Source: https://jmrplens.github.io/phonometry/signals/metrology/random-incidence/ # Random-incidence and diffuse-field response (IEC 61183) A sound level meter is calibrated for sound arriving from one direction, its reference direction, and most of the sound it measures arrives from all of them. Its microphone and its case are not transparent at high frequency, so a meter that reads a plane wave from the front correctly reads a field from every direction low. IEC 61183:1994 gives the two ways of finding by how much: the **free-field method** rotates the instrument in an anechoic room and weights the reading for each direction by the share of the sphere it stands for, and the **diffuse-field method** compares the instrument with a reference instrument in a reverberation room. This page runs both on a synthetic meter: a microphone whose pattern has, at each frequency, the directivity Table B.1 prints for a type LS2aP/LS2F laboratory standard microphone, on a case that narrows it in the plane the case sits in. ## The directivity factor The directivity factor of Formulas (2) and (3) is the ratio of what the meter reads for a plane wave from its reference direction to what it reads, on average over the sphere, for the same wave from every direction: `gamma = 4 pi / integral of 10^(-0,1 [L_rd - L(phi, alpha)]) |sin phi| d alpha d phi`. An omnidirectional meter has `gamma = 1`. The random-incidence sensitivity level is the free-field sensitivity level for the reference direction less the directivity index, `G_RI = G_F - 10 lg gamma` with `G_F = L_rd - L_o` (Formula (1)); `G_F` depends on the individual meter and `gamma` only on its geometry, so one measurement of `gamma` serves every meter of a model (4.2). ### How the measurement goes | Requirement | Value | Clause | | :--- | :--- | :--- | | Anechoic room | Meets ISO 3745; pure tones or random noise, bands no wider than one-third octave, filters of IEC 61260 class 0 or 1 | 4.10 | | Source | Far enough that the level varies by less than ±1 dB within 0,3 m of the microphone | A.2.2 | | Signal | At least 20 dB above the background, and held constant during each rotation | A.2.3 | | Turntable | The acoustical centre of the microphone on the axis of rotation, the reference direction and the source in the plane of rotation | A.2.1, A.4.2 | | Rotations | 360° in the X-Y plane, then the meter turned 90° about its own axis and 360° again, the X-Z plane | A.4.5, A.4.6 | | Angular step | Small enough that no element of the sphere exceeds 3 %; 10° leaves 2,2 % | A.1.6, A.1.7 | | Pure tones | `G_F` and `gamma` may need the rms average of at least eight tones per one-third-octave band, spaced evenly on a logarithmic axis | 4.11 | | Reverberation room | Meets ISO 3741; broadband or filtered random noise, bands no wider than one-third octave | 5.6 | | Integration time | Long enough that repeated results scatter by less than 0,05 dB: 2 min from 500 Hz, 8 min from 250 Hz, 15 min from 125 Hz, longer below | 5.6, B.1.4 | | Microphone path | Both microphones moved in turn along the same circular path, not parallel to any wall, of radius the larger of 1 m and three times the largest dimension of the meter; two uncorrelated omnidirectional sources of about equal power help the diffusivity | B.1.3, B.1.4 | | Reference meter | A directivity factor as near unity as possible; a type LS2aP/LS2F or LS2bP microphone is recommended | B.1.1, B.1.2 | ## Readings in two planes, and what each is worth Annex A rotates the meter through 360° in the X-Y plane and again in the X-Z plane, in equal steps. The elements of the sphere are rings cut into quarters by the planes, so each reading is weighted by the factor of Formulas (A.1) and (A.2), `K(phi) = (1/8)[cos(phi - dphi/2) - cos(phi + dphi/2)]` and `K(0) = K(180°) = (1/4)[1 - cos(dphi/2)]`: ```python from phonometry import metrology k = metrology.adjustment_factors(10.0) # 36 factors, 0° to 350° print(k[[0, 1, 9]].round(5)) # [0.00095 0.00378 0.02179] print(round(2 * k.sum(), 12)) # 1.0 print(round(100 * metrology.largest_element_fraction(10.0), 2)) # 2.18 ``` All ten rows of Table A.1 come out to the five decimals it prints. A.1.6 asks for the largest element to be no more than 3 % of the sphere; at 10° steps it is the one at 90°, 2,18 %, the "approximately 2,2 %" of A.1.7, and a step of 15° (3,26 %) warns with a `SphereDivisionWarning`. `planes=4` gives the four planes at 45° of NOTE 2 of A.6, which halve every factor. The 72 factors sum to one only when the readings at 0° and 180° enter both sums of Formula (A.3). The paragraph under it says those readings "have only to be taken into account once": they have to be measured once. Counted in one sum only, the factors add up to 0,998 097 and `10 lg gamma` comes out high by `-10 lg(1 - gamma K(0) [10^(-0,1 [L_rd - L(0°)]) + 10^(-0,1 [L_rd - L(180°)])])`: 0,008 dB for a meter that reads the same in every direction, about 0,02 dB at `10 lg gamma = 7 dB`. The library counts them in both sums, and the omnidirectional meter has `gamma = 1` exactly. ```python import numpy as np def pattern_db(phi_deg, n, floor=0.02): """L(phi) - L_rd of a meter whose squared pressure is (1 - b)[(1 + cos phi)/2]^n + b.""" lobe = ((1 + np.cos(np.radians(phi_deg))) / 2) ** n return 10 * np.log10((1 - floor) * lobe + floor) phi = np.arange(0, 360, 10) levels = 94.0 + np.vstack((pattern_db(phi, 0.785), # X-Y plane (h), 8 kHz pattern_db(phi, 0.98))) # X-Z plane (v), the case narrows it d = metrology.directivity_factor(levels) print(round(d.gamma, 3), round(d.directivity_index_db, 2)) # 1.845 2.66 d.plot() # the polar response, one curve per plane d.plot(view="weights") # K(phi) of each reading ``` ## One plane, or 38 equal elements A rotationally symmetric meter needs one rotation (Formula (A.4)), and the note to A.1.8 offers 38 directions of equal-area elements, 2,6 % of the sphere each, every reading weighing the same (Formula (A.5)): ```python a = metrology.axisymmetric_directivity_factor(levels[0]) print(round(a.directivity_index_db, 2)) # 2.45 horizontal, vertical = metrology.equal_area_incidence_angles() print(horizontal[:6].round(1)) # [ 0. 32.6 50.8 65.1 77.8 90. ] e = metrology.equal_area_directivity_factor( 94.0 + pattern_db(horizontal, 0.785), 94.0 + pattern_db(vertical, 0.98) ) print(round(e.directivity_index_db, 2)) # 2.66 ``` The X-Y plane alone has the 2,45 dB Table B.1 prints for the microphone at 8 kHz; the case adds the other 0,21 dB. Each equal-area direction halves its element's area in polar angle, which reproduces the list the note prints except 77,9° and 282,1°, where the construction gives 77,85° and 282,15°: all the other pairs of the printed list sum to 180,0° about the grazing direction, and 77,9° + 102,2° is 180,1°. The two are in the [errata register](https://jmrplens.github.io/phonometry/reference/errata/). ## The random-incidence sensitivity level, band by band The synthetic meter at every preferred frequency Table B.1 prints, its X-Y plane given the `10 lg gamma` of the microphone at that band, and its free-field sensitivity level rolling off above 10 kHz: ```python bands = sorted(metrology.IEC61183_TABLE_B1) # the 30 preferred frequencies, 25 Hz to 20 kHz def meter(frequency_hz): """The synthetic meter at one band: the X-Y plane has the 10 lg gamma Table B.1 prints for the microphone, and the case narrows the X-Z plane.""" target = metrology.IEC61183_TABLE_B1[frequency_hz].directivity_index_db n = 0.98 / (10 ** (-target / 10) - 0.02) - 1 # the exponent with that index return metrology.directivity_factor( 94.0 + np.vstack((pattern_db(phi, n), pattern_db(phi, 1.25 * n)))) index = [meter(f).directivity_index_db for f in bands] g_f = -10 * np.log10(1 + (np.array(bands) / 25000) ** 4) # L_rd - L_o at each band r = metrology.random_incidence_sensitivity(bands, g_f, index) shown = np.isin(r.frequencies_hz, [1000, 4000, 8000, 16000]) print(r.correction_db[shown].round(2)) # [-0.06 -0.95 -2.66 -5.62] print(r.random_incidence_level_db[shown].round(2)) # [-0.06 -0.95 -2.71 -6.29] r.plot() r.plot(view="correction") ``` The correction is what a meter calibrated in a free field reads low in a random-incidence field: negligible up to about 1 kHz and several decibels by 10 kHz. It is why a meter is specified for one field or the other: IEC 61672-1:2013 applies its frequency-weighting limits to the free-field or to the random-incidence response, as applicable (5.5.4), and has the random-incidence response determined by the free-field method of IEC 61183 (5.5.5). One meter cannot be flat for both. ## The diffuse-field method The meter under test and a reference meter are placed in turn at the same positions in a reverberation room; the difference of what they indicate, `dG_D = L_D - L_D,ref` (Formula (8)), is added to the diffuse-field sensitivity level of the reference: `G_RI,ref` for a reference calibrated by clause 4 (Formula (9)), `G_F,ref - 10 lg gamma_ref` for a free-field calibrated one (Formula (10)), `G_P,ref + Delta_DP` for a pressure calibrated one (Formula (11)). Annex B recommends a type LS2aP/LS2F or LS2bP microphone as the reference, Table B.1 prints `10 lg gamma` and `Delta_DP` of the first, and Formulas (10) and (11) take their correction from it unless told otherwise: In an 80 dB diffuse field the meter indicates the field plus its random-incidence level (1.2), and a reference LS2aP whose pressure sensitivity level is 0 dB the field plus its `Delta_DP`: ```python print(metrology.IEC61183_TABLE_B1[8000.0]) # ReferenceMicrophoneRow(directivity_index_db=2.45, diffuse_pressure_difference_db=1.2) delta_dp = np.array([metrology.IEC61183_TABLE_B1[f].diffuse_pressure_difference_db for f in bands]) dd = metrology.diffuse_field_sensitivity( bands, 80.0 + r.random_incidence_level_db, # L_D, the meter under test 80.0 + delta_dp, # L_D,ref, a pressure-calibrated LS2aP reference_pressure_level_db=0.0, # G_P,ref; Delta_DP from Table B.1 ) print(dd.reference_correction_db[shown]) # [0. 0.25 1.2 3.05] print(dd.diffuse_field_level_db[shown].round(2)) # [-0.06 -0.95 -2.71 -6.29] dd.plot() ``` The two methods meet: the diffuse-field sensitivity level may be used interchangeably with the random-incidence one (1.2), and the comparison in a room returns the level the anechoic measurement gave. ## What this guide covers Implemented: the calculations of both methods of IEC 61183:1994, the adjustment factors of Formulas (6), (7), (A.1) and (A.2) for any step and any number of planes, with the halving of NOTE 2 of A.6; the 3 % criterion of A.1.6 as a warning; the directivity factor from two or more planes, from one plane under rotational symmetry and from 38 equal-area elements (Formulas (A.3) to (A.5)); the random-incidence sensitivity level of Formulas (1) and (A.6); the diffuse-field sensitivity level by the three routes of Formulas (8) to (11); and Table B.1. Not implemented: the measurements themselves (the ISO 3745 anechoic and ISO 3741 reverberation rooms, the source, the turntable, the signal-to-noise ratio of A.2.3 and the integration times of B.1.4), the averaging of at least eight tones per band (4.11). The corrections of IEC 62585 that bring a measurement made with a sound calibrator, a comparison coupler or an electrostatic actuator to the meter's free-field response are on [their own page](https://jmrplens.github.io/phonometry/signals/metrology/free-field-corrections/); its corrections for the case, the microphone and the windscreen (clauses 9 to 11) are not implemented. ## See also - [Free-field corrections](https://jmrplens.github.io/phonometry/signals/metrology/free-field-corrections/): IEC 62585, the corrections that bring a meter on a calibrator, a coupler or an actuator to its free-field response, for a meter with a free-field microphone. - [Calibration and dBFS](https://jmrplens.github.io/phonometry/signals/metrology/calibration/): the calibrator tone that sets the pressure sensitivity every level of a meter starts from. - [Compliance and verification](https://jmrplens.github.io/phonometry/signals/metrology/compliance-verification/): what the IEC 61672-1 class of a meter asserts, and which of its tests the library runs. - [Microphone characterisation](https://jmrplens.github.io/phonometry/devices/electroacoustics/microphones/): the IEC 60268-4 directivity index of a microphone on its own. - [Errata in published sources](https://jmrplens.github.io/phonometry/reference/errata/): the two equal-area angles. - API reference: [`metrology.random_incidence`](https://jmrplens.github.io/phonometry/reference/api/metrology/random-incidence/).