Low-complexity computer simulation of multichannel room impulse responses

Abstract

The telephone'' model has been, for last one hundred thirty years, base of modern telecommunications with virtually no changes in its fundamental concept. The arise of smaller and more powerful computing devices have opened new possibilities. For example, to build systems able to give to user illusion of being talking to remote party as if both where in same place. To achieve this still many challenges have to be overcome. In this thesis, a part of acoustical signal processing problem is treated. To acoustically create illusion of presence, fast and accurate control over sound field in a room is required. The sound field given one or more sources is subject to different acoustical phenomena, such as reflection and diffraction. Because of these, to model or estimate sound field in a room is in general a difficult task. In particular acoustical reflection poses an important challenge. The sound field reflects on walls, ceiling and floor and a moment later those reflections reflect again, and later these reflect again. This recursive process makes number of reflections as a function of time to increase, in general, at a geometric rate. To synthesize an artificial sound field in real time, one has to be able to model these reflections fast and accurately enough. In this thesis a fast algorithm to model sound field in box-shaped rooms is proposed. Part one of this thesis begins with an introduction to topic, here different acoustical phenomena of interest are explained, and concept of room impulse response (RIR) is introduced. The RIR is defined as time-domain signal sensed at a receiver position as generated by a point source that emits an impulse. Assuming a linear time-invariant (LTI) model, if point source emits not an impulse but an arbitrary signal, actual sound field at a given observation location can then be modeled as a convolution of source signal with RIR. Moreover, since we are assuming a linear model, sound field generated by an arbitrary number of point sources emitting arbitrary signals can be easily computed once RIRs from locations of sources to observation locations are known. Efficient computation of RIR is therefore of theoretical and practical interest. Consequently, this part concludes with a summary of most prominent algorithms to simulate RIR. Part two of this thesis contains relevant papers that make up this work. The analysis is given first for case of fully reflective walls. It is noted that in this case all acoustical reflections can be modeled by a set of virtual sources following a periodic structure over a lattice. The whole set of virtual sources is generated by repetitions of a small set of sources called the mother sources''. On other side, Poisson summation formula establishes relation between periodicity and discretization under Fourier transform. Relating these concepts, it is shown that by carefully discretizing spectral representation of RIR of mother sources in free-field, exact periodic structure that makes up sound field in a room can be obtained. This is key idea behind proposed method. Carefully discretizing all domains, and making use of fast Fourier transform (FFT), a fast multichannel RIR simulation method is obtained. Unfortunately this idea only works for fully reflective walls. By allowing walls to have constant complex-valued reflection coefficients (this is, to model absorption and phase shift at walls) sound field of set of virtual sources is not anymore periodic. A generalization of Fourier transform is then introduced. First, a generalized Poisson summation formula is derived. This formula relates discretization in generalized Fourier domain to a geometrically weighted periodic summation in reciprocal domain. Basic properties of this transform are derived, its application to non zero-padded linear convolution is derived, but moreover a fast implementation, called generalized fast Fourier transform (GFFT), is given. The proposed method is then extended to account for walls with constant complex-valued reflection coefficients. It is shown that by separating sound field of mother sources into its orthant-sided parts (the analogue of single-sided parts of a function of a scalar variable), sound field inside a room can be expressed as a sum of geometrically weighted sound fields generated by periodic set of virtual sources. This summation is then related to a sampling condition on generalized spectrum of orthant-sided parts of sound field of mother sources. Using GFFT method simulates RIR given a source at a dense set of spatial positions with very low complexity. In experiments a comparison with a model called mirror image source method (MISM) is given. In one scenario, time MISM would take to compute RIR at a dense set of positions is estimated to be about one and a half years. The newly proposed method computes RIR at all positions in only forty-eight minutes. This shows contrasting difference in computational complexity, making new method an important step on road to simulate realistic sound fields in real time.