Pole-zero modeling and principal component analysis of head-related transfer functions.

Abstract

Synthesizing over headphones the free-field to eardrum transfer functions, or head-related transfer functions (HRTFs), of the human auditory system poses several difficulties from both signal processing and psychoacoustic perspectives. Viewed as a system approximation problem, an experiment and two analytic approaches are considered for designing digital filter implementations of HRTFs. Compared to an all-zero model of HRTFs, a pole-zero model would seem to promise fewer parameters and, consequently, a lower computational complexity associated with synthesizing 3-D acoustic fields. The number of parameters might be further reduced by using an appropriate error criterion in the design of these HRTF approximations. In order to identify approximation design criteria, results are presented from a psychophysical experiment that measured the ability of listeners to discriminate between measured HRTFs and their pole-zero model approximations. While a significant reduction in the number of parameters is achieved using a pole-zero, model, a least-squares error criterion may not be the most appropriate in designing the HRTF approximations. Instead, error criteria that are sensitive to log-magnitude spectrum differences between the measured and approximated HRTFs are likely to be more subjectively relevant. A pole-zero model design algorithm is presented, incorporating gradient search techniques to minimize both log-magnitude and phase response errors. Application of the algorithm to approximating HRTFs results in smaller pole-zero model orders than those required when using a least-squares error criterion. Moreover, the gradient search techniques provide a means to interpolate pole-zero models. In comparing two proposed interpolation methods to a standard all-zero model interpolation method, one of the proposed methods shows relatively poor performance, and the other shows performance similar to the all-zero model interpolation method. In a separate analysis, principal component analysis (PCA) is used to investigate the common and directional transfer function representation of HRTFs. Based on alternative assumptions from those made previously, results from PCA applied to measured HRTFs show that a common transfer function does not exist. Consequently, an alternative representation of HRTFs is proposed, and comparisons are made to the directional transfer function model.