Digital acoustics: processing wave fields in space and time using DSP tools

Systems with hundreds of microphones for acoustic field acquisition, or hundreds of loudspeakers for rendering, have been proposed and built. To analyze, design, and apply such systems requires a framework that allows us to leverage the vast set of tools available in digital signal processing in order to achieve intuitive and efficient algorithms. We thus propose a discrete space–time framework, grounded in classical acoustics, which addresses the discrete nature of the spatial and temporal sampling. In particular, a short-space/time Fourier transform is introduced, which is the natural extension of the localized or short-time Fourier transform. Processing in this intuitive domain allows us to easily devise algorithms for beam-forming, source separation, and multi-channel compression, among other useful tasks. The essential space band-limitedness of the Fourier spectrum is also used to solve the spatial equalization task required for sound field rendering in a region of interest. Examples of applications are shown.

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

Applications of short space-time Fourier analysis in digital acoustics

Francisco Pereira Correia Pinto, Martin Vetterli

This paper presents a signal processing tool for analyzing and manipulating digitized acoustic wave fields, based on a spatio-temporal extension of the time-frequency representation space. The emphasis is on wave fields acquired with a 1-D linear array of equidistant microphones (representing the spatial samples), but the basic formulation is valid for the three dimensions of space. We start by defining a spatio-temporal extension of the short time Fourier transform, in both continuous and discrete space and time, and then focus on applications of the proposed representation. In particular, we show that acoustic scenes with multiple sources can benefit from the use of space-frequency analysis in applications such as source separation (spatial filtering) and spatial audio coding (wave field coding). The experiments suggest that there is a spatial window size for which the performance of filtering and coding is optimal.

Ieee Service Center, 445 Hoes Lane, Po Box 1331, Piscataway, Nj 08855-1331 Usa2011

Endoscopic low coherence interferometry applied to tri-dimensional contouring of upper airways

Yves Delacrétaz

This thesis presents a novel approach for optical tri-dimensional contouring, requiring only one acquisition to extract a contour depth. It is based on an interferometric setup, using a reduced coherence light source, and is fully adaptable to endoscopic procedure commonly in used in ear-nose-throat (ENT) medicine. It has been demonstrated that the method can be successfully applied to imaging of porcine upper airways. Although this method uses interferometry to obtain depth range measurement, the phase signal is not directly evaluated, rather the location of the coherent superposition of two waves is extracted, and thus provides a robust technique even in harsh environment. The key feature is to extract the coherent area on the image by locating intentionally induced fringes created by off-axis superposition of a smooth reference wave and an object wave coming from a rough or diffusing sample through an optical system with limited aperture. A model for the propagation of the light through the optical system has been developed. It takes into account the broadened emission spectrum of the light source, as well as the statistical speckle effect induced by the illumination of a rough surface with partially coherent light. Simulated results have been carefully compared with measurements, both in the spatial domain and in the Fourier domain (spatial frequencies response), which has permitted to better understand the processes involved in the creation of the interference fringes, and improve the design of the device. Different filters have been investigated in order to extract the depth information from acquired interferograms. By using Gabor filterbanks, which do not need any a priori knowledge of the signal, selectivity of the extracted signal has been enhanced by a factor of 1.3 compared to a simple local Fourier spectrum evaluation, and time of the extraction procedure divided by 6, even when compared to fast iterative calculation in the Fourier spectrum. Moreover, by imposing the frequency and/or orientation of the signal to detect, which can be known by a simple calibration procedure, it has been proved that the selectivity can be further enhanced by a factor of 2.5, keeping the calculation time as short or even shorter. A compact prototype has been designed and built, with a specific multiple fibers arrangement for illumination. The system is separated in two parts: the first one is directly attached to the rigid endoscope, and contains imaging as well as recombining object/reference wave optics, and the detector. It fits into a box with overall dimensions of 115 × 83 × 94 mm. The second part contains the laser source, the scanning arm and the injection optics for the fibered system. It is designed to be left on a small cart. This device can potentially be brought to the bedside, and used during routine endoscopic check-up. Associated with the device, a complete framework has been elaborated in order to simplify the acquisition procedure as well as the signal processing. This results in a fully automated environment taking in charge hardware control, data storage and signal processing, up to the tri-dimensional scene rendering.