Construction of a Multipurpose Spherical Microphone ArrayPedro Santos et al. Construction of a...

Associação Portuguesa de Engenharia de Áudio Secção Portuguesa da Audio Engineering Society

Artigo Apresentado no 13º Encontro da APEA 7 e 8 de Outubro de 2011 ESMAE - IPP

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Construction of a Multi-purpose Spherical

Microphone Array

Pedro Santos1, Gavin Kearney1, Marcin Gorzel1 1 Department of Electronic and Electrical Engineering, Trinity College Dublin

Correspondence should be addressed to: Pedro Santos ([email protected])

ABSTRACT Ambisonics is a surround sound technique developed in the 1970s by Michael Gerzon and colleagues at the Mathematical Institute of Oxford. Ambisonics is often criticized for its small sweet spot area. This can be improved by increasing the order of the Ambisonic microphone through the use of higher-order spherical harmonics.

This project consists of the physical construction of a spherical microphone containing several capsules placed on its surface and a description of how it can be used to generate virtual microphones for multi-channel playback and higher-order Ambisonics recording and reproduction. The microphone’s functionality is then assessed and suggestions are made for future improvements.

RESUMO Ambisonics é uma técnica surround desenvolvida nos anos setenta por Michael Gerzon e seus colegas no Instituto de Matemática da Universidade de Oxford. Esta técnica é frequentemente criticada pela sua reduzida definição espacial. Estes aspectos podem ser melhorados aumentando a ordem do microfone através da utilização de esféricos harmónicos de ordens superiores.

Este projeto consiste na construção de um microfone esférico contendo várias cápsulas colocadas na sua superfície e uma descrição de como pode ser usado para gerar microfones virtuais para a reprodução multi-canal e para gravação e reprodução de ambisonics de ordem superior. A funcionalidade do microfone é posteriormente aferida e são feitas sugestões relativamente a futuros melhoramentos.

1. INTRODUCTION Throughout the years, many different sound recording and reproduction techniques have been experimented in order to give the listener a full spatial immersion, from Fantasound in the 1940s, to the recent Hamasaki 22 (or 22.2), through the current cinema standard known as 5.1.

Unfortunately, surround systems like 5.1, 7.1, 10.2 or 22.2 aren’t easily interchangeable between each other: if the soundtrack of a film is encoded in 5.1, decoding it to 7.1 or 22.2 is not straightforward, nor from 22.2 to 5.1. On the other hand, although some multi-channel recording techniques do exist, like the Double MS (coincident technique) or the OCT Surround (spaced

technique), using different recording techniques for different reproduction systems is impractical, since one will be determining in the early stages how a film or musical piece will be reproduced in cinema or at home.

Ambisonics can be a solution to the problem of interchanging between different recording/reproduction setups. The biggest advantage of Ambisonics is that it is independent of the loudspeaker setup, meaning that it can easily be adjusted to whatever setup the user happens to have at his disposal, though some basic rules in terms of speaker placement should be followed in order to obtain a good spatial impression. The biggest disadvantage, as happens with other coincident recording techniques, is its lack of spaciousness and the reduced size of its sweet

Pedro Santos et al. Construction of a Multi-purpose Spherical Microphone Array

APEA – 13º Encontro, Porto, ESMAE-IPP, 7 e 8 de Outubro de 2011 Página 2 de 6

spot area, making it difficult to be implemented in cinema or other environments with large audiences.

One solution to this, which is explored in this paper, is the creation of a higher-order Ambisonics (HOA) microphone, capable of encoding the sound field with a much finer resolution. Higher-order Ambisonics has the advantage of being backwards compatible, meaning that a recording done, for example, in fourth order, can be reproduced in a third, second or first order playback system. HOA is thus a good option for recording or encoding sounds with a high spatial resolution that can be adapted afterwards to different loudspeaker setups.

2. REVIEW

2.1. Introduction to Ambisonics Ambisonics [1] is an expansion of Blumlein’s MS technique [2], using four coincident microphones: one omnidirectional (the W signal), which measures the pressure of the sound field arriving at the microphone, and three bidirectional (the X, Y and Z signals), that measure the velocity at each axis. The x-axis microphone points towards left, the y-axis towards the front, and the z-axis upwards. The resulting signals W, X, Y and Z are called B-format.

However, it is physically impossible to have four microphone capsules coinciding at the same spatial point, and the distance between the capsules will cause artefacts in the spatial reproduction, known as ‘spatial aliasing’. To minimize this problem, Gerzon developed a microphone that contained four cardioid capsules placed on the four faces of an imaginary tetrahedron, close enough to each other so that spatial artefacts would be minimized. This is called the A-Format, from which the B-format can be easily derived.

Fig. 2.1 Draft of the Soundfield microphone, with the four capsules arranged in a tetrahedral shape, as proposed by Michael Gerzon [3]

2.2. Spherical Harmonics The Spherical Harmonics represent the polar sensitivity of a microphone. The order of a microphone depends on the order of its spherical harmonics. The original B-format Ambisonic microphone technique is first-order,

since it comprises an omnidirectional (zeroth order) and three bidirectional (first order) microphones.

Spherical harmonics are given by the following equation [4]:

!!"! !,! = !!" sin ! ∗cos !" !"!! = 1sin !" !"!! = −1(2.1)

where Pmn is the semi-normalized associated Legendre function of degree m and order n, ! is the azimuth angle, and ! is the elevation angle. Spatial decoding of B-format recordings can be done using the following equation [5]:

!! =!! [! + ! cos !! cos !! + ! cos !! sin !! +! sin !! ] (2.2) where N is the number of output channels and θi and ϕi are the angles of the loudspeaker i.

Decoding can also be done by performing a pseudoinverse [5] [6] [7]. If B is a matrix containing the recorded signals !!"! , !!"! = ! ∗ !!"! !,! (2.3) where p is a column vector containing the desired loudspeaker signals S, and C the re-encoding matrix, containing the coefficients for each loudspeaker calculated using the spherical harmonics equation !!"! !,! , we get: ! = ! ∗ ! (2.4) To calculate the vectors of signals fed to each loudspeaker, the equation needs to be rearranged to:

! = !!! ∗ ! (2.5) with C-1 being the inverse of C. However, it is only possible to invert a matrix when such matrix is square, i.e. when the number of speakers, L, is the same as the number of spherical harmonic coefficients, N. Since usually N>L, the solution is to perform a pseudo-inverse of C to find the best possible results, which translates to the following:

!"#$ ! = !!(! ∗ !!)!! (2.6) where CT is the complex conjugate of C.

2.3. Problems of Ambisonics Critics of Ambisonics affirm that because of the high coherence between the different channels, due to the fact that all the capsules are coincident, the spatial representation of the sound field lacks detail. Some problems that occur when using coincident techniques include comb filtering at high frequencies and dominance of lower frequencies due to high-frequency cancellations. Also, the sweet spot can be quite small – the image collapses to the nearer loudspeaker when the listener is off-centre, due to a low angular resolution caused by the broad polar patterns of the different virtual microphones.



There have been several attempts to improve Ambisonics’ spatial imaging. One hypothesis is to use narrower beams by increasing the number of spherical harmonics, thus increasing the order of the microphone.

2.4. Higher Order Ambisonics The image below shows the 3-D representations of the first four spherical orders.

Fig. 2.1 3D view of spherical harmonics up to order 3 with

usual designation of associated HOA components [7]

As can be seen in figure 2.1, as the order increases, so does the directionality of the polar patterns. This, in turn, makes the sound field more stable, by increasing the size of the sweet spot also making it less prone to comb filtering, since there will be less loudspeakers sharing the same signals (see figure 2.2).

Figure 2.2 First to fourth order encoding functions (horizontal only) [8]

The fourth-order representation on figure 2.2 shows that the angular discrimination of the two sound sources A and B is greater than in the case of first-order rendering, for example, since there is one narrow lobe pointing mostly in the direction of A, and another pointing in the direction of B, making the loudspeakers more selective than when using a first-order setup. It is thus visible that the possibility of confusing the source’s virtual position is reduced.

In conclusion, the higher the order of a microphone, the higher the number of spherical harmonics, and thus the better the directional accuracy of the captured sound field.

3. REVIEW OF PREVIOUS WORK

3.1. Sound Field Recording with Higher Order Ambisonics, by Moreau et al [7] In this paper, Moreau suggests the construction of a fourth order Ambisonics microphone by processing the input signal of different capsules distributed around a sphere and mathematically correlating them to the corresponding spherical harmonics by “deriving HOA signals from a set of acoustic measurements made by usual microphones and a corresponding mathematical model which relates these quantities to HOA signals” [7].

The process used by Moreau to calculate the weightings of each capsule in order to obtain the different spherical harmonics, known as pseudoinverse, is explained in the Background section of this paper.

The minimum number of capsules on an Ambisonics microphone depends on its order: 2M+1 for horizontal recording, and (M+1)2 for periphonic, where M is the order of the microphone. This means that for a fourth-order microphone, the minimum number of capsules needed to record a 3D sound field is 25.

Since it is not possible to equally distribute 25 vertices around the surface of a sphere, the solution found by Moreau and his team was to use the 32 vertices of a Pentakis-Dodecahedron, i.e. a combination of an Icosahedron, with 12 vertices, and a Dodecahedron, with 20 vertices, as the position of 32 capsules. This can be regarded as an over-sampling of the sphere, minimizing the errors that may occur due to the fact that the capsules aren’t exactly at the same distance from one another.

One problem that arises when combining signals from different capsules placed close to each other is what is known as Spatial Aliasing: the maximum frequency !!"# that can be accurately captured is calculated based on the minimum distance between two capsules. !!"# is given by

!!"# = !!! (3.1) where c is the speed of sound in metres per second and d is the minimum distance between two capsules placed on the surface of the microphone sphere, in metres.

Once built, the microphone was measured in IRCAM’s anechoic chamber. The signal used to measure each sensor’s response was a logarithm sine sweep, and the resulting impulse responses were then used to plot the measured polar patterns and compare them with the ideal ones.

The results were fairly satisfactory: the construction of this microphone proved the possibility of building higher-order Ambisonic microphones, despite some fragilities that can be addressed with more research and technical capabilities.



3.2. A Spherical Microphone Array for Synthesizing Virtual Directivity Microphones in Live Broadcasting and in Post Production, by Farina et al [9] In 2010, Angelo Farina presented a paper at the 40th AES International Conference, in Tokyo, which described the process of creating a microphone capable of synthesizing multiple directive virtual microphones from a single microphone body. In this case, the body of the microphone was pre-built, using, similarly to Moreau’s fourth-order microphone, 32 capsules, but the way of distribution of the capsules was not revealed.

Farina’s approach to beamforming does not employ higher-order spherical harmonics, but rather virtual microphones of varying order. In the end, the microphone is capable of rendering seven virtual microphones in real-time, with an interface that gives the user the capability of orienting the beams in real-time.

The implementation of this project was fairly straightforward: the first step was to take a large number of impulse responses around the microphone, using sweep tones. After this, the Fast Fourier Transform of each measurement was taken and stored in a three-dimensional matrix C of size DxMxS where D is the number of angles measured, M is the number of capsules and respective impulse response consisting of S samples, converted to the frequency domain. By storing the FFTs of these measurements it is possible not only to plot the sensitivity of each capsule or of the whole microphone to a given frequency, but also to derive filters that will later be used to create the virtual microphone beams.

The equation for the beams is given by

! !,! = 0.5 + 0.5 ∙ cos ! ∙ cos ! ! (3.2) where ! is the azimuth angle, ! is the elevation, and n is the order, with a first-order beam having a cardioid pattern. A matrix P of size DxV is then created using ! !,! , for all the D measuring angles and V virtual microphones.

The impulse responses were measured in an anechoic room, using a sine sweep tone. The results showed that the capsules were close to omnidirectional when exposed to low frequencies, but became more directional at higher frequencies, as would be expected, due to the interference of the sound waves with the body of the microphone.

This approach of measuring impulse responses around the sphere allowed the team to skip an important step that otherwise would have had to be made previously: matching the frequency response of all the 32 capsules. In theory, in order to be able to use this many capsules at the same time, it would be necessary to measure the individual response of each one, and match it to the response of the remaining, with the objective of avoid unwanted filtering due to imbalances in the different frequency responses. However, by measuring the impulse response of the microphone capsules for sources

placed all around the sphere, and assuming that there is a sufficiently large number of measuring positions, one can filter the response of each capsule directly during the pseudo-inverse stage, where a matrix H of size MxV, containing the filters with which each microphone signal must be convolved with, can be obtained by reordering the equation

! = ! ∗! (3.3) to

! = !!! ∗ ! (3.4) The results of this study showed that the resultant beams had a very good frequency response for frequencies up to 10kHz and orders up to 6th, with a good signal-to-noise ratio.

3.3. Ambisonics Equivallent Panning, by Neukom et al [10] The main objective of AEP is to avoid the complicated functions used to encode and decode ambisonic signals and to allow real-time panning using arbitrary higher-order ambisonic beams.

From the original complex ambisonic panning functions, Neukom derived a simplified equation for in-phase decoding using as variables the angles of the loudspeakers, the angles of the sound being panned and the order of the beam:

! = !!!"# !!!! !"# ! !"# !! !!"# ! !"# !!!! (3.5)

where ! and ! are the azimuth and elevation angles of each loudspeaker, !! and !! are the azimuth and elevation angles of the sound being panned, and p is the order of the beam. In this case p doesn’t have to be an integer number, as happens in higher-order Ambisonics, giving the user more control over the directivity of the sound source.

As the order of the beam is increased, the level of the loudspeakers distant from the source begins to decay, while the level of loudspeakers close to the source remains more or less constant. This means that besides implementing the above equation to get the amplitude coefficients of each speaker, it is also important to normalize the overall volume in order to keep a constant level when varying the order of the beam. This is done by multiplying each speaker signal by !!!! , where p is the order of the beam, and n is the number of loudspeakers.

4. METHODOLOGY AND IMPLEMENTATION

4.1. Microphone construction In terms of capsule placement, it was decided to follow Moreau’s choice of distributing the capsules at the vertices of a Pentakis-Dodecahedron. These coordinates were implemented in a MatLAB script, to simulate a three-dimensional model of the solid, and confirm that all the coordinates were correct and matched the surface of a sphere of radius 1.



Figure 4.1 3D simulation of a Pentakis-Dodecahedron

In order to insert these virtual coordinates into an actual sphere the solution found was to use a technique called UV-unrwapping, which basically is a way of converting a three-dimensional solid into a two-dimensional one, like a piece of paper, and then wrap it around the spherical object.

Fig. 4.2 Uv-unwrapping of ¼ of a sphere, with the vertices of the

Pentakis-Dodecahedron marked on top

The capsules chosen for this project were small back electret omnidirectional Panasonic WM-61A. According to their spec sheet, these capsules have a fairly flat frequency response from 20Hz to 20kHz, and an acceptable signal-to-noise ratio of more than 62dB. They are unbalanced and have a standard operation voltage of 2V.

The sphere used was a plastic ball, with a radius of 4.9 cm, where the smallest distance between two capsules is 3.14 cm, from which we can calculate a spatial aliasing frequency of

!!"# = !"#!∗!.!"#$ = 5414!". The 32 capsules were placed on the sphere, and each was soldered to a pair of cables, one connected to the signal pin and the other to the ground pin. On the other side, each pair was soldered to an XLR plug. However, since the capsules needed to be fed at least with 2V, a way of reducing the 48V Phantom Power of each preamp had to be found, which was done by attaching a 47kΩ resistor to the signal cable (Pin 2). Pins 1 (ground) and 3 (cold) were both connected to ground.

After soldering together all the different elements, the capsules were glued to the sphere and tested.

4.2. Measuring the microphone’s sensitivity Following Farina’s method [9], the microphone was measured using sine sweep tones in order to get a visual representation of the sensitivity of the different capsules for different frequencies.

These measurements were done in an untreated room, using 7-second sweeps created with Voxengo Deconvolver. The capsules were measured for 32 different angles: from 22.5º to -22.5º Azimuth, with intervals of 45 degrees, and from 67.5º to -67.5º Elevation, also with intervals of 45 degrees.

After recording all the 1024 files, these were deconvolved, creating 1024 impulse responses.

The impulse responses were then used to plot 3D models representing the sensitivity of each capsule at a given frequency band.

Overall, the response of each capsule is fairly irregular, although for lower frequencies it does show a tendency to be close to omnidirectional, but with a slightly predominance to sound sources coming from the direction of the capsule. It can also be seen that as the frequency increases the lobes become narrower.

These results were expected: the capsules are not matched, and although no measurements were made to assess their individual frequency responses for a source placed on axis, it was thought that even at lower frequencies the capsules would show a small amount of directional preference.

Although these measurements broadly characterize the microphone’s sensitivity, more measurements should be done in the future, preferably in an anechoic chamber, and with a finer angular resolution.

4.3. AEP implementation using Max/MSP In order to demonstrate the functionality of the microphone, a Max patch was created that allowed the user to pan between the different capsules and change the order of the beam(s) formed, making it possible for the listener to hear the different spatial characteristics of the acoustic space. Three different sound sources were recorded using a loudspeaker, with the sounds placed in three different positions around the microphone.

The gains of each capsule were all set to the maximum level, with the objective of matching the different preamps as closely as possible. However, it is important to emphasize that the capsules weren’t matched in terms of their frequency response and sensitivity.

The three recording angles were: 67.5º azimuth, 50º elevation; 22.5º azimuth, -35,0º elevation; and -90,0 azimuth, 75º elevation. The gain of the speaker was set to a level loud enough without distorting any of the input signals or the speaker itself.

With the aim of providing sources with different spectral contents and also of different styles, the source sounds used were white noise, an anechoic recording of an



orchestra, and the voice of a man counting from one to ten.

Finally, an extra recording was done in the outside, under a tunnel close to a busy street, with cars passing in front of the microphone (at +90º), from the right to the left. This recording was made with the objective of testing the effectiveness of the microphone in ‘real-life’ situations and as a virtual stereo pair.

On the Max patch, the volume of each of the 32 channels is calculated using the formula (3.5). The user can create a virtual stereo pair and define different beam orders for the virtual loudspeakers.

It is difficult to say, without measuring each capsule using a broadband signal positioned on-axis, what influence the frequency sensitivity had on the overall perceptual effect of the acoustic space when using the Ambisonics Equivalent Panner. However, it is quite noticeable that as one pans from the source position to the surrounding capsules, the higher frequencies start to decay, and the reflections from the room’s walls become more predominant than the direct sound. This is physically accurate and was the expected result.

5. CONCLUSIONS The initial goal of this project was to build a fully functional fourth-order microphone following the work done by Farina and colleagues. However, the time-frame wasn’t sufficient to get to the decoding stage, where the microphone signals would be convolved by the respective matrix of coefficients.

A demonstration of the microphone’s functionality was made using the Ambisonics Equivalent Panner applied to the different capsule signals. Informal listening tests showed that even though the capsules aren’t matched, panning a single beam towards and away from a source has an audible effect both in terms of the spectral contents of the sound and on its amplitude, especially when using higher beam orders.

From here it can be affirmed that if the 32 capsules were measured as to their individual response and filtered in order to have the same frequency response, the spherical microphone could be used as a multi-channel recording device, with different virtual beams pointing in different directions, allowing the user to choose different orders for each beam, and then feeding them to their respective loudspeaker.

The measurements taken around the sphere can be further used to create the filters needed to implement virtual directivity beams, following the method of Farina et al., but also, and more interestingly, to implement a weighted sum of higher-order spherical harmonics using this same method. The impulse responses can be used to calculate the inverse filters needed to encode the different input signals into higher-order spherical harmonic components, by simply calculating the FFT of each impulse response and storing the resulting coefficients in a 32x32 matrix.

Further work can also be done in implementing a converter capable of convolving the filters with the input signals and playing them through a loudspeaker setup containing 32 speakers displayed as the vertices of a pentakis-dodecahedron, in the same way as the spherical microphone, to perform objective and subjective listening tests. The placing of these speakers is quite complicated, as is finding an anechoic room where they can be placed. A solution would be to use virtual loudspeakers to simulate each of the 32 sources using headphones.

6. REFERENCES [1] Gerzon, M., "Periphony: With-Height Sound

Reproduction", Mathematical Institute, University of Oxford, 2010, pp. 20-25.

[2] Blumlein, A., “Improvements in and relating to Sound-Transmission, Sound-Recording, and Sound-Reproduction Systems”, Patent no. 394,325, 1931, available at http://www.doramusic.com/patents/394325.htm, last consulted on 04-10-2011.

[3] Craven, P., and Gerzon, M., “Coincident Microphone Simulation Covering Three Dimensional Space and Yielding Various Directional Outputs”, US Patent, 1977.

[4] Malham, D., "Space in Music – Music in Space”, MPhil Thesis, University of York, York, United Kingdom, 2003.

[5] Hollerweger, F., “An Introduction to Higher-Order Ambisonics”, 2005, available at http://flo.mur.at/writings, last consulted on 04-10-2011.

[6] Daniel, J. et al, “Further Investigations of High Order Ambisonics and Wavefield Synthesis for Holophonic Sound Imaging”, Audio Engineering Society, presented at the 114th Convention, Amsterdam, The Netherlands, 2003.

[7] Moreau, S. et al, “3D Sound Field Recording with Higher Order Ambisonics – Objective Measurements and Validation of a 4th Order Spherical Microphone”, Audio Engineering Society, presented at the 30th International Conference, Paris, France, 2006, p. 7.

[8] Moore, J., “The Development of a Design Tool for 5-Speaker Surround Sound Decoders”, PhD Thesis, University of Huddersfield, Huddersfield, United Kingdom, 2005.

[9] Farina, A.et al, “A Spherical Microphone Array for Synthesizing Virtual Directive Microphones in Live Broadcasting and in Post Production”, Audio Engineering Society, presented at the 40th International Conference, Tokyo, Japan, 2010.

[10] Neukom, M., Schacher, J., “Ambisonics Equivallent Panning”, Institute for Computer Music and Sound Technology, University of the Arts, Zurich, Switzerland, 2008.

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