Solution Strategies for Linear Inverse Problems in Spatial Audio Signal Processing

Abstract

Abstract:The aim of this study was to compare algorithms for solving inverse problems generally encountered in spatial audio signal processing. Tikhonov regularization is typically utilized to solve overdetermined linear systems in which the regularization parameter is selected by the golden section search (GSS) algorithm. For underdetermined problems with sparse solutions, several iterative compressive sampling (CS) methods are suggested as alternatives to traditional convex optimization (CVX) methods that are computationally expensive. The focal underdetermined system solver (FOCUSS), the steepest descent (SD) method, Newton's (NT) method, and the conjugate gradient (CG) method were developed to solve CS problems more efficiently in this study. These algorithms were compared in terms of problems, including source localization and separation, noise source identification, and analysis and synthesis of sound fields, by using a uniform linear array (ULA), a uniform circular array (UCA), and a random array. The derived results are discussed herein and guidelines for the application of these algorithms are summarized.

Microphone and Loudspeaker Array Signal Processing to Build a "Radiation Keyboard" for Authentic Samplers

“…The radiated sound P X is recorded but the source sound P Y , which created the recorded sound pressure distribution, is sought. When solving the linear equation system, e.g., by means of Gaussian elimination or a pseudo-inverse matrix of G [22], the resulting sound pressure levels tend to be huge due to small rounding errors and measurement noise. The reason for that is that the propagation matrix is ill-conditioned when microphone positions are close to one another compared to the considered wavelength.…”

“…A regularization method relaxes the matrix and yields lower amplitudes. An overview about regularization methods can be found in [20,22]. For musical instruments, the Minimum Energy Method (MEM) [20,23] is superior.…”

Perceptual sound field synthesis concept for music presentation

“…Widely applied regularization methods for square matrices and over-determined systems, which describe the desired sound field at more locations than loudspeakers present in the setup, are least-squares methods, Tikhonov regularization, and truncated singular value decomposition. These are revisited in [49], together with an alternative approaches named golden section search. Another newly developed regularization method, the radiation method, is not mathematically-motivated but originates in physical assumptions.…”

Guest Editors’ Note—Special Issue on Spatial Audio

“…The paper authored by Bai, Chung, Wu, Chiang, and Yang [9] proposes a general strategy to tackle the inverse problem that is often encountered in solving the source identification and separation problems for the spatial audio signal processing. Various inverse problem solvers, for both underdetermined and overdetermined problems, are investigated and compared in terms of PESQ and segSNR.…”