Basilar membrane models

Transcription

1 Technical University of Denmark Basilar membrane models 2 nd semester project DTU Electrical Engineering Acoustic Technology Spring semester 2008 Group 5 Troels Schmidt Lindgreen s David Pelegrin Garcia s Eleftheria Georganti s Instructor Mørten Løve Jepsen

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3 Technical University of Denmark DTU Electrical Engineering Ørsteds Plads, Building Lyngby Denmark Telephone Title: Basilar membrane models Course: Auditory Signal Processing and Perception Spring semester 2008 Project group: 5 Participants: Troels Schmidt Lindgreen David Pelegrin Garcia Eleftheria Georganti Supervisor: Torsten Dau Instructor: Mørten Løve Jepsen Date of exercise: March 13 th Pages: 23 Copies: 4 Synopsis: Two linear basilar membrane models are studied. The first model ( transmission-line model ) is based on the mechanical properties (mass, stiffness and damping) of different segments along the basilar membrane and it is implemented as an electric circuit taking advantage of the existing electro-mechanical analogies. Thus, different phenomena are deduced from the mechanical properties, such as the travelling wave along the basilar membrane and the forward spread of masking. The second model ( gammatone auditory filterbank ) is an approach to the tuning properties at different places on the basilar membrane with the corresponding delays. It is used to study the energy distribution of a sound at the different regions of the basilar membrane as a function of time. No part of this report may be published in any form without the consent of the writers.

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5 Introduction A physical system can be studied analyzing each one of its parts and the relationship between them. Thus, the properties of one of the parts determine some of the properties of the overall system. Applying this study method to the human hearing system, it can be stated that the outer ear acts as a tube providing resonances and the middle ear provides an impedance matching between different mediums. The inner ear is the place where mechanical vibrations are transformed into neural excitation. Therefore, the study of the elements inside the inner ear can lead to know in which way this transformation takes place, depending on the excitation signal and level. In this report, the basilar membrane is studied by means of computer models. A first model is designed according to its mechanical properties in a simplified way, seeing the basilar membrane as a discrete series of elements with constant mass, but changing stiffness and damping. The differential equations defining electrical and mechanical magnitudes are similar and thus it is possible to establish an analogy between them and in fact, the basilar membrane model is implemented as an electrical circuit. A second model is the gammatone auditory filterbank, implemented as a series of parallel band-pass filters that model the tuning frequency at each point of the basilar membrane and take the delay into account, making possible to analyze the excitation patterns produced by a given input signal. Technical University of Denmark, March 25, 2008 Troels Schmidt Lindgreen David Pelegrin Garcia Eleftheria Georganti

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7 Contents 1 Theory Transmission-line model Gammatone auditory filterbank Results Transmission-line model Gammatone-filter model Discussion Transmission-line model Gammatone filterbank model Excitation patterns Conclusion 21 Bibliography 23 A Matlab Code A1 A.1 GammaIR A1 A.2 Gammatone filterbank A1 A.3 Chirp that compensates for the delay of the flters A2

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9 March 2008 Chapter 1. Theory Chapter 1 Theory 1.1 Transmission-line model The transmission-line model is a simplified model of the basilar membrane (cochlea). The model describes the basilar membrane velocity and the force and fluid volume velocities in the cochlea. The model can be thought of as a chain of independent mechanical filters, representing different segments along the cochlear partition. [Dau et al., 2008, p. 2] Toward a model The form of the cochlea and the structure of the cell aggregates within it are very complicated. Thus, some simplifications are made in order to have a tractable model of the cochlea. These simplifications are made over the basis of the following assumptions: The fluid in the canals is incompressible and will neglect all viscosity effects in it. The movements are so small that the fluid and the basilar membrane operate linearly. The above assumptions result in the cochlea having a linear behavior, which is not the actual case, but in this report the assumption is sufficient. Because the cochlea is assumed to behave linearly, linear mathematical formulas can be used to describe the system. Another assumption is that the various parts of the basilar membrane are not mechanically coupled to each other and that all coupling occurs via the surrounding fluid. This means that the mechanics of the cochlear partition can be described by a single function of x, the acoustic impedance ξ(x). [de Boer, 1980, p. 139] The motion on the basilar membrane is described by a traveling wave, the speed of propagation of which decreases with increasing distance x. The one-dimensional approximation Page 1

10 Chapter 1. Theory Technical University of Denmark will only be valid if the dimensions in the y- and z-directions are small compared with the smallest wavelength in the x-direction. This is assumed in this report as the acoustic impedance is assumed to be a function of only x. A well-known model for the one-dimensional linear cochlear is the wave equation: Where: p(x) is pressure. ρ is the density of the fluid. h(x) is the effective height. ξ(x) is the acoustic impedance. 2 p(x) x jωρ p(x) = 0 [Dau, 2008, p. 19] (1.1) h(x)ξ(x) It is clear that the solution p(x) critically dependence on the product h(x)ξ(x). h(x) varies slowly with x, so for simplicity it is regarded as a constant, hence the impedance ξ(x) is the critical factor. The acoustic impedance is composed of three parts, a mass part, a damping part and a stiffness part and is given by: ξ(x) = jω m(x) + r(x) + c(x) jω [de Boer, 1980, p. 141] (1.2) Where: m(x) is the mass of the considered basilar membrane segment at position x. r(x) is the damping of the considered basilar membrane segment at position x. c(x) is the stiffness of the considered basilar membrane segment at position x. The stiffness is responsible for the occurrence of the traveling wave propagating along the cochlear partition. Due to the variations of the stiffness term, the velocity of propagation will greatly depend on x Mechanical circuit The transmission-line model described by (1.1) can be modeled mechanically as seen on figure 1.1 [Dau et al., 2008, p. 3]. The fluid pipe in figure 1.1 consists of distributed masses, m 1, m 3,.... A series of resonators are connected to taps of the fluid pipe. Each resonator represents a segment on the basilar membrane and consists of a mass m i, a spring k i and a damper b i. Page 2 Basilar membrane models

11 Explain the differences in the excitation pattern of the frequencies in the gammatone filter bank in response to a rising chirp stimulus compared to a click stimulus State one characteristic where the transmission line model and the gammatone filter bank differ from each other with respect to the shape of the auditory filters 2 Transmission-line model March 2008 Chapter 1. Theory A strongly simplified mechanical circuit of the cochlea mechanics is shown in Fig. 2. The fluid F m 1 m 3 m N-1 m 2 k 2 m 4 k 4 m N k N b 2 b 4 b N Figure Figure 1.1: Mechanical 2.1: Mechanical equivalent equivalent circuit circuit of the cochlea. of the cochlea. The driving The driving force Fforce is applied F is applied at the stapes. at the The scala vestibuli stapes. The is modeled scale vestibuli as a fluid is modelled pipe and as the a fluid basilar pipe. membrane The basilar membrane independent as segments. independent segments. The pressure (force) difference between scala vestibuli and tympani driving the segments of the basilar membrane is simplified to a force from the scala vestibuli only. The model is driven by a force F applied to each resonator segments and the masses along filled the pipe. pipe consists The force of distributed appliedmasses to a msegment 1, m 3,... awill series cause of resonators all elements is connected of the to taps specific of the fluid pipe. Each of the resonators is meant to reflect a segment of the basilar membrane. segment Resonator to move i consists at a certain of a mass velocity. m i, a spring k i and a damper b i. As a good first approximation, the fluid pipe can be thought of as the scala vestibuli. The model is driven by a force F applied to the stapes. This will result in forces applied to each of the resonator segments and the masses 1.1.3along Electric the pipe. circuit The force applied to a segment will cause all elements of the segment to move at a certain velocity. Assuming F varies harmonically as F 0 e jωt, the velocity v i of element i can To implement be obtained and fromanalyze the forcethe F i applied transmission-line to element i and model the mechanical it is advantageous impedance of tothe translate element the mechanical model into an equivalent electric 3circuit model by using an electro-mechanical analogy. The relationship between mechanical and electric elements can be seen in table 1.1. Element type Electrical Mechanical Name Impedance Name Impedance Effort source Voltage (V) Force (F) Flow source Current (I) Velocity (v) Flow storage Inductor (L) jωl Mass (m) jωm Effort storage Capacitor (C) 1/( jωc) Spring (k) k/( jω) Dissipator Resistor (R) R Damper (b) b Table 1.1: Summery of equivalent elements in the electro-mechanical analogy. From [Dau et al., 2008, p. 4]. By using table 1.1 to replace the mechanical elements with equivalent electric models the circuit becomes an electric model as shown on figure 1.2. The parallel circuit of the mass/spring/damper system in each segment becomes a serial LCR element. This is because all elements that have the same velocity in the parallel circuit have to draw the same current in the electrical equivalent, which is the case for a serial circuit [Dau et al., 2008, p. 4]. Page 3

12 Effort source voltage[u] force[f] Flow source current[i] velocity[v] Flow storage inductor[l] Ls mass[m] ms Effort storage capacitor[c] 1/Cs spring[k] k/s Dissipator resistor[r] R damper[b] b Table 2.1: Summary of equivalent elements in the first electro-mechanical analogy. s represents the complex frequency variable jω. Chapter 1. Theory Technical University of Denmark L 1 L 3 L N-1 U L 2 L 4 L N C 2 C 4 C N R 2 R 4 R N Figure 1.2: Figure Equivalent 2.2: Equivalent electric model electrical for the circuit mechanical for the mechanical model in figure model 1.1. of the From cochlea [Dau et in al., Fig. 2008, 2. p. 4]. masses are replaced by inductors L i, the compliances of springs are replaced by capacitors C i and 1.2 the Gammatone dampers are replacedauditory by resistors R i. The filterbank parallel circuit of the mass/spring/damper system becomes a serial LCR element. This is because all elements that have the same velocity in the parallel mechanical circuit have to draw the same current in the electrical equivalent, which is The gammatone the case for aauditory serial circuit. filterbank is a set of band-pass filters designed to characterise the nerve impulse responses to sounds, originally studied with experiments in cats [Dau 4 et al., 2008, p. 6]. Each filter (centered at frequency f c ) has an impulse response given by: g(t) = a 1 e(t) cos(2π f c t + φ), t > 0, (1.3) where φ is the phase of the carrier (cosine) relative to e(t), the envelope function of the filter, and a is a normalization constant, calculated as follows: e(t) = t n 1 e 2πbt (1.4) a = 0 e(t)dt (1.5) In these equations, n is the order of the filter, which indicates the slope of the skirts of the filter and b is a parameter that determines its time duration and the delay of the response. A value of n = 4 is assumed in this exercise, because in this range the shape of the filter is similar to the round exponential filter, which is used to approximate the magnitude of the human auditory filter characteristic. The equivalent rectangular bandwidth (ERB) is a way to quantify the bandwidth of human filters at center frequencies f c. The ERB is the bandwidth of a rectangular filter which has the same energy as a given auditory filter. The ERB as a function of the center frequency of the filter is estimated to be: ERB = f c (1.6) In the present laboratory exercise, b = ERB. Page 4 Basilar membrane models

13 March 2008 Chapter 2. Results Chapter 2 Results 2.1 Transmission-line model Analytical analysis of a single segment In figure 2.1 it can be seen the electrical circuit that corresponds to the first segment of the basilar membrane mechanical model, designed under the electronics simulation program PSpice. L 1 Z 1 Z 2 + L 2 + V i V 0 - C - R Figure 2.1: First segment of the equivalent electrical circuit for the mechanical model of the cochlea. V i is assumed as equal to V i e jωt. The analytical voltage transfer function (node between L 1 and L 2 ) according to electrical circuits theory would be: V O V i = Z 2 Z 1 + Z 2 = R + j(l 2 ω 1 Cω ) R + j((l 1 + L 2 )ω 1 Cω ) (2.1) Page 5

14 Chapter 2. Results Technical University of Denmark Additionally the analytical transfer function for the current equals to: I 0 V i = 1 R + j ( (L 1 + L 2 ) ω 1 ωc ) (2.2) Equation (2.1), which is the transfer function for the voltage equals a low pass filter, whereas the transfer function of the current equals a bandpass filter. As can be seen from equations 2.3 and 2.4 the characteristics of the voltage are like a low-pass filter because when ω goes to the response is attenuated although it does not go to zero. When ω goes to zero then the response is not attenuated. This can be seen in figure 2.2. V O V i V O V i = 1 (2.3) ω 0 L 2 = (2.4) ω L 1 + L 2 This filter circuit boosts certain frequencies around ω 1 with a maximum magnitude of the transfer function given by equation 2.5, whereas the frequencies around ω 2 are attenuated. V O V i V O V i L 1 = 1 j ω=ω1 R (2.5) C(L 1 + L 2 ) ( ) 1 L 1 = 1 + j ω=ω2 R (2.6) CL 2 1 ω 1 = (2.7) C(L 1 + L 2 ) 1 ω 2 = (2.8) CL 2 The current transfer function acts as a band-pass filter as seen in equations 2.9 and The maximum response is obtained at ω 0 = 1/((L 1 + L 2 )C) and the magnitude is given by equation This can be seen on figure 2.2. I O V i I O V i I O V i = 0 (2.9) ω 0 ω ω0 = 1 R (2.10) ω = 0 (2.11) Page 6 Basilar membrane models

15 March 2008 Chapter 2. Results Figure 2.2: Plots of the transfer functions of the voltage and the current (multiplied with 1000) of the first element of the equivalent electrical circuit of the mechanical model of cochlea Electrical Circuit Simulation In this part of the exercise the full transmission line-model of the cochlea was established by adding the necessary elements of the circuit (figure 2.7). The values of the capacitors, described in section 1.1.3, are varied according to equation: C i = C o e ( i ), i= (2.12) where C o is assumed to be 100 nf [Dau et al., 2008, p. 5]. In order to achieve a constant Q-value, the resistors were also varied according to [Dau et al., 2008, p. 5]: δ = 1 C Q = R (2.13) L ( ) L R i = δ (2.14) where R 1 = 150Ω. The values of the resistances and capacitors according to equations 2.12 and 2.14 can be seen in table 2.1 and the loss factor δ was calculated from the first segment of the circuit. The loss factor is assumed to be constant and is calculated to be The voltages for segments 4 (L75), 6 (L79) and 8 (L83) as a function of frequency and time appear at figures 2.3 and 2.4, and the currents respectively at figures 2.5 and 2.6. C i Page 7

16 Chapter 2. Results Technical University of Denmark Number Capacitance [nc] Resistance [Ω] Table 2.1: Values of the resistances and capacitors of the equivalent electrical circuit of the mechanical model of cochlea. Figure 2.3: Voltage as a function of frequency for segments 4 (L75), 6 (L79) and 8 (L83) of the electric circuit that appears at figure 2.7. Plotted with double log axis. Page 8 Basilar membrane models

17 March 2008 Chapter 2. Results Figure 2.4: Voltage as a function of time for segments 4 (L75), 6 (L79) and 8 (L83) of the electric circuit that appears at figure 2.7. Plotted with double linear axis. Figure 2.5: Current as a function of frequency for segments 4 (C39), 6 (C37) and 8 (C35) of the electric circuit that appears at figure 2.7. Plotted with double linear axis. Page 9

18 Chapter 2. Results Technical University of Denmark Figure 2.6: Current as a function of time for segments 4 (C39), 6 (C37) and 8 (C35) of the electric circuit that appears at figure 2.7. Plotted with double linear axis. Figure 2.7: Equivalent electrical circuit of the mechanical model of the cochlea Page 10 Basilar membrane models

19 March 2008 Chapter 2. Results 2.2 Gammatone-filter model Analysis of a single gammatone filter A Matlab function was composed (see appendix A) in order to generate an impulse response that corresponded -to a gammatone auditory filter for different center frequencies. In figure 2.8 it can be seen the impulse responses (IR) of the gammatone filters that correspond to center frequencies 250 Hz, 500 Hz and 1000 Hz in the time domain and additionally their magnitude spectra in the frequency domain. Amplitude Time [ms] 250 Hz 500 Hz 1000 Hz Magnitude [db] Frequency [Hz] Figure 2.8: Impulse responses of filters with center frequencies 250 Hz, 500 Hz and 1000 Hz in the time domain and the magnitude spectra in the frequency domain. In figure 2.9, the magnitude frequency response of three filters that correspond to center frequencies of 300 Hz and 6000 Hz, plotted in logarithmic scale, is shown. In figure 2.10 the magnitude frequency response for three filters centered at 900 Hz, 1300 Hz and 2000 Hz is presented in linear frequency scale. An analytical formula for the time t max ( f c ), where the envelope of the gammatone IR has its maximum was derived from equation (1.3). The t max can be calculated by taking the derivative of the envelope in equation (1.4). de(t) dt = 0 3tmaxe 2 2πbt max 2πbtmaxe 3 2πbt max = 0 3tmax 2 2πbtmax 3 = 0 t max = 3 t=tmax 2πb (2.15) Page 11

20 Chapter 2. Results Technical University of Denmark The primary parameter of the filter b is ERB and t max equals to: t max = 3 2π b = 3 2π 1.018ERB = 3 2π ( f c ) (2.16) Magnitude [db] Hz 6000 Hz Frequency [Hz] Figure 2.9: Magnitude frequency response of filters that correspond to center frequencies of 300 Hz and 6000 Hz. The frequency axis is plotted in logarithmic scale Magnitude [db] Hz 1300 Hz 2000 Hz Frequency [Hz] Figure 2.10: Magnitude frequency response for gammatone filters centered at 900 Hz, 1300 Hz and 2000 Hz. The frequency axis is plotted in linear scale. Page 12 Basilar membrane models

21 March 2008 Chapter 2. Results Analysis of the gammatone auditory filterbank In this section the gammatone auditory filter bank was analysed. In order to demonstrate the excitation along the basilar membrane in response to different stimuli the built-in function gammafb of Matlab was used [Dau et al., 2008, p. 7]. As input signals a delta pulse and a chirp signal (figure 2.11) were used Amplitude Time [ms] Figure 2.11: Chirp signal in the time domain. It was used as an input stimulus for the gammatone filterbank. The resulting excitation patterns as a function of time from a gammatone filterbank within the range of 100 Hz to 8000 Hz, when a delta function or a chirp signal were used, can be seen in figures 2.12 and 2.13 respectively Composition of a chirp that compensates for the delay of the filters In this part of the exercise a new chirp signal was composed in order to compensate for the delay of the filters. A Matlab function was designed in order to make the new chirp that would excite the basilar membrane model more than the prior chirp. The code can be found in Appendix A. The resulting chirp is presented in figure It was inserted as an input stimuli to the gammatone filterbank, and the resulting excitation pattern is shown in figure m Page 13

22 Chapter 2. Results Technical University of Denmark Center frequency (Hz) Time (ms) Figure 2.12: Excitation pattern of the gammatone filterbank for a range of 100 Hz to 8000 Hz when a delta function is used as an input signal. Darker areas indicate more excitation Center frequency (Hz) Time (ms) Figure 2.13: Excitation pattern of the gammatone filterbank for a range of 100 Hz to 8000 Hz when a chirp is used as an input signal. Darker areas indicate more excitation. Page 14 Basilar membrane models

23 March 2008 Chapter 2. Results Amplitude Time [ms] Figure 2.14: Chirp signal in the time domain designed to compensate for the time delay of the filters Center frequency (Hz) Time (ms) Figure 2.15: Time frequency distribution of the excitation of a gammatone filterbank when a chirp that compensates for the delay is used. Darker areas indicate more excitation. Page 15

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25 March 2008 Chapter 3. Discussion Chapter 3 Discussion 3.1 Transmission-line model The two models of the basilar membrane are based on different approaches in their premises. The transmission line model is based on the mechanical properties of the basilar membrane and implemented taking advantage of the electro-mechanical analogies. The designed circuit has several segments or filters, modeling areas of the basilar membrane with different impedances. The changes on the values of R and C, equivalent to the damping and the compliance, result in different center frequencies of these filters. A single segment of the model is analyzed and the resulting transfer functions for the voltage and current (or force and velocity) are shown in figure 2.2. The force presents a low-pass filter characteristics, showing that no further segments will be highly excited with the highest frequencies. When several segments are connected in cascade, the output of each one is the input for the next, and it is possible to see that at each output, the frequency range is reduced. This shows that the lowest frequencies travel longer distance along the basilar membrane. The current observed at figure 2.2 presents a band-pass filter characteristics and this is equivalent to say that the velocity of a segment on the basilar membrane has a maximum amplitude which depends on the stimulus frequency and that there is one particular area at the basilar membrane that responds maximally for each given frequency. By analyzing different segments on the circuit (see figures 2.5 and 2.6), it can be seen that the maximum amplitude of the current impulse response (or velocity of the basilar membrane segments) is delayed inversely proportional to the center frequency of the filter. From the first figure, it can also be seen that the slope towards low frequencies of each filter Page 17

26 Chapter 3. Discussion Technical University of Denmark is steeper than the slope towards high frequencies. This proves that the upward spread of masking is related to the mechanical properties of the basilar membrane. It should also be noticed in figure 2.3 that the cut-off frequency of the low-pass filters becomes lower at the latter segments of the circuit, so the excitations for the latest segments lack of high-frequency content (only low-frequencies propagate). 3.2 Gammatone filterbank model The gammatone filterbank is designed to be more efficient and faster in computational terms than the transmission-line model. This model is able to take into account the delays at the different parts of the basilar membrane as well as the associated tuning frequencies, as it is shown in figure 2.8. As it happened with the transmission-line model, the filters with lower center frequencies present higher delay at the maximum position of the impulse response, given by equation (2.16). At the same time, the filters with high center frequencies have a shorter overall duration in comparison to the ones with lower center frequencies. The frequency response of these filters is symmetric in a linear frequency scale and not in a logarithmic one, as it can be observed by comparing figures 2.9 and In figure 3.1, the responses of three filters obtained by the transmission line model and the gammatone filterbank are compared. It can be observed that the gammatone filters are much narrower, but despite this fact, the slopes at the both sides of the center frequency appear to be equally steep, in opposition to the transmission-line model. The peaks in the frequency responses of the gammatone filters are equal amongst them, because they have been calculated as independent filters, whereas the transmission-line filters represent the result of a cascade system. 3.3 Excitation patterns The excitation pattern of the gammatone auditory filterbank when a delta function is used as input can be seen on figure 2.12 on page 14. It can be seen that the delta pulse excites the basilar membrane almost instantaneously and degrades fast. It is clearly seen that the high frequencies are excited far more than the low frequencies which have a longer traveling time and will therefore be excited at a later time. E.g. frequencies above 1 khz are excited within 1 ms and frequencies below 300 Hz are first excited after 5 ms. The reason for the high frequencies being excited far more than the low frequencies is because the high frequencies are excited in a short time period, and the low frequencies are excited for a much longer period but with the same energy. This means that the high Page 18 Basilar membrane models

27 March 2008 Chapter 3. Discussion Figure 3.1: Comparison of three filters with different center frequencies obtained by the transmission line model and the gammatone model. The thin lines correspond to the filters obtained from the transmission line model and the thick ones to the filters obtained from the gammatone filterbank. frequencies are very excited at once and the low frequencies are only slightly excited but for a long period. This is not an optimal solution, because it is difficult to measure the neural activity without the whole basilar membrane being excited at once. Chirp signals are used to enhance excitation from the basilar membrane, in order to produce maximum neural activity in the brain. The excitation pattern of the gammatone auditory filterbank when a chirp function (see figure 2.11 on page 13) is used as input can be seen on figure 2.13 on page 14. From figure 2.11 it can be seen that the chirp consists of a sinusoid with rising frequency meaning that the low frequencies are presented first to the basilar membrane. The chirp is meant to compensate for the delay of frequencies in such a way that all frequencies will be excited at once, giving a higher combined excitation and therefore will be easier to extract in real measurements. When comparing figure 2.12 to figure 2.13 it is easy to see that the chirp in fact will excite all frequencies at once. (In this case the frequency range is from 100 Hz to 8 khz.) This results in a high combined excitation despite that the chirp does not excite a specific region of the basilar membrane more than the delta stimuli does. A drawback of using a chirp is that there will be a longer time period before the basilar membrane will be excited so this should be taken into account when analyzing measurement results where a chirp is used as stimuli. Page 19

28 Chapter 3. Discussion Technical University of Denmark From figure 2.13 it can be seen that despite using a chirp and exciting all frequencies at once, each filter in the model is not maximum excited at the same time. The high frequencies have maximum excitation around 11 ms whereas the low frequencies have maximum excitation around 14 ms - 15 ms. To solve this, a new chirp is developed using knowledge of the model. This means that the new chirp is developed specifically for the gammatone filterbank model. Figure 2.14 on page 15 shows the chirp in time domain and the excitation pattern can be seen on figure 2.15 on page 15. Now all frequencies have maximum excitation at once (around 10 ms). It is also clearly seen that the model predicts much higher excitation of the basilar membrane for each filter. Page 20 Basilar membrane models

29 March 2008 Chapter 4. Conclusion Chapter 4 Conclusion Two basilar membrane models are investigated in this report. The models are implemented using Pspice and Matlab. Both methods assume linearity in the basilar membrane transfer function. It is possible to model the basilar membrane, just by taking into account its mechanical properties: mass, compliance and damping. This model is called the transmission-line model and it can be implemented by using electric components, taking advantage of the electro-mechanical analogies, where mass is represented by inductors, compliance by capacitors and damping by resistances. The voltage of the electrical circuit corresponds to force and the current to velocity. Another approach based on the gammatone filterbank model offers a range of independent filters that describe the shape of the impulse response function of the auditory system for different inputs. When both models are compared, the transmission line model seems more realistic than the gammatone filterbank, because it explains several effects that take place in the basilar membrane, such as the upward spread of masking and the traveling wave (or distribution of the frequencies along the cochlea), assuming that the basilar membrane behaves in a linear way. However, it requires a considerable computational load and it is not suitable for real-time analysis of signals, where the gammatone filterbank model is preferred. To maximize the excitation from the basilar membrane, a chirp stimuli is better than a delta impulse. This is because the chirp is designed to stimulate all frequencies on the basilar membrane at once, compensating for the non-homogeneous delay of different Page 21

30 Chapter 4. Conclusion Technical University of Denmark frequency components of a signal. Page 22 Basilar membrane models

31 March 2008 Bibliography Bibliography [Dau, 2008] Dau, T. (2008). Cochlear transformation. Slide show. [Dau et al., 2008] Dau, T., Favrot, S., and Jepsen, M. L. (2008). membrane models edition. Exercise 3: Basilar [de Boer, 1980] de Boer, E. (1980). Auditory Physics. Physical principles in hearing theory I. Page 23

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33 Project group 5 March 2008 Appendix A Matlab Code A.1 GammaIR 1 function [t,ir, spectrum ]= gammair ( duration, fc, fs, flags ) 2 % Function t h a t g e n e r a t e s an Impulse Response corresponding to a gammatone 3 % a u d i t o r y f i l t e r o f order 4. 4 % The phase phi i s assumed to be zero 5 fftsize =fs /2; 6 n =4; 7 ERB = * fc; 8 b = 1.018* ERB ; 9 a = 6/( -2* pi *b) ^4; 10 t = [1/ fs :1/ fs: duration ]; 11 IR = a ^( -1) *t.^(n -1).*exp( -2* pi *b*t).* cos (2* pi *fc*t) ; % s u b p l o t ( 2, 1, 1 ) ; 14 % p l o t (1000 t, IR, f l a g s ) ; 15 % xlim ( [ ] ) ; 16 % y l a b e l ( Amplitude ) ; 17 % x l a b e l ( Time [ ms ] ) ; 18 % h o l d on ; f = fs/ fftsize *[0: fftsize /2-1]; 21 spectrum = 20* log10(abs( f f t (IR, fftsize ))); A.2 Gammatone filterbank 1 %A n a l y s i s o f t h e gammatone a u d i t o r y f i l t e r b a n k 2 clear a l l ; 3 close a l l ; 4 fs = 32000; 5 low_f = 100; 6 up_f = 8000; 7 Page A1

34 APPENDIX 8 % duration = fs * 125e -3; 10 chirp = genbmchirp ( low_f,up_f,fs); 11 delta = dpulse ( duration,1) ; 12 [out, cfs ] = gammafb ( delta, low_f,up_f,fs); 13 timefreqdistribution ( out, cfs, fs, [0 30]) 14 [out, cfs ] = gammafb ( chirp, low_f,up_f,fs); 15 timefreqdistribution ( out, cfs, fs, [0 30]) figure ; 18 t = [1/ fs :1/ fs :125e -3]; 19 plot (t *1000, chirp ); 20 xlim ([0 12]) ; 21 xlabel ( Time [ms] ); 22 ylabel ( Amplitude ); % Optional 25 figure ; 26 tdur =10e -3; % d u r a t i o n o f t h e c h i r p (10 ms l i k e t h e p r e v i o u s one ) 27 t = [1/ fs :1/ fs :125e -3]; % time v e c t o r 28 int_matrix = triu ( ones (length(t),length(t))); % Matrix in order to c a l c u l a t e t h e i n t e g r a l 29 f = min((1/0.108) *(3./(1.018*2* pi *( tdur -t)) -24.7),20000) ; % C a l c u l a t i o n o f f ( t ) 30 phase = 2* pi *(f* int_matrix ) *1/ fs; % C a l c u l a t i o n o f phase 31 chirp_opt = sin ( phase ); % C a l c u l a t i o n o f o p t i o n a l c h i r p 32 chirp_opt ( tdur *fs:length( chirp_opt )) =0; % Zero padding a f t e r 10 ms 33 [out, cfs ] = gammafb ( chirp_opt, low_f,up_f,fs); 34 timefreqdistribution ( out, cfs, fs, [0 30]) %P l o t t h e TimeFrequencyDistribution A.3 Chirp that compensates for the delay of the flters 1 close a l l ; 2 clear a l l ; 3 4 fs =32000; 5 tdur =10e -3; 6 7 t = [1/ fs :1/ fs :125e -3]; 8 int_matrix = triu ( ones (length(t),length(t))); 9 f = min((1/0.108) *(3./(1.018*2* pi *( tdur -t)) -24.7),20000) ; 10 phase = 2* pi *(f* int_matrix ) *1/ fs; 11 chirp2 = sin ( phase ); 12 chirp2 ( tdur *fs:length( chirp2 )) =0; 13 % p l o t ( t 1000, c h i r p 2 ) ; 14 % xlim ( [ 0 tdur 1000]) ; Page A2 Basilar membrane models