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 (HRTF's) 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 HRTF's, which also explains monaural localization. We 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. In this paper, we investigate the problem of defining the design constraints which when optimized yield a set of HRTF's for hearing and monaural vertical localization in an attempt to better understand, and if possible, duplicate nature's design. We pursue an engineer's design perspective and formulate a constrained optimization problem, where the desired set of HRTF's 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 HRTF's such as nonnegativity, energy etc. The value of the cost function for a candidate set of HRTF's is an indication of the similarity of that set of HRTF's with respect to the ideal solution (measured HRTF data). The final optimization results we present are similar to the actual HRTF's 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 relevant 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 HRTF's in an HRTF-based localization system, and could lead to a better understanding and modeling of the auditory system.