The Plenacoustic Function and its Sampling

We study the spatialization of the sound field in a room, in particular the evolution of room impulse responses as a function of their spatial positions. We observe that the multidimensional spectrum of the solution of the wave equation has an almost bandlimited character. Therefore sampling and interpolation can easily be applied using signals on an array. The decay of the spectrum is studied on both temporal and spatial frequency axes. We study how this decay influences the performance of the interpolation. Based on the support of the spectrum, we determine the number and the spacing between the microphones needed to reconstruct the sound field up to a certain temporal frequency. The optimal sampling pattern for the microphone positions is given for the linear and the planar case. Existing techniques usually make use of room models to recreate the sound field present at some point in the space. Our technique simply starts from the measurements of the sound field in a finite number of positions and with this information the total sound field can be recreated. Finally, simulations and experimental results are presented and compared with the theory.

Thibaut Ajdler, Luciano Sbaiz, Martin Vetterli

The spatialization of the sound field in a room is studied, in particular the evolution of room impulse responses as a function of their spatial positions. It was observed that the multidimensional spectrum of the solution of the wave equation has an almost bandlimited character. Therefore, sampling and interpolation can easily be applied using signals on an array. The decay of the spectrum is studied on both temporal and spatial frequency axes. The influence of the decay on the performance of the interpolation is analyzed. Based on the support of the spectrum, the number and the spacing between the microphones is determined for the reconstruction of the sound pressure field up to a certain temporal frequency and with a certain reconstruction quality. The optimal sampling pattern for the microphone posi- tions is given for the linear, planar and three-dimensional case. Existing techniques usually make use of room models to recreate the sound field present at some point in the space. The presented technique simply starts from the measurements of the sound pres- sure field in a finite number of positions and with this information the sound pressure field can be recreated at any spatial position. Finally, simulations and experimental results are presented and compared with the theory.

2006

The Plenacoustic Function, Sampling and Reconstruction

Thibaut Ajdler, Martin Vetterli

In the present paper, we study the spatialization of the soundfield in a room, in particular the evolution of room impulse responses as function of their spatial positions. The presented technique allows us to completely characterize the sound field in any arbitrary location if the sound field is known in a certain finite number of positions. The existing techniques make usually use of room models to recreate the sound field present at some point in the space. Our technique simply starts from the measurements of impulse responses in a finite number of positions and with this information the total sound field can be recreated. An analytical solution of the problem is given for different cases of spaces. Further, we determine the number and the spacing between the microphones needed to perfectly reconstruct the sound field up to a certain temporal frequency. The optimal sampling pattern for the microphone positions is given. Applications are also discussed.

2003

Signal Processing in Space and Time

Francisco Pereira Correia Pinto

Sound waves propagate through space and time by transference of energy between the particles in the medium, which vibrate according to the oscillation patterns of the waves. These vibrations can be captured by a microphone and translated into a digital signal, representing the amplitude of the sound pressure as a function of time. The signal obtained by the microphone characterizes the time-domain behavior of the acoustic wave field, but has no information related to the spatial domain. The spatial information can be obtained by measuring the vibrations with an array of microphones distributed at multiple locations in space. This allows the amplitude of the sound pressure to be represented not only as a function of time but also as a function of space. The use of microphone arrays creates a new class of signals that is somewhat unfamiliar to Fourier analysis. Current paradigms try to circumvent the problem by treating the microphone signals as multiple "cooperating" signals, and applying the Fourier analysis to each signal individually. Conceptually, however, this is not faithful to the mathematics of the wave equation, which expresses the acoustic wave field as a single function of space and time, and not as multiple functions of time. The goal of this thesis is to provide a formulation of Fourier theory that treats the wave field as a single function of space and time, and allows it to be processed as a multidimensional signal using the theory of digital signal processing (DSP). We base this on a physical principle known as the Huygens principle, which essentially says that the wave field can be sampled at the surface of a given region in space and subsequently reconstructed in the same region, using only the samples obtained at the surface. To translate this into DSP language, we show that the Huygens principle can be expressed as a linear system that is both space- and time-invariant, and can be formulated as a convolution operation. If the input signal is transformed into the spatio-temporal Fourier domain, the system can also be analyzed according to its frequency response. In the first half of the thesis, we derive theoretical results that express the 4-D Fourier transform of the wave field as a function of the parameters of the scene, such as the number of sources and their locations, the source signals, and the geometry of the microphone array. We also show that the wave field can be effectively analyzed on a small scale using what we call the space/time-frequency representation space, consisting of a Gabor representation across the spatio-temporal manifold defined by the microphone array. These results are obtained by treating the signals as continuous functions of space and time. The second half of the thesis is dedicated to processing the wave field in discrete space and time, using Nyquist sampling theory and multidimensional filter banks theory. In particular, we show examples of orthogonal filter banks that effectively represent the wave field in terms of its elementary components while satisfying the requirements of critical sampling and perfect reconstruction of the input. We discuss the architecture of such filter banks, and demonstrate their applicability in the context of real applications, such as spatial filtering and wave field coding.

EPFL2010

Signal Processing in Space and Time

Francisco Pereira Correia Pinto

EPFL2010