Journal of Sound and Vibration (2002) 258(3), 405 417 doi:10.1006/jsvi.5264, available online at http://www.idealibrary.com on CORRELATION FACTORS DESCRIBING PRIMARY AND SPATIAL SENSATIONS OF SOUND FIELDS Y. Ando Graduate School of Science and Technology Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan. E-mail: andoy@kobe-u.ac.jp (Accepted 30 May 2002) The theory of subjective preference of the sound field in a concert hall is established based on the model of human auditory brain system. The model consists of the autocorrelation function (ACF) mechanism and the interaural crosscorrelation function (IACF) mechanism for signals arriving at two ear entrances, and the specialization of human cerebral hemispheres. This theory can be developed to describe primary sensations such as pitch or missing fundamental, loudness, timbre and, in addition, duration sensation which is introduced here as a fourth. These four primary sensations may be formulated by the temporal factors extracted from the ACF associated with the left hemisphere and, spatial sensations such as localization in the horizontal plane, apparent source width and subjective diffuseness are described by the spatial factors extracted from the IACF associated with the right hemisphere. Any important subjective responses of sound fields may be described by both temporal and spatial factors. # 2002 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION The human ear sensitivityto a sound source in front of the listener is essentiallyformed by the physical system from the source point to the oval window of the cochlea [1, 2]. Due to specific characteristics in electro-physiological responses from both the left and right human cerebral hemispheres [3 10], the workable model may be proposed as shown in Figure 1. In this figure, a sound source pðtÞ is located at r0 in a three-dimensional space and a listener is sitting at r which is defined by the location of the center of the head, hl;rðrjr0; tÞ being the impulse responses between r0 and the left and right ear-canal entrances. The impulse responses of the external ear canal and the bone chain are el;rðtÞ and cl;rðtÞ; respectively. The velocities of the basilar membrane are expressed by Vl;rðx; oÞ; x being the position along the membrane. The action potentials from the hair cells are conducted and transmitted to the cochlear nuclei, the superior olivary complex including the medial superior olive, the lateral superior olive and the trapezoid body, and to the higher level of two cerebral hemispheres. The input power density spectrum of the cochlea Iðx0Þ can be roughly mapped at a certain nerve position x0 [11, 12], as a temporal activity. Amplitudes of waves (I IV) of the auditory brainstem response (ABR) reflect the sound pressure levels as a function of the horizontal angle of incidence to a listener [3]. Such neural activities, in turn, include sufficient information to attain the autocorrelation function (ACF), probablyat the lateral lemniscus as indicated by FllðsÞ and FrrðsÞ: In fact, the time domain analysis of firing rate from the auditory nerve of a cat reveals a pattern of ACF rather than the frequency 0022-460X/02/$35.00 # 2002 Elsevier Science Ltd. All rights reserved. 406 Y. ANDO Figure 1. Model of the auditory brain system with autocorrelation and interaural crosscorrelation mechanisms and specialization of human cerebral hemispheres [2]. domain analysis [13]. Pooled interspike interval distributions resemble the short time or the running ACF for low-frequency component. Also, pooled interval distributions for sound stimuli consisting of the high-frequency component resemble the envelope to the running ACF [14]. From the viewpoint of the missing fundamental or pitch of complex components judged by humans, the running ACF must be processed in the frequency components below about 5 kHz [15]. Due to the absolute refractory or resting period of a single neuron (about 1 ms), the missing fundamental or pitch may be perceived to be less than about 1 2 kHz [16]. A model of running ACF processor is illustrated in Figure 2, which is dominantly connected with the left cerebral hemisphere. As also discussed [3], the neural activity (wave V together with waves IVl and IVr) may correspond to the IACC as shown in Figure 3. Thus, the interaural crosscorrelation mechanism may exist at the inferior colliculus. It is concluded that the output signal of the interaural crosscorrelation mechanism including the IACC may be dominantly connected to the right hemisphere. Also, the sound pressure level may be expressed by a geometrical average of the ACFs for the two ears at the origin of time (s ź 0) and in fact appears in the latency at the inferior colliculus, which may be processed in the right hemisphere. It was discovered that the listening level (LL) and the IACC are dominantly associated with the right cerebral hemisphere and the temporal factors, Dt1 and Tsub; and the sound field in a room is associated with the left (Table 1). The specialization of the human cerebral hemisphere may relate to the highly independent contribution between the spatial and temporal criteria on any subjective attributes. It is remarkable that, for example, cocktail party effects may well be explained by such specialization of the human brain because speech is processed in the left hemisphere, and independently the spatial information is mainly processed in the right hemisphere. Based on the model, one can describe primary and spatial sensations, and thus any subjective attributes of sound fields in terms of processes in the auditory pathways and the specialization of two cerebral hemispheres. PRIMARY AND SPATIAL SENSATIONS 407 Figure 2. Neural processing model of the running ACF. Figure 3. Relationship between values of IACC and P ź A2 =½AIV;lAIV;rŠ; as a function of horizontal angle (x) V of sound incidence to a listener, where AIV;l and AIV;r are amplitudes of ABR waves IV and AV is that of wave V averaged. A linear relationship between the IACC and the value P is observed ( p50 01). Note that the diameter of full circles corresponds to a number of available data obtained in recording ABR (1 4) from four subjects. 408 Y. ANDO Table 1 Hemispheric specialization obtained by analyses of AEP (SVR), EEG and MEG[ 3 10] Factors AEP (SVR) EEG, ratio of ACF te MEG, ACF te value varied AðP12N1Þ values of a-wave of a-wave Temporal Dt1 L > R (speech) L > R (music) L > R (music) Tsub } L > R (music) } Spatial LL R > L (speech) }} IACC R > L (vowel/a/) R > L (music) } R > L (band noise) Sound sources used in the experiments are indicated in the brackets. 2. ORTHOGONAL FACTORS 2.1. FACTORS EXTRACTED FROM ACF The ACF is defined by Z þT 1 FPðtÞ Åº p0ðtÞp0ðt þ tÞ dt; ð1Þ 2T T where p0ðtÞ ÅºpðtÞ*sðtÞ; sðtÞ being the ear sensitivity, which is essentially formed by the transfer function of physical system to the oval of the cochlea. For practical convenience, sðtÞ may be chosen as the impulse response of an A-weighted network [1, 2]. The ACF and the power density spectrum mathematically contain the same information. There are three significant items, which can be extracted from the ACF: (1) energy represented at the origin of the delay, Fpð0Þ: Note that the definition of LL is given by equation (11); (2) fine structure, including peaks and delays (Figure 4(a)). For instance, t1 and f1 are the delay time and the amplitude of the first peak of ACF, tn and fn being the delay time and the amplitude of the nth peak. Usually, there are certain correlations between tn and tnþ1; and between fn and fnþ1; (3) effective duration of the envelope of the normalized ACF, te; which is defined by the ten-percentile delay and which represents a repetitive feature or reverberation containing the sound source itself. The normalized ACF is defined by FpðtÞ ÅºFpðtÞ=Fpð0Þ: ð2Þ Similar to the manner shown in Figure 4(b), this value is obtained by fitting a straight line for extrapolation of delay time at 10 dB, if the initial envelope of ACF decays exponentially. Therefore, orthogonal and temporal factors that can be extracted from the ACF are Fpð0Þ; t1; f1; and the effective duration, te: 2.2. AUDITORY-TEMPORAL WINDOW In the analysis of the running ACF, the so-called auditory-temporal window 2 T in equation (1) must be carefully determined. The initial part of ACF within the effective duration te of the ACF contains important information of the signal. In order to determine the auditory-temporal window, successive loudness judgements in pursuit of the PRIMARY AND SPATIAL SENSATIONS 409 Figure 4. Definition of independent factors other than Fpð0Þ extracted from the normalized ACF. (a) Values of t1 and f1 for the first peak; (b) the effective duration of the ACF te; which is defined bythe 10 percentile delay (at 10 dB) and which is obtained practicallybythe extrapolation of the envelope of the normalized ACF during the decay, 5 dB initial. running LL have been conducted. Results show that the recommended signal duration ð2TÞr to be analyzed is approximately given by ð2TÞr ź 30ðteÞmin; ð3Þ where ðteÞmin is the minimum value of te obtained byanalyzing the running ACF [17]. This implies that the time constant represented by fast or slow of the sound level meter is deeply related to such a temporal window depending on the effective duration of ACF. The running step ðRsÞ which signifies a degree of overlap of the signal to be analyzed is not critical. It may be selected as K2ð2TÞr; K2 being chosen, say, in the range of 1/4 1/2. 2.3. FACTORS EXTRACTED FROM IACF The IACF is given by Z þT 1 FlrðtÞ Åº p0lðtÞp0rðt þ tÞ dt; ð4Þ 2T T where p0 ðtÞ ÅºpðtÞl;r *sðtÞ; pðtÞl;r is the sound pressure at the left- and right-ear entrances. l;r The normalized IACF is given by flrðtÞ ÅºFlrðtÞ=½Fllð0ÞFrrð0ÞŠ1=2; ð5Þ where Fllð0Þ and Frrð0Þ are autocorrelation functions ðt ź 0Þor sound energies arriving at the left- and right-ear entrance respectively. Spatial factors extracted from the IACF, IACC, tIACC and WIACC are defined in Figure 5 [2]. Note that the listening level is given by Equation (11). 410 Y. ANDO Figure 5. Definition of independent factors IACC, tIACC and WIACC extracted from the normalized IACF, d ź 0 1: In analyzing the running IACF, 2T is also selected by equation (3). For the purpose of spatial design for sound fields, however, longer values of ð2TÞr are recommended. 3. PRIMARY SENSATIONS 3.1. PITCH First of all, consider the pitch or the missing fundamental of the sound signal, which can be given by sP ź fPðFpð0Þ; t1; f1; DÞ; ð6Þ where D is the duration of sound signal as is represented by musical notes. When a sound signal contains only a number of harmonics without the fundamental frequency, one hears the fundamental as a pitch. This phenomenon is mainly explained by the delay time of the first peak in the ACF fine structure, t1; in the condition that the missing fundamental is less than about 1 2 kHz [16]. According to experimental results on the pitch perceived when listening to the bandpass noises without any fundamental frequency, the pitch sp is expressed by equation (6) as well. The strength of the pitch sensation is described bythe magnitude of the first peak of the ACF, f1 [18]. For the signal of a short duration, factor D might be taken into account. 3.2. LOUDNESS Next, consider the loudness sL which may be given by sL ź fLðFpð0Þ; t1; f1; te; DÞ: ð7Þ Since the sampling frequency of the sound wave is more than twice that of the maximum audio frequency, the value 10 log Fð0Þ=Fð0Þref is far more accurate than the Leq which is measured by the sound level meter, Fð0Þref being the reference. This fact is the most significant for an impulsive sound. Scale values of loudness within the critical band were obtained in paired-comparison tests using sharp filters with the slope of 1080 2068 dB/octave under the condition of a constant Fpð0Þ [19]. Obviously, when a sound signal has a similar repetitive feature, te becomes a large value, like a pure tone, then the greater loudness results are as shown in PRIMARY AND SPATIAL SENSATIONS 411 Figure 6. Loudness as a function of the bandpass noise byuse of filters with the slope of 1080 2068 dB/octave: (a) Bandpass noise centered on 1 kHz; (b) complex noises with the fundamental centered on 1 kHz. Figure 6(a). Thus, a plot of loudness versus bandwidth is not flat in the critical band centered at 1 kHz. This contradicts previous results of the frequency range centered on 1 kHz [20]. Figure 6(b) shows results of complex noises with the fundamental centered at 1 kHz. Comparing figures (a) and (b) of Figure 6, scale values of loudness are similar to each other, when the pitch is the same as given by t1: 3.3. TIMBRE The third primary sensation, timbre, that includes pitch, loudness and duration is assumed to be given by sT ź fT½Fpð0Þ; te; t1; f1; DŠ: ð8Þ Any experimental results on timbre according to equation (8) are not available at present. 3.4. DURATION The fourth-primitive sensation is introduced here because information in musical notes includes loudness, pitch and duration. It is a perception of signal duration, which is given by sD ź fD½Fpð0Þ; te; t1; f1; DŠ: ð9Þ Experimental results have been described in relation to t1; f1; and D in references [21, 22]. 412 Y. ANDO Table 2 Primary sensations, which may be described in relation to factors, extracted from the autocorrelation function and the interaural crosscorrelation function Factors Primitive sensations Loudness Pitch Timbrey Duration ACF LL Xx X X t1 XX X X f1 xX X X te Xx X x D xz xz Xz X X and x: Major and minor factors influencing the corresponding response. D: Physical duration of sound signal. LL ź 10 log½Fð0Þ=Fð0Þref Š; where Fð0Þ Åº½Fllð0ÞFrrð0ÞŠ1=2: y In order to describe timbre, additional factors ti and fi (i ź 2; 3; . . . ; N) must be taken into account on occasions. z It is recommended that loudness; pitch and timbre should be examined in relation to the signal duration, D as well. Table 2 summarizes the possible relation between the four primary sensations and the factors extracted from the ACF and the physical signal duration D: 4. SPATIAL SENSATIONS 4.1. DIRECTIONAL SENSATION The perceived direction of a sound source in the horizontal plane is described as s ź f ðLL; IACC; tIACC; WIACCÞ; ð10Þ where LL ź 10 log½Fllð0ÞFrrð0ÞŠ1=2=Fref ð0Þ: ð11Þ Fllð0Þ and Frrð0Þ signify sound energies of the signals arriving at the left and right ear entrances, and Fref ð0Þ is the reference. In these four spatial and orthogonal factors in Equation (10), the interaural delay time, tIACC; is well known as a significant factor in determining the perceived horizontal direction of the source. A well-defined direction is perceived when the normalized interaural crosscorrelation function has one sharp maximum, a high value of the IACC and a narrow value of the WIACC; due to high- frequency components. On the other hand, subjective diffuseness or no spatial directional impression corresponds to a low value of IACC (50 15) [23]. Apart from these four spatial factors, of particular interest is the perception of a sound source located in the median plane. The temporal factors extracted from the ACF of sound signal arriving at the ear entrances may act as cue [24] because little changes in spatial factors in the median plane [25]. Figure 7(a) shows that significant differences in the three factors, te; t1; and f1; as a parameter of the incident angle are found. Few differences, however, may be found in the head-related transfer functions as shown in Figure 7(b) [10]. PRIMARY AND SPATIAL SENSATIONS 413 Figure 7. (a) Three-dimensional illustration of t1; f1; and te extracted from the normalized ACF at each incident angle in the median plane to a listener for sound localization. The number inside the circles is the vertical angle in the median plane. (b) Amplitudes of the head-related transfer function at each incident angle in the median plane [25], which are used to obtain the normalized ACF. 4.2. SUBJECTIVE DIFFUSENESS The scale value of subjective diffuseness is assumed to be given byEquation (10) also. In order to obtain the scale value of subjective diffuseness, paired-comparison tests with bandpass Gaussian noise, varying the horizontal angle of two symmetric reflections have been conducted. Listeners judged which of two sound fields were perceived as more diffuse, under the constant conditions of LL, tIACC; and WIACC [26]. The strong negative relationship between the scale value and the IACC can be found in the results with frequency bands between 250 Hz and 4 kHz. The scale value of subjective diffuseness may be well formulated in terms of the 3/2 power of the IACC in a manner similar to the 414 Y. ANDO Table 3 Spatial sensations in relation to factors extracted from the ACF and the IACF Factors Spatial sensations ASW Subjective diffuseness Image shift Horizontal direction Vertical direction ACF t1 X f1 X te X IACF Fllð0Þ }} X (X) x Frrð0Þ }} X (X) x LL XX } X } tIACC xx X X x WIACC XX X X x IACC XX X X x X and x: Major and minor factors influencing the corresponding response. LL ź 10 log½Fð0Þ=Fð0Þref Š; where Fð0Þ Åº½Fllð0ÞFrrð0ÞŠ1=2; ASW: Apparent source width. subjective preference for the sound field, i.e., Sdiffuseness ź aðIACCÞb; ð12Þ where coefficients a ź 2 9 and b ź 3=2: 4.3. APPARENT SOURCE WIDTH (ASW) It is considered that the scale value of apparent source width (ASW) is given by Equation (10) as well. For a sound field with a predominant low-frequency range, the long-term IACF has no sharp peaks in the delay range of jtj51 ms, and thus a wide value of WIACC results. Clearly, the ASW may be well described by both factors, IACC and WIACC [27], under the conditions of a constant LL and tIACC ź 0: The scale values of ASW were obtained by paired-comparison tests with a number of subjects. The listening level affects ASW [28]; therefore, in this experiment the total sound pressure levels at the ear- canal entrances of sound fields were kept constant at a peak of 75 dBA. Listeners judged which of the two sound sources they perceived to be wider. The results of the analysis of variance for the scale values SASW indicate that both of factors IACC and WIACC are significant (p50 01), and contribute to the SASW independently, thus SASW ź aðIACCÞ3=2 þ bðWIACCÞ1=2; ð13Þ where coefficients a 1 64 and b 2 44: Calculated scale values sASW by equation (13) and measured scale values are in good agreement (r ź 0 97; p50 01) [27]. These formulas also hold for complex noise [29]. Table 3 indicates a list of spatial sensations with their significant factors extracted from the IACF. 5. SUBJECTIVE PREFERENCE AND ANY SUBJECTIVE RESPONSES FOR SOUND FIELDS The most preferred conditions for the sound field in a concert hall are briefly described here by both temporal and spatial factors [2]. PRIMARY AND SPATIAL SENSATIONS 415 5.1. TEMPORAL CRITERIA (1) The most preferred initial time delay gap between the direct sound and the first reflection is expressed by ½Dt1Šp ½1 log10 AŠðteÞmin; ð14Þ where ðteÞmin is the minimum value of the effective duration of the running ACF of the source signal, and A is the total amplitude of reflections given by ( )1=2 1 X A ź A2 ; ð15Þ n nź1 where An is the pressure amplitude of the nth reflection, n ź 1; 2; . . . . (2) The most preferred-subsequent-reverberation time is approximately expressed by ½TsubŠp ź 23ðteÞmin: ð16Þ 5.2. SPATIAL CRITERIA (3) The typical spatial factor is the IACC. The consensus preference is obtained at the small value of the IACC, so that signals arriving at both ears should be dissimilar. But, the peak value of the IACF must be maintained at the origin of the delay time, i.e., tIACC ź 0 ð17Þ so that the sound field should be well balanced. (4) The listening level LL in a room is classified into a spatial factor because of its right hemisphere dominance (Table 1). This is calculated at each seat, such as LL ź PWL þ 10 logð1 þ AÞ 20 log d0 11ðdBÞ; ð18Þ where PWL is the power level of sound source, and d0 is the distance between the source and a listener, which is related to the direct sound. In the design stage of a concert hall and an opera house, the most preferred listening level ½LLŠp is assumed at the center part of seating area because performers can control to some extent PWL to the listeners. Important subjective responses of sound fields in relation to all the above-mentioned all of orthogonal factors are listed in Table 4. These include preferred conditions of performers [30], speech intelligibility [18], and reverberance of sound fields [31]. 6. CONCLUSIONS (i) Primary sensations, pitch, loudness, timbre and duration may well be described by factors extracted from the ACF of the signal (Table 2). 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