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Development of Ambisonic Microphone Design Tools – Part 1
Authors : Charles J Middlicott and Bruce J Wiggins
University of Derby, UK
Correspondence should be addressed to Charles J Middlicott (c.middlicott1@derby.ac.uk)
ABSTRACT
In recent years an increase in the capture and production of ambisonic material has occurred as a result of
companies such as YouTube and Facebook utilising ambisonics for spatial audio playback. There is now a greater
need for affordable higher order microphone arrays. This work details the development of a set of tools which can
be used to simulate and evaluate such microphone arrays, The ‘Ambisonic Array Design Tool’ for simulation and
‘Ambisonic Array Evaluation Tool’ for evaluation. The microphone capsules’ position and directivity can be
changed, with the effects on the synthesised spherical harmonics frequency and polar responses observed within
the GUI. These scripts written in MatLab have been packaged within a GUI and will be available online.
1 Introduction
The capture of ambisonic signals was developed, to
1st order, by Gerzon and Craven in 1977 [1], where
the notion of using a tetrahedral array of capsules was
described. More recently, with the advent of spatial
audio for virtual reality and 360° video, software tools
have enabled users to synthesize higher order
ambisonic material in Digital Audio Workstations
(DAWs) in order to create immersive, full-sphere,
sound fields. Yet the availability of microphones
capable of capturing such signals has not grown at the
same pace, or affordability.
A number of tools are available to design and
simulate spaced microphone arrays such as MMAD
[2] [3] and MARRS [4] but few tools are currently
available for the design of coincident Ambisonic
microphones.
This e-brief presents the development of a set of tools
that can be used to simulate and evaluate the design
of higher order microphone arrays, giving useful
insight and performance metrics to use as a
benchmark, prior to developing physical prototypes.
2 Ambisonics
2.1 Background
Ambisonics is a full-sphere isotropic surround format
based around the decomposition of a 3D sound field
into Spherical Harmonics (SH). These are an
orthonormal set of basis functions on a sphere.
Figure 1 - Spherical Harmonics up to Order N = 2
The SH shown in Figure 1. are defined by the
following equation [5].
!"#$
q
%
f
&'()*+,
-. $*/0&1
$*+0&12"#$345
q
&67#
f
(1)
Where
n is the order (0 to the max order N)
m is the degree (-n to n)
(·)! is the factorial function
2"#
is the associated legendre polynomial
Middlicott & Wiggins Ambisonic Microphone Design Tools
Page 2 of 6
2.2 An Ideal Ambisonic Microphone
For an ambisonic array to exhibit an ideal response,
the capsules must be coincident. In practice, it isn’t
possible for them to occupy the same physical space.
Additionally, a typical microphone may exhibit a flat
on axis response frequency response – but for a
capsule to be useful when developing an ambisonic
array ideally it needs to exhibit an even response
regardless of the angle of incidence. Capsules, even
when matched, won’t typically exhibit this desired
response. This is especially true of capsules that have
a low cost per unit, such as electrets.
2.3 Stages of Array Optimization
2.3.1 Pre-Filtering
Pre-filtering must occur by means of capsule
calibration. To compensate for a less than ideal
frequency/polar response. This optimisation also
aims to correct for capsule mismatch. This has been
implemented in various forms in the AAET.
2.3.2 Post-Filtering
Post filtering is the process of applying filters to the
generated SH signals, typically to help compensate
for the effects on the frequency response due to the
non-coincident nature of the capsules and spatial
aliasing
Filtering of the low frequency (LF) content varies
depending on the order of the SH in question, this is
especially useful as it limits excessive gain at LF
which would otherwise increase the noise floor.
Spatial Aliasing occurs as a product of the sampling
process, most typical schemes are alias free for order
limited functions or exhibit negligible aliasing.
However, in practice a sound field composed of plane
waves isn’t order limited. Therefore, due to the
existence of higher order elements in a sound field
(infinite number of spherical harmonics) spatial
aliasing will occur [5].
The simplest way to overcome aliasing problems
would be to increase the number of microphones,
however this may not be possible. A more feasible
approach would be to implement anti-aliasing filters
or alias minimisation. This will be implemented in the
applications at a later date.
3 Array Design / Evaluation Criteria
3.1 Current Functionality
Below a set of performance parameters have been
defined, these are shown as plots that can be
evaluated within the current incarnation of the AADT
and AAET applications.
Individual Capsule Responses
• Time Domain Response
• Polar Response
• Frequency Response
Generated B-Format Responses
• Polar Response
• Frequency Response
These five responses were chosen as to aid the
evaluation of an array at multiple stages in various
domains.
4 Simulation Tool – Overview
The Ambisonic Array Design Tool (AADT) simulates
the response of an array that has specific set of design
attributes, chosen by the user, these are shown in
Table 1.
These attributes are utilised to calculate a set of
impulse responses (IRs) that can be used to evaluate
an arrays theoretical performance characteristics
listed in section 3.1.
As a tool, this gives the user a ‘best case scenario’, in
regard to array performance, which subsequently
developed prototype arrays can be evaluated against.
4.1 Assignable Parameters
To simulate an array using a graphical user interface
(GUI) a set of user assignable parameters has to be
defined, covering all the relevant array design
features needed. These are used in various
calculations detailed throughout this work. Table 1
details the parameters used to facilitate the generation
of these responses.
Middlicott & Wiggins Ambisonic Microphone Design Tools
Page 3 of 6
List of User Assignable Parameters
Ambisonic Order
Order ‘N’
Array Radius
(in millimetres)
Array Type
2D or 3D (3D to be added)
Sampling Scheme
a
b
c
Equal Angle
Gaussian
(To Be Added)
T-Design
(To Be Added)
Capsule Directivity
Directivity ‘d’
a
b
c
Omni
d = 0
Cardioid
d = 1
Figure of 8
d = 2
IR Filter Length
(in Samples)
Sampling Frequency
(in kHz)
Source Test Distance
(in metres)
Source Increment
(in degrees ‘°’)
Table 1 – User Assignable Parameters in GUI
4.2 Generating Simulated Impulses
4.2.1 Capsule Directivity and Offset
Each capsule that makes up the array will have a
stated 1st order polar pattern based on the directivity
factor ‘d’ as shown below in equation 2. The
direction of orientation of the microphone is set by
the parameter ‘M’.
89:;<=$)/>&+;>?<345;$@A/;BA&
(2)
Where
d is the microphone capsule directivity factor
S
q
is a given angular source position in radians
M
q
is a given angular capsule position in radians
This generates gains for ‘M’ microphones at ‘S’
source positions, shown in Figure 2.
Figure 2 - Offset Directivity Patterns at M
q
positions
4.2.2 Simulating Distance Effects
In order to simulate the distance effects, such as array
radius, a fractional delay line is necessary. This can
be implemented as a Sinc pulse that can be used to
represent an impulse occurring in the free field at
given time ‘t’.
To generate the required fractional delay line, a time
delay needs to be calculated. To derive this, first we
must generate both the microphone and source
positions in cartesian coordinates, see Eq. 3 & 4.
BCDEF>GHI;<345$BA&
BJDEF>GHI;<5KL;$BA&
@CD@MNOGIP;<345$@A&
@JD@MNOGIP;<5KL;$@A&
(3)
(4)
Where the SrcDist is the distance between the centre
of the array and the cartesian coordinates at each
angular source location. The Radius is the radius of
the array in metres.
These coordinates are then used in the numerator of
Eq. 5 to derive the time delay by generating the
distances from the source minus the microphone
location. The resulting distance is then divided by ‘c’
Middlicott & Wiggins Ambisonic Microphone Design Tools
Page 4 of 6
thus generating a time delay in seconds, with ‘c’
being the speed of sound at 343m/s.
Q$@C/;BC&R+;$@J/;BJ&R;
N
(5)
A nominal acoustic delay is calculated by multiplying
the sampling frequency by the source distance (the
distance from a given source position to the centre of
the array) in metres, this is divided by ‘c’. This is used
so the simulated impulses can be centred relative to
the specified filter length.
As previously mentioned once the time delay is
calculated the Sinc pulse can then be utilised to
implement the fractional delay line. A Sinc is defined
in the following equation using the sine function.
IG*N$S&'T,
5KL;$S&
S
(6)
which has the following normalisation [6].
UIG*N$S&>SD;
¥
V
¥
p
(7)
A set of dynamically generated Sinc pulses are used
(minus the calculated time delay for source to
microphone) to give the fractional delay (inter-
sample) needed for that source location. A hamming
window is then applied to fix ripple issues.
Using Eq. 2 and applying the aforementioned Sinc
pulses results in the packed matrix ‘C’ containing
simulated impulses per capsule / source direction.
4.2.3 Generating B-Format Signals
To generate the desired B-Format signals a set of SH
coefficients must be calculated and applied to each
microphone capsule. The coefficients for each SH are
generated using the following equation modified
from [7].
!"#$
q
%
f
&'W"X#X2"X#X$5KL$B
f
&&;Y345$X0XB
q
&;;;;;KZ;[;8;
5KL$X0XB
q
&;;;;;KZ;\;8]
(8)
Where
M
q
is a given angular capsule azimuth in radians
M
F
is a given angular capsule elevation in radians
n is the order (0 to the max order N)
m is the degree (-n to n)
2"#;
is the associated Legendre polynomial
W"#;
is a gain value for a given normalisation scheme
Once calculated for a given SH they must be
convolved with each capsules simulated IRs. These
are then summed together to create the desired SH
signal.
5 Evaluation Tool - Overview
The Ambisonic Array Evaluation Tool (AAET) differs
to its counterpart in that its purpose is to
evaluate/validate a physical arrays performance, be it
a prototype or a commercially available product,
against the performance of a comparable simulation.
This application takes in the same user assignable
parameters as the AADT and generates its simulated
impulses in the same manner shown above. It needs
to be supplied with a set of measured IRs, The GUI
takes in a .MAT file containing the array parameters
and measured IRs. These are then both plotted in the
same manner as the AADT but with the simulated /
measured responses being displayed side by side for
direct comparison.
The main difference in terms of functionality is the
ability to generate calibration filters from the
measured responses utilising various approaches.
This pre-filtering of the capsules can be evaluated in
the same manner as the raw responses. This is useful
as you can instantly view any change in response.
Implementation and evaluation of the post-filtering
stage will be considering in Part 2 of this work.
5.1 Prototype Circular Array
While creating the AAET application this work
necessitated the development of a physical prototype.
In this instance, it was decided to develop a 2nd order
circular array. The decision was taken to focus solely
on the azimuth plane. This was so factors such as
Middlicott & Wiggins Ambisonic Microphone Design Tools
Page 5 of 6
spacing, calibration and filtering could be evaluated
with greater ease and findings could be used to inform
the development of a 3D spherical array. Secondary
to this, with the current measurement apparatus only
offering the ability to measure responses in the
azimuth plane, it was decided a circular array was
appropriate as an initial prototype.
The minimum number of capsules needed to capture
the horizontal only 2nd order SH signals using an
equal-angle sampling distribution is 2(N + 1), where
N is the order [5]. This led to a circular array
prototype that consists of five 16.5mm electret
capsules with a capsule spacing of 72° and an array
radius of 20mm. This radius was calculated
considering the physical size of the TSB165A-T
capsules utilised [8]. The initial prototype was used
to validate visually against the simulation data
generated in the AAET.
5.1.1 Objective Measurement
Objective measurement of the prototype array took
place in a hemi-anechoic chamber, a total set of 360
measured IRs were captured. Measurements were
taken on the horizontal plane in 5° increments, using
an automated turntable, at a 2-metre distance from the
acoustic centre of the prototype array.
The measurement source material was a 15 second
20Hz to 20kHz logarithmic sine sweep, this was fed
to a KRK Systems ROKIT RP8 loudspeaker. An
omnidirectional measurement microphone [9] was
used to capture the response of the loudspeaker used.
Nelson/Kirkeby inversion was utilised to generate an
inverse of this speaker response using Eq. 9 [10]. The
resultant filter is convolved with the captured IRs thus
compensating for the frequency response of the
loudspeaker.
^7"_$
w
&D; ^`abcd`;;<;;ef*g$^$
w
&&;;
ef*g=^$
w
&?;<;^$
w
&+;6$
w
&
(9)
Where H is the measured response at the frequency
index
w
, Htarget is the desired target response,
e
is a
regularisation parameter.
6 UI Design
The GUI was designed using Graphical User
Interface Design Environment (GUIDE) a drag-and-
drop environment for laying out user interfaces (UIs).
The interactive behavior of the applications is then
coded separately in MATLAB [11]. See Figure 3 for
a view of the AADT layout
Figure 3 – AADT GUI Layout
7 Future Work
A set of additional features are to be implemented,
such as full 3D simulation capability, additional array
construction type (rigid/open, dual sphere), frequency
dependant capsule directivity, propagation loss and
additional sampling schemes. Further to this the
AAET will be updated to consider both the generation
and evaluation of pre/post filters in future revisions.
8 Conclusions
This e-brief has described the development of a set of
ambisonic array design and evaluation tools that can
be used to rapidly simulate a range of array designs.
Array performance data can be generated to evaluate
an array at various stages such as the pre/post
processing. This can be used to aid the development
of future array prototypes.
Middlicott & Wiggins Ambisonic Microphone Design Tools
Page 6 of 6
References
[1] M. A. Gerzon, P. Craven, “Coincident
Microphone Simulation Covering Three-
Dimensional Space and Yielding Various
Directional Outputs,” US4042779A (1977).
[2] M. Williams, G. Le Du, “The Quick Reference
Guide to Multichannel Microphone Arrays
Design Part I : using Cardioid Microphones”.
In Audio Engineering Society Convention 110,
5335, (2001)
[3] M. Williams, G. Le Du, “The Quick Reference
Guide to Multichannel Microphone Arrays
Design Part II : using Supercardioid and
Hypocardioid Microphones”. In Audio
Engineering Society Convention 116, 6059,
(2004)
[4] H, Lee. D. Johnson, M. Mironovs, “An
Interactive and Intelligent Tool for
Microphone Array Design”. In Audio
Engineering Society Convention 143, e-Brief
390, (2017)
[5] B, Rafaely. “Fundamentals of Spherical
Array Processing” (Springer Topics in Signal
Processing) Springer (2015).
[6] Weisstein, E W. "Sinc Function." From
MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/SincFunction.
html
[7] J, Daniel. “Représentation de champs
acoustiques, application à la transmission et à
la reproduction de scènes sonores complexes
dans un contexte multimedia” PhD Thesis
(2000)
[8] JLI Electronics LLC. "TSB165A-T Capsule."
http://www.jlielectronics.com/content/JLI-
165A-T.pdf
[9] Earthworks Inc. "M30BX Omni Microphone."
https://earthworksaudio.com/wp-
content/uploads/2018/07/M30BX-Data-
Sheet-2018.pdf
[10] H. Tokuno, O. Kirkeby, P. Nelson, “Inverse
filter of sound reproduction systems using
regularization” IEICE Transactions on
Fundamentals of Electronics,
Communications and Computer Sciences, Vol.
80, No. 5. 809-820. (1997).
[11] Mathworks Inc, “
Bhijhkl
App Building -
R2018a”
https://www.mathworks.com/help/pdf_doc/m
atlab/buildgui.pdf
