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.