Model-based Bayesian direction of arrival analysis for sound sources using a spherical microphone array Christopher R. Landschoot and Ning Xiang a) Graduate Program in Architectural Acoustics, School of Architecure, Rensselaer Polytechnic Institute, Troy, New York 12180, USA (Received 9 February 2019; revised 12 April 2019; accepted 17 April 2019; published online 31 December 2019) In many room acoustics and noise control applications, it is often challenging to determine the directions of arrival (DoAs) of incoming sound sources. This work seeks to solve this problem reli- ably by beamforming, or spatially filtering, incoming sound data with a spherical microphone array via a probabilistic method. When estimating the DoA, the signal under consideration may contain one or multiple concurrent sound sources originating from different directions. This leads to a two- tiered challenge of first identifying the correct number of sources, followed by determining the directional information of each source. To this end, a probabilistic method of model-based Bayesian analysis is leveraged. This entails generating analytic models of the experimental data, individually defined by a specific number of sound sources and their locations in physical space, and evaluating each model to fit the measured data. Through this process, the number of sources is first estimated, and then the DoA information of those sources is extracted from the model that is the most concise to fit the experimental data. This paper will present the analytic models, the Bayesian formulation, and preliminary results to demonstrate the potential usefulness of this model-based Bayesian analysis for complex noise environments with potentially multiple concur- rent sources. V C 2019 Acoustical Society of America. https://doi.org/10.1121/1.5138126 [KTW] Pages: 4936–4946 I. INTRODUCTION The purpose of the research presented in this paper is to offer a solution to the problem of localizing multiple concurrent sound sources through a model-based probabilis- tic approach. This work demonstrates that, given a set of sound signals recorded on a spherical microphone array (Meyer and Elko, 2002; Rafaely, 2015), the number of sound sources, as well as the directions in which they arrive, can be predicted algorithmically. This requires a process known as beamforming, or the spatial filtering of a sound signal using spherical harmonics theory based on a spherical microphone array (Williams, 1999). This is combined with the probabil- istic methods of analysis called Bayesian model selection and parameter estimation (Knuth et al., 2015; Xiang and Fackler, 2015). Localizing multiple concurrent sound sources simulta- neously in complex sound environments can be a challenge as there may be variations in the number of sources, along with their locations, characteristics, and strengths (Blandin et al., 2012; Bush and Xiang, 2018; Escolano et al., 2014). In addition to these variations, there can be unwanted inter- ference through fluctuating background noise as well. Various solutions to this problem (Mohan et al., 2008) have begun to be explored, particularly in recent years with growing interest in immersive auditory virtual reality and augmented reality applications (Vorl€ ander et al., 2015). Whereas the interest in spatial sound has been growing greatly in recent history, the ideas surrounding it are not completely new. Theoretical work and early applications manifested themselves in the form of ambisonics (Craven and Gerzon, 1977; DuHamel, 1952), which began to provide some spatial information about a soundscape. Although these methods did contain spatial information about a sound signal, they did not address the localization of the sound sources or their characterization in any way. Without employing microphone array technology, the spatialization of sound was inherent to the recorded audio signals them- selves rather than gleaned via post-processing (Furness, 1990). Using microphone array technology (Madhu and Martin, 2008), specifically spherical microphone arrays, an entire sound field could be analyzed without traditional microphone directionality ignorance or bias (Meyer and Elko, 2002). One aspect of decomposing complex soundscapes is performing a direction of arrival (DoA) analysis on the recorded signals. There are many methods that have been implemented to attempt to solve this problem. Recent exam- ples can be seen through the efforts of utilizing various different microphone arrays, including sparse linear micro- phone arrays (Bush and Xiang, 2018; Nannuru et al., 2018) and a two-microphone array used in a room-acoustic study (Escolano et al., 2014). A spherical microphone array, or spherical array, is sim- ply a sphere with microphone capsules sampling its surface that can record sound signals simultaneously. Because of the principles of spherical microphone arrays, there are no inherent directional constraints. The recorded signals can be processed to simulate any orientations of directionality desired. This has allowed for researchers to experiment with a) Electronic mail: [email protected]4936 J. Acoust. Soc. Am. 146 (6), December 2019 V C 2019 Acoustical Society of America 0001-4966/2019/146(6)/4936/11/$30.00
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Model-based Bayesian direction of arrival analysis for soundsources using a spherical microphone array
Christopher R. Landschoot and Ning Xianga)
Graduate Program in Architectural Acoustics, School of Architecure, Rensselaer Polytechnic Institute, Troy,New York 12180, USA
(Received 9 February 2019; revised 12 April 2019; accepted 17 April 2019; published online 31December 2019)
In many room acoustics and noise control applications, it is often challenging to determine the
directions of arrival (DoAs) of incoming sound sources. This work seeks to solve this problem reli-
ably by beamforming, or spatially filtering, incoming sound data with a spherical microphone array
via a probabilistic method. When estimating the DoA, the signal under consideration may contain
one or multiple concurrent sound sources originating from different directions. This leads to a two-
tiered challenge of first identifying the correct number of sources, followed by determining the
directional information of each source. To this end, a probabilistic method of model-based
Bayesian analysis is leveraged. This entails generating analytic models of the experimental data,
individually defined by a specific number of sound sources and their locations in physical space,
and evaluating each model to fit the measured data. Through this process, the number of sources is
first estimated, and then the DoA information of those sources is extracted from the model that is
the most concise to fit the experimental data. This paper will present the analytic models, the
Bayesian formulation, and preliminary results to demonstrate the potential usefulness of this
model-based Bayesian analysis for complex noise environments with potentially multiple concur-
rent sources. VC 2019 Acoustical Society of America. https://doi.org/10.1121/1.5138126
[KTW] Pages: 4936–4946
I. INTRODUCTION
The purpose of the research presented in this paper is to
offer a solution to the problem of localizing multiple
concurrent sound sources through a model-based probabilis-
tic approach. This work demonstrates that, given a set of
sound signals recorded on a spherical microphone array
(Meyer and Elko, 2002; Rafaely, 2015), the number of sound
sources, as well as the directions in which they arrive, can be
predicted algorithmically. This requires a process known as
beamforming, or the spatial filtering of a sound signal using
spherical harmonics theory based on a spherical microphone
array (Williams, 1999). This is combined with the probabil-
istic methods of analysis called Bayesian model selection
and parameter estimation (Knuth et al., 2015; Xiang and
various configurations and methods of data processing in
attempts to determine the best ways to filter sound and
decompose complex soundscapes. This includes methods
such as spherical harmonic beamforming in combination
with optimal array processing, frequency smoothing methods
(Khaykin and Rafaely, 2012), spherical harmonics smooth-
ing (Jo and Choi, 2017), and modal smoothing methods
(Morgenstern and Rafaely, 2018). Sun et al. (2012) applied a
spherical microphone array to localize reflections in rooms,
while Nadiri and Rafaely (2014) localized multiple speakers
under a reverberant environment. The array configurations do
not even have to be fully spherical. Hemispherical micro-
phone arrays (Li and Duraiswami, 2005) can be more suitably
deployed on a table top in conference room applications,
mounted in the ceiling, or deployed on the ground for outdoor
sound source DoA estimation and tracing of flight objects.
Zuo et al. (2018) have formulated the theory of spatial sound
intensity vectors in a spherical harmonic domain applicable
for a variety of acoustic scenarios. Jo and Choi (2018) pro-
posed a solution to avoid ill-conditioned singularity when
solving least-squares and eigenvalue problems to estimate the
DoAs. Wong et al. (2019) discovers rules-of-thumb on how
the estimation precision for an incident source’s azimuth-
polar DoA depends on the number of identical isotropic
sensors.
Each method tested with spherical arrays helps improve
the ability to determine the DoAs of sound sources, but some
still rely on the basic concept of predicting source locations by
correlating them directly with high sound energy levels. A
bulk of recent work also exists using spherical harmonics in
wave-field synthesis (Ahrens and Spors, 2012), sound radia-
tions (Shabtai and Vorl€ander, 2015), or noise analysis (Zhao
et al., 2018). Torres et al. (2013) applies a cylindrical micro-
phone array in room-acoustic studies. Fernandez-Grande
(2016) reconstructs an arbitrary sound field based on measure-
ments with a spherical microphone array. Richard et al. (2017)
applies a spherical microphone array to measure acoustic sur-
face impedance at oblique incidence.
As for complex sound/noise source analysis, the ability
to determine the likelihood of discrete source locations has
not been well investigated unless they are clearly separated
in space. This work applies 16 microphones uniformly
mounted flush on a rigid full sphere. The array has a spatial
resolution up to order two of spherical harmonics. With a
large number of noise sources, the signals to be analyzed can
blend together if the sound sources are located too closely to
each other in physical space. This situation requires model-
based analysis to resolve it or even higher order spherical
arrays to accurately determine noise sources, which, in turn,
requires more microphone channels.
In addition to the parameter estimation problems, which
are solely associated with the DoA estimation given the
known number of sound sources, there is a need to solve the
overarching question of how to reliably determine the
number of sound sources without having to use brute force
by adding more microphones. The answer resides in model-
based Bayesian analysis, which is a method that can estimate
the number of sources and their attributes through probabil-
istic analysis rather than just correlating high sound energy
levels to sound source locations. This method leverages
machine learning through an iterative process, allowing for a
more reliable and consistent DoA analysis. To answer this
overarching question of determining the correct number of
concurrent sound sources, this work applies Bayesian model
selection to the DoA estimation tasks when the number of
sound sources is unknown prior to the analysis. This
Bayesian formulation for model selection problems starts
with application of Bayes theorem, followed by incorpora-
tion of prior information, and then marginalization (Xiang
and Goggans, 2001). Any interest in directional parameter
values will be deferred into the background of the current
problem. This allows attention to be focused on estimating
the probabilities for the number of concurrent sound sources.
There have been many recent efforts to apply Bayesian
model selection to acoustics problems. Xiang and Goggans
(2003) apply Bayesian model selection to determine the
number of exponential decays present in acoustic enclosures
by analyzing sound energy decay functions. Bayesian model
selection has also been applied to room-acoustic modal anal-
ysis (Beaton and Xiang, 2017). Previous studies (Bush and
Xiang, 2018; Escolano et al., 2014; Nannuru et al., 2018) of
DoA analysis have also employed two levels of Bayesian
inference. However, to the best of the authors’ knowledge,
the model-based Bayesian inference has not yet been suffi-
ciently studied using spherical microphone arrays. This
paper demonstrates that the model-based Bayesian probabil-
istic approach can be applied to spatial sound field analysis
with a set of sound signals recorded on a spherical micro-
phone array, resulting in estimations of the number of sound
sources as well as DoAs. This requires two levels of
Bayesian inference.
The remainder of this paper is as follows. Section II for-
mulates the spherical harmonic models of potentially multi-
ple sound sources. Section III briefly introduces a unified
Bayesian framework. Section IV discusses experimental
results using the two levels of Bayesian inference. Section V
further discusses results pertaining to the Bayesian imple-
mentation. Section VI concludes the paper.
II. BEAMFORMING DATA AND MODELS
There are two important concepts at the heart of this
DoA analysis method. They are spherical harmonic beam-forming using spherical microphone arrays and Bayesiananalysis. Spherical harmonic beamforming is the way in
which the recorded sound signals can be processed to map
the sound energy around the spherical array. Bayesian analy-
sis is the process of solving for the number of the sound
sources and the DoAs of these sources.
A. Spherical harmonics
This work utilizes principles of spherical harmonics to
beamform spherical sound signals in order to map and model
the sound energy of a soundscape. Spherical harmonics can
be formulated by solving the spherical wave equation. They
can be mathematically represented (Williams, 1999) as
J. Acoust. Soc. Am. 146 (6), December 2019 Christopher R. Landschoot and Ning Xiang 4937
FIG. 5. (Color online) Directional responses of three sound sources in the
form of two-dimensional sound energy distribution. The directions of the
sound sources are ð5�; 60�Þ; ð135�; 140�Þ and ð270�; 90�Þ. (a) Experimentally
measured beamforming data. Three solid dots indicate the source DoAs. (b)
Bayesian model predicted sound energy distribution. Three solid dots indicate
the estimated DoAs.
FIG. 6. (Color online) Mean evidence estimates along with variances given
the experimental data. The data contain three sound sources at
ð5�; 60�Þ; ð135�; 140�Þ and ð270�; 90�Þ. (a) The mean evidence in decibans,
evaluated over 15 individual random sampling runs using nested sampling.
(b) The Bayes’s factor in decibans, comparing the evidence of the current
number of sources to the previous number. (c) Magnified view of the evi-
dence for four sources with variations.
TABLE II. Experimentally measured and predicted DoAs for three concur-
rent sound sources. The variations are estimated using the Bayesian methodover 15 runs, The errors are the differences between the experimental andpredicted data. Both data sets are analyzed with an angular resolution of3:6�.