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Implications from Sound Diffraction About a Rigid Sphere forum Short Papers Sound Localization Cues for a Magnified Head: Implications from Sound Diffraction about a Rigid Sphere I Introduction Rigid Sphere Results Auditory displays for teleoperation and virtual en- To obtain insight into the frequency scalability of vironments are currendy receiving great attention. While HRTFs within a framework that is theoretically and some applications require displays that recreate the nor- computationally straightforward, we have analyzed the mal acoustic inputs to the ears, others may benefit from following simplified problem. HRTFs were computed as modifications designed to enable localization that ex- the pressure Ps along the surface of a rigid spherical ceeds normal performance. Systems for so-called "super- "head" of radius a = 9 cm arising from a point source at auditory localization" can be realized in several ways distance r from the sphere's center. Due to sound diffrac- (Durlach, Shinn-Cunningham, & Held, 1993). Most tion, there are frequency- and angle-dependent changes systems focus on magnifying only the cues for a source's in the magnitude and phase of the sound that would direction; cues for source's distance are not intentionally otherwise exist were the sphere absent. Following the modified. One such approach that is directly applicable derivation given in Morse and Ingard (1968, Sec. 7.2; to virtual-environment (VE) systems presents localiza- for related review of head effects, see Kuhn, 1977; tion cues that would correspond to having an enlarged Blauert, 1983), Ps is given by head. From basic acoustic such head principles, scaling pcc70 will alter the head-related transferfunctions (HRTFs) from sound sources to each ear, producing increases in (1) Hm(kr) both interaural amplitude and time (or phase) differ- 2 \m + ;(cos 9) eJ(2TTß-TT/2) ences, as well as increases in pinna cues.1 m=0 H'm(ka) In this paper, we quantify this notion for a range of where of interest. In we assess the parameters particular, impact p = density of air =1.18 kg/m3 of conditions simulating magnified-head listening by c = speed of sound = 344.8 m/sec measured HRTFs that are fre- using normal, empirically / = sound frequency scaled the inverse of the desired quency by head-magnifi- k = wave number = 2trf/c = 2tt/\, with X = cation factor. For example, to simulate a head of four sound wavelength times normal a normal would be fre- size, HRTF(f) a = radius of spherical head = 0.09 m scaled to the interaural quency HRTF(f/4); resulting r = distance from source position to the cen- differences for the scaled head at 1 kHz would then be ter of the sphere the differences that occur at 4 represented by normally 9 = angle between radii from the sphere's kHz. Insofar as this is simulations can be scaling valid, center to the sound source location and normal HRTFs. We conveniently implemented using to the measurement position on the shall see that the key variable of importance is the dis- sphere's surface tance from the head to the sound source (s). General im- plications of the results for magnified-head VE systems and other related systems are given at the end of the W. M. Y. and M. Wei paper. Rabinowitz, j. Maxwell, Shao, Research Laboratory of Electronics Massachusetts Institute Presence, Vol. 2. No. 2. Spring 1993. 125-129 of Technology © 1993 The Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Rabinowitz et al. 125 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/pres.1993.2.2.125 by guest on 01 October 2021 126 PRESENCE: VOLUME 2, NUMBER 2 c70 = volume velocity strength of the sound sources, perfect frequency scalability ofHRTFs is ob- source (of infinitesimal extent) tained. However, as sound sources are positioned close Lra(cos 9) = Legendre polynomial functions in cos 9 to the scaled sphere (r —* ßa), the differences between P* Hm(kr) = spherical Hankel functions \nkr and Ps/ß2 increase, with consequent errors in HRTF H'm(ka) = derivatives of spherical Hankel functions scalability. The size and form of these differences are with respect to ka illustrated below. Observe that theform ofP„ i.e., the dependence on/ Polar sensitivity results for three frequencies (0.4, 2, and 9, is determined by the arguments kr and ka. and 10 kHz), two source distances (r = 1 and 4 m), and Scalings that do not change these arguments will, there- one scaling value (ß = 4) are given in Figure 1. To em- fore, leave the form of Ps unchanged. In particular, scal- phasize the angular dependencies independent of overall ing the sphere to a new (larger) radius a' = ßa (with magnitude changes, the plots have been normalized to ß > 1), while simultaneously scaling the source distance unity at 9 = 0°. Also, note that the specified frequency r' = ßr, and inverse scaling the sound frequency (/', applies to the normal sphere (i.e., ß = 1); for the scaled *') = (//ß> */ß)> results in P's = Ps/ß2, which is un- sphere, the results apply to the inverse-scaled frequency changed except for an overall scaling by 1/ß2 that is in- f/ß. As one expects, the patterns generally become more dependent offand 9. This result is consistent with the directional with increasing/and with decreasing r. Of well-known phenomenon that the solution of a geomet- more relevance here, as r decreases, the angular differ- ric acoustics problem at a particular frequency equals the ences between P* (dashed lines) andPs (solid lines) in- solution associated with scaling up (or down) all geo- crease but the forms ofP*(9) and Ps(9) still remain simi- metric dimensions of the problem and evaluating at a lar. The main differences are summarized by the amount frequency that is scaled down (or up) by the same factor. of fall-off in going from 0° to 180°. For r = 1 m and ß = This concept is usefully exploited in architectural acous- 4, this fall-off is about 7 dB more than that which occurs tics when the sound characteristics of large halls are normally. For less extreme cases, such as ß = 2 and/or evaluated in advance using small-scale physical models r = 20 m (not shown), the differences are substantially tested at appropriately scaled up frequencies. smaller. In the case of superauditory localization, we wish to Frequency responses as functions of source distance, magnify interaural differences by simulating an enlarged scaling factor, and the angle of source incidence were head. However, we probably will not want the external also computed. For normalization purposes, Ps (andP*) world to change. In other words, we will want the sound was divided by the "free-field" pressure Pff at the sphere- sources (and other objects) to remain at the same posi- center location with the sphere absent. Pff is simply the tions and, therefore, r' will remain at r rather than scal- spherical radiation pressure at a distance r from a point ing to ßr. The resulting sound pressure will be source: we shall denote as Because Ps(r, ßa,f/ß, 9) (which P*). ockUn same as = = = H-2 this is not the p eJ(2Trfi~kr+Tr/2) n)v ' pressure Ps(ßr, ßa,f/ß, 9) P's 4-nr Ps/ß2, perfect scalability ofHRTFs does not obtain. But how large are the differences between P* and Ps/ß2? As above, results at frequency//ß with the scaled sphere If the sound sources are far from the sphere, then were compared to those at/with the normal sphere. P* = Ps/ß2- This follows because as kr —> oo5 the acoustic Phase effects were examined via the group delay Tg = waves incident on both the normal and scaled spheres —d<f>(f)/df where <t>(/) is the phase of the normalized become planar, the effect ofr (other than an overall am- sound pressure. For the scaled sphere, Tg(//ß)/ß was plitude scaling) disappears, and keeping ¿'a' = ka gives used (and plotted below) to remove effects due to the the above sound pressure equality. Thus, for simulations overall change in distance (and Tg) associated with the that either ignore source distance or consider only distal scaling. Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/pres.1993.2.2.125 by guest on 01 October 2021 Rabinowitz et al. 127 a. 400 Hz b. 2 kHz 180 c. 10 kHz 180 ___^/\/Va»^ -e- -Z^rr._^r\J^/xz& FREQUENCY (Hz) FREQUENCY (Hz) Figure 2. Frequency responses for normal (solid line) and scaled Figure I. Polar responses for the normal sphere, Ps (solid lines), and (ß = 4, dashed line) conditions, for source distances of r = 4 m (a, c) for the sphere scaled by a factor ofß = 4, Pf (dashed lines). Results and I m (b, d), with magnitude (a, b) and group delay (c, d), and for are given for normal (unsealed) frequencies of (a) 400, (b) 2000, and two angles of incidence (8 = 0° and 120", as labeled). Results are given (c) 10,000 Hz and corresponding I /ß-sco/ed frequencies for the scaled relative to the reference free-field sphere-center sound pressure. The conditions. At each frequency, results are given for source distances of specified frequency abscissae refer to the normal (unsealed) conditions. r = 4 m (upper half circle) and I m (lower half circle); actual results are For the scaled conditions, the results were computed at corresponding symmetric in 8. All results are normalized to 0 dB at 6 = 0° and radial I /ß-sco/ed fs and the group delays are shown divided by ß. Symbols sensitivity is 10 dB per division. indicate asymptotic model results (normal = squares, scaled = diamonds) at low frequencies, after Rayleigh (see text), plotted near 40 Hz, and at high frequencies, after Woodworth (see text), plotted near Frequency responses for r = 1 and 4 m, ß = 4, and 14 kHz.
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