Psychoacoustics. Author: Nejc Rosenstein. Advisor: Simon irca

Transcription

1 Psychoacoustics Author: Nejc Rosenstein Advisor: Simon irca Abstract We introduce psychoacoustics as a branch of physics which explores the link between physical properties of sound and listener's perception. We describe experiments which measure threshold, localization and frequency selectivity. We introduce simple physical models of perception of sound in human pinna and cochlea. The results of experiments in psychoacustics, which are in agreement with prediction of both models, are presented.

2 Contents 1 Introduction 2 2 Thresholds 2 3 Frequency selectivity Physical model Auditory lters Localization Physical model of pinna Psychoacoustical investigation Experimentation using headphones Conclusion 11 1 Introduction Two major disciplines that dominate the eld of research of the human auditory system are auditory physiology, which focuses on functions of biological systems that aect the hearing process, and psychoacoustics [1]. The latter deals with relation between sound i.e. physical stimuli and the sensation in the listener which occurs as a result of sound perception. Psychoacoustics therefore encompasses research of thresholds, frequency analysis and masking, perception of pitch, timbre and loudness, analysis of auditory scene, temporal processing and other properties of the auditory system. Psychophysical experiments involve listeners as test subjects who usually have to make some judgments about the sound they detected and produce a certain response. Those experiments have to be carefully planned as much eort is needed to perform them in a proper way, especially when scientists aim to reproduce a certain sound phenomena by the means of some device, for example headphones. 2 Thresholds The sound pressure level (SPL) is dened as L = 20 log(p/p 0 ) with decibels, db, as units, where p is the measured sound pressure and p 0 is the absolute threshold, i.e. the lowest sound level which can be detected by humans [2]. Since it is impossible to determine its exact value, the threshold is usually dened as the sound level at which the probability of detection equals a certain value, which is determined experimentally. In a common experiment where test subject have to tell, whether they perceived a sound or not (two-choice task), that probability usually equals 75% [2]. The threshold is frequency-dependent and is dierent if the listener is using only one or both ears (Fig. 1). Absolute thresholds are also important in psychoacoustical experiments, where quiet, but still detectable signals need to be produced. One such experiment is described in section

3 Figure 1: [Left] Measurements of absolute thresholds for binaural (both-ears) and monaural (one-ear) listening. Shape of the ear and other physiological factors aect our hearing in many ways. Therefore, monaural listening leads to dierent perception of loudness and dierent frequencies in the detected spectrum. This results in dierent frequency dependencies of the threshold. [Right] The dierence between the measurement at the eardrum and the measurement at the position of the center of the listener's head once he is removed from his position. We see that the ear shape aected the detected frequency spectrum. More about the role of ear shape in auditory perception will be told in chapter 4. Pictures are taken from [3]. 3 Frequency selectivity The probability of detecting a sound from a certain source can decrease dramatically if the listener is also exposed to other sounds at the same time; this phenomenon is called masking [2]. While the most obviuous example of masking is the unability to detect low-volume sound in the presence of the loud one, masking can also occur if two sound signals are having similar or the same freqency components. Discoveries in the eld of physiology shed some light on this phenomenon; in 1961, the Nobel prize in physiology was awarded to Georg von Békésy [4] for the discovery of mechanical properties of cochlea that provide an explanation of human frequency analysis and help to explain masking of sounds with similar frequencies. 3.1 Physical model By developing a new method to dissect the human ear, Békésy could perform experiments on a partly intact human chochlea (Fig. 2). He found that each part of the basilar membrane corresponds to a certain frequency; high frequencies of sound stimulate the parts of membrane which are close to the outer ear and low frequencies stimulate response of the membrane at the end of the cochlea. We are not going to present psychiological structure of the inner ear at this point; instead we will focus on a basic mechanical model, which was used by Békésy himself when he planned his physiological observations [5]. 3

4 Figure 2: [Left] A cross-section of the human chochlea [6]. [Right] The schematic representation of cochlea's shape [7]. The inside is divided into two uid compartments which are separated by a cochlear partition (CP), where the basilar membrane (BM) is located. The physical model we will present here is based on a very simplied structure of the cochlea and its sketch is included in Fig. 3. Figure 3: [Top] Simplied model of the cochlea interior, proposed by Bekesy [5]. [Bottom] The coordinate system for modeling the unrolled cochlea [7]. The interior of the cochlea is divided into two separate compartments which are lled with uid and are 4

5 separated by a wall. The large opening in the separating wall is covered by the basilar membrane which is a membrane with known mass, stiness and damping coecient (all these properties were measured by Békésy). Vibrations enter the cochlea through the oval window, covered by the membrane; the round window which is located in the other compartment is also covered by a membrane and its purpose is that it allows the movement of the incompressible uid in the cochlea. Even though Békésy included another opening, which connects both compartments (helicotrema), the exchange of uids is neglected in uid modeling. Even though the Reynolds number of the ow inside both compartments is low, we can treat the ow inside the cochlea as inviscid in the rst order of approximation [5]. If the pulses of sound are short, we may assume that uid ow inside the cochlea is irrotational [5], therefore its velocity can be expressed as v = Φ, where Φ is velocity potential. Since compartments are separated, the velocity potentials in both of them are dierent and we denote them by Φ 1 (in upper compartment) and Φ 2 (in lower compartment). The equations we write below are valid for the upper compartment. The uid is assumed to be incompressible, therefore we write: 2 Φ 1 = 0 (1) Since irrotational ow is assumed, Bernoulli's law applies [8]. We can neglect the term which contains the square of the uid velocity, since velocities are low. In term of irrotational ow due to a velocity potential, Bernoulli's equation is therefore written as: p 1 + ρ Φ 1 t = const. (2) where ρ denotes the density of the uid and p the pressure. The velocity of the uid in the direction perpendicular to the membrane is expressed as Φ/ z and it equals the velocity of the membrane: η t = Φ 1 z. (3) The three equations above describe the motion of the uid in the rst, upper compartment. The same set is then written for the second compartment, which is physically separated from the rst one. The next equation connects the dierence of the uid pressures on both sides of the membrane with the force on the membrane due to its displacement µ(x, t): m (x) 2 η t 2 + β (x) η t + κ (x)η = p 2 (x, 0, t) p 1 (x, 0, t). (4) Values for membrane mass per unit area m and functions for damping coecient per unit area β and stiness per unit area κ that were used to calculate the results, were chosen to represent actual properties of the basilar membrane. In addition to the above equations, two sets of equations similar to (3) and (4) are needed to describe the displacement of the membrane covering the round window, ξ r, and displacement of the oval window membrane, ξ o (see Fig. 3). The last equation, ξ o = F, (5) states that the displacement ξ of the membrane covering the oval window is equal to the incoming signal F which is transferred mechanically to the membrane. We are not going to describe the methods required to solve the model here and will instead focus only on the results. The calculated membrane positions along the length of the cochlea are presented in Fig. 4. Despite many assumptions and simplications, the calculated movement of the basilar membrane possesses the most important property of Bekesy's observations the position of the largest excitation on the basilar membrane is frequency-dependent, as shown in Fig. 4. Békésy also discovered that the membrane excitations only propagate up to a certain 5

6 distance, which is frequency-dependent. The higher the input frequency is, the shorter that distance gets. The membrane movements, calculated by the physical model, possess that attribute as well (see Fig. 4). Figure 4: [Top] The curves represent the shape of the membrane at dierent times. The wave envelope is drawn with dashed line and it shows that the displacement of the membrane has an absolute maximum at some point. [Bottom] Wave envelopes of membrane shapes. The maximum displacement appears on a frequency-dependent position along the length of the membrane (dierent curves were calculated using input signals with dierent frequencies) [5]. 3.2 Auditory lters The discovery of the way in which the basilar membrane acts as frequency analyzer led to a suggestion that the entire peripheral auditory system (which consists of the outer, the middle and the inner ear) acts as a bank of bandpass lters, with passbands that overlap [3]. Since dierent sections of the basilar membrane correspond to dierent frequencies, those sections actually act as auditory lters, centered around those frequencies. The overlapping of the passbands occurs because an excitation at a certain point of the membrane also causes the movement of neighbouring points, i.e. lters. The shape of auditory lters is determined through psychophysical experiments rather than by physical 6

7 models. One of the experiments where the auditory lter shape was determined, was carried out in the following way. The listener had to detect a sound signal (the testing signal) with xed volume in the presence of another, background sound signal (the masker signal). Both signals were sinusoidal with xed frequencies and were coming from the same source. Researchers experimented with dierent volumes of masker signal and determined the minimal masker volume at which the listener could no longer detect the testing signal. The described experiment was repeated many times, each time with dierent masker frequency. The minimal masker volume was the lowest when the dierence of frequencies of both signals was also low. The results are represented in the form of so-called psychophysical tuning curves (see Fig. 5). Figure 5: Psychophysical tuning curves [3] show the level of masker signal which is needed to successfully mask a signal, which is xed at low level (it is represented by dots). The dashed line represents the absolute threshold of a signal at a given frequency. Solid lines connect the measurements of minimal masking signal. The resulting shapes are rough estimates of the shape of auditory lters for dierent frequencies. The experiments were carried out at low volume, because researchers wanted to ensure that the excitation of the membrane would be as low as possible [2]. If the sound with a certain frequency causes the movement of the membrane, which is too strong, the excitation also triggers neighbouring auditory lters on the membrane; this is dened as an o-frequency listening [1]. In order to nd out the accurate shape of the auditory lter at a certain frequency, the o-frequency listening must be reduced as much as possible. This can be achieved by notched-noise technique experiments [1]. In these experiments, the listener is exposed to pure tone signal with added bands of noise above and below the tone frequency (Fig. 6). Both the pure tone and the noise are coming from the same source (either headphones or loudspeakers). If we use this technique, the observed lter is exposed to the least ammount of noise compared to the neighbouring lters which therefore do not contribute much to the detection of tone frequency, due to the increased masking. The minimum audible level is a function of the spectral gap between the edges of each of the noise bands and the tone frequency. The shape of the auditory lter, obtained by the described method is presented in Fig. 6. 7

8 Figure 6: Notched-noise technique [1]. Grey areas represent bands of noise. The auditory lter is centered at the signal frequency. 4 Localization Another important area which the psychoacoustic research is focused on is sound localization, i.e. dening the position of the sound source [2]. The sound from the source usually reaches one ear before the other (except when the sound is emerging from the front or the back of the listener). If the sound wave is sinusoidal and its frequency is low, the dierence in time of arrival of sound to both ears is proportional to the phase dierence, which can be used to determine the direction from which the sound is coming. This, however, is not true for frequencies above 1500 Hz, because in that case the wavelength of sound is small in comparison to the dimensions of the head, and hence the brain can not determine the phase dierence correctly. At high frequencies, another option of localization is more suitable. Since the head of the listener represents an obstacle for high frequency waves, they bend and diract around the head. That causes small sound intensity dierences between both ears. The human auditory system can then determinine the angle from which the sound is coming. Such combination of localization at low and high frequencies is called the duplex theory [2]. However, it has several aws for example, a single non-periodic click can be easily localised using only information about interaural time delays, regardless of the frequencies it contains. Furthermore, localization is also possible when listener is using only one ear (monaural listening). 4.1 Physical model of pinna Since duplex theory does not provide a satisfactory explanation of sound localization, the physical properties of human body must also be taken into consideration. The outer ear, i.e. the pinnae, plays a major role in localization. Its function is to transform the incoming sound into a signal which contains information about the direction from which the sound is coming [9]. That information is then extracted from the signal in the neural system. Incoming sound is reected at dierent parts of the pinna and because of that, eardrums detect not only the original signal, but also reected signals, which arrive to the eardrum with delays τ n. We assume that the incoming signal arrives to the eardrum via M dierent paths due to reections from dierent parts of the pinna. Each reected (delayed) signal is attenuated 8

9 after the reection occurs. We mark the attenuation coecient with a n < 1 and the original sound signal is represented by the function F. We will write the equations in the notation of Laplace transformation. The signal which arrives to the eardrum is [9] where H(s) = F (s)t (s) (6) M T (s) = a n e sτn (7) n=1 is the transfer function of the pinna. The sound signal F is transformed into signal H because of time delays, which are dependent of the direction from which the signal is coming. The transfer function T (s) has an inverse, and according to the research in the eld of physiology, such inverse could be performed in the neural system. Batteau [9] also proposed a so-called attention transformation function which could also theoretically be performed by the neural system in the inner ear: M R(s) = a n e (s(τ M τ n)). (8) n=0 As a result of the above transformation, the following signal is produced: M P (s) = R(s)H(s) = F (s)e sτ M [ a 2 n + n=0 M M a k a j e s(τ k τ j ) ]. (9) k j k The above result indicates that the processed signal has a dierent amplitude and a slightly altered spectrum in comparison to the original signal [9]. The received signal is also delayed by the maximum delay time τ M. According to Batteau's theory, the brain could be able to determine the direction of the incoming sound via the changes in the spectrum and intensity. It has to be mentioned that all delays are not caused by reections from the pinna, but also by reverberation from other objects, such as walls. But reections from the environment do not necessarily cause trouble when it comes to localization. It is easy to treat them separately, because time delays caused by the pinna, are short (2 300 µs) and delays due to the reverberation in environment are signicantly longer (usually more than 10 ms). But that does not mean that reections from the environment do not play a role in localization. Because of small dimensions of the pinna, sounds with high frequencies cannot be localised only by analyzing short delays which are caused by the reections from pinna. 4.2 Psychoacoustical investigation We have shown that the information about localization can be extracted from interaural time delays, interaural intensity dierences and from the delays in the pinna, but nothing has been said yet about our actual ability to localize sounds. The following psychoacoustic experiment was carried out in order to nd out how good we are at determining directions from which sounds are coming [1]. Researches decided to look for minimum audible angles, i.e. angles at which the listeners could no longer determine whether the two sounds with the same frequency were coming from the same direction or not. Two loudspeakers were used in the experiment the rst loudspeaker was placed in front of the listener's head and the second one was placed at the same distance from the center of the head, but at a dierent angle. The researchers then moved the second speaker around and the listener had to determine whether two tones were coming from the same direction or not. The signals used for experimenting were pure tones and the experiment was repeated at dierent frequencies. Minimum audible angles were determined for each frequency and the results are shown in Fig. 7. 9

10 Figure 7: Minimum audible angles at dierent frequencies [1]. White dots are measurements taken when the rst loudspeker was placed in front of the listener and black dots represent measurements when the rst loudspeaker was at 30 angle. The minimum audible angle is the lowest at low frequencies, i.e. at wavelengths which are much bigger than the dimensions of the human head [1]. In that case the information needed for localization is most likely provided to the brain by interaural time dierencies. In 1956, Klumpp and Eady found out that the smallest interaural time dierence which can be detected by a human is approximately 10 µs [1]. Since we know the diameter of the human head and the speed of sound in air, we can estimate the minimum detectable change in direction, which equals roughly 1. This value matches the experimental results at low frequency, shown in Fig. 7. This indicates that the auditory system really processess information about interaural delays. However, the localization is possible also when the listener can only use one ear (monaural listening), as shown in Fig. 8, even though both time-dierence and phase-dierence are impossible in that case. We can therefore conclude that the reections of sound in the pinna, described in section 4, really play a major role in localization. Figure 8: Histograms show results of experiment, where listeners had to determine the direction from which the sound was coming [9]. When listeners were only using one ear [Right], the results were visibly worse than in the case of binaural listening [Left]. 4.3 Experimentation using headphones So far, we have only discussed experiments which were carried out by the use of loudspeakers. When researchers tried to reproduce external sounds with headphones, the listeners often reported that the voices appeared inside, instead of outside their heads [2]. In order to investigate what caused the brains 10

11 to determine that sounds originated from inside the head, careful calibration of headphones had to be performed. The goal was to produce a signal at the eardrum with the headphones, which would have the same amplitude and phase as the signal at the eardrum due to a tone played from a loudspeaker, set at a certain angle. Experiments were carried out where the researchers reproduced signals in headphones, as if they were coming from dierent directions. The same sounds were then also played through loudspeakers placed at the position which corresponded to the position determined by the sound played by headphones. Amplitudes and phases had to be corrected for that purpose and calibration was conrmed as successful after listeners could no longer distinguish between loudspeaker and headphone signals [10]. The sound coming from loudspeaker is dened as the real signal and the one that comes from headphones is dened as the virtual signal. After it was ensured that headphones could satisfactorly reproduce the signal coming from loudspeakers, researchers began to investigate which property of the sound caused externalization, i.e. the property of auditory system to determine that sound does not originate inside the head. They focused on both main aspects of sound localization, described at the beginning of this chapter - time and level dierences [7]. In this chapter, we will only focus on one specic experiment which investigated the eect of interaural time dierencies (ITD) on externalization. The loudspeaker was positioned at the angle of 37 with respect to the directon the listener was facing. The headphones were calibrated to resemble the sound coming from the same angle. If the frequency of the sound is low, we can calculate the ITD from the low-frequency limit of diraction around a sphere [7]: IT D = 3r c sin θ. (10) In the above equation, c is speed of sound in the air and r is radius of the head (which is approximated by a sphere). For a head with a radius of 8 cm we get IT D 420 µs, since θ = 37. But after calibration on the test subjects it turned out that the optimal interaural time dierence was IT D 375 µs. The experimental sessions were divided into three stages - calibration, training and trial. During the training phase, listeners were exposed to sequence of four signals: real - virtual - real - virtual. The purpose of this stage of the experiment was to distinguish real signals from virtual signals; the listeners could repeat the sequence as many times as they wanted in order to learn and remember the dierence. The next stage consisted of 20 trials where the listeners had to decide whether the signal they were exposed to was real or virtual. The experiment was repeated at dierend IT Ds and at dierent frequencies. Figure 9 shows the results for experiment with sounds with fundamental frequency ν = 125 Hz. 5 Conclusion We have presented some basic experiments, which were at one point or another cruicial in the progress in the eld of psychophysics. Both of the physical models included in this paper have been introduced in 1970s and both have seen many developments by dierent authors. In one of the recent articles [7], the authors report that they have succesfully implemented the cochlea model which is still based on the same equations we presented in chapter 2, but the model also encompassess curvature in addition to the basic model. It turns out that the consequence of curvature in cochlea (see Fig. 2) is an increased perception of low frequencies [7]. All psychoacoustic experiments have also been updated up to this day and psychoacoustical models are also useful in the commercial sector. Psychophysical models are used in software such as lossy signal converters (for example MP3-converters), in planning and designing of noise reduction systems and high-end audio systems, as well as in loudspeaker and headphone industry. 11

12 Figure 9: Fractions of correct identications at dierent IT Ds [10]. At IT D = 525µs and IT D = 300µs, listeners almost never identied real sources as virtual sources or vice versa. In contrast, at IT Ds close to the optimal time dierence, test subjects often made a wrong decision, which means that the virtual signal was such a good reproduction of the real signal, that they were too hard to distinguish. It is interesting that for IT D = 525 µs all listeners reported that the virtual signal was coming from the right side of the loudspeaker. On the other hand, at IT D = 300 µs, all listeners claimed that the sound originated from the inside of their head. References [1] Plack, C. J., Sense of hearing (Lawrence Erlbaum Associates Inc, Manwah, 2005). [2] Rossing, T., Springer Handbook on Acoustics (Springer Science+Business Media, New York, 2007). [3] Moore, B. C. J., An Introduction to the P sychology of Hearing (Emerald Group Publishing Limited, Bingley, 2012). [4] (Cited on 5. January, 2014). [5] M. B. Lesser and D. A. Berkley, J. Fluid Mech. 51, (1972). [6] (Cited on 5. January, 2014). [7] H. Chai, D. Manoussaki and R. Chadwick, Phys. Rev. Lett. 96, (2006). [8] Morrison, F. A., An Introduction to F luid M echanics (Cambridge University Press, New York, 2013). [9] D. W. Batteau, Proc. Roy. Soc. B. 168, (1976). [10] W. M. Hartmann and A. Wittenberg, J. Acoust. Soc. Am 99, (1996). 12