We propose a straightforward and cost-effective method to perform diffuse
soundfield measurements for calibrating the magnitude response of a microphone
array. Typically, such calibration is performed in a diffuse soundfield created
in reverberation chambers, an expensive and time-consuming process. A method is
proposed for obtaining diffuse field measurements in untreated environments.
First, a closed-form expression for the spatial correlation of a wideband
signal in a diffuse field is derived. Next, we describe a practical procedure
for obtaining the diffuse field response of a microphone array in the presence
of a non-diffuse soundfield by the introduction of random perturbations in the
microphone location. Experimental spatial correlation data obtained is compared
with the theoretical model, confirming that it is possible to obtain diffuse
field measurements in untreated environments with relatively few loudspeakers.
A 30 second test signal played from 4-8 loudspeakers is shown to be sufficient
in obtaining a diffuse field measurement using the proposed method. An
Eigenmike is then successfully calibrated at two different geographical
locations.Comment: Accepted to appear in IEEE ICASSP 202

Abhayapala, Thushara

Akbar, Noman

Dickins, Glenn

Samarasinghe, Prasanga

Thomas, Mark R. P.

English

arXiv

We propose a straightforward and cost-effective method to perform diffuse
soundfield measurements for calibrating the magnitude response of a microphone
array. Typically, such calibration is performed in a diffuse soundfield created
in reverberation chambers, an expensive and time-consuming process. A method is
proposed for obtaining diffuse field measurements in untreated environments.
First, a closed-form expression for the spatial correlation of a wideband
signal in a diffuse field is derived. Next, we describe a practical procedure
for obtaining the diffuse field response of a microphone array in the presence
of a non-diffuse soundfield by the introduction of random perturbations in the
microphone location. Experimental spatial correlation data obtained is compared
with the theoretical model, confirming that it is possible to obtain diffuse
field measurements in untreated environments with relatively few loudspeakers.
A 30 second test signal played from 4-8 loudspeakers is shown to be sufficient
in obtaining a diffuse field measurement using the proposed method. An
Eigenmike is then successfully calibrated at two different geographical
locations.This research work is funded by Australian Research Council Linkage
Project LP16010037

The Australian National University

A NOVEL METHOD FOR OBTAINING DIFFUSE FIELD MEASUREMENTS FORMICROPHONE CALIBRATIONNoman Akbar∗, Glenn Dickins∗‡, Mark R. P. Thomas†, Prasanga Samarasinghe∗, Thushara Abhayapala∗∗Australian National University, ‡Audinate Sydney, †Dolby Laboratories USA{noman.akbar, prasanga.samarasinghe, thushara.abhayapala}@anu.edu.aug@dickins.com, mark.r.thomas@ieee.orgABSTRACTWe propose a straightforward and cost-effective method to performdiffuse soundfield measurements for calibrating the magnitude re-sponse of a microphone array. Typically, such calibration is per-formed in a diffuse soundfield created in reverberation chambers,an expensive and time-consuming process. A method is proposedfor obtaining diffuse field measurements in untreated environments.First, a closed-form expression for the spatial correlation of a wide-band signal in a diffuse field is derived. Next, we describe a practicalprocedure for obtaining the diffuse field response of a microphonearray in the presence of a non-diffuse soundfield by the introductionof random perturbations in the microphone location. Experimen-tal spatial correlation data obtained is compared with the theoreticalmodel, confirming that it is possible to obtain diffuse field measure-ments in untreated environments with relatively few loudspeakers.A 30 second test signal played from 4 − 8 loudspeakers is shownto be sufficient in obtaining a diffuse field measurement using theproposed method. An Eigenmike R© is then successfully calibrated attwo different geographical locations.Index Terms— Spherical arrays, microphone calibration, dif-fuse field measurements, and spatial correlation.1. INTRODUCTIONDiffuse field measurements are important for calibrating and testingthe magnitude response of acoustic devices. Reverberation chambersare often used to create a diffuse field [1, 2, 3, 4]. However, they areexpensive and may have limited availability. Alternatively, a dif-fuse field measurement can be derived from a dense set of anechoicmeasurements [5]. Such measurements are also difficult to obtainas anechoic chambers and robotic loudspeaker mounts are expensiveand not very common. A practical solution for obtaining a diffusefield response in regular rooms is therefore desirable.A diffuse field is defined as an acoustic field where the energydensity is uniform in all directions [4, 6]. Spatial correlation isan important metric for characterizing diffuse fields [3, 7, 8, 9].While acoustic signals are wideband in nature, most existing re-search works analyzes spatial correlation of narrowband signals ina diffuse field, which may be restrictive in several practical scenar-ios. Instead, we consider the spatial correlation of wideband signalsin diffuse fields.We propose a novel method for obtaining the diffuse field re-sponse of a microphone array in the presence of a non-diffuse sound-field. The main contributions of this work are:This research work is funded by Australian Research Council LinkageProject LP160100379.1. Deriving a closed-form expression for correlation of wide-band signals in diffuse fields.2. Proposing a method to produce diffuse field measurements ina non-diffuse soundfield by perturbing the microphone arraylocation during data capture. Experimental results using theproposed method are shown to agree with the theoretical re-sults.3. Using the proposed method for diffuse field measurement tosuccessfully calibrate an Eigenmike R© at two different geo-graphical locations demonstrating the efficiency and repro-ducibility of the proposed method.2. SPATIAL CORRELATION OF WIDEBAND SIGNALS INA DIFFUSE FIELDConsider an array with M microphones placed at an arbitrary loca-tion in 3D space. The p-th and q-th microphones, denoted by mpand mq , are placed at location xp and xq in the 3D space. The spa-tial correlation ρ (xp,xq) between the signals captured by the twomicrophones is defined as [4, 7]ρ (xp,xq) =E{sp (t) s∗q (t)}E{sp (t) s∗p (t)} , (1)where sp (t) and sq (t) are the signals received at mp and mq , re-spectively. Furthermore, we assume that sp (t) is a wideband signalrepresented bysp (t) =1∆ω∫ ωmaxωmin∫RA (ŷ) e−i(ωc )xp•ŷeiωtdŷdω, (2)where • is the dot product between two vectors, ∆ω = ωmax − ωminspecifies the bandwidth of the signal, R represents a unit sphere,c is the speed of sound in air, ŷ is the unit vector in the directionof signal propagation, and A (ŷ) is the gain of the signal arrivingfrom the direction of ŷ and is assumed to be frequency independent.Using (2) and (1), we obtainρ (xp,xq) =∫ ωmaxωmin∫R|E {A (ŷ)}|2 ei(ωc )(xq−xp)•ŷdŷdω∫ ωmaxωmin∫R|E {A (ŷ)}|2 dŷdω, (3)which can be rewritten asρ (xp,xq) =∫ ωmaxωmin∫Rξ (ŷ, ω) ei(ωc )(xq−xp)•ŷdŷdω,=1∆ω∫ ωmaxωmin∫RG (ŷ) ei(ωc )(xq−xp)•ŷdŷdω, (4)whereξ (ŷ, ω) =|E {A (ŷ)}|2∫ ωmaxωmin∫R|E {A (ŷ)}|2 dŷdω,=1∆ω(|E {A (ŷ)}|2∫R|E {A (ŷ)}|2 dŷ)=G (ŷ)∆ω, (5)and G (ŷ) represents the average power gain of the signal receivedfrom an arbitrary direction ŷ. We highlight that (4) is a general-ized expression valid for any microphone pair and bandwidth of thesignal. The exponential term in (4) can be rewritten by using thespherical harmonic expansion as [10]ei(ωc )(xq−xp)•ŷ = 4π∞∑n=0injn(ωc‖xq − xp‖)×n∑m=−nYnm(xq − xp‖xq − xp‖)Y ∗nm (ŷ) , (6)where jn (.) is the spherical Bessel function, Ynm (.) represents thespherical harmonics, n is the order and m is the degree of the spher-ical harmonics. Substituting the value from (6) in (4), we obtainρ (xp,xq) =1∆ω∫ ωmaxωmin∫RG (ŷ) 4π∞∑n=0injn(ωc‖xq − xp‖)×n∑m=−nYnm(xq − xp‖xq − xp‖)Y ∗nm (ŷ) dŷdω. (7)We next simplify (7) asρ (xp,xq) =4π∆ω∫ ωmaxωmin∞∑n=0injn(ωc‖xq − xp‖)×n∑m=−nYnm(xq − xp‖xq − xp‖)βnmdω, (8)whereβnm =∫RG (ŷ)Y ∗nm (ŷ) dŷ. (9)Assuming that the plane wave is received by the microphones mpand mq from all directions, i.e., the field is diffuse, and β00=1, ex-pression (8) is further simplified asρ (xp,xq) =1∆ω∫ ωmaxωminj0(ωc‖xq − xp‖)dω,=1∆ω∫ ωmaxωminsin(ωc‖xq − xp‖)(ωc‖xq − xp‖) dω. (10)Defining ωc = ωmin+ωmax/2, (10) can be rewritten asρ (xp,xq) =1∆ω∫ ∆ω2−∆ω2sin(ω+ωcc‖xq − xp‖)ω+ωcc‖xq − xp‖dω, (11)which is expanded by using the identity sin (a+ b) = sin (a) cos (b)+cos (a) sin (b) to obtainρ (xp,xq) =1∆ω∫ ∆ω2−∆ω2sin(ωc‖xq − xp‖)cos(ωcc‖xq − xp‖)ω+ωcc‖xq − xp‖+cos(ωc‖xq − xp‖)sin(ωcc‖xq − xp‖)ω+ωcc‖xq − xp‖dω. (12)26 16 26 2616Mic 1 Mic16Fig. 1: Linear microphone array with 16 microphones used in theexperiments. The microphone spacing is in millimeters.Next, the integral in (12) is evaluated. Using the Taylor series expan-sion of the term ω+ωcc‖xq − xp‖ and neglecting the higher powersof ∆ωω, (12) is simplified asρ (xp,xq) =sin(∆ω2c‖xq − xp‖)∆ω2c‖xq − xp‖[sin(ωcc‖xq − xp‖)ωcc‖xq − xp‖− 2cos(ωcc‖xq − xp‖)(ωcc‖xq − xp‖)2 sin2(∆ω‖xq − xp‖4c)]. (13)The second term in (13) can be ignored as it decreases proportionalto ‖xq − xp‖3 as ‖xq − xp‖ increases [6]. Accordingly, we getρ (xp,xq) =sin(∆ω2c‖xq − xp‖)∆ω2c‖xq − xp‖[sin(ωcc‖xq − xp‖)ωcc‖xq − xp‖], (14)which is further simplified toρ (xp,xq) = sinc(∆ω2c‖xq − xp‖)sinc(ωcc‖xq − xp‖).(15)The generalized spatial correlation expression (15) is valid for wide-band signals. Note that narrowband signals are a special case ofthe expression (15), where ∆ω ≈ 0 and ωc = ω. Thus, utilizingsinc(0) = 1, the spatial correlation function of a narrowband signalis obtained from (15) asρNB (xp,xq) = sinc(ωc‖xq − xp‖). (16)3. DIFFUSE FIELD MEASUREMENT WITH FINITELOUDSPEAKER ARRAYConsider a compact loudspeaker array placed in an acoustically un-treated room where each speaker produces uncorrelated bandpasswhite noise. It is natural to ask how many loudspeakers are requiredto produce a diffuse soundfield such that measured spatial correla-tion (1) agrees with the theoretical result (15). A novel method isproposed whereby the location of the microphone array is randomlyperturbed and rotated within the loudspeaker array with a view toproducing a response closer to that of a highly diffuse soundfield1 .3.1. Experimental SetupThe loudspeaker array consisted of 26 loudspeakers mounted on thevertices of a rhombic tricontrahedron with radius 1.8 m.2 Loud-speakers produced uncorrelated bandpass white noise with parame-ters ∆ω = 2π × (4.5 − 0.5) kHz and ωc = 2π × (4.5+0.5) kHz/2.1A demo of the random motion for the proposed method can be viewedat https://vimeo.com/367536785 and https://vimeo.com/367536766.2Additional details about the rhombic triacontrahedron loudspeaker arraymodel “DAARRT26 1318” can be found at https://vimeo.com/361493511.0 0.05 0.1 0.15 0.2 0.25 0.3-0.200.20.40.60.81Microphone dataInterpolation of microphone dataTheoretical expression eq. (15)(a) Experiment 1: Fixed mic array with 2 loudspeakers playing test signal.0 0.05 0.1 0.15 0.2 0.25 0.3-0.200.20.40.60.81Microphone dataInterpolation of microphone dataTheoretical expression eq. (15)(b) Experiment 2: Proposed method with 2 loudspeakers playing test signal.0 0.05 0.1 0.15 0.2 0.25 0.3-0.200.20.40.60.81Microphone dataInterpolation of microphone dataTheoretical expression eq. (15)(c) Experiment 1: Fixed mic array with 26 loudspeakers playing test signal.0 0.05 0.1 0.15 0.2 0.25 0.3-0.200.20.40.60.81Microphone dataInterpolation of microphone dataTheoretical expression eq. (15)(d) Experiment 2: Proposed method with 26 loudspeakers playing test signal.Fig. 2: Microphone distance versus spatial correlation obtained from microphone data and theoretical expression for 2 and 26 loudspeakers.A 16-element linear microphone array with inter-element spacingshown in Fig. 1 was used to produce pairs of spatial correlation esti-mates with spacing between 0.016 and 0.32 m.3.2. Experiment 1: Fixed Microphone ArrayThe linear microphone array was placed at the center of the loud-speaker array and 30 seconds of data was captured using 2 and26 loudspeakers. Fig. 2a and Fig. 2c depict microphone distanceversus spatial correlation for all microphone pairs. While the 26-loudspeaker case follows the theoretical curve more closely than the2-loudspeaker case, large variances are observed in the correlationdata in both cases, suggesting that 26 loudspeakers are insufficientto produce a diffuse soundfield with a fixed microphone array.3.3. Experiment 2 (Proposed Method): Moving ArrayThe conditions of Experiment 1 were repeated, except the micro-phone array location was perturbed and rotated within the loud-speaker array during capture. Fig. 2b and Fig. 2d shows the spatialcorrelation for the proposed method with 2 loudspeakers and 26loudspeakers, respectively. Comparing Fig. 2a and Fig. 2c withFig. 2b and Fig. 2d, we clearly observe that the variance of the spa-tial correlation and deviation from the theoretical model is greatlyreduced with the proposed method. For example, for the pro-posed method with microphone distance 0.15 m, the variance ofthe microphone data is reduced by 99% compared to the fixed case.Importantly, even with 2 loudspeakers, we observe a large reductionin variance. In order to verify the repeatability of the method, theexperiment was repeated with a 30-loudspeaker dodecahedron arrayat a different geographical locations with three different participantswho received limited instructions about movement, and similarresults were obtained. This demonstrates that the measurement ap-proximates that of a diffuse soundfield using the proposed methodat multiple locations. Additionally, the method appears robust tothe pattern of movement since it has little impact on the results.The test signal is received at the microphones from multiple anglesand directions using the proposed method. We obtain diffuse fieldmeasurements as a result.3.4. Experiment 3: Restricted Loudspeaker CountThe sensitivity of diffuseness to loudspeaker count was investigated.Table. 1 depicts the sum of variances of spatial correlations for differ-Table 1: Sum of Variances for All Microphone DistancesFixed Proposed Method1 Speaker 1.2311 0.01352 Speakers 0.2356 0.00454 Speakers 0.1006 0.00258 Speakers 0.1418 0.002616 Speakers 0.0697 0.001726 Speakers 0.0467 0.0017ent number of loudspeakers. It is observed that the proposed methodusing one loudspeaker outperforms the fixed microphone array with26 loudspeakers. Furthermore, the proposed method achieves an ap-proximate diffuse field measurement with as little as 4 − 8 loud-speakers.4. CALIBRATING AN EIGENMIKE R©Spherical microphone arrays are widely used for 3D soundfield cap-ture [11, 12, 13, 14, 15, 16]. Equipment calibration is an importantstep in soundfield capture, reproduction, and validation of soundfieldduplication for consumer device testing [17, 18, 19]. A variant ofthe proposed calibration method was used in [17] to achieve a broadspectral alignment within 1 dB. We next demonstrate the practicalsignificance of the proposed method by calibrating an Eigenmike R©(SN37).Uncorrelated pink noise was played in the 1.8 m rhombic tria-contrahedron loudspeaker array and a diffuse field measurement wasobtained using the proposed method in Section. 3. We asked a par-ticipant to repeat the experiment twice, once by using 2 loudspeakersin a normal study room and once by using 26-loudspeaker array ina semi-anechoic room. The movement pattern of the Eigenmike R©was completely different in the two experiments. Fig. 3 depicts theconsistency and repeatability of the two measurements. For this plot,the mean magnitude response of all 32 microphones was subtractedfrom individual microphone magnitude responses. Importantly, themagnitude responses in the two experiments are within 0.2 dB. Thesame behaviour were observed in all the remaining 30 microphones.The Eigenmike R© release notes specify that the magnitude re-sponses from all the microphones are trimmed at 1 kHz [20]. Ourtests indicate that the Eigenmike R© trim at 1 kHz has a drift of ap-proximately 1.7 dB. After calibration, the magnitude responses at1 kHz for all the microphones were within 0.26 dB. The proposedmethod applies calibration throughout the spectrum rather than trim-ming at a specific frequency. After calibration, the magnitude re-sponses of all microphones were within 0.5 dB for majority of thespectrum. We calibrated the Eigenmike R© at a different geographicallocation and obtained similar results.5. CONCLUSIONWe proposed a method for microphone calibration by taking adiffuse field measurement with a few loudspeakers. We demon-strated the effectiveness of the proposed method by calibrating anEigenmike R© with a 26-speaker array in a semi-anechoic room andwith 2 speakers in a study room and achieving similar results. Withthe proposed method, it is possible to calibrate a microphone arraysuch that the magnitude responses of all microphones are within0.5 dB for majority of the spectrum. We demonstrated that as littleas 4 − 8 loudspeakers playing uncorrelated noise are sufficient to102 103 104-1-0.8-0.6-0.4-0.200.20.40.60.81Mic 11, 26 speakers in semi-anechoic labMic 11, 2 speakers in a study  roomMic 27, 26 speakers in semi-anechoic labMic 27, 2 speakers in a study  roomFig. 3: The Eigenmike R© magnitude response offset for Mic 11 andMic 27 using 2 loudspeakers and 26 loudspeakers demonstrating re-peatability of the proposed method in different environments.achieve a diffuse field measurement for microphone calibration.The proposed microphone calibration method is simple and cost-effective and can be performed in non-anechoic environments. Theproposed method can greatly reduce the time and effort required incalibrating microphone arrays.6. REFERENCES[1] D. T. Bradley, C. Diaz, E. Snow, “Improved sound field rever-berance and diffusivity in a reverberation chamber through im-plementation of resonant-diffusing wall panels,” Acta Acusticaunited with Acustic, vol. 101, pp. 181-189, 2015.[2] C. T. Morrow, “Point-to-point correlation of sound pressuresin reverberation chambers,” J. Sound and Vibration, vol. 16,no. 1, pp. 29–42, 1971.[3] I. Chun, B. Rafaely, and P. Joseph, “Experimental investigationof spatial correlation in broadband reverberant sound fields,” J.Acoust. Soc. Amer., vol. 113, pp. 1995-1998, 2003.[4] R. K. Cook, R. V. Waterhouse, R. D. Berendt, S. Edelman, andM. C. Thompson, “Measurement of correlation coefficientsin reverberant sound fields,” J. Acoust. Soc. Amer., vol. 27,pp. 1072–1077, 1955.[5] A. Politis and H. Gamper, “Comparing modeled andmeasurement-based spherical harmonic encoding filters forspherical microphone arrays,” in Proc. IEEE Workshop Appli-cations Signal Process. Audio Acoust. (WASPAA), New Paltz,NY, Oct. 2017, pp. 224–228.[6] H. 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A Novel Method for Obtaining Diffuse Field Measurements for Microphone Calibration

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