The plenacoustic function and its applications

This thesis is a study of the spatial evolution of the sound field. We first present an analysis of the sound field along different geometries. In the case of the sound field studied along a line in a room, we describe a two-dimensional function characterizing the sound field along space and time. Calculating the Fourier transform of this function leads to a spectrum having a butterfly shape. The spectrum is shown to be almost bandlimited along the spatial frequency dimension, which allows the interpolation of the sound field at any position along the line when a sufficient number of microphones is present. Using this Fourier representation of the sound field, we develop a spatial sampling theorem trading off quality of reconstruction with spatial sampling frequency. The study is generalized for planes of microphones and microphones located in three dimensions. The presented theory is compared to simulations and real measurements of room impulse responses. We describe a similar theory for circular arrays of microphones or loudspeakers. Application of this theory is presented for the study of the angular sampling of head-related transfer functions (HRTFs). As a result, we show that to reconstruct HRTFs at any possible angle in the horizontal plane, an angular spacing of 5 degrees is necessary for HRTFs sampled at 44.1 kHz. Because recording that many HRTFs is not easy, we develop interpolation techniques to achieve acceptable results for databases containing two or four times fewer HRTFs. The technique is based on the decomposition of the HRTFs in their carrier and complex envelopes. With the Fourier representation of the sound field, it is then shown how one can correctly obtain all room impulse responses measured along a trajectory when using a moving loudspeaker or microphone. The presented method permits the reconstruction of the room impulse responses at any position along the trajectory, provided that the speed satisfies a given relation. The maximal speed is shown to be dependent on the maximal frequency emitted and the radius of the circle. This method takes into account the Doppler effect present when one element is moving in the scenario. It is then shown that the measurement of HRTFs in the horizontal plane can be achieved in less than one second. In the last part, we model spatio-temporal channel impulse responses between a fixed source and a moving receiver. The trajectory followed by the moving element is modeled as a continuous autoregressive process. The presented model is simple and versatile. It allows the generation of random trajectories with a controlled smoothness. Application of this study can be found in the modeling of acoustic channels for acoustic echo cancellation or of time-varying multipath electromagnetic channels used in mobile wireless communications.

Dynamic Measurement of Room Impulse Responses using a Moving Microphone

Thibaut Ajdler, Luciano Sbaiz, Martin Vetterli

A novel technique for the recording of large sets of room impulse responses or head-related transfer functions is presented. The technique uses a microphone or a loudspeaker moving with constant speed. Given a setup (e.g. length of the room impulse response), a careful choice of the recording parameters (excitation signal, speed of movement) is shown to lead to the reconstruction of all impulse responses along the trajectory. In the case of moving element along a circle, the maximal angular speed is given in function of the length of the impulse response, its maximal temporal frequency, the speed of sound propagation and the radius of the circle. As result of this theory, it is shown that head-related transfer functions sampled at

44.1~

kHz can be measured at all angular positions along the horizontal plane in less than one second. The presented theory is compared with a real system implementation using a precision moving microphone holder. The practical setup is discussed together with its limitations.

2007

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

Fourier uniqueness in even dimensions

Maryna Viazovska

In recent work, methods from the theory of modular forms were used to obtain Fourier uniqueness results in several key dimensions (d = 1, 8, 24), in which a function could be uniquely reconstructed from the values of it and its Fourier transform on a discrete set, with the striking application of resolving the sphere packing problem in dimensions d = 8 and d = 24. In this short note, we present an alternative approach to such results, viable in even dimensions, based instead on the uniqueness theory for the KleinGordon equation. Since the existing method for the Klein-Gordon uniqueness theory is based on the study of iterations of Gauss-type maps, this suggests a connection between the latter and methods involving modular forms. The derivation of Fourier uniqueness from the Klein-Gordon theory supplies conditions on the given test function for Fourier interpolation, which are hoped to be optimal or close to optimal.

2021