SPATIAL FREQUENCY RESPONSE SURFACES (SFRS S): AN ALTERNATIVE VISUALIZATION AND INTERPOLATION TECHNIQUE FOR HEAD-RELATED TRANSFER FUNCTIONS (HRTF S)

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1 Cheng and Wakefield Page 1 of 13 Pre-print: AES16: Rovaniemi, Finland SPATIAL FREQUENCY RESPONSE SURFACES (SFRS S): AN ALTERNATIVE VISUALIZATION AND INTERPOLATION TECHNIQUE FOR HEAD-RELATED TRANSFER FUNCTIONS (HRTF S) COREY I. CHENG 1 AND GREGORY H. WAKEFIELD 2 1 University of Michigan, Department of Electrical Engineering and Computer Science Ann Arbor, Michigan, U.S.A. coreyc@eecs.umich.edu 2 University of Michigan, Department of Electrical Engineering and Computer Science Ann Arbor, Michigan, U.S.A. ghw@eecs.umich.edu This paper introduces Spatial Frequency Response Surfaces (SFRS s), an alternative, spatial-domain visualization tool for head-related transfer functions (HRTF s). Qualitative analysis of SFRS s is easier than direct comparison of HRTF magnitude responses, and reveals many cross-subject similarities in HRTF data corresponding to well-known HRTFrelated psychophysical phenomena. We present an SFRS-based HRTF interpolation algorithm, along with results of a psychophysical experiment comparing the perceptual quality of interpolated and non-interpolated HRTF s. 1. INTRODUCTION Previous psychophysical studies have concluded that knowledge of the acoustic filtering properties of the pinna, head, and torso are important in human localization of sounds in space. The combined effects of such filtering are summarized by a single set of spatially dependent filters known as Head-Related Transfer Functions (HRTF s). HRTF s are can be empirically measured, and are commonly approximated as FIR filters. The directional component of the HRTF is called the Directional Transfer Function (DTF) and is the quantity often used in synthesizing virtual auditory space. DTF s can be computed from HRTF s and the HRTF measurement apparatus transfer function [10]. This paper presents an alternative visualization of HRTF data by processing existing HRTF data sets into Spatial Frequency Response Surfaces (SFRS s). The purpose of this paper is to describe this spatial domain visualization method, and to show that this representation of HRTF data is both qualitatively and quantitatively useful. Specifically, we show that a simple visual inspection of SFRS s provides insight into many features of HRTF data which are difficult to see with a traditional visual inspection of raw HRTF magnitude frequency responses. Also, we show that SFRS s can be used to detect gross calibration errors in the HRTF measurement apparatus. Finally, we propose a spatial, SFRS-based HRTF interpolation algorithm along with the results of a perceptual evaluation of the algorithm. The present work focuses on several subjects HRTF data sets, where each data set consists of left and right ear magnitude responses measured at 400 different azimuth-elevation locations. Although irregularly spaced, these locations are roughly degrees apart in the azimuth and elevation directions. The sampling rate was 50 khz, the resolution of the data taken was 16 bits, and a 512-point FFT was used to compute the frequency response at each location. The data for this paper was taken at John C. Middlebrooks laboratory at the Kresge Hearing Research Institute of the University of Michigan. 2. SFRS S AND HRTF S 2.1 Traditional HRTF visualization techniques Many previous studies have attempted to visualize HRTF data by examining how certain macroscopic properties of HRTF sets, such as peaks and notches in particular locations of the magnitude frequency responses, associate and/or systematically vary with the perception of azimuth (left-right discrimination) and/or elevation (up-down discrimination) [1][4][5][9]. Typically, one attempts to see how these features change when azimuth changes and elevation is held constant, or vice-versa. For example, Figure 1 and Figure 2 present one subject s HRTF s in this style, where Figure 1 presents HRTF s in the horizontal plane, and Figure 2 presents HRTF s in the median plane. Cheng and Wakefield Page 1 of 13 Pre-print: AES16: Rovaniemi, Finland

2 Cheng and Wakefield Page 2 of 13 Pre-print: AES16: Rovaniemi, Finland Figure 1. Tradiditional -style visualization of several HRTF s in the horizontal plane (elevation 0). Although some patterns are evident in the raw HRTF magnitude frequency responses as azimuth changes, crosscomparison of HRTF data is difficult. Cheng and Wakefield Page 2 of 13 Pre-print: AES16: Rovaniemi, Finland

3 Cheng and Wakefield Page 3 of 13 Pre-print: AES16: Rovaniemi, Finland Figure 2. Tradiditional -style visualization of several HRTF s in the median plane (azimuth 0). Although some patterns are evident in the raw HRTF magnitude frequency responses as elevation changes, cross-comparison of HRTF data is difficult. Visualization using this technique does highlight some macroscopic features of HRTF s, as there are noticeable patterns to the changes in HRTF s as elevation and azimuth change continuously. For example, in Figure 1, one can see that the HRTF s on the contralateral side seem to be less smooth than those on the ipsilateral side, suggesting that SNR for ipsilateral HRTF s is generally higher than that for contralateral HRTF s. This agrees with the basic physics of head-shadowing. In Figure 2, one can see that there is a notch at 7 khz that migrates upwards in frequency as elevation increases. A shallow peak can be seen at 12 khz for lower elevations in the median plane, and this peak flattens out for higher elevations. However, interpretation of these graphs is fairly difficult, as comparison with many graphs is needed to generalize findings for different azimuths and elevations. It is difficult to draw cross-subject conclusions from these graphs as well, since the shapes of individualized HRTF s do not always match well [4]. 2.2 Constructing SFRS s from existing HRTF data SFRS s present the same HRTF magnitude response information, except in a different coordinate system. Specifically, one SFRS is constructed for every frequency bin in the HRTF left or right magnitude response, where magnitude is plotted against azimuth and elevation. In this manner, every HRTF is used in the computation of a single SFRS. Since the spatial sampling pattern is irregular, linear interpolation is used to construct a surface that approximates the continuous spatial frequency response of the ear at the specified frequency. Thus, the value of the SFRS for frequency f at the coordinate (θ,φ) represents how much power at frequency f the right or left ear receives at (θ,φ) compared with other locations. Figure 3 describes the procedure for computing a SFRS for a given ear and frequency bin, while Figure 4 and Figure 5 show representative SFRS s. Cheng and Wakefield Page 3 of 13 Pre-print: AES16: Rovaniemi, Finland

4 Cheng and Wakefield Page 4 of 13 Pre-print: AES16: Rovaniemi, Finland Individual HRTF s for all available spatial locations H H H H f f f Bin 72 = 7 khz f Azimuth 0, elevation 20 Azimuth 20, elevation 50 Azimuth -60, elevation -30 Azimuth 120, elevation 20 Spatial frequency response graph for bin 72 = 7 khz Azimuth Elevation Figure 3. Construction of SFRS s from HRTF s 3. QUALITATIVE ANALYSIS OF SFRS S 3.1 Cross-subject similarity of SFRS s The spatial frequency graphs are an immediate, visual, and very useful presentation of many qualitative properties of directional hearing. One can readily see general trends in these graphs across subjects which are harder to identify by analyzing raw magnitude frequency responses alone. Initial observations reveal that for a given frequency, different subjects SFRS s very similar to each other. The SFRS s do differ in their absolute magnitudes, but the general cross-subject features of the surfaces are the same. Table 1 summarizes these general, cross-subject observations for several frequencies, and highlights the important changes in the features as frequency increases from 1-13 khz. Figure 4 shows a single subject s SFRS s for several frequencies corresponding to each entry in Table 1. Figure 5 illustrates the cross-subject similarity of SFRS s among three subjects for two different frequencies. The most striking quality of these graphs is the location, and apparent motion of the hotspots, or well-defined peaks, of the SFRS s as a function of frequency. The hotspots well defined existence starting from 1-2 khz coincides with the well-known frequency of 1.5 khz, at which IID (interaural intensity difference) cues become more important for spatial perception than ITD (interaural time difference) cues in traditional duplex theory [1]. Figures 4e-t show that torso effects, predicted to be pronounced between 1-2 khz by the physical models [9], actually occur over a larger frequency range, though in a very limited area around the head. This can be seen by the shallow null on the contralateral side at lower elevations in many frequencies below 13 khz. Note also the symmetry of many of the hotspots about the horizontal plane. The hotspots in most graphs containing more than one hotspot are located on the same azimuth, and are separated by roughly equal elevations, as can be seen in Figures 4e-j. The only asymmetries in this respect occur between 6-8 khz and 8-10 khz in Figures 4o-r, where the magnitude of one of two salient peaks is clearly greater than that of the other. Frequencies Figure Description of corresponding SFRS s Hz 5a,b Roughly equal power is received from all directions. No salient features present..6-1 khz 5c,d Head shadowing can be seen, as the ipsilateral ear receives more energy than the contralateral ear. Diffraction effects due to the head can be seen on the contralateral side of the head, near ±100 degrees. 1-2 khz 5e,f Head shadowing becomes more prominent; diffraction effects are clearly seen on the contralateral side of the head. Two to three distinct peaks at ipsilateral azimuth 100, elevations +30 and 30 are starting to form. Attenuating effects of the torso can be seen on the contralateral side, lower elevations khz 5g,h Three peaks in the surface can be seen at ipsilateral azimuth 70-80, elevations 30, 10, and 50. Diffraction effects can still be seen on the contralateral side near contralateral azimuth 100. Torso effects can be seen khz 5i,j The three peaks on the ipsilateral side have moved closer to the median plane and slightly higher in elevation. A fourth peak is starting to form beneath the other three, at ipsilateral azimuth 40, elevation 50. Diffraction effects are starting to lessen, as the contralateral peak at contralateral azimuth 100 is beginning to fade. There are some nulls in the ipsilateral side, and torso effects can still be seen. 4-5 khz 5k,l The three peaks on the ipsilateral side have become one, large peak centered near ipsilateral azimuth 50, elevation 0. Diffraction effects are nearly gone, but torso effects can still be seen in lower elevations. 5-6 khz 5m,n The large ipsilateral hotspot has moved farther away from the median plane, and upwards in elevation. The spot is now at ipsilateral azimuth 75, elevation 20. Torso effects can still be seen. 6-8 khz 5o,p The single 5-6 khz peak has become two smaller peaks at ipsilateral azimuth 75, elevations 40, +40. Torso effects can still be seen. 8-10kHz 5q,r The two ipsilateral hotspots are still present, but the lower elevation peak is more prominent than the higher elevation peak. A third hotspot is beginning to form on the median plane at azimuth 0, elevation 30. Torso effects can still be seen kHz 5s,t Four hotspots are now apparent, one on the median plane at azimuth 0 elevation 20, and the other three at ipsilateral azimuth 100, elevations 40, 0, Torso effects can still be seen. Table 1. Summary of cross-subject similarity in SFRS s from 1-13 khz. Cheng and Wakefield Page 4 of 13 Pre-print: AES16: Rovaniemi, Finland

5 Cheng and Wakefield Page 5 of 13 Pre-print: AES16: Rovaniemi, Finland Each contour line represents a 1 db change in magnitude. Grayscale in db: Figure 4. Subject CC s SFRS s for several frequencies. See Table 1 for details. Cheng and Wakefield Page 5 of 13 Pre-print: AES16: Rovaniemi, Finland

6 Cheng and Wakefield Page 6 of 13 Pre-print: AES16: Rovaniemi, Finland Each contour line represents a 1 db change in magnitude. Grayscale in db: Figure 4 (continued). Subject CC s SFRS s for several frequencies. See Table 1 for details. Cheng and Wakefield Page 6 of 13 Pre-print: AES16: Rovaniemi, Finland

7 Cheng and Wakefield Page 7 of 13 Pre-print: AES16: Rovaniemi, Finland Each contour line represents a 1 db change in magnitude. Grayscale in db: Figure 5. Cross-subject comparison of SFRS s for three subjects at two different frequencies. Although there are some differences in absolute magnitudes among subjects, the general features of the SFRS s remain the same. Cheng and Wakefield Page 7 of 13 Pre-print: AES16: Rovaniemi, Finland

8 Cheng and Wakefield Page 8 of 13 Pre-print: AES16: Rovaniemi, Finland 3.2 Visualizing diffraction effects with SFRS s Given a sphere and an incident monotonic plane wave, it is a well-known fact that for some frequencies and incident angles, the sphere has an amplifying effect on the wave at certain points near the sphere [7]. Thus, in using a rigid spherical model of the head to calculate HRTF s analytically, there are some locations on the contralateral side of the head where this effect occurs, even though these locations lie in the direct shadow of the head. Shaw refers to this spot as a bright spot [9] in his analyses, and an analytical derivation of the effect can be found in [1]. Diffraction effects are clearly visible in SFRS s from 1-5 khz, and agree well with theoretical predictions. Shaw s bright spot can easily be interpreted in these SFRS s as hotspots or peaks in the surfaces directly opposite the contralateral ear (azimuth 90 for right ear SFRS s; azimuth +90 for left ear SFRS s). Figures 4c-h show this effect. 3.3 Using SFRS s to investigate the theory of directional bands The location of an SFRS s hotspots may also support some aspects of the theory of directional bands, which hypothesizes that certain narrowband sounds are correlated with perceptually preferred spatial directions. More specifically, an SFRS with one dominant hotspot may suggest that the auditory system favors the location of that hostpot perceptually when presented with narrowband noise centered at that frequency. For example, Figure 8 shows that the dominant hotspot at 6 khz is clearly positive in elevation, while the dominant hotspot at 8.5 khz is clearly negative in elevation. A psychophysical study designed to test the theory of directional bands for several frequencies found that subjects tended to localize narrowband sounds centered at 6 and 8 khz as originating from positive and negative elevations respectively, regardless of the actual free-field location of the source [6]. Therefore, comparison of the SFRS hotspots locatioons to Middlebrooks psychophysical data at these frequencies shows that in this limited case, there is a correlation of a dominant hotspot s spatial location with that frequency s perceptually preferred spatial directions. 4. APPLICATIONS OF SFRS S 4.1 Using SFRS s to detect HRTF measurement calibration errors SFRS s were also useful in detecting calibration problems in the HRTF measurement process. As shown in Figure 6, HRTF measurements used for this paper were taken at two locations for one movement of the test apparatus. Therefore, since two different speakers were used in presenting the stimuli, calibration of the two signal paths used in stimuli presentation was essential to obtaining accurate results. In another set of HRTF data recorded from Middlebrooks lab, subjects reported excessive trouble perceiving spatial location correctly when listening to sounds processed with their own personalized HRTF s (Wakefield, personal communication). Examination of these subjects data sets using SFRS s shows immediately that there was a calibration problem in the two different paths used for stimuli presentation. In Figure 8, simple visual inspection reveals that at a given frequency, the magnitude of one half of the data differs from the magnitude of the other half of the data by a constant value. Because of the geometry of the test apparatus, we can conclude that the gain of one signal path was greater than that of the other, and that therefore, the signal paths gains were not correctly calibrated. Possible set of apparatus trajectories Speaker for stimuli presentation Speaker for stimuli presentation Subject, with probe microphones inserted in right and left ear canals Figure 6. Geometry of HRTF data acquisition apparatus. two measurements were taken for one movement of the apparatus by using two speakers for stimuli presentation. The locations of the speakers are polar opposites on a fictional sphere. The bar connecting the two speakers has two degrees of freedom, corresponding to changes in azimuth and elevation. Each speaker recorded data from roughly one hemisphere of the domain space. Cheng and Wakefield Page 8 of 13 Pre-print: AES16: Rovaniemi, Finland

9 Cheng and Wakefield Page 9 of 13 Pre-print: AES16: Rovaniemi, Finland Each contour line represents a 1 db change in magnitude. Grayscale in db: Figure 7. Using SFRS s to support the theory of directional bands. In this figure, the dominant hotspots positive and negative elevations correspond well to preferred perceptual directions for 6 and 8 khz, respectively. Each contour line represents a 2.3 db change in magnitude. Grayscale in db: Figure 8. SFRS s from two subjects HRTF sets taken with a miscalibrated measurement apparatus. One half of the data clearly differs from the other half of the data by roughly a constant offset. 4.2 SFRS-based interpolation of HRTF s and perceptual evaluation The goal of synthesizing virtual auditory space is to perceptually place a sound at any location in free space. However, due to the complexity of HRTF measurement along with practical signal processing limitations, only a finite number of spatial locations are measured in practice. Therefore, the process of computing perceptually acceptable HRTF s for arbitrary spatial locations from existing data is of central importance for creating virtual auditory spaces. SFRS s suggest an alternative interpolation method that alleviates some of the limitations of other proposed methods. The present algorithm constructs a DTF magnitude response for a desired spatial location one frequency at a time, according to a weighted average of values taken directly from each SFRS. Therefore, this method allows different frequency components of different DTF s to contribute to the interpolated DTF to varying degrees, unlike simpler spatial methods [6]. No internal parameters, such as model order, need be predetermined, as is the case in pole-zero interpolation [2]. Cheng and Wakefield Page 9 of 13 Pre-print: AES16: Rovaniemi, Finland

10 Cheng and Wakefield Page 10 of 13 Pre-print: AES16: Rovaniemi, Finland Specifically, the algorithm first performs a triangulation of the azimuth elevation coordinate system in order to create a grid for the available, irregularly spaced data. The vertices of the triangulation are the locations at which DTF s are known. In order to minimize the effect of the irregularity in spatial sampling, interpolated locations are taken only from where the triangulation is most uniform. For each SFRS, a plane is constructed using the three magnitude response values associated with the three vertices of the triangle enclosing the desired spatial location. The interpolated value for that surface is taken as the value of the plane evaluated at the desired spatial location. This process is repeated for each SFRS, and each interpolated value is placed into the appropriate frequency bin of the interpolated magnitude DTF. The interpolation algorithm is shown in Figure 9. In order to test the perceptual validity of this interpolation algorithm, an informal psychophysical experiment was conducted involving two subjects whose HRTF s were measured. In this experiment, the subjects were presented with three sounds which either 1) shared the same approximate azimuth but differed in elevation by 10 (same azimuth testing), or 2) shared the same approximate elevation but differed in azimuth by 10 (same elevation testing). The subjects were then asked to select which of the three sounds appeared to lie spatially between the other two. In all trials, the leftmost, rightmost, highest elevation, and/or lowest elevation sounds were derived from DTF s at known locations. However, in one half of the trials, the middle sound was derived from an SFRS-interpolated DTF, while in the other half of the trials, the middle sound was derived from a non-interpolated DTF. The test shows spatial perceptions differ between interpolated and noninterpolated DTF s corresponding to the same spatial location. Furthermore, it reveals the extent to which perceptual accuracy differs in determining spatial location in the azimuth and elevation directions. In the listening task, subjects heard all three sounds once and were then allowed to hear each sound individually for as many times as needed to make a judgment. Each sound was approximately 750 ms in duration and was separated from other sounds by approximately ms of silence. The source sound for all trials was a 32k pseudo-random binary sequence. Sounds from interpolated DTF s were tested at 104 spatial locations. Of these 104 locations, 48 were taken from three different elevations (-30, 0, and 30 degrees) for same azimuth testing, and 56 were taken from eight different azimuths (0, 45, 90, 135, 180, 225, 270, and 315 degrees) for same elevation testing. Some locations were repeated among the same elevation and same azimuth tests. Ten trials were performed for each interpolation test condition, using both interpolated and noninterpolated DTF-derived middle conditions. In all, this experiment required hours of observation time and the subject was allowed to proceed at their own pace. Sounds were presented over headphones (Sennheiser 265) in a double-walled sound-proof booth. Test results for both subjects are shown in Figures 10-11, and are given in percentage correct responses for both subjects, where a correct response refers to correctly identifying the middle sound. In general, same-elevation testing was more successful than sameazimuth testing. For same-elevation conditions, both subjects performed better with interpolated, rather than non-interpolated, DTF s. Specifically, CC s performance was 94% (interpolated) and 87% (non-interpolated) and MM s performance was 93% (interpolated) and 83% (non-interpolated). For same-azimuth conditions, interpolated DTF s appeared to produce better performance than non-interpolated as well. In this case, CC s performance was 79% (interpolated) and 66% (non-interpolated) and MM s performance was 77% (interpolated) and 69% (non-interpolated). Plane containing all three values of the spatial frequency response evaluated at the vertices of the triangle SFRS for bin 0 SFRS for bin 1 SFRS for bin 2 SFRS for bin 256 Value of spatial frequency response at interpolated location Desired location SFRS for bin 2 azimuth Figure 9. Using SFRS s to compute interpolated HRTF s. Value of spatial frequency response at known location Vertices of the triangle are locations at which spatial frequency response is exactly known via measurements elevation Cheng and Wakefield Page 10 of 13 Pre-print: AES16: Rovaniemi, Finland

11 Cheng and Wakefield Page 11 of 13 Pre-print: AES16: Rovaniemi, Finland Figure 10. Perceptual evaluation of interpolated and non-interpolated HRTF s for subject CC. Same elevations and same azimuths testing for interpolated and non-interpolated HRTF s. Each shaded box represents a perceptual test location. Cheng and Wakefield Page 11 of 13 Pre-print: AES16: Rovaniemi, Finland

12 Cheng and Wakefield Page 12 of 13 Pre-print: AES16: Rovaniemi, Finland Figure 11. Perceptual evaluation of interpolated and non-interpolated HRTF s for subject MM. Same elevations and same azimuths testing for interpolated and non-interpolated HRTF s. Each shaded box represents a perceptual test location. Cheng and Wakefield Page 12 of 13 Pre-print: AES16: Rovaniemi, Finland

13 Cheng and Wakefield Page 13 of 13 Pre-print: AES16: Rovaniemi, Finland The difference in overall performance between sameelevation and same-azimuth testing is consistent with measures of the just-noticeable difference (JND) in elevation or azimuth of a given source [1]. For any given spatial location, listeners, in general, show greater acuity for changes in azimuth than changes in elevation. For same-elevation testing, there were a larger number of errors at azimuths of 90 and +90 degrees. These results are consistent with the fact that in general, the just-noticeable difference (JND) for azimuth is smaller than the JND for elevation [1]. In contrast, there is less of a clear pattern of errors for same-azimuth testing. In general, errors using interpolated DTF s occur in roughly the same spatial locations as those using noninterpolated DTF s, but there appears to be little clustering in their locations. It is surprising that listeners perform better using interpolated, rather than non-interpolated, DTF s. The source of this effect is not exactly known; however, one possible explanation is that subjects develop memory for the spectral coloration associated with a certain region of space, and learn to respond on to this cue alone. For example, both subjects reported that pitch cues could be used for elevation discrimination in the same azimuth tests. Another important result is that localization with exact DTF s is far from 100%, even in the absence of interpolation. These results are often reported in the literature and may reflect errors in HRTF measurement and DTF computation [10]. 5. CONCLUSIONS AND FUTURE DIRECTIONS This paper presented spatial frequency response surfaces (SFRS s) as an alternative, spatial-domain visualization tool for existing head-related transfer function (HRTF) data. The cross-subject similarity of SFRS representations allowed several cross-subject patterns in HRTF data to be seen quickly, such as diffraction and attenuation effects due to head shadowing and the torso, respectively. SFRS s were useful in detecting gross calibration errors in the HRTF measurement process. The structure of SFRS s suggested a spatial, SFRS-based HRTF interpolation algorithm, and a psychophysical experiment was performed to compare the perceptual quality of SFRS-interpolated HRTF s and noninterpolated HRTF s. This experiment showed that the interpolation algorithm produced interpolated HRTF s with perceptual quality comparable to exact HRTF s. Future directions include more formal psychophysical testing of the interpolation algorithm for smaller angles of separation; principal components analyses of SFRS s for parameter reduction; and parametrizations of the hotspots in selected SFRS s with a beamformer model. 6. ACKNOWLEDGMENTS The authors thank Dr. John C. Middlebrooks at the Kresge Hearing Research Institute of the University of Michigan for providing the data used in this research. We also thank Dr. Michael A. Blommer for his early investigations of the interpolation problem. This research was supported by a grant from the Office of Naval Research in collaboration with the Naval Submarine Medical Research Laboratory, Groton, Connecticut. 7. REFERENCES [1] Blauert, Jens. Spatial Hearing. The MIT Press, Cambridge: [2] Blommer, A. and Wakefield, G. A comparison of head related transfer function interpolation methods IEEE ASSP Workshop on Applications of Signal Processing to Audio and Acoustics. (IEEE catalog number: 95TH8144). [3] Cheng, Corey I. and Wakefield, Gregory H. Spatial Frequency Response Surfaces: An Alternative Visualization Tool for Head-Related Transfer Functions (HRTF s) IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP99). Phoenix, Arizona: [4] Kendall, Gary S. A 3-D sound primer: Directional hearing and stereo reproduction. Computer Music Journal, 19(4): Winter [5] Kistler, Doris J. and Wightman, Frederic L. A model of head-related transfer functions based on principal components analysis and minimum-phase reconstruction. Journal of the Acoustical Society of America, 91(3): March [6] Middlebrooks, John C. Narrow-band sound localization related to external ear acoustics. Journal of the Acoustical Society of America, 92(5): November [7] Morse, Philip M. Vibration and Sound. McGraw- Hill Book Company, Inc., New York: [8] Rao, K. Raghunath and Ben-Arie, Jezekiel. Optimal head related transfer functions for hearing and monaural localization in elevation: A signal processing design perspective. IEEE Transactions on Biomedical Engineering, 43(11): November [9] Shaw, E.A.G. The External Ear. Handbook of Sensory Physiology V/1: Auditory System, Anatomy Physiology(Ear). Springer-Verlag, New York: [10] Wightman, Frederic L. and Kistler, Doris J. Headphone simulation of free-field listening. I: Stimulus synthesis. Journal of the Acoustical Society of America, 85(2): February Cheng and Wakefield Page 13 of 13 Pre-print: AES16: Rovaniemi, Finland