Theory and investigation of acoustic multiple-input multiple-output systems based on spherical arrays in a room

TL;DR

This work addresses the need for spatially informed room acoustics analysis beyond energy-based metrics by introducing a MIMO system built from spherical loudspeaker and microphone arrays. It develops a closed-form, SH-based far-field transfer model and extends it to enclosed spaces via the image-source method, proving key properties such as unit rank ($rank=1$) in free field and rotation/mirroring invariance of the singular-value spectrum. The room-modeling analysis shows the effective rank grows with the number of significant reflections, suggesting richer spatial control and analysis as reverberation increases. Both simulations and an experimental study validate the theory, demonstrating how the effective rank evolves with room characteristics and how field synthesis performance improves when the spatial complexity of the field is higher.

Abstract

Spatial attributes of room acoustics have been widely studied using microphone and loudspeaker arrays. However, systems that combine both arrays, referred to as multiple-input multiple-output (MIMO) systems, have only been studied to a limited degree in this context. These systems can potentially provide a powerful tool for room acoustics analysis due to the ability to simultaneously control both arrays. This paper offers a theoretical framework for the spatial analysis of enclosed sound fields using a MIMO system comprising spherical loudspeaker and microphone arrays. A system transfer function is formulated in matrix form for free-field conditions, and its properties are studied using tools from linear algebra. The system is shown to have unit-rank, regardless of the array types, and its singular vectors are related to the directions of arrival and radiation at the microphone and loudspeaker arrays, respectively. The formulation is then generalized to apply to rooms, using an image source method. In this case, the rank of the system is related to the number of significant reflections. The paper ends with simulation studies, which support the developed theory, and with an extensive reflection analysis of a room impulse response, using the platform of a MIMO system.

Figures (5)

• Figure 1: Mirroring about the x-z plane, viewed from the z-axis (top view)
• Figure 2: (color online) (a) System effective rank at $700$Hz as a function of $\tau$, time-window duration, for a room with adjustable reflection coefficients, as detailed in the figure caption. (b) Simulated impulse response between omnidirectional loudspeaker and microphone.
• Figure 3: Reproduced sound field around the microphone array: (a)-(c) Simulated rooms, see table 1. (d)- Target incident field.
• Figure 4: (color online) (a) System effective rank as a function of time-window length $\tau$. (b) Measured impulse response between omnidirectional directivity patterns at loudspeaker and microphone arrays.
• Figure 5: (color online) Singular value distribution of $\bm G[0.016\text{sec}, 700\text{Hz}]$, $\bm G[0.029\text{sec}, 700\text{Hz}]$, and $\bm G[1.011\text{sec}, 700\text{Hz}]$, with effective ranks of 1.88, 3.92, and 6.09, respectively.