Range dependence of the response of a spherical head model

Abstract

The head-related transfer function ͑HRTF͒ varies with range as well as with azimuth and elevation. To better understand its close-range behavior, a theoretical and experimental investigation of the HRTF for an ideal rigid sphere was performed. An algorithm was developed for computing the variation in sound pressure at the surface of the sphere as a function of direction and range to the sound source. The impulse response was also measured experimentally. The results may be summarized as follows. First, the experimental measurements were in close agreement with the theoretical solution. Second, the variation of low-frequency interaural level difference with range is significant for ranges smaller than about five times the sphere radius. Third, the impulse response reveals the source of the ripples observed in the magnitude response, and provides direct evidence that the interaural time difference is not a strong function of range. Fourth, the time delay is well approximated by well-known ray-tracing formula due to Woodworth and Schlosberg. Finally, except for this time delay, the HRTF for the ideal sphere appears to be minimum-phase, permitting exact recovery of the impulse response from the magnitude response in the frequency domain. a radius of the sphere ͑m͒ c ambient speed of sound ͑m/s͒ f frequency ͑Hz͒ h head-related impulse response h m mth-order spherical Hankel function h m Ј the derivative of h m with respect to its argument Hhead-related transfer function relative to free field H head-related transfer function relative to source i ͱϪ1 j m mth-order spherical Bessel function k acoustic wave number ͑/m͒ n m mth-order spherical Neumann function p f f free-field pressure at the center of the sphere ͑kg/m 2 ) p s pressure on the surface of the sphere ͑kg/m 2 ) p pressure at a small sphere surrounding the source ͑kg/m 2 ) P m Legendre polynomial of degree m Q m mth-order modified spherical Hankel function r distance from the center of the sphere to the source ͑m͒ r radius of a small sphere surrounding the source S magnitude of flow from an ideal point source ͑m 3 /s͒ t time ͑s͒ ⌬t time between arrival at observation point and sphere center ͑s͒ ⌬ normalized ⌬t angle of incidence ͑rad͒ 0 angle for tangent incidence ͑rad͒ wavelength ͑m͒ normalized frequency normalized distance to the source 0 density of air ͑kg/m 3 ) normalized time radian frequency ͑rad/s͒

Concurrent Development of the Head and Pinnae and the Acoustical Cues to Sound Location in a Precocious Species, the Chinchilla (Chinchilla lanigera)

“…It has been shown in the cat that the measured ILDs are considerably different from the predictions computed from a spherical head model across various ages of development (Tollin and Koka 2009a). Here we also show that the ILD prediction from the spherical head model of Duda and Martens (1998) with an adult head diameter of 35.5 mm (dashed line, Fig. 6D) vastly underestimates the actual ILD magnitudes experienced by the chinchilla particularly for mid-to high-frequencies.…”

Objective perceptual quality assessment for self-steering binaural hearing aid microphone arrays

“…Both, d[m, Θ] and ΓNN[m] depend on to the assumed wave propagation model which may differ from the true (and generally unknown) wave propagation from the source to the microphones. We distinguish four models, free-field (FF), two head models (HM1 [5], HM2 [6]) and the measured anechoic transfer functions from the source to the head-mounted hearing aid microphone array (HRTF). The simplest approach is to use a free-field / farfield assumption (FF), i.e., the sound propagation is modeled as a plane wave without interfering objects in the propagation path.…”

“…For headworn arrays it is beneficial to include knowledge about head shadow and diffraction effects [1,11], especially for lateral target signal sources. Thus, head models by Duda et al [5,6] are applied which are effective parametric models that are based on the characteristics of a sphere. In HM1, the interaural time difference (ITD) cues are modeled by Woodworth and Schlosberg's frequency independent ray-tracing formula.…”

“…The gross magnitude characteristics of the HRTF spectrum, namely the interaural level difference (ILD) cues, are covered by a first order IIR head shadow filter which also accounts for an additional frequency dependent delay at low frequencies [5]. In HM2, near-field effects and interference effects that introduce ripples in the frequency response which are quite prominent on the shadowed side are incorporated as described in [6]. For both head models (HM1, HM2) the frequency dependent group delay τ [m, Θ] and magnitude have to be calculated for each microphone and angle of incidence due to [5,6].…”

“…In HM2, near-field effects and interference effects that introduce ripples in the frequency response which are quite prominent on the shadowed side are incorporated as described in [6]. For both head models (HM1, HM2) the frequency dependent group delay τ [m, Θ] and magnitude have to be calculated for each microphone and angle of incidence due to [5,6]. For HRTF, the propagation vector d[m, Θ] equals the measured anechoic 6-channel HRTF for the angle of incidence Θ. ΓNN [m] can be estimated for a cylindrical isotropic diffuse noise field by integrating the propagation vectors over all directions Θ.…”