Embed Size (px)
Citation preview
Beamforming using a spherical microphone array
based on legacy microphone characteristics
Nils Peters1,2 and Andrew W. Schmeder1
1Center for New Music and Audio Technologies (CNMAT), UC Berkeley, USA2International Computer Science Institute (ICSI), Berkeley, USA
Email: [email protected], [email protected]
AbstractThis paper presents a numerical approach that projectsfrequency-dependent directivity patterns of classicrecording microphones onto steerable beams createdwith a spherical microphone array. In an anechoicchamber, the spatial and timbral characteristics of alegacy recording microphone and the characteristics of a120-channel spherical microphone array were measured.Using a least-square matching approach, the measuredfrequency responses were used to calculate the setof filters that synthesize the desired legacy recordingmicrophone characteristic from the 120-channel sphericalmicrophone array. Synthesized microphone-beams areshown and compared with the measured characteristicsof the original legacy microphones.
IntroductionSpherical microphone array technology allows for sound-field analysis and beamforming, and the creation ofdynamic beams in any desired direction. Althoughflexible beamforming features may be desired in audioproduction, Tonmeisters tend to prefer first-order legacyrecording microphones over novel microphone arrays.This preference is probably related to the familiarityin working with recording microphones. Contraryto engineering measurement applications that requireneutral microphones with a flat frequency response, insound recording, Tonmeister often desire microphoneswith “character”, expressed through parameters such assensitivity, nonlinear distortions and off-axis frequencyresponse. Consequently, the selection and placement ofthe “right” microphone becomes a highly subjective andirreversible decision.
Because beamforming technology can capture a sound-field with a flexible directivity and steering orientation,these kinds of preproduction decisions can be moved tothe production process.
Unfortunately, beamforming technology has a reputationfor not capturing the sound with the same character asrecording microphones. This might be due to the factthat beamforming algorithms often prioritize the creationof highly directed beams while timbral features are ofsecondary importance. Although the authors are notaware of studies on the importance of specific microphoneparameters on overall preference, research in soundreproduction quality suggests that timbral attributes areof higher importance than spatial attributes [1]. Also,
an array’s performance and sound character may beaffected by manufacturing differences of a large numberof material components, such as capsules, microphone-preamps, or AD-converters.
In this paper we use a spherical microphone array(the SAM-array) to synthesize the frequency-dependentdirectivity response of recording microphones. Wefirst measured the frequency response of a recordingmicrophone and the frequency response of our sphericalmicrophone array. Then, using a numerical approach,we compute the best-fit filters for a filter-and-sumbeamforming algorithm that matches the frequencyresponse of the recording microphone. Theoreticalresults are compared with real-measurements. Wealso show how this virtual recording microphonecan dynamically be re-oriented and how the desiredmicrophone’s directivity pattern can be modified at runtime.
One might wonder why we we aim to simulate a recordingmicrophone using a microphone array. As with many oldsynthesizers, tube amplifiers and other electro-acousticalinstruments, there is a desire to simulate the complexbehavior of rare, expensive and sensitive musical devicesto enable their use in today’s digital audio workstationenvironments.
A decade ago, the company Antares1 released theMicrophone Modeler. This audio effect strived tovirtually change the brand and type of the microphoneused in a mono-channel recording. The user defines a)what microphone model was used within a recordingand b) what microphone model is desired. Then thealgorithm aimed to match the frequency response ofboth microphones to create the desired effect. Sincethe Microphone Modeler is discontinued, it was probablycommercially not very successful. The algorithm onlymodeled the on-axis frequency response of the desiredmicrophones using a one-channel input signal, neglectingother crucial microphone parameter such as the off-axisfrequency response.
1http://www.antarestech.com/products/amm.shtml
A recording microphoneMicrophones are categorized according to their idealdirectivity Γ in classes as listed in Table 1. Thedirectivity can be computed with Equation 1 with theangle of incidence δ.
Omni Cardioid Hyper- Figure-8directional cardioid
a = 1.0 0.5 0.3 0.0
Table 1: Common first-order microphone directivity pattern
Γi = a+ (1− a) · cos δi 0 ≤ a ≤ 1. (1)
Manufacturers usually supply octave-smoothed, two-dimensional cross-sections of the directivity pattern as areference, because the directivity depends on frequencyand microphone model. We measured the directivity of apopular large-diaphragm cardioid condenser microphone(2200 US$) from 576 directions in an anechoic chamber.The measurement procedure is described later inthis paper. The horizontal polar patterns of themeasurement compared with the specification sheet bythe manufacturer shows a lot of resemblance across theentire frequency range (see Figure 1). However, thesehorizontal polar pattern do not reveal common axialasymmetries in the directivity, which are only visible inthe spherical ballon plots (Figure 2).
A transformation of this directivity pattern into sphericalharmonics is visualized in Figure 3. One can seethat most prominent contributions are of zero and firstorder spherical harmonics. This is expected, since therecording microphone has a cardioid characteristic whichcan be expressed with zero and first order sphericalharmonics. Also clearly visible are the omnidirectionalcharacteristic in low frequency bands due to the highcontribution of the zero order harmonics, and an
increasing contribution of the first order harmonics in themid-frequencies around 1 kHz. Second order harmonicsstart to contribute at around 500 Hz to the the measureddirectivity pattern. It can be observed in Figure 2 howthe recording microphone becomes increasingly directedfor higher frequencies; the increase in directivity isexpressed through additional contributions of higherspherical harmonics.
10010 20
100
1020
20
0
20
125 Hz
100 10 2010
010
2020
0
20
250 Hz
0 10 2010
010
2020
0
20
500 Hz
0 10 2010
010
2020
0
20
1000 Hz
100 10 20
200
2020
0
20
2000 Hz
10010 20
100
102040
20
0
4000 Hz
10010 20
200
2020
10
0
10
8000 Hz
100
1020
5051020
10
0
10
16000 Hz
Figure 2: Measured spherical directivity of the recordingmicrophone at different frequencies. Projection contour plotsat the floor show directivity at different cross-sections. Redcross indicates projected center of origin.
100 250 500 1000 2500 5000 10000
0
1
2
3
4
5
6
7
8
9
Frequency [Hz]
Ord
er
of
sp
he
rica
l h
arm
on
ics
Le
ve
l co
ntr
ibu
tio
n in
[d
B]
−30
−25
−20
−15
−10
−5
0
Figure 3: Frequency dependent contribution of differentorders of spherical harmonics to the overall directivity of themeasured recording microphone.
125 250 500 1k 2k 4k 8k 16k
mea
sure
dd
ata
shee
t
5
10
15
20
25
125Hz
5
10
15
20
25
250Hz
5
10
15
20
25
500Hz
5
10
15
20
25
1000Hz
5
10
15
20
25
2000Hz
5
10
15
20
25
4000Hz
5
10
15
20
25
8000Hz
5
10
15
20
25
16000Hz
1
Figure 1: Comparison of manufacturer’s directivity specification and measured directivity in horizontal plane (normalized and3rd octave-smoothed)
The SAM-arrayThe SphericAl Microphone array (SAM), a co-development of CNMAT and Meyer Sound Laboratories,consists of 144 microphones. The majority (120) arecardioid microphones (DPA 4080), normally placed on arigid sphere whose radius is about 4.5 cm. The distancebetween the rigid sphere and the actual microphonediaphragms is about 2.5 cm, resulting in an array radiusof about 7 cm. The spherical array order is aboutN = 9, estimated by the number of cardioid microphones(N =
√120 − 1). With that array order, the spatial
aliasing frequency is about 7 kHz, defined by kr ≈ N ,where k is the wavenumber and r the array radius. TheRayleigh resolution limit, which estimates the smallestangle where two proximate plane waves can be separated,is about π/N = 20◦. Because accurate capturingand processing of low frequencies is challenging witha small array diameter, additional 24 omnidirectionalmicrophones (DPA 4060) are mounted as outriggersaround the inner sphere in 3 different radii, ranging from200 cm to 1 m.
All microphone pre-amps and AD-converters are embed-ded inside SAM, permitting distribution of all 144 audiochannels over an ethernet connection in 24 Bit/96 kHz[2]. Additional field-programmable gate arrays (FPGA)for onboard signal processing are also available.
The matching approachTo synthesize the directivity of a recording microphone,we adopted a filter-and-sum beamformer approach byFarina et al. [3]. This approach is based on computinga set of FIR filters (hn) for the M microphones of amicrophone array. These filters modify the microphonesignals of the SAM-array so that its sum results in theoutput signal y (see Figure 5). The output signal y isa microphone beam whose directivity approximates thedirectivity pattern q of the desired recording microphone.
The frequency responses of the best-fit FIR filters (hn)are estimated in the spectral domain by solving the least-square equation 2 for each spectral bin k in a half-sided FFT spectrum. This approach requires that thefrequency responses of all M microphones from the SAM-array are measured from D source positions around asphere. These frequency responses are stored in thematrix C, and C∗ indicates its complex conjugate. Thesensitivity of the recording microphone with respect tothe same D source positions is represented by the vectorQ. If L different microphone beams are synthesized,Q become a matrix of size M × L. In contrast to [3],Q can vary across spectral bins k to account for thefrequency-dependent directivity pattern of the recordingmicrophone. One advantage of this numerical approach isthat C comprises the effects of sensor misplacements andindividual component irregularities of the microphonearray. Therefore the resulting FIR filters hn ought tocompensate these irregularities.
Figure 4: The SAM-array with outriggers (upper picture),and close-ups of the onboard electronics (left) and of the innersphere (right).
h1(t)
h2(t)
h120 (t)
+.:
x1(t)
x2(t)
x120(t)
.:
y(t)
Figure 5: Filter-and-Sum beamforming principle
[Hk]M =[Ck]∗M×D ·WM×M · [Qk]D · e−jπk
[Ck]∗M×D ·WM×M · [Ck]D×M + βk · [I]M×M(2)
WeightingWe added the weighting matrix W to Equation 2 toaccount for an unevenly spherical distribution of themeasurement positions D. The measurements (seenext section) employed an equi-angle sampling gridcharacterized by a relative increase in sampling densityfrom the equator to the poles. In estimating the best-fit filter response without weighting, the least-squaresolution would favor the oversampled poles comparedto the undersampled equator region. Therefore, a sinweighting function was used. Other sampling schemesrequire different weighting functions (see [4] for anoverview).
RegularizationThe regularization parameter βk controls the tradeoffbetween accuracy and robustness of an ill-conditionedinversion of (C∗
k ·W ·Ck). Setting βk = 0 producesthe best accuracy but has the potential that smallsampling errors cause instabilities and excessive signalamplification in Hk. This is especially common on asmall array for low frequencies due to sampling errors.Several approaches to estimate an optimal regularizationparameter β are known [5]. We implemented analgorithm that selects β based on the condition numberκk (Equation 3). The condition number κk is known tobe a measure of a matrix’s ease of invertibility, and is 1for an orthogonal matrix and >> 1 for an ill-conditionedmatrix. For each spectral bin k and starting at βk = 0,our iterative algorithm increases βk until the conditionnumber κ reaches a predefined value. This predefinedvalue can depend on frequency-dependent, similarly towhat is suggested in 2.
κk = cond([Ck]
∗M×D ·WM×M · [Ck]D×M
)(3)
Steering of the synthesized microphoneChanging the orientation of the synthesized microphonebeam can be expresses as modifying Q with regardsto the steering angle. The correct modification can beperformed via coordinate transforms of the microphone’sspherical base solutions (Figure 3) as described in [6] or[7]. Another less efficient strategy is to compute Q byrotating and interpolating the sampled polar pattern.
Recomputing entirely Equation 2 every time Q changeswould be too expensive for real-time applications.Because a majority of this computation is independentfrom Q, Equation 2 can be simplify to:
[Hk]M = [Ak]M×D · [Qk]D · e−jπk (4)
with
[Ak]M×D =[Ck]∗M×D ·WM×M
[Ck]∗M×D ·WM×M · [Ck]D×M + βk · [I]M×M(5)
The matrix Ak in Equation 5 is independent fromthe desired beam directivity Q and can therefore beprecomputed. Thus, when Q changes, the computationalcost of Hk can be reduced to one matrix-vectormultiplication per spectral bin k. The directivityvector Q can also manipulated in different ways: byraising Q to the power of a number larger than 1increases the directivity of the beam. As a side note,this algorithm scales easily on many-core processorarchitectures because computing Hk is independent foreach k.
MeasurementsTo apply the matching approach discussed in theprevious section, impulse responses from the SAM-arrayand the recording microphone were measured from 576positions in an anechoic chamber. Both measurementswere facilitated by using a robot arm that rotates themicrophone device under test in azimuth and elevationangle with regards to a fixed reference loudspeaker, aMeyer Sound HD-1. The distance of the loudspeaker tothe microphone device was 4 m, sufficiently satisfying thefar-field assumption. The measurement setup is depictedin Figure 6. For measuring the recording microphone, themic-preamp and AD converter of the RME Fireface wereused. The 120 audio channels of the SAM-array weredigitally streamed directly over an ethernet connectionto the computer.
For both the SAM-array and the recording microphone,an equi-angle sampling scheme was used. The spatialsampling resolution was 10 degrees in both azimuth andelevation angle. Unfortunately, both devices could notbe measured from all spherical directions, because at 2elevation angles, the robot arm would have interferedwith the propagation path. Therefore, only 36 x 16instead of 36 x 18 were collected. In the case of the SAM-array, for each of these 576 measurements, the responseof all 120 cardioid microphones were captured. Noneof the 24 outriggers were measured. For both devices,the impulse responses were computed in 24 Bit/96 kHzfrom a 1 second exponential sine sweep [8] ranging from100 Hz to 16 kHz. The high sampling rate of 96 kHzis beneficial for accurately measuring the relative time-of-arrival differences between the array microphones.We did not measure frequency components lower than100 Hz because beamforming processing with a smallarray diameter is prone to low frequency measurementerrors. A future measurement session will include the 24outrigger microphones and address these frequencies.
Anechoic Chamber
DACRME Fireface
Sine Sweep
SAMLoudspeaker
computer
HD
Robot Arm
Control Data
Digital Audio
wordclock sync
Figure 6: Sketch of the measurement setup
ResultsThis section presents four results using the matchingapproach. Two different microphones are re-created:first, an ideal cardioid microphone whose frequency-independent directivity pattern was computed withEquation 1; second, the real-world recording microphonewhose measured frequency-dependent directivity patternis shown in Figure 2. We first study the matchingalgorithm under ideal conditions, using a simulatedSAM-array. Afterward, the frequency responses of thereal SAM-array are used. For objective comparisons,all results are computed with a regularization parameterβ = 0.0. The FFT size is 2048 samples at 96 kHz.
Simulated SAMWe simulate the 120 cardioid capsules of the SAM-array using the ViMiC system [9]. ViMiC is a 3Dauditory virtual environment where sound sources andsinks in the form of microphones can be freely placed in avirtual room. According to parameters for characterizingsources, room, and microphones, the virtual microphonesignals are computed with the proper time-of-flightdelays and level attenuation of direct sound andoptional early reflections. The simulation approximatesthe SAM-array with frequency-independent cardioidsmicrophones and without the inner rigid sphere. Thelatter approximation is reasonable because the modalmagnitude responses between an open cardioid array anda cardioid array with a rigid sphere are reasonably similar[10].
Figure 7 shows the matching results of an ideal cardioidmicrophone onto the simulated microphone array. Atfirst, one can see in Figure 7(b) that the ideal cardioidmicrophone is very well re-created up the 4 kHz. Abovethis frequency, the computed beam continues to bedirectional, but matching errors start to distort thesmooth cardioid directivity pattern. Figure 7(a) showsthe normalized magnitude response of the 120 arrayfilters h computed with Equation 2. These filters havetheir largest amplification around 10 kHz - a region wherethe matching approach according to Figure 7(b) fails towork. Also, for low frequencies, a few filters attenuatethe input signal by about 80 dB and can be attributed tothe high condition number of the ill-conditioned matrix(see Equation 3). A frequency-dependent regularizationas shown in Equation 2 will prevent those extremeattenuation and amplification values at the expense ofa less accurate directivity matching at those frequencies.
Figure 8 shows the matching results of the measureddirectivity response of the legacy recording microphoneonto the simulated microphone array. A visualcomparison of the measured directivity pattern (Figure2 with Figure 8(b)) indicates that the character of therecording microphone is very well re-created up the4 kHz. Beginning with 8 kHz, the directivity pattern isgetting more and more distorted, also observable in thefloor projections of the directivity cross-sections in Figure8(b).
102
103
104
−80
−60
−40
−20
Magnitude and Phase of all Mics
Ma
gn
itu
de
[dB
]
102
103
104
−8
−6
−4
−2
0
2x 10
4
Frequency [Hz]
Ph
ase
[d
eg
]
Frequency [Hz]
010
20
−20
−10
0
10
20
−40
−30
−20
−10
0
10
20
x
Target
y−10
010
20
−20
−10
0
10
20
−40
−30
−20
−10
0
10
20
x
Result
y
0 1020−20
020
−40
−20
0
20
x
125 Hz, Bin 4
y0 1020
−200
20
−40
−20
0
20
x
250 Hz, Bin 6
y0 1020
−200
20
−40
−20
0
20
x
500 Hz, Bin 12
y0 1020
−200
20
−40
−20
0
20
x
1000 Hz, Bin 22
y
−1001020
−200
20
−40
−20
0
20
x
2000 Hz, Bin 44
y −100102030
−200
20
−40
−20
0
20
x
4000 Hz, Bin 86
y−100
102030
−200
20
−40
−20
0
20
x
8000 Hz, Bin 172
y−100
1020−100
10
−40
−20
0
20
x
16000 Hz, Bin 342
y
(a) Magnitude responses of the computed filters
010
20
−20
0
20
−40
−20
0
20
125 Hz
010
20
−20
0
20
−40
−20
0
20
250 Hz
010
20
−20
0
20
−40
−20
0
20
500 Hz
010
20
−20
0
20
−40
−20
0
20
1000 Hz
−100
1020
−20
0
20
−40
−20
0
20
2000 Hz
−100
1020
30
−20
0
20
−40
−20
0
20
4000 Hz
−100
1020
30
−20
0
20
−40
−20
0
20
8000 Hz
−100
1020
−100
10
−20
0
20
16000 Hz
(b) Resulting directivity plots for different frequencies. The colorindicates the degree of error between desired directivity and achieveddirectivity pattern.
Figure 7: Synthesizing an ideal cardioid microphone withthe simulated SAM-array
102
103
104
−100
−80
−60
−40
−20
0Magnitude and Phase of all Mics
Ma
gn
itu
de
[dB
]
102
103
104
−8
−6
−4
−2
0
2x 10
4
Frequency [Hz]
Ph
ase
[d
eg
]
Frequency [Hz]
−100
1020
−10
0
10
−20
−10
0
10
x
Target
y −10
0
10
20
−10
0
10
−20
−10
0
10
x
Result
y
−505 1015
−100
10
−20
−10
0
10
x
125 Hz, Bin 4
y −1001020
−100
1020
−40
−20
0
20
x
250 Hz, Bin 6
y0 1020
−200
20
−40
−20
0
20
x
500 Hz, Bin 12
y−100
1020−20
020
−40
−20
0
20
x
1000 Hz, Bin 22
y
−1001020
−200
20
−40
−20
0
20
x
2000 Hz, Bin 44
y−100
1020
−200
20
−40
−20
0
x
4000 Hz, Bin 86
y−100
1020−20
020
−40
−20
0
20
x
8000 Hz, Bin 172
y−5
05
10
−50
5−10
−5
0
5
x
16000 Hz, Bin 342
y
(a) Magnitude responses of the computed filters
−50
510
15
−10
0
10
−20
−10
0
10
125 Hz
−100
1020
−100
1020
−20
0
20
250 Hz
010
20
−20
0
20
−40
−20
0
20
500 Hz
−100
1020
−20
0
20
−40
−20
0
20
1000 Hz
−100
1020
−20
0
20−20
0
20
2000 Hz
−100
1020
−20
0
20
−40
−20
0
4000 Hz
−100
1020
−20
0
20
−20
0
20
8000 Hz
05
10
−5
0
5
−10
−5
0
5
16000 Hz
(b) Resulting directivity plots for different frequencies. The colorindicates the degree of error between desired directivity and achieveddirectivity pattern. Floor projections show directivity pattern atdifferent cross-sections.
Figure 8: Synthesizing the measured recording microphonewith the simulated SAM-array
Measured SAMIn this section we create microphone beams based on themeasured frequency responses of the SAM-array.
Figure 9 depicts the results of creating a beam withan ideal cardioid characteristic. A visual evaluation ofFigure 9(b) shows that the matching algorithm workswell for frequencies up to 2 kHz. Above this frequency,the results degrade in the same way as described inthe previous section: the beam is still directed andsteers into the right direction, but the smoothness ofthe cardioid directivity pattern increasingly distorts withhigher frequencies. Figure 9(a) shows the normalizedfrequency response of the 120 generated array filters. Upto about 1.5 kHz, the magnitude responses are between−40 dB and 0 dB (despite a few filters which have a largerattenuation at low frequencies). From 1.5 kHz to 4 kHz,the magnitude responses steadily decrease from −40 dBto −80 dB. Remarkably, this magnitude drop happens inthe frequency range where the matching errors start toaffect the re-created directivity pattern (see Figure 9(b)).
Figure 10 depicts the results of re-creating the legacyrecording microphone. Up to 2 kHz, the legacy recordingmicrophone is reasonably well re-created with the SAM-array, see Figure 10(b). Again, at higher frequencies,the matching errors increase and the directivity patternbecomes distorted. The magnitude responses of thecomputed filters, shown in Figure 10(a), are similar, yetnot equal, compared to those for re-creating the idealcardioid microphone in Figure 9(a). Compared to thoseof the simulated SAM-array in Figure 8(a), the filterslook different, which could be due to the frequency-dependent sensitivity of the SAM-array sensors.
DiscussionThe matching results of the simulated SAM-array(Figure 7 and 8) compared to those achieved with themeasured SAM-array (Figure 9 and 10) are similar inaccuracy for low and mid frequency up to about 2 kHz.Above 2 kHz, the matching accuracy using the measuredSAM-array data decreases. With the simulated SAM-array, such decrease in matching accuracy starts oneoctave later at about 4 kHz.
As a measure of overall matching accuracy, we introducethe mean error εk, which we define as the differencebetween the desired directivity Q and the computeddirectivity of the beam, averaged over all D samplingpositions (Equation 6). This mean error ε is frequencydependent as denoted through the index k.
εk =1
D
[D=576∑d=1
∣∣∣∣∣Qk(d)−M=120∑i=1
Ck(d, i) ·Hk(i)
∣∣∣∣∣]
(6)
Figure 11 shows ε as a function of the frequency forall four matching results. Both simulated results havea mean error below 1 dB up to 4 kHz. For higherfrequencies, the errors increase fast. The mean errorsof the results using the measured SAM-array data aresimilar to those of the simulated array up to about 1 kHz.
103
104
−100
−80
−60
−40
−20
0Magnitude and Phase of all Mics
Ma
gn
itu
de
[dB
]
103
104
−5
−4
−3
−2
−1
0
1x 10
4
Frequency [Hz]
Ph
ase
[d
eg
]
Frequency [Hz]−10
010
20
−20−10
010
20
−40
−30
−20
−10
0
10
20
Target
−100
1020
−20
−10
0
10
20
−40
−30
−20
−10
0
10
20
Result
−1001020
−200
20
−40
−20
0
20
150 Hz, Bin 4
−1001020
−200
20
−40
−20
0
20
250 Hz, Bin 6
−1001020
−200
20
−40
−20
0
20
500 Hz, Bin 12
−1001020
−200
20
−40
−20
0
20
1000 Hz, Bin 22
−100102030
−200
20
−40
−20
0
20
2000 Hz, Bin 44
−100102030
−200
20
−40
−20
0
20
4000 Hz, Bin 86
−1001020
−200
20
−40
−20
0
20
8000 Hz, Bin 172
−1001020
−200
20
−40
−20
0
20
16000 Hz, Bin 342
(a) Magnitude response of the 120 computed filters h
−100
1020
−20
0
20
−40
−20
0
20
150 Hz
−100
1020
−20
0
20
−40
−20
0
20
250 Hz
−100
1020
−20
0
20
−40
−20
0
20
500 Hz
−100
1020
−20
0
20
−40
−20
0
20
1000 Hz
−100
1020
30
−20
0
20
−40
−20
0
20
2000 Hz
−100
1020
30
−20
0
20
−40
−20
0
20
4000 Hz
−100
1020
−20
0
20
−40
−20
0
20
8000 Hz
−100
1020
−20
0
20
−40
−20
0
20
16000 Hz
(b) Resulting directivity plots for different frequencies. The colorindicates the degree of error between desired directivity and achieveddirectivity pattern.
Figure 9: Synthesizing an ideal cardioid microphone withthe SAM-array
103
104
−100
−80
−60
−40
−20
0Magnitude and Phase of all Mics
Ma
gn
itu
de
[dB
]
103
104
−5
−4
−3
−2
−1
0
1x 10
4
Frequency [Hz]
Ph
ase
[d
eg
]
Frequency [Hz]
−100
1020
−10
0
10
−20
−10
0
10
Target
−10
010
20
−10
0
10
−20
−10
0
10
Result
010
20
−100
10
−20
−10
0
10
150 Hz, Bin 4
−1001020
−100
1020
−40
−20
0
20
250 Hz, Bin 6
−1001020
−200
20
−40
−20
0
20
500 Hz, Bin 12
−1001020
−200
20
−40
−20
0
20
1000 Hz, Bin 22
−1001020
−200
20
−40
−20
0
20
2000 Hz, Bin 44
−1001020
−200
20
−40
−20
0
4000 Hz, Bin 86
−1001020
−200
20
−40
−20
0
20
8000 Hz, Bin 172
−50
510
−50
5
−10
−5
0
5
16000 Hz, Bin 342
(a) Magnitude responses of the 120 computed filters hn
010
20
−10
0
10
−20
−10
0
10
150 Hz
−100
1020
−100
1020
−20
0
20
250 Hz
−100
1020
−20
0
20
−40
−20
0
20
500 Hz
−100
1020
−20
0
20
−40
−20
0
20
1000 Hz
−100
1020
−20
0
20
−20
0
20
2000 Hz
−100
1020
−20
0
20
−40
−20
0
4000 Hz
−100
1020
−20
0
20
−20
0
20
8000 Hz
−50
510
−50
5
−10
−5
0
5
16000 Hz
(b) Resulting directivity plots for different frequencies. The colorindicates the degree of error between desired directivity and achieveddirectivity pattern.
Figure 10: Synthesizing the measured recording microphonewith the SAM-array
Then, the matching error of the results based on themeasured SAM-array increases to about 4 dB at 4 kHz.
Besides sampling errors due to spatial aliasing, whichare expected for frequencies above 8 kHz, it is unclearwhat the cause is for the divergence of the mean
102
103
104
0
1
2
3
4
5
6
7
8
9
10
Mean e
rror
[dB
]
Frequency [Hz]
Ideal mic−array synthesizing ideal cardioid
Ideal mic−array synthesizing measured reference microphone
Measured mic−array synthesizing ideal cardioid
Measured mic−array synthesizing measured reference microphone
Simulated IR gain error ± 3dB
Simulated IR alignment error ± 1 sample @ 44.1 kHz
Figure 11: Mean error comparison of the matching resultsusing simulated and real measured data
error. To find an answer to that question, we simulatedtwo different errors which could have occurred in theprocess of measuring SAM’s frequency responses: first, a±3 dB sensitivity error randomly added to the frequencyresponses in c; second, a temporal jitter of 1 samplerandomly introduced to c which corresponds to about0.8 cm flight distance at the speed of sound. The effectof these simulated errors on ε are shown with dashedlines in Figure 11. Compared to the mean error ofthe simulated SAM-array, the random sensitivity errorconstantly increases the mean error by about 2 dB fromlow to high frequencies. At 5 kHz both error curves startto converge again. A temporal jitter of ±1 sample cansignificantly affect the matching accuracy starting at midfrequencies of about 500 Hz and rises to 5.5 dB at 4 kHz.This error simulation suggests that the time alignmentof the measured frequency responses is important andneeds to be carefully controlled to achieve the desiredbeam directivity for higher frequencies. The mean errorof matching results using the measured SAM-array hasa similar progression, but starts one octave later. Thiscould indicate a small jitter error in the measurements.
Conclusion and future workThis paper described a numerical matching approach tosynthesize legacy recording microphones with a sphericalmicrophone array. We explained the motivation for thiswork and described the mathematical concepts to findthe best filters for driving the microphone array. Using a120-channel microphone array, results of this approachshow that it is possible to synthesize the frequency-dependent directivity pattern of a recording microphonefor a wide frequency band. For high frequencies, thisapproach starts to fail. Simulated measurement errorssuggest that for best matching results, it is crucialto capture the frequency responses of the sphericalmicrophone array with best temporal accuracy.
Future work will focus on increasing the matchingaccuracy for higher frequencies and to synthesize othermicrophone types. The discussed results assume thatthe recording microphone is used as a main microphonewith sources in the far-field. Because many recordingmicrophones are used as spot microphones with sound
source in close proximity, it would be interesting tosynthesize microphones for this use case too.
Clearly, the flexible beamforming and beamsteeringcapabilities of microphone arrays can be useful insound recording applications. When evolving from first-order microphones to the new generation of higher-order spherical microphone arrays for sound recordingpurposes, we need to understand what the salientparameters of recording microphones are. Is it thefrequency-dependent directivity, the non-linearities, or isit the wooden box inside which an expensive microphoneis packaged? Maintaining or even improving any of thesesalient parameters in the design of microphone arrays andbeamforming algorithms will help to increase the use ofmicrophone arrays in musical applications.
AcknowledgementThanks to Meyer Sound Laboratories for access to theiranechoic chamber and technical support, and to LorenzoChiesi for clarifying [3]. Nils Peters is supported by theGerman Academic Exchange Service (DAAD).
References[1] F. Rumsey, S. Zielinski, R. Kassier, and S. Bech, “On
the relative importance of spatial and timbral fidelitiesin judgments of degraded multichannel audio quality,” J.Audio Eng. Soc., vol. 118, no. 2, pp. 968–976, 2005.
[2] R. Avizienis, A. Freed, T. Suzuki, and D. Wessel,“Scalable connectivity processor for computer musicperformance systems,” in International Computer MusicConference, pp. 523–526, 2000.
[3] A. Farina, A. Capra, L. Chiesi, and L. Scopece,“A spherical microphone array for synthesizing virtualdirective microphones in live broadcasting and in postproduction,” in Proc. of the AES 40th InternationalConference, (Tokyo, Japan), 2010.
[4] F. Zotter, “Sampling strategies for acoustic holog-raphy/holophony on the sphere,” in Fortschritte derAkustik: DAGA Tagungsband, (Rotterdam, Nether-lands), 2009.
[5] P. Hansen, Rank-deficient and discrete ill-posed prob-lems: numerical aspects of linear inversion. No. 4,Society for Industrial Mathematics, 1998.
[6] N. Gumerov and R. Duraiswami, “Fast, exact, andstable computation of multipole translation and rotationcoefficients for the 3-d helmholtz equation,” 2001.
[7] F. Zotter, Analysis and Synthesis of Sound-Radiationwith Spherical Arrays. PhD thesis, University for Musicand Dramatic Arts, Graz, Austria, 2009.
[8] A. Farina, “Simultaneous measurement of impulseresponse and distortion with a swept-sine technique,” in108th AES Convention, Preprint 5093, (Paris, France),2000.
[9] J. Braasch, N. Peters, and D. L. Valente, “A loudspeaker-based projection technique for spatial music applicationsusing virtual microphone control,” Computer MusicJournal, vol. 32, no. 3, pp. 55–71, 2008.
[10] P. Plessas and F. Zotter, “Microphone arrays aroundrigid spheres for spatial recording and holography,” inFortschritte der Akustik: DAGA Tagungsband, (Berlin,Germany), 2010.
