Swiss National Science Foundation
The project aimed at developing Compressive sensing techniques to room acoustics, especially to retrieve a precise reconstruction of the Sound field in a room using the smallest number of measurements possible. This precise understanding of the sound field in a room can lead to multiple applications, including but not limited to, equalization of room modes at low frequency. From this understanding, it is possible to create different practical tools in order to assess critically the performance of different noise reduction/acoustic control methods. Moreover, the complete reconstruction and rendering of the spatial sound field of the room could possibly provide precious information for the designs of different room modes equalization methods.
Room modes can significantly impair the quality of sound diffusion, in particular at low-frequencies and in small auditoria. These low-frequency resonances are the most difficult to control as passive sound absorbers are unable to damp them efficiently. The ability to control low-frequency room modes is of tremendous interest to audio engineers and architects working in and designing recording studios, auditoria, concert halls or home cinemas. To reduce these resonances, acoustic simulations on simplified models of the room can be run to obtain either the Room Frequency Response (RFR) or the Room Impulse Response (RIR), that help spotting causes and places where resonances are the loudest. The assessment of the effect of potential sound absorbers is also achieved trough the measurement of the RFR/RIR with a few microphones in the room.
Recently, a new concept of Electroacoustic Absorber (EA) developed at EPFL has allowed a step forward in the effective damping of such room resonances, thus providing a solution for equalizing the sound pressure field within the whole space. However, to broaden the frequency damping range and improve further the efficiency of the control, a precise knowledge of the RFR/RIR is mandatory, but it is difficult to measure with accuracy. At the moment it requires arrays of microphones regularly spaced. This quickly becomes impractical with non-rectangular rooms, or as the room dimensions increase. New methods for measuring accurately the RFR/RIR are therefore needed.
In this project we propose to apply a new signal processing approach called "Compressive Sensing" (CS) to the problem of room acoustics. This research topic has recently become very popular among mathematicians and signal processing scientists. It is strongly related to the concept of sparsity in signal processing, where the information is assumed sparse within measured signals, provided we choose the appropriate signal representation. Although it is a powerful technique on the theoretical level, it is not widely spread and up to now has been scarcely developed in the application side. In particular, in acoustics, where signals are often sparse linear combinations of simple signals, the potential applications are numerous. The problem of standing waves in rooms is one of the most likely to take advantage of CS. This project aims at developing CS to sound fields in closed spaces, and give practical experimental proof of concept for future instrumentation for acoustic practitioners.
A precise measurement of the RFR/RIR by mean of CS opens the door to a better control of room modes and the reduction of noise pollution at lowfrequency. It can potentially lead to awide range of new practical tools for assessing room characteristics such as in situ acoustic impedance retrieval, wall absorption estimation, source directivity measurements, or even room anechoicity measurements. The outcome of the project can have a wide impact in the domain of room acoustics, acoustic measurements and control. It could lead to new configurations with passive and/or active control for more efficient anechoic room for scientific experiments, improved audio experience in auditoria, or reduced low-frequency noise in factories.
The project structure will be organized along two main axes:
- development of microphone arrays signal processing techniques based on CS for room acoustic characterization (first in the low-frequency range, then extended to medium-high frequency range;
- development of optimal global 3D active control techniques based on sparse sound field representation
The achieved research outcomes are the following structure:
Signal recovery and its relevance with room acoustics at low frequencies
At the beginning, several studies were conducted to investigate traditional signal recovery methods such as Nyquist, spline recovery and their limitation when it comes to measurement in space. From there, several more recent and advanced techniques such as compressive sensing, greedy algorithm, matching pursuit are investigated to understand their relevance to the spatio-temporal Helmholtz equation in room acoustics and how each methods can be used to reduce the number of measurement points without reducing the faithfulness of the signal reconstruction.
Sparse representation and analysis of the wave equation in room acoustics
From the previous stage, it becomes clear that the wave solution for the sound field of the room at low frequencies will require certain reformulation to highlight the sparsity that inherently exists within itself. Thanks to the plane waves approximation technique, the wave solution can be further approximated by a closed form solution. This approximation presents the wave solution in the form of a finite sum of damped harmonic plane waves. The convenience in this representation form is that it gives an estimation to the spatiotemporal solution of the sound field in the room regardless of the types of room modes. This means that the formulation can be used to depict the sound field of a non-rectangular room.
Reconstruction framework design of the sound field reconstruction algorithms
Based on the reformulation of the governing equation for the sound field in the room at low frequencies, various algorithms including matching pursuit and least squares approximation can be used to recover the unknown parameters in the governing equations from a limited set of random spatial measurements in the room. With these parameters known, it is now possible to reconstruct the entire sound field of the room using a limited set of microphones.
Reconstruction results: validation and assessment
Thanks to the reconstruction framework from the previous stage, the sound field of the room can now be recovered using a small set of sample points randomly placed in the room. In order to investigate the performance of the framework, it is first tested on a numerical models using FEM simulation. The model of a 6-sided non-rectangular room was introduced which represent an existing reverberation chamber in the facilities. Inside this model, multiple measurement points and evaluation points were spread randomly. Using the simulation results, multiple aspects of the reconstruction frameworks have been tested, including average accuracy (based on the average correlation values between the reconstructed signals and the original signals), spatial faithfulness of the reconstruction (based on the graphical reconstruction of the room spatial responses where the mode shapes can be clearly visible) as well as further analysis regarding the number of microphones, room modes density and wall dampings. All analysis produced highly accurate results which validate the design of the sound field reconstruction framework and the algorithm used. Keeping this in mind, the framework was expanded to real experiments where real microphones were placed in a non-rectangular reverberation chamber to assess its performance. The analysis once again showed satisfying results where the correlation are kept between 97-99% using as low as 15 microphones.
Assessment of room absorption devices using the reconstruction framework
From the analysis in the previous stage, it is clear that the reconstruction framework is highly effective in recovering the sound field of an empty non-rectangular room. This stage focuses more on the applicability of the framework in cases where there are active absorption elements in the room. In order to investigate this aspect, the electroacoustic absorbers (EAs) which are active room modes absorbing elements are placed in the corner of the room. The reconstruction framework are then repeated in this environment. The analysis shows that the framework still performs with high reconstruction accuracy (>97%) even when there are active elements in the room. This eventually means that the framework can be effectively used to assess the spatial performance of not just passive but also active sound absorption elements in room acoustics.
Sound field reconstruction application in application in room modes equalization
This final stage of the project focuses on a higher level application of the framework, which is to use reconstruction results to influence the design of active room modes equalization devices. In order to perform this, first, two new equalization metrics were proposed to analyze the equalization aspects of the modes of the room using spatial information. The target was to compare the room responses in the frequency domain and assess how much deviation the responses are from a flat line. Using this target in combination with the reconstruction framework from the previous stage, it is possible to optimize the built-in parameters of the aforementioned Electroacoustic Absorber (EA). Through an automatic parametric sweeping process, different combination of the 3 built-in parameters of the EAs can be evaluated based on the metrics, which, can be calculated from the outcome of the sound field reconstruction framework using 10-12 microphones. By ranking the combinations based on the metrics, we were able to find the most optimized combination for a 1-degree of freedom resonator model of the EA. This opens door to future application where a more complex combination system can be used to optimized the settings of a multi degree of freedom model where a higher number of built-in control parameters is needed.
- T.Pham Vu, H. Lissek, Low frequency sound field reconstruction in a non-rectangular room using a small number of microphones, Acta Acustica 4(2), 2020 (open access)
- H. Peic Tukuljac et al, Joint estimation of the room geometry and modes with compressed sensing, IEEE ICASSP. Calgary, Alberta, Canada, pp. 6882–6886. https://doi.org/10.1109/ICASSP.2018.8462655
- H. Peic Tukuljac et al, Localization of sound sources in a room with one microphone, Proc. SPIE 10394, Wavelets and Sparsity XVII, 103940I (24 August 2017); https://doi.org/10.1117/12.2271249
This project has been funded by the Swiss National Science Foundation under grant agreement No. 200021_169360.