Problem Set 3.2  Spring 2009
Computer-Based Model of the Conductive Auditory System:
Sound Transmission/Transfer through the Ear

A. Introduction

This exercise involves creating a computer-based model of the conductive auditory system as a first step in studying sound transfer through the auditory system. To do so, the systems analysis approach will be used, as detailed in Module 3.  Students should turn in a completed Excel spreadsheet, a printout of each of the Excel charts requested, as detailed below, and a brief answer to the questions at the end of the assignment.

B.

1. The following diagram is a simple model of the transformation (gain and loss) of an acoustic signal through the auditory system of a normal-hearing human. As labeled, the model includes five basic elements: an input sound source (I), the external ear system (E), the middle ear system (M), the cochlea (C), and the total output (O).
   

   

   Figure PS3.2.1
   Each separate element as a function of frequency, f. The functions themselves are continuous, although each system may be evaluated at a specific frequency, f.

   Question 1: Consider the entire series of systems. Using this functional notation, write an equation representing the output of the entire system as a composite function of its input. A general example of the form of this equation is: A(B(C(x))).

2. Using (1) as an outline and the Excel template PS32TEMP.xls, let's create a computer-based model of the auditory system that allows an examination of the external ear and middle ear subsystems. The goal here is to combine the amplitude responses or transfer functions of multiple systems, as we did in Section 3.2.  For simplicity, the phase response will be considered to be linear and will be ignored.  Thus, the terms amplitude response and transfer function will be used synonymously, and changes in the amplitude of input signals will be discussed in terms of the "gain" of the system.  Recall that when representing the amplitude response of a system in terms of gain, a transfer function can have positive gain (amplification), negative gain (attenuation), or zero gain.

   The necessary transfer functions are provided in the Excel template. Column B of the spreadsheet is labeled "FREQ (Hz)" and ranges from 250 to 8000 Hz in 256 logarithmic steps. Column C labeled "EETF (lin)" represents the external ear transfer function. Here, the gain of the system is in linear units of sound pressure. Column L labeled "METF (lin)" represents the middle ear transfer function, and the gain is also in linear units. Following the steps outlined below, represent each of those transfer functions graphically in Excel charts.  This hopefully will lead to a better appreciation of the contribution of each system to the transmission of sound from the outside environment into the cochlea.

   Assume that each of the systems above is a separate LTI subsystem. In addition:

   • Assume that the auditory nerve (AN) and central nervous system (CNS) do not produce gain (+ or -) over the audible spectrum, so we can treat the output of the cochlea as the audible signal (this is not entirely accurate, but it is a reasonable simplification for our purposes).  A signal that is just barely audible is at a level known as absolute threshold.  If we determine absolute threshold over a range of different sound frequencies, the resulting transfer function describes the sensitivity of the auditory system.
     
   • Assume that there is no gain (+ or -) between the sound source and the entrance to the external ear.
     
   • Note that the ear canal transfer function is taken from Shaw (1974) and is measured as the change in sound pressure from a point near the entrance to the ear canal to a point very close to the eardrum. This function includes the influence of the ear canal, concha, and pinna-flange, as well as the effects of sound diffraction from the head, neck, and torso. The middle ear transfer function is taken from Kurokawa and Goode (1995) and is measured as the change in amplitude of vibration from the eardrum to the oval window. This function includes the influence of the tympanic membrane, the middle ear cavity and air spaces, the eustachian tube, the ossicles, and the middle ear muscles, ligaments, and tendons. It does not take into account the impedance of the cochlea.
   • In this homework assignment, the various system transfer functions and their influence on any possible signal will be considered in general terms.  The output of the system for specific input signals will not be measured.
   1. Create an Excel chart representing the gain of the external ear transfer function in linear units as a function of frequency. The x axis or abscissa should represent frequency on a linear scale. Place the chart between columns E and K.

   2. Create a second chart of the external ear transfer function with gain in linear units. In this case, the x axis or abscissa should represent frequency on a logarithmic scale from 100 to 10,000 Hz. To change the axis from linear to logarithmic scale, use the mouse to click on the axis and change the scale using selections in the pop-up menu. Place this chart below the previous one.

      NOTE: From this point on, each chart should have a logarithmic frequency scale on the abscissa.

   3. Convert the external ear transfer function from linear gain to gain in dB (use column D). Since the function represents sound pressure, use the 20 log equation. Create a chart representing this function in dB and place it between the charts in steps (a) and (b) above.

   4. Create a chart of the middle ear transfer function with gain in linear units. Place this chart to the right of column L.

   5. Create a chart of the middle ear transfer function with logarithmic gain (dB) and place the chart below the chart in step (d) above.

   6. Combine the ear canal and middle ear transfer functions in both linear and decibel units (use columns S and T). Plot the combined functions and place them to the right of column T with the linear-gain plot on top.

    

3. Print each of the six charts with log axes and arrange them in front of you from left to right. Note that the charts represent each of the systems shown in Figure PS3.2.1. Provide brief answers to the following questions on a separate sheet. You may rely on your intuition, text readings, and information presented in lectures or lecture notes.
   1. Describe the external ear transfer function shown in 2(c). What is the resonant frequency of the transfer function? Provide a possible explanation for the transfer function shape.

   2. Describe the difference between the graphs with linear gain and decibel gain.  Why would decibel gain sometimes be preferred?

   3. Why would a logarithmic abscissa be preferable to a linear abscissa in these charts?

   4. Describe the middle ear transfer function shown in 2(e) and a possible explanation for its shape. 

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References

Kurokawa, H., and Goode, R. L. (1995). "Sound pressure gain produced by the human middle ear," Otolaryngology - Head and Neck Surgery, 113, 349 – 355.

Shaw, E. A. G. (1974). "Transformation of sound pressure level from the free field to the eardrum in the horizontal plane," Journal of the Acoustical Society of America, 56, 1848 – 1861.