Evaluation of decay times in coupled spaces: Bayesian parameter estimation

Neural network for multi-exponential sound energy decay analysis

An established model for sound energy decay functions (EDFs) is the superposition of multiple exponentials and a noise term. This work proposes a neural-network-based approach for estimating the model parameters from EDFs. The network is trained on synthetic EDFs and evaluated on two large datasets of over 20 000 EDF measurements conducted in various acoustic environments. The evaluation shows that the proposed neural network architecture robustly estimates the model parameters from large datasets of measured EDFs while being lightweight and computationally efficient. An implementation of the proposed neural network is publicly available.

The Optimal Determination of the Truncation Time of Non-Exponential Sound Decays

The noise effects in the room impulse response (RIR) make the decay range of the integrated impulse response insufficient for reliable determination of reverberation time (RT). One of the preferred techniques to minimize noise effects is based on noise subtraction, RIR truncation, and correction for the truncation. The success of RT estimation through the method depends critically on the accurate estimation of the truncation time (TT). However, noise fluctuation and RIR irregularities can lead to discrepancies in the determined TT from the optimal value. The general goal of this paper is to improve RT estimates. An iterative procedure based on a non-exponential decay model consisting of a double-slope decay term and a noise term is presented to estimate the TT accurately. The model parameters are generated until the iterative procedure converges to a minimum difference between the energy decay curve (EDC) generated by the model and the Schroeder decay function. The decay rates of the EDCs with added pink noise levels are compared to those of the EDCs with low background noise. In addition, the detected TTs and the corresponding RTs are compared with the existing method and the noise compensation method (subtraction–truncation–correction method).

Model-based Bayesian analysis in acoustics-A tutorial

Bayesian analysis has been increasingly applied in many acoustical applications. In these applications, prediction models are often involved to better understand the process under investigation by purposely learning from the experimental observations. When involving the model-based data analysis within a Bayesian framework, issues related to incorporating the experimental data and assigning probabilities into the inferential learning procedure need fundamental consideration. This paper introduces Bayesian probability theory on a tutorial level, including fundamental rules for manipulating the probabilities, and the principle of maximum entropy for assignment of necessary probabilities prior to the data analysis. This paper also employs a number of examples recently published in this journal to explain detailed steps on how to apply the model-based Bayesian inference to solving acoustical problems.

Model-based Bayesian direction of arrival analysis for sound sources using a spherical microphone array

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 reliably 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 concurrent sources.

Bayesian decay time estimation in a reverberation chamber for absorption measurements

This work investigates the use of the initial decay time to obtain the Sabine absorption coefficient from measurements conducted in a reverberation chamber. Due to non-uniform distribution of sound absorption in the test chamber, measured energy decay functions exhibit multiple slopes, which cannot be evaluated unambiguously using linear regression as prescribed in the current standard (ISO 354, International Organization for Standardization, Geneva, Switzerland, 2003). As an alternative, this study proposes a Bayesian framework that allows estimating multiple decay parameters, hence capturing more accurately the energy decay features. Measurements are carried out in a reverberation chamber with and without diffusing elements to investigate the influence of diffusers on the absorption coefficient and on the decay process. Measured absorption coefficients of a porous sample are compared to theoretical values estimated with a transfer matrix model. The results show that the Sabine absorption coefficient calculated using the shortest decay time agrees well with the size-corrected theoretical absorption coefficient.

Bayesian Inference for Acoustic Direction of Arrival Analysis Using Spherical Harmonics

This work applies two levels of inference within a Bayesian framework to accomplish estimation of the directions of arrivals (DoAs) of sound sources. The sensing modality is a spherical microphone array based on spherical harmonics beamforming. When estimating the DoA, the acoustic signals may potentially contain one or multiple simultaneous sources. Using two levels of Bayesian inference, this work begins by estimating the correct number of sources via the higher level of inference, Bayesian model selection. It is followed by estimating the directional information of each source via the lower level of inference, Bayesian parameter estimation. This work formulates signal models using spherical harmonic beamforming that encodes the prior information on the sensor arrays in the form of analytical models with an unknown number of sound sources, and their locations. Available information on differences between the model and the sound signals as well as prior information on directions of arrivals are incorporated based on the principle of the maximum entropy. Two and three simultaneous sound sources have been experimentally tested without prior information on the number of sources. Bayesian inference provides unambiguous estimation on correct numbers of sources followed by the DoA estimations for each individual sound sources. This paper presents the Bayesian formulation, and analysis results to demonstrate the potential usefulness of the model-based Bayesian inference for complex acoustic environments with potentially multiple simultaneous sources.

Diffusion equation modeling for sound energy flow analysis in multi domain structures

This study investigates reliable models and methods to be applied in sound field analysis of multi-domain structures. The case structures are two monuments, namely, Süleymaniye Mosque and Hagia Sophia in İstanbul. These are both multi-volume spaces with many smaller sub-volumes coupled to each other by coupling apertures in form of arches. A key concern of the study is to examine energy flow decays and understand the mechanism of multi-slope sound energy decays. The methodology involves diffusion equation model (DEM) application in a finite-element scheme for sound energy flow analysis. Energy flow decays, energy flow dips, and spatial flow vectors are examined for single versus multi-domain DEM solutions. It is concluded that specification of different domains with individual diffusion coefficients is a critical setting such that, if not assigned correctly, may mislead the results. The energy flow vector analysis has enabled us to comprehend the architectural features in relation to such energy flow decay dip occurrence. The computational efficiency of DEM is also discussed. The DEM application in this study has proved to be a powerful and practical method in room acoustics applications, specifically for multi-rate decay investigations.

Statistics of natural reverberation enable perceptual separation of sound and space

In everyday listening, sound reaches our ears directly from a source as well as indirectly via reflections known as reverberation. Reverberation profoundly distorts the sound from a source, yet humans can both identify sound sources and distinguish environments from the resulting sound, via mechanisms that remain unclear. The core computational challenge is that the acoustic signatures of the source and environment are combined in a single signal received by the ear. Here we ask whether our recognition of sound sources and spaces reflects an ability to separate their effects and whether any such separation is enabled by statistical regularities of real-world reverberation. To first determine whether such statistical regularities exist, we measured impulse responses (IRs) of 271 spaces sampled from the distribution encountered by humans during daily life. The sampled spaces were diverse, but their IRs were tightly constrained, exhibiting exponential decay at frequency-dependent rates: Mid frequencies reverberated longest whereas higher and lower frequencies decayed more rapidly, presumably due to absorptive properties of materials and air. To test whether humans leverage these regularities, we manipulated IR decay characteristics in simulated reverberant audio. Listeners could discriminate sound sources and environments from these signals, but their abilities degraded when reverberation characteristics deviated from those of real-world environments. Subjectively, atypical IRs were mistaken for sound sources. The results suggest the brain separates sound into contributions from the source and the environment, constrained by a prior on natural reverberation. This separation process may contribute to robust recognition while providing information about spaces around us.

Investigations on sound energy decays and flows in a monumental mosque

This work investigates the sound energy decays and flows in the Süleymaniye Mosque in İstanbul. This is a single-space superstructure having multiple domes. The study searches for the non-exponential sound energy decay characteristics. The effect of different material surfaces and volumetric contributions are investigated using acoustic simulations and in situ acoustical measurements. Sound energy decay rates are estimated by Bayesian decay analysis. The measured data reveal double- or triple-slope energy decay profiles within the superstructure. To shed light on the mechanism of energy exchanges resulting in multi-slope decay, spatial sound energy distributions and energy flow vectors are studied by diffusion equation model (DEM) simulations. The resulting sound energy flow vector maps highlight the contribution of a sound-reflective central dome contrasted with an absorptive carpeted floor in providing delayed energy feedback. In contrast, no multi-slope energy decay pattern is observed in DEM simulations with a bare marble floor, which generates a much more diffuse sound field than in the real situation with a carpeted floor. The results demonstrate that energy fragmentation, in support of the non-exponential energy decay profile, is due to both the sound absorption characteristics of materials and to their distributions, as well as to relations between the subvolumes of the mosque's interior.

Extraction of the envelope from impulse responses using pre-processed energy detection for early decay estimation

The Schroeder's backward integration method and its applications have been widely studied in the literature; some papers analyze the performance of the method, some others suggest various enhancement techniques. In spite of these findings, there exist several cases where the energy decay curve extracted using the classical backward integration method and the parameters computed from it seem not always representative of the phenomenon under study. Among them, the cases where the early decay is dependent on strong, distinct reflections occurring just after the direct wave, as in most Italian opera houses. Other cases are measured impulse responses with a very low signal-to-noise ratio or missing the direct wave. In the literature, alternatives to the Schroeder's method have been proposed, ranging from Hilbert transform to non-linear processing techniques. In this work a method for the extraction of the envelope based on pre-processed energy detection for early decay estimation is proposed. It is shown that it returns an envelope well matching the first part of the decay even in non-linear cases, returning detailed information on the first part of the decay. The performance of the proposed method is presented and discussed for some exemplary impulse responses measured in historical opera houses. A preliminary study on the perceptive relevance of the method is finally presented.

The effects of stage absorption on reverberation times in opera house seating areas

The effects of stage absorption on reverberation times in opera houses were investigated using computer simulations and scale model measurements. The reverberation time (RT) was measured in stalls seating with and without variable stage elements (e.g., fly curtains, side curtains, cycloramas, and stage sets). The absorption coefficients of the walls and ceiling of the stage houses were varied accordingly. It was found that variable stage elements have a significant influence on reverberation times in seating areas, particularly for a reverberant stage house, due to the low absorption of the walls and ceiling in the stage house. It was also found that the absorption coefficients of the walls and ceiling should be over 0.5 to avoid RT decreases of over 10% due to the absorption of the variable stage elements. In addition, coupled room effects were investigated both with and without variable stage elements and the results show that double slope was not found in the opera houses investigated in this study.

Investigation of the effective aperture area of sliding and hinged doors between coupled spaces

Acoustical coupling between architectural spaces can be implemented by sliding or hinged doors. This study compares the effects of these variable coupling area designs on the sound field using temporal energy decay curve analysis. Varying the aperture size alters the multi-slope decay curve properties such as the decay rate of each slope and their point of intersection (time and level). A predictive model is proposed, based on a geometrical approach and statistical theory for coupled volumes. Differences between scale model measurements and analytical predictions are quantified by means of deviations of acoustical parameters; reasonable agreement is found.

Nested sampling applied in Bayesian room-acoustics decay analysis

Room-acoustic energy decays often exhibit single-rate or multiple-rate characteristics in a wide variety of rooms/halls. Both the energy decay order and decay parameter estimation are of practical significance in architectural acoustics applications, representing two different levels of Bayesian probabilistic inference. This paper discusses a model-based sound energy decay analysis within a Bayesian framework utilizing the nested sampling algorithm. The nested sampling algorithm is specifically developed to evaluate the Bayesian evidence required for determining the energy decay order with decay parameter estimates as a secondary result. Taking the energy decay analysis in architectural acoustics as an example, this paper demonstrates that two different levels of inference, decay model-selection and decay parameter estimation, can be cohesively accomplished by the nested sampling algorithm.

Moth wing scales slightly increase the absorbance of bat echolocation calls

Coevolutionary arms races between predators and prey can lead to a diverse range of foraging and defense strategies, such as countermeasures between nocturnal insects and echolocating bats. Here, we show how the fine structure of wing scales may help moths by slightly increasing sound absorbance at frequencies typically used in bat echolocation. Using four widespread species of moths and butterflies, we found that moth scales are composed of honeycomb-like hollows similar to sound-absorbing material, but these were absent from butterfly scales. Micro-reverberation chamber experiments revealed that moth wings were more absorbent at the frequencies emitted by many echolocating bats (40–60 kHz) than butterfly wings. Furthermore, moth wings lost absorbance at these frequencies when scales were removed, which suggests that some moths have evolved stealth tactics to reduce their conspicuousness to echolocating bats. Although the benefits to moths are relatively small in terms of reducing their target strengths, scales may nonetheless confer survival advantages by reducing the detection distances of moths by bats by 5–6%.

Different sound decay patterns and energy feedback in coupled volumes

Different non-exponential decays such as the concave and the convex double sloped decays in the coupled rooms provide distinct sound qualities. These are commonly considered to occur in the less reverberant sub-room and the more reverberant sub-room, respectively. However, numerical simulations and experiments in this paper show that the demarcation line is not located along the physical boundaries (e.g., the partition and the coupling aperture), but in the more reverberant sub-room. The sound field with the concave double sloped decay penetrates into the auxiliary sub-room to an extent which is influenced by the difference between the two natural reverberations of the sub-rooms. Furthermore the sound energy flows in different regions are investigated, demonstrating how energy feedback leads to the concave double sloped decay.

Bayesian characterization of multiple-slope sound energy decays in coupled-volume systems

Due to recent developments in concert hall design, there is an increasing interest in the analysis of sound energy decays consisting of multiple exponential decay rates. It has been considered challenging to estimate parameters associated with double-rate (slope) decay characteristics, and even more challenging when the coupled-volume systems contain more than two decay processes. To meet the need of characterizing energy decays of multiple decay processes, this work investigates coupled-volume systems using acoustic scale-models of three coupled rooms. Two Bayesian formulations are compared using the experimentally measured sound energy decay data. A fully parameterized Bayesian formulation has been found to be capable of characterization of multiple-slope decays beyond the single-slope and double-slope energy decays. Within the Bayesian framework using this fully parameterized formulation, an in-depth analysis of likelihood distributions over multiple-dimensional decay parameter space motivates the use of Bayesian information criterion, an efficient approach to solving Bayesian model selection problems that are suitable for estimating the number of exponential decays. The analysis methods are then applied to a geometric-acoustics simulation of a conceptual concert hall. Sound energy decays more complicated than single-slope and double-slope nature, such as triple-slope decays have been identified and characterized.

Comment on "Optimum absorption and aperture parameters for realistic coupled volume spaces determined from computational analysis and subjective testing results" [J. Acoust. Soc. Am. 127, 223-232 (2010)]

A recent paper [D. T. Bradley and L. M. Wang, J. Acoust. Soc. Am. 127, 223-232 (2010)] has reported inconsistencies between the results of two different approaches for characterizing non-exponential decays in coupled-volume systems. This letter aims to expose the origin of these inconsistencies, which are due to a limitation in the methodology utilized for the analysis presented in the paper referenced above.

Optimum absorption and aperture parameters for realistic coupled volume spaces determined from computational analysis and subjective testing results

This project utilizes computational modeling to study the effects of varying two architectural parameters, absorption ratio and aperture size, in a realistic coupled volume concert hall. Coupled volumes have been shown to exhibit non-exponential sound energy decay profiles, referred to as double slope effect. A number of objective metrics (T30/T15, LDT/T10, decay ratio, and DeltaL) have been used to quantify the double slope effect of the profiles generated in the virtual hall. T30/T15 and LDT/T10 showed similar trends across all hall configurations, indicating decreasing double slope effect with increasing coupled volume absorption ratio for each aperture size, and producing highest values at a specific aperture size for each absorption ratio. Generally, LDT/T10 provides finer resolution than T30/T15 when analyzing the decay profiles in this study. Results from the two metrics derived from Bayesian analysis, decay ratio and DeltaL, seem less consistent. Subjective testing has also been conducted to determine the effect of varying the two architectural parameters in the hall, and multidimensional scaling analysis shows that, in general, listener preference is inversely proportional to the level of double slope effect, with the highest levels of preference occurring at low and medium levels of double slope effect. Recommended design guidelines for coupled volume halls are provided based on these computational and subjective results.

Identifying acoustical coupling by measurements and prediction-models for St. Peter's Basilica in Rome

St. Peter's Basilica is one of the largest buildings in the world, having a huge volume resulting from the addition of different parts. Consequently, sound propagation cannot be interpreted using a conventional approach and requires experimental measures to be compared with statistical-acoustics and geometrical predictions in order to explain the interplay between shape, materials, and sound waves better. In previous research one of the most evident effects, the surprisingly low reverberation time, was believed to result from acoustical coupling phenomena. Taking advantage of more refined measuring techniques available today an acoustic survey was carried out and the results were analyzed using different methods, including Bayesian parameter estimation of multiple slope decays and directional energy plots, which showed that coupling effects actually take place, even though measured reverberation times were longer than those given in previous studies. In addition, experimental results were compared with geometrical- and statistical-acoustic models of the basilica, which showed that careful selection of input data and, in statistical models, the inclusion of phenomena such as direct sound radiation and non-diffuse energy transfer, allow obtaining accurate results. Finally, both models demonstrated that reduced reverberation depends more on increased absorption of decorated surfaces than on coupling effects.

Investigation of acoustically coupled enclosures using a diffusion-equation model

Recent application of coupled-room systems in performing arts spaces has prompted active research on sound fields in these complex geometries. This paper applies a diffusion-equation model to the study of acoustics in coupled-rooms. Acoustical measurements are conducted on a scale-model of two coupled-rooms. Using the diffusion model and the experimental results the current work conducts in-depth investigations on sound pressure level distributions, providing further evidence supporting the valid application of the diffusion-equation model. Analysis of the results within the Bayesian framework allows for quantification of the double-slope characteristics of sound-energy decays obtained from the diffusion-equation numerical modeling and the experimental measurements. In particular, Bayesian decay analysis confirms sound-energy flux modeling predictions that time-dependent sound-energy flows in coupled-room systems experience feedback in the form of energy flow-direction change across the aperture connecting the two rooms in cases where the dependent room is more reverberant than the source room.

Efficient estimation of decay parameters in acoustically coupled-spaces using slice sampling

Room-acoustic energy decay analysis of acoustically coupled-spaces within the Bayesian framework has proven valuable for architectural acoustics applications. This paper describes an efficient algorithm termed slice sampling Monte Carlo (SSMC) for room-acoustic decay parameter estimation within the Bayesian framework. This work combines the SSMC algorithm and a fast search algorithm in order to efficiently determine decay parameters, their uncertainties, and inter-relationships with a minimum amount of required user tuning and interaction. The large variations in the posterior probability density functions over multidimensional parameter spaces imply that an adaptive exploration algorithm such as SSMC can have advantages over the exiting importance sampling Monte Carlo and Metropolis-Hastings Markov Chain Monte Carlo algorithms. This paper discusses implementation of the SSMC algorithm, its initialization, and convergence using experimental data measured from acoustically coupled-spaces.

Evaluation of decay times in coupled spaces: an efficient search algorithm within the Bayesian framework

This paper discusses an efficient method for evaluating multiple decay times within the Bayesian framework. Previous works [N. Xiang and P. M. Goggans, J. Acoust. Soc. Am. 110, 1415-1424 (2001); 113, 2685-2697 (2003); N. Xiang, P. M. Goggans, T. Jasa, and M. Kleiner, 117, 3707-3715 (2005)] have applied the Bayesian inference to cope with demanding tasks in estimating multiple decay times from Schroeder decay functions measured or calculated in acoustically coupled spaces. Since then a number of recent works call for efficient estimation methods within the Bayesian framework. An efficient analysis is of practical significance for better understanding and modeling the sound energy decay process in acoustically coupled spaces or even in single spaces for reverberation time estimation. This paper will first formulate the Bayesian posterior probability distribution function (PPDF) in a matrix form to reduce the dimensionality as applied to the decay time evaluation. Based on existence of only global extremes of PPDFs as observed from extensive experimental data, this paper describes a dedicated search algorithm for an efficient estimation of decay times.

Evaluation of decay times in coupled spaces: reliability analysis of Bayeisan decay time estimation

This paper discusses quantitative tools to evaluate the reliability of "decay time estimates" and inter-relationships between multiple decay times for estimates made within a Bayesian framework. Previous works [Xiang and Goggans, J. Acoust. Soc. Am. 110, 1415-1424 (2001); 113, 2685-2697 (2003)] have applied Bayesian framework to cope with the demanding tasks in estimating multiple decay times from Schroeder decay functions measured in acoustically coupled spaces. A parametric model of Schroeder decay function [Xiang, J. Acoust. Soc. Am. 98, 2112-2121 (1995)] has been used for the Bayesian model-based analysis. The relevance of this work is that architectural acousticians need to know how well determined are the estimated decay times calculated within Bayesian framework using Schroeder decay function data. This paper will first address the estimation of global variance of the residual errors between the Schroeder function data and its model. Moreover, this paper discusses how the "landscape" shape of the posterior probability density function over the decay parameter space influences the individual decay time estimates, their associated variances, and their inter-relationships. This paper uses experimental results from measured room impulse responses in real halls to describe a model-based sampling method for an efficient estimation of decay times, and their individual variances. These parameters along with decay times are relevant decay parameters for evaluation and understanding of acoustically coupled spaces.

Evaluation of decay times in coupled spaces: Bayesian decay model selection

This paper applies Bayesian probability theory to determination of the decay times in coupled spaces. A previous paper [N. Xiang and P. M. Goggans, J. Acoust. Soc. Am. 110, 1415-1424 (2001)] discussed determination of the decay times in coupled spaces from Schroeder's decay functions using Bayesian parameter estimation. To this end, the previous paper described the extension of an existing decay model [N. Xiang, I. Acoust. Soc. Am. 98, 2112-2121 (1995)] to incorporate one or more decay modes for use with Bayesian inference. Bayesian decay time estimation will obtain reasonable results only when it employs an appropriate decay model with the correct number of decay modes. However, in architectural acoustics practice, the number of decay modes may not be known when evaluating Schroeder's decay functions. The present paper continues the endeavor of the previous paper to apply Bayesian probability inference for comparison and selection of an appropriate decay model based upon measured data. Following a summary of Bayesian model comparison and selection, it discusses selection of a decay model in terms of experimentally measured Schroeder's decay functions. The present paper, along with the Bayesian decay time estimation described previously, suggests that Bayesian probability inference presents a suitable approach to the evaluation of decay times in coupled spaces.