Robust Sound Field Control for Audio Reproduction : A Polynomial Approach to Discrete-Time Acoustic Modeling and Filter Design
Abstract: This thesis is concerned with the design and analysis of robust discrete-time filters for audio equalization and sound field control in real reverberant environments. Inspired by methods in polynomial control theory, a unified framework for acoustic modeling and filter design is developed. The work on modeling is centered around three main themes: First, the acoustic channel between a loudspeaker and a point in space is studied in time, frequency and space, and a polynomial matrix fraction description with diagonal denominator is selected as a physically motivated channel model. As a means for representing channel uncertainties, a probabilistic design model is proposed. Second, the concept of sound field dimensionality, based on the Karhunen-Loève expansion of the sound field, is explored and integrated into the polynomial systems context. Third, a method for spatial interpolation of acoustic transfer functions is proposed and evaluated. Interpolation errors are accounted for by applying the probabilistic uncertainty model to the interpolated data. The work on filter design can be categorized into single- and multichannel methods. The single-channel problem concerns the improvement of the impulse and frequency responses of a single loudspeaker over a region in space, by means of a scalar prefilter. This problem is posed in a SIMO (single-input multiple-output) feedforward control setting, and is solved using polynomial methods. The solution offers several useful insights and results. In particular, new results are derived regarding the adverse pre-ringing problem associated with mixed phase filters. Based on the new results, a refined mixed phase method is proposed that is practically free from pre-ringing artifacts. In the multichannel problem, a desired spatio-temporal sound field is approximated by the joint use of several loudspeakers. This problem is initially formulated and solved by feedforward control over a continuous spatial domain, assuming full knowledge of the spatial field. To obtain a practically feasible design, the control criterion is then spatially discretized, resulting in a standard MIMO(multiple-input multiple-output) linear quadratic feedforward control problem. Since information is generally lost in the discretization process, a robust design based on spatial interpolation and probabilistic error modeling is proposed. The multichannel designs are assessed in an automotive setting, using practical measurements of a nine-channel sound system in a car.
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