Directional properties of the sound transformation at the ear of four intact echolocating bats, Eptesicus fuscus, were investigated via measurements of the head-related transfer function (HRTF). Contributions of external ear structures to directional features of the transfer functions were examined by remeasuring the HRTF in the absence of the pinna and tragus. The investigation mainly focused on the interactions between the spatial and the spectral features in the bat HRTF. The pinna provides gain and shapes these features over a large frequency band (20-90 kHz), and the tragus contributes gain and directionality at the high frequencies (60 to 90 kHz). Analysis of the spatial and spectral characteristics of the bat HRTF reveals that both interaural level differences (ILD) and monaural spectral features are subject to changes in sound source azimuth and elevation. Consequently, localization cues for horizontal and vertical components of the sound source location interact. Availability of multiple cues about sound source azimuth and elevation should enhance information to support reliable sound localization. These findings stress the importance of the acoustic information received at the two ears for sound localization of sonar target position in both azimuth and elevation.

An efficient method for head-related transfer function (HRTF) measurement is presented. By applying the acoustical principle of reciprocity, one can swap the speaker and the microphone positions in the traditional (direct) HRTF measurement setup, that is, insert a microspeaker into the subject's ear and position several microphones around the subject, enabling simultaneous HRTF acquisition at all microphone positions. The setup used for reciprocal HRTF measurement is described, and the obtained HRTFs are compared with the analytical solution for a sound-hard sphere and with KEMAR manikin HRTF obtained by the direct method. The reciprocally measured sphere HRTF agrees well with the analytical solution. The reciprocally measured and the directly measured KEMAR HRTFs are not exactly identical but agree well in spectrum shape and feature positions. To evaluate if the observed differences are significant, an auditory localization model based on work by J. C. Middlebrooks [J. Acoust. Soc. Am. 92, 2607-2624 (1992)] was used to predict where a virtual sound source synthesized with the reciprocally measured HRTF would be localized if the directly measured HRTF were used for the localization. It was found that the predicted localization direction generally lies close to the measurement direction, indicating that the HRTFs obtained via the two methods are in good agreement.

Spherical microphone arrays have attracted attention for analyzing the sound field in a region and beamforming. The analysis of the recorded sound has been performed in terms of spherical wavefunctions, and recently the use of plane-wave expansions has been suggested. We show that the plane-wave basis is intimately related to the spherical wave-functions. Reproduction in terms of both representations satisfies certain band-limit criteria. We provide an error bound that shows that to reproduce the spatial characteristics of a sound of a certain frequency we need to be able to accurately represent sounds of up to a particular order, which establishes a Nyquist-like criterion. The order of the sound field in turn is related to the number of microphones in the array necessary to achieve accurate quadrature on the sphere. These results are illustrated with simulations. INTRODUCTIONMany authors have proposed the use of spherical microphone arrays for beamforming and for capturing the spatial structure of the sound field in the region of space occupied by the array and in the nearby area. In [1] the sound field was captured using an open spherical microphone array. Microphones can also be positioned on the surface of a rigid sphere to make use of the scattering [2]. The design of the spherical arrays was considered in [3], and the analysis of the recorded sound field using plane-wave expansion was described in [4]. We present further details of the mathematical theory here, adding the effects of the finite number of microphones, the influence of the sound frequency, and error bounds. While plane-waves are often used to represent sound sources in the far-field, they also constitute a remarkable basis for the wave equation that can represent the sound field in both the near and farfields. This representation of the wave field is being exploited in developing the fast multipole method for the Helmholtz equation [5]. Using error bounds presented in [6], we build on all the papers mentioned above, show how analysis in terms of both wavefunctions and plane-waves can be used interchangeably, and provide explicit error bounds and frequency dependence. ANALYSIS OF THE SOUND FIELDSound propagation and sound scattering off the spherical array are governed by the wave equation (1, left). Using the Fourier transform (2, left), we can convert the wave equation to the Helmholtz equation in the frequency domain (1, right)where k is the wavenumber and ω = 2π f is the circular frequency. The wave propagation and scattering problem can be reduced then to solving the Helmholtz equation for a number of frequencies. To obtain the time domain solution, we take the inverse Fourier Transform (2, right) of ψ. For scattering problems the total (measured) field can be decomposed asIn this paper, we consider two problems: sound field analysis and beamforming. The goal of sound field analysis is to build an approximation to ψ in at the array location (center) using the sound recorded on the surface at the microphones, ψ, which is a combinat...

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