 * np.exp(-(np.log2(np.geomspace(20, 20000, 260) / 52.0) ** 2) / 0.12)), distortion=(np.geomspace(50, 5000, 140), 0.3 + 2.6 * np.exp(-(np.log2(np.geomspace(50, 5000, 140) / 70.0) ** 2) / 0.45)), directivity=radiating_piston(0.075, np.array([1000.0, 2000.0, 4000.0]), angles=np.radians(np.linspace(0, 90, 46))), polar_frequency=2000.0, ) print(round(result.sensitivity_level_db, 1)) # dB, 1 W / 1 m print(tuple(round(x) for x in result.effective_range)) # Hz result.report("loudspeaker.pdf", metadata=ReportMetadata(measurement_standard="IEC 60268-5")) ``` `loudspeaker_characteristics` returns a `LoudspeakerCharacteristics` with the computed `sensitivity_level_db`, `effective_range`, `reference_level_db`, `characteristic_sensitivity_pa` and `minimum_impedance`, and a `.report()` that writes the fiche. The on-axis response is drawn with its tolerance band and the effective-range markers to the IEC 60263 proportion (one frequency decade equal to 25 dB), and the polar directivity on the IEC 60263 25 dB reference circle. The impedance modulus (with the 80 %-of-rated line), the total-harmonic-distortion curve and the directivity feed the secondary panels, reusing the radiating piston directivity of section 6 and a [swept-sine THD](https://jmrplens.github.io/phonometry/guides/swept-sine-distortion/) result for the distortion curve. The rated characteristics are also available interactively through `.plot()`, which draws **one concept per figure** with the same panel code the report composes, selected by `quantity`. Passing an axes draws on it: ```python result.plot() # on-axis response (default) result.plot(quantity="impedance") # |Z| modulus with the rated / 80 % lines result.plot(quantity="thd") # total harmonic distortion vs frequency result.plot(quantity="directivity") # polar response on the 25 dB circle ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import loudspeaker_characteristics freqs = np.geomspace(30, 24000, 320) spl = 87.0 + 1.2 * np.sin(2 * np.log2(freqs / 900.0)) spl -= 10 * np.log10(1 + (50.0 / freqs) ** 6) # low-frequency roll-off spl -= 10 * np.log10(1 + (freqs / 16000.0) ** 7) # high-frequency roll-off result = loudspeaker_characteristics(freqs, spl, rated_impedance=8.0, sensitivity_band=(200.0, 4000.0)) result.plot() # quantity="response" (the default) plt.show() ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import loudspeaker_characteristics freqs = np.geomspace(30, 24000, 320) spl = 87.0 - 10 * np.log10(1 + (50.0 / freqs) ** 6) fz = np.geomspace(20, 20000, 260) result = loudspeaker_characteristics( freqs, spl, rated_impedance=8.0, sensitivity_band=(200.0, 4000.0), impedance=(fz, 6.6 + 24 * np.exp(-(np.log2(fz / 52.0) ** 2) / 0.12)), ) result.plot(quantity="impedance") plt.show() ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import loudspeaker_characteristics freqs = np.geomspace(30, 24000, 320) spl = 87.0 - 10 * np.log10(1 + (50.0 / freqs) ** 6) thd_f = np.geomspace(50, 5000, 140) result = loudspeaker_characteristics( freqs, spl, rated_impedance=8.0, sensitivity_band=(200.0, 4000.0), distortion=(thd_f, 0.3 + 2.6 * np.exp(-(np.log2(thd_f / 70.0) ** 2) / 0.45)), ) result.plot(quantity="thd") plt.show() ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import loudspeaker_characteristics, radiating_piston freqs = np.geomspace(30, 24000, 320) spl = 87.0 - 10 * np.log10(1 + (50.0 / freqs) ** 6) result = loudspeaker_characteristics( freqs, spl, rated_impedance=8.0, sensitivity_band=(200.0, 4000.0), directivity=radiating_piston(0.075, np.array([1000.0, 2000.0, 4000.0]), angles=np.radians(np.linspace(0, 90, 46))), polar_frequency=2000.0, ) result.plot(quantity="directivity") plt.show() ```

The example fiche, regenerated with `make reports`, is kept rendered in the repository. Click the preview to open the PDF: [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iec60268_5_loudspeaker_example.pdf) *Loudspeaker characteristics fiche (`LoudspeakerCharacteristics.report`), the IEC 60268-5 rated-characteristics table beside the on-axis response, impedance, THD and polar panels drawn to the IEC 60263 25 dB-per-decade and 25 dB reference-circle conventions.* ## 8. Microphone characteristics report (IEC 60268-4) The microphone companion of section 7: the rated characteristics IEC 60268-4 defines around a measured free-field frequency response gather into a single **microphone characteristics** result that renders the standard's rated-characteristics data sheet. Four of the numbers are computed from the standard's own definitions rather than merely repeated: - **Sensitivity level** (11.1). The rated free-field sensitivity `M` (mV/Pa, at the 1 kHz reference frequency of 11.3) as a level, `LM = 20 lg(M / 1 V/Pa)`: 12.5 mV/Pa is −38.1 dB re 1 V/Pa. - **Effective frequency range** (12.2). The band over which the response, normalized to 0 dB at the reference frequency, stays within the stated ± tolerance; the edges are the interpolated tolerance crossings. - **Directivity index** (13.2.2). `D = 20 lg(M0/M_diff)` with the diffuse-field sensitivity from the 11.2.2 a) integral over a rotationally symmetric pattern; the ideal cardioid returns `10 lg 3 = 4.8` dB. - **Equivalent noise level** (17.2). The weighted inherent-noise voltage over the rated sensitivity as a sound pressure level, `LN = 20 lg((UN/M)/20 µPa)`, with the signal-to-noise ratio re 1 Pa (94 dB SPL) derived from it. The overload sound pressure level (15.2) is read from a distortion-against-level curve at the stated THD limit. ```python import numpy as np from phonometry import microphone_characteristics, ReportMetadata freqs = np.geomspace(20, 20000, 400) response = -10 * np.log10(1 + (30.0 / freqs) ** 4) # low-frequency roll-off response -= 10 * np.log10(1 + (freqs / 19000.0) ** 8) # high-frequency roll-off response += 2.0 * np.exp(-(np.log2(freqs / 9000.0) ** 2) / 0.3) # presence region angles = np.linspace(0, 179, 359) cardioid = 20 * np.log10((1 + np.cos(np.radians(angles))) / 2) result = microphone_characteristics( freqs, response, 12.5, tolerance_db=3.0, # 12.5 mV/Pa at 1 kHz rated_impedance=150.0, minimum_load_impedance=1000.0, noise_voltage=1.25e-6, # A-weighted, V max_spl_thd_percent=0.5, distortion=(np.linspace(100, 140, 81), 0.5 * 10 ** ((np.linspace(100, 140, 81) - 130.0) * 0.08)), polar=(angles, cardioid), polar_frequency=1000.0, powering="Phantom P48 (IEC 61938)", supply_current_ma=3.1, ) print(round(result.sensitivity_level_db, 1)) # -38.1 dB re 1 V/Pa print(tuple(round(x) for x in result.effective_range)) # Hz print(round(result.directivity_index_db, 1)) # 4.8 dB (cardioid) print(round(result.equivalent_noise_level_db, 1)) # dB(A) result.report("microphone.pdf", metadata=ReportMetadata(measurement_standard="IEC 60268-4")) ``` `microphone_characteristics` returns a `MicrophoneCharacteristics` with the computed `sensitivity_level_db`, `effective_range`, `directivity_index_db`, `equivalent_noise_level_db`, `max_spl_db`, `signal_to_noise_ratio_db` and `diffuse_field_sensitivity_level_db`, and a `.report()` that writes the fiche. The free-field response is drawn with its tolerance band, the reference-frequency marker and the effective-range markers to the IEC 60263 proportion (one frequency decade equal to 25 dB), and the directional pattern on the IEC 60263 25 dB reference circle; the inherent-noise spectrum and the distortion-against-level curve (with the THD limit and the overload level marked) feed the secondary panels. A `requirement` in the metadata is checked as a maximum permitted equivalent noise level. The rated characteristics are also available interactively through `.plot()`, which draws **one concept per figure** with the same panel code the report composes, selected by `quantity`. Passing an axes draws on it: ```python result.plot() # free-field response (default) result.plot(quantity="directivity") # polar pattern on the 25 dB circle result.plot(quantity="noise") # inherent-noise band spectrum result.plot(quantity="distortion") # THD vs sound pressure level ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import microphone_characteristics freqs = np.geomspace(20, 20000, 400) response = -10 * np.log10(1 + (30.0 / freqs) ** 4) # low-frequency roll-off response -= 10 * np.log10(1 + (freqs / 19000.0) ** 8) # high-frequency roll-off response += 2.0 * np.exp(-(np.log2(freqs / 9000.0) ** 2) / 0.3) # presence region result = microphone_characteristics(freqs, response, 12.5, tolerance_db=3.0) result.plot() # quantity="response" (the default) plt.show() ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import microphone_characteristics freqs = np.geomspace(20, 20000, 400) response = -10 * np.log10(1 + (30.0 / freqs) ** 4) angles = np.linspace(0, 179, 359) cardioid = 20 * np.log10((1 + np.cos(np.radians(angles))) / 2) result = microphone_characteristics( freqs, response, 12.5, tolerance_db=3.0, polar=(angles, cardioid), polar_frequency=1000.0, ) result.plot(quantity="directivity") plt.show() ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import microphone_characteristics freqs = np.geomspace(20, 20000, 400) response = -10 * np.log10(1 + (30.0 / freqs) ** 4) noise_f = np.geomspace(20, 20000, 31) result = microphone_characteristics( freqs, response, 12.5, tolerance_db=3.0, noise_voltage=1.25e-6, noise_spectrum=(noise_f, 6.0 + 12.0 * np.log10(1000.0 / noise_f)), ) result.plot(quantity="noise") plt.show() ```

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```python import matplotlib.pyplot as plt import numpy as np from phonometry import microphone_characteristics freqs = np.geomspace(20, 20000, 400) response = -10 * np.log10(1 + (30.0 / freqs) ** 4) spl_axis = np.linspace(100, 140, 81) result = microphone_characteristics( freqs, response, 12.5, tolerance_db=3.0, max_spl_thd_percent=0.5, distortion=(spl_axis, 0.5 * 10 ** ((spl_axis - 130.0) * 0.08)), ) result.plot(quantity="distortion") plt.show() ```

The example fiche, regenerated with `make reports`, is kept rendered in the repository. Click the preview to open the PDF: [](https://raw.githubusercontent.com/jmrplens/phonometry/main/.github/reports/iec60268_4_microphone_example.pdf) *Microphone characteristics fiche (`MicrophoneCharacteristics.report`), the IEC 60268-4 rated-characteristics table beside the free-field response, with the cardioid directional pattern, inherent-noise and THD-against-level panels drawn to the IEC 60263 25 dB-per-decade and 25 dB reference-circle conventions.* ## References - Beranek, L. L., & Mellow, T. J. (2012). *Acoustics: Sound fields and transducers*. Academic Press. ISBN 978-0-12-391421-7. [doi:10.1016/C2011-0-05897-0](https://doi.org/10.1016/C2011-0-05897-0). The transducer physics behind section 5: loudspeaker radiation, on-axis pressure and the near-field to far-field transition. ## Standards IEC 60268-3:2013, *Sound system equipment – Part 3: Amplifiers* (clauses 14.12.2–14.12.11): total harmonic distortion `THD_F`/`THD_R` (the 14.12.3.2 formula defines the R form), nth-order harmonic distortion `dₙ`, the per-order modulation (`d_m,n`) and difference-frequency (`d_d,n`) intermodulation, total difference-frequency distortion, dynamic intermodulation (DIM) and the ITU-R BS.468-4 / IEC 60268-1 weighted THD. AES17-2015, *Measurement of digital audio equipment* (clauses 5.2.5, 5.2.8 and 6.3.1): the THD+N ratio via the standard notch filter and the standard measurement bandwidth; SINAD is derived from it. ITU-R BS.468-4: the weighting-network nominal response. Bendat & Piersol (2010), *Random Data: Analysis and Measurement Procedures* (4th ed., Wiley): the `H1` and `H2` frequency-response estimators and the ordinary coherence `γ²`. IEC 60268-4:2014, *Sound system equipment – Part 4: Microphones*: the rated microphone characteristics of sections 5 and 8 — the free-field sensitivity and its level re 1 V/Pa (11.1/11.3), the frequency response and effective frequency range (12.1/12.2), the directional pattern and the directivity index through the 11.2.2 a) diffuse-field integral (13.1/13.2), the overload sound pressure level (15.2), the equivalent sound pressure level due to inherent noise (17) and the rated impedances and power supply (9/10); since revised as IEC 60268-4:2018, the 2014 edition is the implemented one. IEC 60268-5:2003+A1:2007, *Sound system equipment – Part 5: Loudspeakers*: the rated loudspeaker characteristics of section 7 — the rated impedance (16), the rated frequency range (19.1), the characteristic sensitivity and its level referred to 1 W at 1 m (20.3/20.4), the effective frequency range against the −10 dB band (21.2), the directivity index (23.3) and the total harmonic distortion against frequency (24.1). IEC 60263:1982, *Scales and sizes for plotting frequency characteristics and polar diagrams*: the scale proportions of the section 7/8 characteristic graphs — one frequency decade equal to 25 dB on the ordinate (clause 2) and the polar diagram on a 25 dB reference-circle radius (clause 3). All quantities are verified against exact analytic oracles (synthetic signals with known harmonic/intermodulation amplitudes, a clipped-sine Fourier oracle, a full DIM test-signal synthesis, and a known LTI path).