Spatial extrapolation of early room impulse responses in local area using sparse equivalent sources and image source method

Abstract

The room impulse response (RIR) is important in most acoustic applications, such as the design of concert halls and sound field control, because it characterizes the sound propagation. The measurement of RIRs at multiple points is challenging, as it requires a huge microphone array or repeating the experiment by microphone replacement. Several RIR interpolation and extrapolation methods of RIRs have been developed for obtaining RIRs from multiple measurement points efficiently. Extrapolation methods offer more efficient RIR measurement compared to interpolation. However, previous studies focused on extrapolation at frequencies below 1 kHz, and the extrapolation at higher frequencies was difficult. In this study, we propose an extrapolation method for RIRs of direct sound and early reflections in a local area using a small number of measurement points. The proposed method represents the RIRs around the microphones using superpositions of sparse equivalent sources located around the loudspeaker and image sources. We conducted a measurement experiment in an anechoic chamber to estimate RIRs around microphones using sound-reflecting boards. From the experimental results with 2.5 dimensional conditions, the proposed method achieved about above 10 dB of signal to noise ratio (SNR) near the microphone array from 0.5–8.5 kHz. For the extrapolation accuracy over the entire evaluation area (0.6 × 0.54 m²), the proposed method improved the SNR by about 5–6 dB compared to results using the plane wave decomposition.

Introduction

Measurement of room impulse response (RIR) is essential to understand sound propagation in a room. For example, the RIRs at multiple points are useful for sound field control/synthesis [4], [23], [6], and visualization of sound field, etc [10], [29]. Sound propagation is considered as a linear time-invariant system, and the system considers the loudspeaker as the input and the microphone as the output. The RIR depends on the positions of the loudspeaker and the microphone because the sound propagation path can change based on those positions. Thus, to obtain the RIRs from the loudspeaker to other points, we must repeat measurements after relocating the microphone. Another approach is to use a microphone array that has multiple microphones located at all measurement points. However, as the number of microphones increases, the sound reflections from the microphones become larger and the calibration of the microphones become more complicated.

In recent studies, several RIR interpolation methods have been proposed that are able to measure multiple RIRs efficiently. In [19], proposed an RIR interpolation method for an entire room using the sparsity of the early reflections in the time domain.

In [12], [13], the RIRs at grid points were recovered by solving the inverse problem of interpolating the signal of the microphone on a moving path from RIRs at the grid points.

Alternatively, the equivalent source method (ESM) is well known for the reconstruction of the radiated and scattered sound fields [14], [11], [27]. Based on the Kirchhoff–Helmholtz integral equation [30], the sound field can be represented by the superposition of point sources surrounding the domain of interest. Because the number of point sources exceeds the number of measurement points, it is treated as an undecided problem and is solved by the least squares (LS) method.

With the recent developments in compressed sensing [3], a sparse representation of the acoustic field has been shown to be effective for some applications, such as multi-sound field control [21], [22], sound field decomposition [15], and RIR interpolation [19], [2]. In ESM, the sparse expression is also more effective than the LS method to represent the near field [5].

In addition, the extrapolation method helps us to obtain the RIRs more efficiently compared to the interpolation, since it is simple and easy to place the microphones. In [9], the room transfer function is modeled by the common acoustical poles and their residues corresponding to the eigenfrequencies of the room. In [17], neural networks are applied for the sound field reconstruction, and sound field variations are predicted based on the observations by a small number of microphones. Furthermore, in the work presented in [28], the RIRs at arbitrary points were extrapolated by a limited number of plane waves using compressed sensing. These previous studies have shown the effectiveness of RIR extrapolation at frequencies below 1 kHz. However, extrapolation of RIRs at higher frequencies is required for various applications such as sound field control and concert hall design. We previously proposed extrapolation methods for direct sound and primary reflections based on sparse ESM and image source method [24], [25], and evaluated the effectiveness of our proposed method by simulation experiments.

In this study, to estimate the sound reflections around the microphones using the decomposition of the reflection components on each wall, we propose the extrapolation method of RIRs including not only primary reflection but also the other early reflections. In the measurement experiment in an anechoic chamber, we evaluate the estimation accuracies at frequencies up to 8.5 kHz.

Our proposed method can determine the reflection components for each wall using the image source method [1]. This decomposition can help in various applications such as sound visualization and room acoustics design in architectural acoustics, based on the relationship between each reflection component and the reflecting object. Furthermore, since it is known that the early part of RIRs affects the timbre and sound localization [7]; therefore, it is necessary to design and control the early part of RIRs in most acoustic applications. Thus, in this study, we focus on extrapolation of the early part of RIRs in the frequency band 0.5–8.5 kHz. We conduct an experimental evaluation with sound-reflecting boards in an anechoic chamber. We evaluate extrapolation accuracies of RIRs including primary and secondary sound reflections with two different configurations of the microphone array.

The outline of this paper is as follows: In Section 2, we present the estimation method of RIRs with sparse ESM and the image source method. In Section 3, experimental results are reported to evaluate the proposed method. Finally, we conclude this paper in Section 4.