Room acoustic modal analysis using Bayesian inference a) Douglas Beaton and Ning Xiang b) Graduate Program in Architectural Acoustics, School of Architecture, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180, USA (Received 15 January 2017; revised 22 April 2017; accepted 28 April 2017; published online 16 June 2017) Strong modal behavior can produce undesirable acoustical effects, particularly in recording studios and other small rooms. Although closed-form solutions exist to predict modes in rectangular rooms with parallel walls, such solutions are typically not available for rooms with even modest geometrical complexity. This work explores a method to identify multiple decaying modes in experimentally measured impulse responses from existing spaces. The method adopts a Bayesian approach working in the time domain to identify numerous decaying modes in an impulse response. Bayesian analysis provides a unified framework for two levels of inference: model selec- tion and parameter estimation. In this context model selection determines the number of modes present in an impulse response, while parameter estimation determines the relevant parameters (e.g., decay time and frequency) of each mode. The Bayesian analysis in this work is implemented using an approximate numerical technique called nested sampling. Experimental measurements are performed in a test chamber in two different configurations. Experimentally measured results are compared with simulated values from the Bayesian analyses along with other, more classical calculations. Discussion of the results and the applicability of the method is provided. V C 2017 Acoustical Society of America.[http://dx.doi.org/10.1121/1.4983301] [JFL] Pages: 4480–4493 I. INTRODUCTION Modal behavior appears in a wide variety of areas including room acoustics, noise control, vibration analysis and more. Xu and Sommerfeldt used a hybrid modal expan- sion to study sound fields in enclosed spaces. 1 Huang applied modal analysis in the development of a drumlike silencer for controlling duct noise. 2 Modal analysis has also seen numer- ous applications in the study of musical acoustics. 3–5 In small rooms, widely spaced modal frequencies in the lower range of human hearing can color sound in a notice- able, often undesirable manner. This effect can be particu- larly problematic in rooms devoted to recording or listening to music. Considerable effort has been focused on developing design guidelines and approaches to mitigate these effects. 6–9 Room modes can also exacerbate noise issues. If nearby equipment creates noise, particularly narrowband noise, near the frequency of a room mode the level of noise in the room can be greater than might be predicted by theories based on diffuse sound fields. Rooms exhibiting strong modal behavior will not have a diffuse sound field. Over a range of frequencies with widely spaced modes the sound in a room may exhibit multiple slope decay, and must be characterized using multiple energy decays. In cases such as these, architectural acousti- cians are faced with the problem of determining the correct number of decay rates present in the data, which can be a challenging task. 10 This situation also creates a problem when estimating room characteristics, as a large number of equations and approaches commonly applied in room acous- tics are based on the assumption of a diffuse sound field. Estimations of reverberation time are almost invariably based on this assumption. In such situations it can be useful to determine the modes present in a field-measured impulse response of a sys- tem. Analytical solutions exist to predict modal frequencies in rooms with parallel walls based on solutions to the wave equation. 11 As geometries grow more complex, closed-form solutions become less tractable and acousticians must look to numerical methods to identify and characterize the modes in a room. This work applies Bayesian analysis to identify and characterize the modes present in an experimentally measured room impulse response. The impulse response is simulated in the time domain using the Prony Model. 12 This same model, with an additional noise term, was used by Bhuiyan et al. to simulate audible clinical percussion sig- nals. 13 Although an impulse response measured in a room is used as an example, the methods used herein are equally applicable anywhere modal behavior may be present in an impulse response. To the authors’ knowledge, the applica- tion of Bayesian techniques to modal analysis has not been sufficiently reported in the literature. The Bayesian algorithms employed here were imple- mented in MATLAB with minimal use of specialized librar- ies. Open source packages for Bayesian analysis are also available in other popular languages, like R and Python, for readers that may wish to perform their own experiments. This paper is organized as follows. Section II gives background information on the problem and its modeling. Section III introduces Bayesian analysis in a general sense. Section IV gives an overview of the implementation of Bayesian analysis used in this work. Section V outlines a) Aspects of this work have been presented at the 171st ASA Meeting in Salt Lake City, UT, 2016. b) Electronic mail: [email protected]4480 J. Acoust. Soc. Am. 141 (6), June 2017 V C 2017 Acoustical Society of America 0001-4966/2017/141(6)/4480/14/$30.00
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Room acoustic modal analysis using Bayesian inferencea)
Douglas Beaton and Ning Xiangb)
Graduate Program in Architectural Acoustics, School of Architecture, Rensselaer Polytechnic Institute,110 8th Street, Troy, New York 12180, USA
(Received 15 January 2017; revised 22 April 2017; accepted 28 April 2017; published online 16June 2017)
Strong modal behavior can produce undesirable acoustical effects, particularly in recording studios
and other small rooms. Although closed-form solutions exist to predict modes in rectangular
rooms with parallel walls, such solutions are typically not available for rooms with even modest
geometrical complexity. This work explores a method to identify multiple decaying modes in
experimentally measured impulse responses from existing spaces. The method adopts a Bayesian
approach working in the time domain to identify numerous decaying modes in an impulse
response. Bayesian analysis provides a unified framework for two levels of inference: model selec-
tion and parameter estimation. In this context model selection determines the number of modes
present in an impulse response, while parameter estimation determines the relevant parameters
(e.g., decay time and frequency) of each mode. The Bayesian analysis in this work is implemented
using an approximate numerical technique called nested sampling. Experimental measurements
are performed in a test chamber in two different configurations. Experimentally measured results
are compared with simulated values from the Bayesian analyses along with other, more classical
calculations. Discussion of the results and the applicability of the method is provided.VC 2017 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4983301]
[JFL] Pages: 4480–4493
I. INTRODUCTION
Modal behavior appears in a wide variety of areas
including room acoustics, noise control, vibration analysis
and more. Xu and Sommerfeldt used a hybrid modal expan-
sion to study sound fields in enclosed spaces.1 Huang applied
modal analysis in the development of a drumlike silencer for
controlling duct noise.2 Modal analysis has also seen numer-
ous applications in the study of musical acoustics.3–5
In small rooms, widely spaced modal frequencies in the
lower range of human hearing can color sound in a notice-
able, often undesirable manner. This effect can be particu-
larly problematic in rooms devoted to recording or listening
to music. Considerable effort has been focused on developing
design guidelines and approaches to mitigate these effects.6–9
Room modes can also exacerbate noise issues. If nearby
equipment creates noise, particularly narrowband noise, near
the frequency of a room mode the level of noise in the room
can be greater than might be predicted by theories based on
diffuse sound fields.
Rooms exhibiting strong modal behavior will not have a
diffuse sound field. Over a range of frequencies with widely
spaced modes the sound in a room may exhibit multiple
slope decay, and must be characterized using multiple
energy decays. In cases such as these, architectural acousti-
cians are faced with the problem of determining the correct
number of decay rates present in the data, which can be a
challenging task.10 This situation also creates a problem
when estimating room characteristics, as a large number of
equations and approaches commonly applied in room acous-
tics are based on the assumption of a diffuse sound field.
Estimations of reverberation time are almost invariably
based on this assumption.
In such situations it can be useful to determine the
modes present in a field-measured impulse response of a sys-
tem. Analytical solutions exist to predict modal frequencies
in rooms with parallel walls based on solutions to the wave
equation.11 As geometries grow more complex, closed-form
solutions become less tractable and acousticians must look
to numerical methods to identify and characterize the modes
in a room. This work applies Bayesian analysis to identify
and characterize the modes present in an experimentally
measured room impulse response. The impulse response is
simulated in the time domain using the Prony Model.12 This
same model, with an additional noise term, was used by
Bhuiyan et al. to simulate audible clinical percussion sig-
nals.13 Although an impulse response measured in a room is
used as an example, the methods used herein are equally
applicable anywhere modal behavior may be present in an
impulse response. To the authors’ knowledge, the applica-
tion of Bayesian techniques to modal analysis has not been
sufficiently reported in the literature.
The Bayesian algorithms employed here were imple-
mented in MATLAB with minimal use of specialized librar-
ies. Open source packages for Bayesian analysis are also
available in other popular languages, like R and Python, for
readers that may wish to perform their own experiments.
This paper is organized as follows. Section II gives
background information on the problem and its modeling.
Section III introduces Bayesian analysis in a general sense.
Section IV gives an overview of the implementation of
Bayesian analysis used in this work. Section V outlines
a)Aspects of this work have been presented at the 171st ASA Meeting in
where nx, ny, nz, are modal indices in each of the respective
Cartesian axes and Lx, Ly, Lz, are the dimensions of the
chamber in the corresponding axes. Table IV lists the modal
frequencies predicted for the empty test chamber over the
frequency range considered in this work.
It is interesting to note that the Bayesian analysis
selected the 5-mode model as the most appropriate match to
the measured impulse response in the empty chamber, while
the closed-form solution predicted six modes over the fre-
quency range considered. Two of the predicted modes are
degenerate modes occurring near 339 Hz. The selected 5-
mode model has a single mode near this frequency, while
the 6-mode model from the Bayesian analysis does indicate
two degenerate modes occurring near this frequency.
However, as shown in Fig. 3 and Fig. 7, the addition of the
sixth mode does not create a sufficient increase in evidence
to warrant the added complexity of the model. This result
indicates that the implemented nested sampling, as an
approximation method of exploring the multidimensional
parameter space, is able to carry out parameter estimations
based on models with 5 and 6 modes as accurately as classi-
cal modal theory while simultaneously yielding estimations
of modal decay times. On the other hand, in the model with
6 modes, two degenerate models in the vicinity of 339 Hz do
not create a significant increase in Bayesian evidence. The
magnitude frequency content spectrum in Fig. 9 may reveal
the reason. The degenerate modes near 339 Hz occupy a sin-
gle peak in the magnitude spectrum. As shown in Fig. 9, the
5-mode model places a mode near this frequency and cap-
tures the contribution of that content. The amount of new
frequency content captured by the addition of a sixth mode
is small by comparison to that captured by the addition of
the fifth mode, thus resulting in only a marginal increase in
Bayesian evidence. However, in most practical scenarios,
FIG. 5. (Color online) Comparison of decay times and frequencies from the
various analyses and experimental measurements for the test chamber with
applied foam. Only results corresponding to measured frequencies are
shown.
4488 J. Acoust. Soc. Am. 141 (6), June 2017 Douglas Beaton and Ning Xiang
FIG. 6. (Color online) Normalized likelihood surface over all parameter pairs from the 3rd and 5th modes of a 5-mode model. The model is centered at the
point of maximum likelihood from an analysis of the impulse response measured in the empty chamber. (a)–(ab) Normalized likelihoods plotted over {A3,
over the course of nested sampling iterations. As shown in
the detail window, likelihood values increase asymptotically
as the nested sampling converges. The lowest likelihood val-
ues in the early iterations are picked from the initial popula-
tion. At this stage in the sampling the population covers the
largest area of the parameter space. As the iterations pro-
gress, the nested sampling algorithm explores the multi-
dimensional parameter space. Each successive iteration
moves one sample in the population to a position of higher
likelihood. The area of parameter space covered by the sam-
ple population shrinks as sampling progresses and the sam-
ples are restricted to higher and higher likelihood values.
FIG. 9. (Color online) Frequency content of the measured impulse response
in the empty test chamber, along with the modal frequencies from the model
selected by Bayesian inference.
FIG. 10. (Color online) Decay times and frequencies in a population of 400 samples, with five modes each, over the course of an analysis using the empty
chamber data. Each point on the plots represents a single mode within a single sample model. The five blue �’s in (a) represent the decay times and frequen-
cies of five modes within a single sample. (a) Initial population. (b) After 10 000 iterations. (c) After 30 000 iterations. (d) After 60 000 iterations. (e) After
90 000 iterations. (f) After 120 000 iterations.
J. Acoust. Soc. Am. 141 (6), June 2017 Douglas Beaton and Ning Xiang 4491
When approaching the peak of the likelihood surface the
parameter space has been thoroughly explored with mono-
tonically increasing likelihood proxy values as indicated by
the magnified window in Fig. 11. The likelihood function in
Fig. 11 is combined with approximation techniques outlined
by Skilling18 to evaluate Eq. (10) and determine the evidence
for a given model.
E. Caveats
Several caveats complicated the analyses performed in
this work. The most salient among them are outlined below.
1. Truncating direct sound
As the direct sound of an impulse response cannot be
accurately modeled by a decaying sinusoid it was necessary
to remove the direct sound from the measured impulse
responses before analysis. To accomplish this, the measured
impulse responses were truncated at 30 ms after the approxi-
mate arrival of the direct sound. Given the source and
receiver positions shown in Fig. 2, the direct sound and first
reflections arrive very close to one another. The limit of
30 ms was set sufficiently long to provide certainty that the
direct sound had been removed.
Analyses were performed both with and without
removal of the direct sound. While the resulting number of
modes and modal frequencies were essentially the same,
analyses that removed the direct sound produced decay times
in much better agreement with measured values. Figure 4
shows these decay times for analyses where the direct sound
was removed. The resulting decay times show good agree-
ment for those modal frequencies selected for the switch-off
measurements, which are proper measurements of steady-
state modal energy decays.
With analyses using the truncated signal producing
more accurate results in the areas of interest, the 30 ms trun-
cation time was deemed appropriate. All results reported in
this work consider the measured impulse responses with this
30 ms truncation. Possible modifications to the Prony model
to capture the direct sound, and the sensitivity of early signal
truncation on parameter estimates are two potential areas of
future study arising from these results.
2. Length of signal considered
The length of signal considered was set at 470 ms, to
include the bulk of the signal measured in the test chamber.
Initial analysis runs in early investigations of this work con-
sidered a 200 ms long portion of the impulse response. Using
this shorter signal length improved run times by reducing the
number of points in the signal. However, it was found that
results produced using this shorter signal length overesti-
mated decay times compared to the measured values. For
this reason the length of impulse response selected for analy-
sis was increased to include the bulk of the modal energy
process. This adjustment increased analysis run times but
also resulted in improved decay time estimates. The
increased number of signal points used for analysis also pro-
duced some of the numerical issues outlined below.
3. Numerical considerations
The error term in Eq. (13), equal to one half the sum of
the squared errors between the measured and modeled
impulse responses, is affected by how the measured impulse
response’s amplitude is normalized. Numerical issues can
arise if the sum of the squared error terms becomes too large
or too small. To avoid such issues a scale factor can be
applied to the error term. This approach is equivalent to scal-
ing the measured impulse response and modal amplitudes by
the same amount. The same scale factor must be applied
across all analyses to ensure results are comparable. The fact
that amplitudes and error terms can be scaled in this manner
also implies that the exact values of evidence for each model
in Fig. 3 are irrelevant—only their relative amounts are of
interest. This also implies that a constant can be added or
subtracted from the resulting logarithmic evidence values
(Fig. 3) to facilitate comparing and ranking the evidence
results over a convenient range. The same constant must be
added or subtracted from all evidence values determined by
evaluating Eq. (10).
VIII. CONCLUDING REMARKS
The 5-mode Prony model selected using Bayesian infer-
ence agrees well with the measurements taken in the test
chamber. A plot of the model signal in the time domain
shown in Fig. 1 exhibits good agreement with the measured
impulse response. Estimates of modal decay times from the
simulations also agree well with decay times estimated from
the sine tone switch-off measurements, as shown in Fig. 4.
This work has demonstrated the suitability of Bayesian
inference for selecting an appropriate model in a context of
room acoustic modal analysis. The approach adopted here is
by no means limited to room acoustics. It could, in principal,
be applied anywhere modal behavior occurs.
While the method is sound the current implementation
of Bayesian analysis used in this work was computationally
expensive. There are numerous aspects of this approach that
could be made more efficient. The ability to determine
appropriate population sizes before running an analysis
would be of great benefit, and could reduce the number of
analyses required. Alternative methods of exploring the
FIG. 11. (Color online) Likelihood profile over the course of analysis itera-
tions. Results are shown for a 5-mode model with a sample population of
400 using data from the empty test chamber. The analysis shown here was
also used to produce Fig. 10.
4492 J. Acoust. Soc. Am. 141 (6), June 2017 Douglas Beaton and Ning Xiang
parameter space for points of higher likelihood could reduce
the number of iterations required, rendering each analysis
more efficient. Such methods might consider varying multi-
ple parameters in a single iteration, or determining step sizes
using alternative distributions or even deterministic rules.
One intriguing aspect of this work centers on the need to
truncate the initial portion of the impulse response so as to
remove the direct sound. Further investigation could deter-
mine if the Prony model can be modified to effectively cap-
ture the direct sound and eliminate the need for signal
truncation.
Perhaps the most interesting question raised in this work
concerns why the 6-mode model, which captured the two
degenerate modes shown in Table IV, did not produce suffi-
cient Bayesian evidence to warrant selection over the 5-
mode model. The close spacing of modal frequencies in the
degenerate mode pair is likely a contributing factor. Further
investigation could identify the required separation between
two modal frequencies that results in the selection of more
complex models. Whether this separation is also a function
of the contribution of the degenerate modes as compared to
the other modes in the signal is also an item of interest.
ACKNOWLEDGMENT
The authors are grateful to Dr. Paul Goggans for many
discussions that motivated the authors to continue this work.
Thanks go to Wesley Henderson, who made the first effort to
explore Bayesian model selection and parameter estimation
in room acoustic modal analysis. The authors would also
like to thank Dr. Cameron Fackler and Dane Bush for
exchanging experience and ideas in implementing the nested
sampling algorithm.
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