Optimal head related transfer functions for hearing and monaural localization in elevation: a signal processing design perspective

Localization of sound sources by human listeners has been widely studied and theories and various models of the localization and hearing mechanism have been constructed. In the classical "duplex" theory, sound localization in azimuth is explained by interaural time or equivalently, phase differences at low frequencies, and by interaural amplitude differences at higher frequencies. Head related transfer functions (HRTFs) present a linear system approach to modeling localization by representing the direction-dependent transformation the sound undergoes at each ear, Localization in elevation is explained by directional differences in the HRTFs, which also explains monaural localization. The authors conjecture that the HRTF\´s evolved during the course of nature (due to the evolution of the shape and structure of the ear etc.) are optimal with respect to several physically realizable criteria. Here, they investigate the problem of defining the design constraints which when optimized yield a set of HRTFs for hearing and monaural vertical localization in an attempt to better understand, and if possible, duplicate nature\´s design. The authors pursue an engineer\´s design perspective and formulate a constrained optimization problem, where the desired set of HRTFs is optimized according to a cost function based on several criteria for localization, hearing and smoothness, and also by imposing physically realizable constraints on the HRTFs such as nonnegativity, energy etc. The value of the cost function for a candidate set of HRTFs is an indication of the similarity of that set of HRTFs with respect to the ideal solution (measured HRTF data). The final optimization results the authors present are similar to the actual HRTFs measured in human subjects, and the associated cost function values are found to be almost equal. This points to the fact that the optimization criteria defined are quite relevant. The significant outcome of this research is the identification of a relev- - ant set of mathematical criteria that could be optimized in the human auditory system to facilitate good hearing and localization. These criteria along with the associated constraints represent the desirable characteristics of the HRTFs in an HRTF-based localization system, and could lead to a better understanding and modeling of the auditory system.