Convention Paper Presented at the 126th Convention 2009 May 7 10 Munich, Germany

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1 Audio Engineering Societ Convention Paper Presented at the th Convention 9 Ma 7 Munich, German The papers at this Convention have been selected on the basis of a submitted abstract and etended precis that have been peer reviewed b at least two qualified anonmous reviewers. This convention paper has been reproduced from the author s advance manuscript, without editing, corrections, or consideration b the Review Board. The AES takes no responsibilit for the contents. Additional papers ma be obtained b sending request and remittance to Audio Engineering Societ, East nd Street, New York, New York 5-5, USA; also see All rights reserved. Reproduction of this paper, or an portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Societ. Design and Limitations of Non-Coincidence Correction Filters for Soundfield Microphones Christof Faller and Mihailo Kolundžija Illusonic LLC, Chemin du Trabandan A, Lausanne, Switzerland Audiovisual Communications Laborator, EPFL, 5 Lausanne, Switzerland Correspondence should be addressed to Christof Faller (christof.faller@illusonic.com) ABSTRACT The tetrahedral microphone capsule arrangement in a Soundfield microphone captures a so-called A-format signal which is then converted to a corresponding B-format signal. The phase differences between the A- format signal channels due to non-coincidence of the microphone capsules cause errors in the corresponding B-format signals and linear combinations thereof. Various strategies for designing B-format non-coincidence correction filters are compared and limitations are discussed.. INTRODUCTION A Soundfield microphone [,] captures four signals with sub-cardioid microphone capsules arranged as a tetrahedral microphone arra []. These subcardioid signals (often denoted A-format) are then converted to the well known B-format, which corresponds to signals with an omnidirectional characteristic (W signal) and three signals with dipole characteristic (, Y, and Z signals). The sub-cardioid microphone capsules in a Soundfield microphone are arranged as coincidentall as possible, but nevertheless gain and directional response errors result from the phase differences between the microphone capsule signals when converted to B-format. It has been claimed previousl that errors due to non-coincidence can be corrected [ ]. While a - b- filter matri could be optimized for the task, it has been argued that a -b- scalar matri followed b an equalization filter for each of the B-format signals (W,, Y, and Z) is sufficient and simpler to determine []. In this paper, we derive the theor of the noncoincidence correction proposed b Gerzon []. Based on simulation of a Soundfield microphone, the effect of the non-coincidence is illustrated and

2 least mean squares optimal non-coincidence correction filters are proposed. Practical considerations are discussed and it is shown that nearl optimal non-coincidence correction filters can be computed based on a small number of microphone measurements. The theor and simulation data presented in this paper also clarif the limitations of non-coincidence correction for Soundfield microphones. It is shown that while in the =, =, and z = planes errors due to non-coincidence can be corrected up to high frequencies, in other planes the performance of the microphone is worse despite of non-coincidence correction. The paper is organized as follows. Section describes a Soundfield microphone, A-format, and B- format signals in detail. It also derives the A- format (microphone capsule signals) to B-format scalar conversion matri assuming perfect coincidence. Non-coincidence correction is discussed from various points of view in Section, including theoretical results, simulations, and practical was of determining non-coincidence correction filters. The conclusions are given in Section.. SOUNDFIELD MICROPHONE, A-FORMAT, AND B-FORMAT A Soundfield microphone uses four cardioid capsules whose membrane centers are arranged at the tips of a tetrahedron pointing outwards [], as is illustrated in Figure. According to [], the cardioid capsules are configured to ield sub-cardioid polar pattern ( (+cosφ)), which has the advantage that off ais response is better than for the cardioid case. Figure illustrates the microphone capsule center positions relative to the,, and z aes. Note that the aes are defined such that the B-format signal corresponds to a dipole pointing towards the positive -ais. Similarl, the B-format Y and Z signals are oriented towards the positive and z aes, respectivel. The coordinates of the microphone capsule center positions are p LF = dv LF p RB = dv RB p LB = dv LB p RF = dv RF, () Fig. : Sub-cardioid microphone capsule arrangement in a Soundfield microphone (picture source: []). where d in meter is the distance of the origin to the edges of the tetrahedron, v LF = [ ] v RB = [ ] v LB = [ ] v RF = [ ], () are unit vectors defining the look directions of the microphone capsules, and is the transpose operation. Throughout this paper we use d =.7 cm []. The A-format is converted to B-format b W(t) S LF (t) (t) Y (t) = M S RB (t) AB S LB (t), () Z(t) S RF (t) where t is the time variable and S LF (t), S RB (t), S LB (t), and S RF (t) are the microphone capsule signals. M AB is a -b- matri which is derived in the following for the coincident case (d = ). AES th Convention, Munich, German, Ma 7 Page of

3 z LF (Up) RF (Down) LF (Up) RB (Up).... LB (Down) RB (Up) RF (Down) LB (Down) (a) Fig. : Microphone capsule center positions: (a) top view and (b) side view. A large dot indicates a position above and a small dot a position below the planes defined b the shown aes. (b) A signal with a directional response of a+( a)cos φ pointing towards azimuth θ and elevation φ can be obtained b combining the B-format signals: S(t) = aw(t) + a v θ (t), φ Y (t), () Z(t) where v θ, φ is a unit vector pointing towards azimuth θ and elevation φ, i.e. v θ, φ = [cosθ cosφ sin θ cosφ sin φ ]. (5) From () it follows that the matri converting B- format signals to the microphone capsule signals (Aformat) is M BA = = a a a a a a a a ( a) v LF ( a) v RB ( a) v LB ( a) v RF a a a a a a a a a a a a, () where the constant a is chosen according to the specification of the microphone capsules. The original Soundfield microphone uses a = [], which corresponds to the sub-cardioid directional response shown in linear scale in Figure. Fig. : Sub-cardioid directional response in linear scale. The matri for converting the microphone capsule signals (A-format) to B-format signals () is the inverse of (), i.e. with M AB = M BA = b = a a a a b b b b b b b b b b b b, (7) ( a). () Figure shows ideal directional responses of the W and signals. Note that all dipole directional responses shown in this paper are normalized such that the maimum gain is db ( in the linear scale which is shown) as opposed to db according to B-format definition. In the following, it is shown that noncoincidence of the microphone capsules will impair the directional responses of the B-format signals.. NON-COINCIDENCE CORRECTION Due to the non-coincidence the directional responses of the B-format signals deviate from the ideal case shown in Figure. Note that from the smmetr of the setup (microphone capsule positions, coordinate sstem) it follows that: AES th Convention, Munich, German, Ma 7 Page of

4 .5.5 W Fig. : Polar plots of the directional responses of the omni-directional W and dipole signals in linear scale. The directional responses of W measured in the = and = planes are identical to the directional response of W measured in the horizontal (z = ) plane. The directional response of measured in the = plane is identical to the directional response of measured in the z = plane. The directional response of Y measured in the = plane is identical to the directional response of Y measured in the z = plane. The directional response of Z measured in the = plane is identical to the directional response of Z measured in the = plane. The D directional responses of Y and Z are identical to the directional response of, if Y and Z are rotated accordingl. Gerzon proposed [] appling a scalar matri (7) for A-format to B-format conversion, followed b a filter for each B-format channels, i.e. W(t) = h W (t) W(t) (t) = h (t) (t) Ỹ (t) = h Y (t) Y (t) Z(t) = h Z (t) Z(t), (9) where denotes linear convolution and h W (t), h (t), h Y (t), and h Z (t) are the non-coincidence correction filters. Assuming ideal identical microphone capsules positioned precisel, Gerzon showed that h Z (t) = h Y (t) = h (t) holds, which also follows from the previousl mentioned B-format properties due to smmetr. Section. gives a detailed derivation of Gerzon s proposed non-coincidence correction filters in a generalized form (not onl for cardioid capsules) and discusses the results. In Section., a Soundfield microphone is simulated and least mean squares optimal non-coincidence correction filters are estimated. Considerations on what is a good approach to use in practice, based on microphone measurement, are discussed in Section.... Theoretical derivation of non-coincidence correction A monochromatic far-field sound with temporal frequenc ω, propagating from the direction defined b a unit vector u = [cosθ cosφ sin θ cosφ sin φ], can be described b a plane wave p(r, t) = Ae j(ωt k r), () where A is a comple amplitude (conveing the information about both the real amplitude and the initial phase), and k = ku is the wave vector that points toward the direction of sound propagation with intensit k = ω c, where c is the speed of sound. The signals captured b the four microphone capsules of a Soundfield microphone can be epressed as S LF (t) = (a + ( a)v LF u)p(p LF, t) S RB (t) = (a + ( a)v RB u)p(p RB, t) S LB (t) = (a + ( a)v LB u)p(p LB, t) S RF (t) = (a + ( a)v RF u)p(p RF, t).() Using the decomposition of a plane wave into a sum of spherical waves (e.g., see []) e jkd(u v) = j n (n + )P n (u v)j n (kd), () n= where P n () is the nth-degree Legendre polnomial and j n () is the nth-order spherical Bessel function of the first kind, together with the A-format to B- format conversion formula given in (), one can epress signals W and as W = Ae jωt n= = Ae jωt n= On W (kd) amn W (u) + ( a)nn W (u) On (kd) amn (u) + ( a)nn (u), () AES th Convention, Munich, German, Ma 7 Page of

5 n Mn W(u) NW n (u) M n (u) N n (u) cosθ cosφ cosθ cosφ sin θ cosφsin φ sin θ cos φsin φ sinθ cosφsin φ 5 cosθ cosφ cosθ cos φ + 5 cosθ cosφ(5 sin φ ) Table : Values of functions M W n (u), N W n (u), M n (u) and N n (u) up to the second order, epressed as linear combinations of spherical harmonics. where and On W n + () = jn j n () a (n + ) On () = j n j n (), ( a) M W n (u) = P n(u v LF) + P n(u v RB) + P n(u v LB) + P n(u v RF) N W n (u) = u v LF P n(u v LF) + u v RB P n(u v RB) + u v LB P n(u v LB) + u v RF P n(u v RF) M n (u) = P n(u v LF) P n(u v RB) P n(u v LB) + P n(u v RF) N n (u) = u v LF P n(u v LF) u v RB P n(u v RB) u v LB P n(u v LB) + u v RF P n(u v RF). O n W [db] O n [db] n= n= n= n= n= n= n= n= n= n= (a) Fig. 5: Magnitude frequenc responses of functions O W n (kd) (a) and O n (kd) (b) for different orders n. (b) The directional responses of W and are direction dependent linear combinations of On W (kd) and On (kd), respectivel. Figure 5 shows the magnitude frequenc characteristics of On W (kd) and On (kd) up to order n =. The data shown in the figure indicate that the signals W and are mostl determined b the terms of orders n up to two for frequencies up to about khz. Furthermore, for orders higher than two, the functions Mn W (u), Nn W (u), Mn (u), and Nn (u), whose values are given in Table, contain onl spherical harmonics of orders higher than one. Since the signals W and should be proportional to zeroth (constant) and first order (cos θ cosφ) spherical harmonics, respectivel, higher than first order spherical harmonics can be considered as aliasing distortions. Note also that, due to orthogonalit, high-order spherical harmonic terms do not affect Tables of spherical harmonics can be found in [5]. the non-coincidence correction of the zeroth and first order spherical harmonic responses of signals W and. Following these observations, the signals W and for a monochromatic far-field sound of frequenc ω that propagates from the direction defined b unit vector u can be simplified as follows: W(t) = (F W (ω) + R W (ω, θ, φ))p (t) (t) = ( cosθ cosφf (ω) + R (ω, θ, φ))p (t), () AES th Convention, Munich, German, Ma 7 Page 5 of

6 magnitude [db] phase [degree] W (a) Fig. : Magnitude (a) and phase (b) frequenc characteristics of non-coincidence correction filters H W and H according to Gerzon s theor. where F W (ω) = j ( ωd c F (ω) = j ( ωd c ) (b) ( ) ωd + j j c ) + j a a j ( ) ( ωd ωd j c c ) (5) are the Soundfield microphone s frequenc responses of zeroth- and first-order harmonics, respectivel, and the remaining terms R W (ω, θ, φ) and R (ω, θ, φ) contain onl higher-order spherical harmonic terms. For non-coincidence correction, one needs to invert the frequenc characteristics F W (ω) and F (ω). The non-coincidence correction filters h W (t) and h (t) are the inverse Fourier transform of H W (ω) = H (ω) = F W (ω) (ω). () F H W (ω) and H (ω) are shown for a = in Figure. The responses given in (5) are in accordance with the frequenc responses of zeroth- and first-order spherical harmonics of a Soundfield microphone using cardioid capsules (a = ), given b Gerzon in []. It should also be noted that non-coincidence correction b filters h W (t) and h (t) is effective up to some limiting frequenc at which aliasing terms, i.e. high-order spherical harmonics, start to dominate the directional responses of the signals W and. In [], Gerzon gives an approimation of the limiting frequenc in the form f l = /d khz (where d is epressed in cm), which for d =.7 cm gives a limiting frequenc f l = 7.5 khz. Above the limiting frequenc f l, Gerzon suggests correcting nominal zeroth- and first-order harmonic signals, W and, to have flat responses to diffuse random sound fields. It is also worth noting that even though the signal contains strong second-order spherical harmonic terms N (u) = cosθ cosφsin φ and M (u) = cosθ cosφsin φ (as observed from the last two rows of Table ), its directional response in the plane z = is not affected b them, since the spherical harmonic cosθ cosφsin φ equals zero in this plane. Due to smmetr, the same holds for the = plane. Also, the directional response of W is better in the =, =, and z = planes, since the large third order spherical harmonic term N W (u) is zero in these planes. Simulations shown in the following section confirm these observations... Non-Coincidence correction determined b simulation In this section, it is shown how to compute noncoincidence correction filters based on simulating the Soundfield microphone described in Section. Ideal sub-cardioid capsules are assumed with aiall smmetric directional responses. Directional responses are simulated considering far field sound. The noncoincidence correction filters are determined such that the B-format directional response errors are minimized in a least mean squares sense. The top two panels of Figure 7 show the directional responses of the B-format signals W and, respectivel, measured in the horizontal plane (z = ). There is no need to show the directional responses of Y and Z because the are identical up to a rotation to the directional response. Note that the omnidirectional W gain is decreasing and the dipole gains are increasing as frequenc increases, which is quantitativel in line with the theoretical analsis, the data shown in [], and the Soundfield microphone measurements given in [7]. Note that the gain changes relativel quickl as frequenc increases, but the shape of the directional responses is rather correct up to about khz. AES th Convention, Munich, German, Ma 7 Page of

7 W (z= plane).5.5 W (=z plane).5.5 Hz Hz Hz Hz Hz =z (z= plane) (=z plane) Fig. 7: Polar plots at various frequencies of directional responses of W and in the z = (top two panels) and = z (bottom two panels) planes. W phase error [degree] (z= plane) phase error [degree] (z= plane) W phase error [degree] (=z plane) phase error [degree] (=z plane) angle [degree] Fig. : The phase error of and W in the z = (top two panels) and the = z (bottom two panels) planes. The bottom two panels of Figure 7 show the directional responses of the same B-format signals but measured in the = z plane. Note that in the = z plane the shape of the directional responses of the W and signals degrade much more quickl than in the horizontal plane as frequenc increases. This is in line with the previousl derived theoretical results, which showed that large third and second order spherical harmonic terms in the W and signals, respectivel, vanish in the z = plane. The phase differences between the simulated W and signals and ideal corresponding signals are shown in Figure. Ecept at the lowest frequencies, there is a significant phase error. Figure 9 shows eamples of B-format decodings measured in the horizontal (z = ) plane where a cardioid is steered to different directions in the horizontal plane. The data shown impl that the gain and directional response are highl frequenc-dependent at and above khz. As mentioned previousl, the data shown in the top panels of Figure 7 indicate that the shapes of the B- format magnitude responses in the horizontal plane are close to desired, ecept for the highest frequencies. Thus, b means of a frequenc-dependent gain Center Front Right Hz.5 Hz Hz Hz Hz Front Left Rear Left Fig. 9: Polar plots at various frequencies of directional responses of horizontal (z = ) B-format decodings. AES th Convention, Munich, German, Ma 7 Page 7 of

8 correction one can obtain B-format signals with a magnitude response much more similar to the desired one. However, as has been shown, the directional responses measured in other planes can degrade much more quickl as frequenc increases. Noncoincidence correction b means of B-format frequenc-dependent gain correction will onl result in directional responses approimating the ideal responses at those frequencies and in those planes where the shape of the B-format directional response is correct. A good magnitude response alone is not enough, since phase differences between the B-format channels will result in errors in the magnitude responses of B-format decoded signals. Thus, phase differences between W and the, Y, and Z dipoles should be minimized if possible. The data shown in Figure indicate that the phase error of the B-format signals has usuall a significant non-zero mean for each frequenc, which if subtracted will reduce the average phase error. The Fourier transform of the filters h W (t) and h (t), which minimize the mean square error between the desired and actual directional responses, are magnitude [db] phase [degree] W (a) Fig. : Magnitude (a) and phase (b) frequenc characteristics of the least mean squares non-coincidence correction filters H W and H. of H W (ω) at high frequencies of Gerzon s filter is significantl larger than that of the corresponding least mean squares filter. The deviation of Gerzon s filters from the least mean squares filters above khz implies that the former are sub-optimal. (b) H W (ω) = H (ω) = R π R π π π d W(θ, φ)d W (θ, φ, ω) cos φdθdφ R π R π π π D W(θ, φ, ω)dw (θ, φ, ω)cos φdθdφ R π R π π π d (θ, φ)d (θ, φ, ω)cos φdθdφ R π R π, π π D (θ, φ, ω)d (θ, φ, ω)cos φdθdφ (7) Figure shows the magnitude of the corrected B- format directional responses corresponding to the responses shown in Figure 7. The directional responses in the horizontal (z = ) plane are similar to the desired responses (shown in Figure ) up to high frequencies. The directional responses in the = z plane also improve in terms of gain. respectivel, where θ is azimuth, φ is elevation, d W (θ, φ) and d (θ, φ) are the desired omnidirectional (W) and dipole () directional responses d W (θ, φ) = d (θ, φ) = cosθ cosφ, () and D W (θ, φ, ω) and D (θ, φ, ω) are the actual frequenc dependent comple directional responses. The magnitude and phase responses of H W (ω) and H (ω) are shown in Panels (a) and (b) of Figure, respectivel. Comparison with Figure indicates that above khz Gerzon s filters start to deviate from the least mean squares filters. The magnitude Figure shows the phase error of the corrected B- format signals. The data shown indicate that the phase correction eliminates the phase errors nearl ideall up to more than khz in the horizontal plane. The average phase error is also reduced in the = z plane. The eample decodings in the horizontal plane of the corrected B-format, shown in Figure, show that the directional response gains and shapes are greatl As mentioned earlier in the paper, Gerzon proposed correction of B-format at higher frequencies (> 7.5 khz) to have flat response to diffuse random sound fields, which corresponds to filters which deviate less from the least mean squares filters at these frequencies. AES th Convention, Munich, German, Ma 7 Page of

9 Center Front Left W (z= plane) (z= plane) W (=z plane).5 Hz Hz Hz Hz Hz =z (=z plane).5.5 Front Right.5.5 Hz Hz Hz Hz Hz Rear Left Fig. : Polar plots at various frequencies of directional responses of non-coincidence corrected W and in the z = (top two panels) and = z (bottom two panels) planes. W phase error [degree] (z= plane) phase error [degree] (z= plane) W phase error [degree] (=z plane) phase error [degree] (=z plane) angle [degree] Fig. : The phase error of non-coincidence corrected and W in the z = (top two panels) and the = z (bottom two panels) planes Fig. : Polar plots at various frequencies of directional responses of non-coincidence corrected horizontal (z = ) B-format decodings. improved compared to the non-corrected case shown in Figure 9. Similar decodings, but with 5 degrees elevation (φ = π ), are shown in Figure, indicating that decoded B-format signals are worse in planes other than the =, =, or z = planes. But note that the least mean squares non-coincidence correction filter also improves this case compared to not using non-coincidence correction filters... Practical Considerations Determination of B-format correction filters based on theor is problematic for various reasons. Practical constraints such as non-ideal microphone capsules, non-matching microphone capsules, limited positioning precision, and diffraction effects make filters determined under ideal assumptions deviate substantiall from the desired ideal filters. An advantage of the proposed least mean squares filters (7) is that the can be computed from actual measurements, i.e. D W (θ, φ, ω) and D (θ, φ, ω). Perfect smmetr can not be assumed in practice and it is favorable to compute separate noncoincidence correction filters for Y and Z, h Y (t) and h Z (t), based on measurement of D Y (θ, φ, ω) and D Z (θ, φ, ω). AES th Convention, Munich, German, Ma 7 Page 9 of

10 Center =z.5.5 Front Right Hz Hz Hz Hz Hz.5.5 Front Left =z Rear Left magnitude [db] phase [degree] 5 5 W (a) (b) =z.5 =z Fig. : Polar plots at various frequencies of directional responses of non-coincidence corrected elevated (z = ) B-format decodings. Farina proposed computation of B-format noncoincidence correction filters based on a single onais forward measurement per capsule []. This approach is much simpler in terms of measurement effort. The procedure is as follows: Measure magnitude and phase response of W relative to sound arriving from the positive - direction. Measure magnitude and phase responses of, Y, and Z relative to sound arriving from the positive,, and z directions, respectivel. Design the B-format correction filters such that their magnitude and phase responses are the inverse of the measured magnitude and phase responses and multipl the, Y, and Z filters with. Figure 5 shows the magnitude and phase responses of the non-coincidence correction filters obtained with a single measurement per filter. Comparison with Figure indicates that up to 7 khz these filters are ver similar to the least mean squares filters. Above 7 khz the magnitude correction of the single measurement based filter for W is larger than Fig. 5: Magnitude (a) and phase (b) frequenc characteristics of the single measurement based noncoincidence correction filters H W and H. the magnitude correction of the least mean squares filter. Thus, it is favorable to reduce the single measurement based filter magnitude at high frequencies to prevent ecessive high frequenc gain. Using the same Soundfield microphone simulation as previousl, Figure shows the directional responses of W and at various frequencies when using single measurement based non-coincidence correction. Corresponding B-format decodings in the z = and z = planes are shown in Figures 7 and, respectivel. Comparison of these B-format decodings with the corresponding ones using least mean squares correction filters, shown in Figures and, confirms too high gains at high frequencies (in the z = plane). As mentioned, this effect ma be reduced b reducing the gain of the W correction filter at high frequencies.. CONCLUSIONS Non-coincidence correction for Soundfield microphones was investigated in this paper. B-format non-coincidence correction filters determined based on Gerzon s theor were compared to filters determined b simulation and optimization. The best correction filters minimize the mean square error between the achieved and desired B-format directional responses. The practicall most useful approach is, as proposed b Farina, correction based on a single AES th Convention, Munich, German, Ma 7 Page of

11 W (z= plane) (z= plane) Center Front Left W (=z plane) Hz Hz Hz Hz Hz (=z plane) =z Front Right Hz Hz Hz Hz Hz =z Rear Left.5 =z Fig. : Polar plots at various frequencies of directional responses of non-coincidence corrected W and in the z = (top two panels) and = z (bottom two panels) planes. =z =z Fig. : Polar plots at various frequencies of directional responses of non-coincidence corrected elevated (z = ) B-format decodings. measurement per B-format channel. This simplified correction filter is as good as the least mean squares filter up to about 7 khz. Center Front Right.5.5 Hz Hz Hz Hz Hz Front Left.5.5 Rear Left Analsis and simulations presented in this paper clarif fundamental limitations of non-coincidence correction. The non-coincidence corrected B-format W signal measured in the =, =, and z = planes ields nearl ideal directional responses up to about khz. Similarl, the signal ields ver good results in the z = and = planes, the Y signal in the = and z = planes, and the Z signal in the = and = planes. In other planes, the errors of the directional responses of the B-format signals are higher. This can be eplained theoreticall b the absence of strong second and third order spherical harmonic terms in these planes Fig. 7: Polar plots at various frequencies of directional responses of non-coincidence corrected horizontal (z = ) B-format decodings. For B-format decoding, a Soundfield microphone performs best for sources in the z = plane and simultaneousl decoding in the same plane. Similar good performance is achieved for sources in the = and = planes when decoding in these planes. The results are worse for sources in other planes or when using elevated decoding (e.g. z = plane). AES th Convention, Munich, German, Ma 7 Page of

12 5. REFERENCES [] K. Farrar, Soundfield microphone: Design and development of microphone and control unit, Wireless World, pp. 5, Oct [] K. Farrar, Soundfield microphone - : Detailed functioning of control unit, Wireless World, pp. 99, Nov [] M. A. Gerzon, The design of precisel coincident microphone arras for stereo and surround sound, in Preprint 5th Conv. Aud. Eng. Soc., Mar [] P. G. Craven and M. A. Gerzon, Coincident microphone simulation covering three dimensional space and ielding various directional outputs, U.S. Patent Specification 779, Aug [5] Eric W. Weisstein, Spherical harmonic, [] M. A. Gerzon, Ambisonics in multichannel broadcasting and video, J. Aud. Eng. Soc., vol., no., pp. 59 7, Nov. 95. [7] A. Farina, Anechoic measurement of the polar plots of a soundfield st-5 b-format microphone, format/test-soundfield- Microphones/Probe Comparison.PDF, undated. [] A. Farina, A-format to b-format conversion, Oct.. AES th Convention, Munich, German, Ma 7 Page of