Yost Chapter 5: Sound analysis pp 53-62 [24 January 2001]

The important concept here is that of the "resonator:" a vibrating object that is vibrating at its natural frequency. This is the basis for the Fourier analysis done by the ear, and also forms the basis of the important concept of "filtering" which is also done by the ear: in fact it is the way that the ear accomplishes a Fourier analysis, the result of which is converted in neural signals at different places in the brain. Also in this chapter is a more extended discussion of linearity and non-linearity in auditory signals and receivers (particularly the ear).

The important concept running through this chapter is that of resonance, which describes the degree of affinity between two vibrating objects. The initial distinction that Yost makes is between the "free" vibration of chapter 2, which is produced by an impulse response, that is, an initial movement and then a relaxation of the driving force, and the "forced" vibration in which the driving force is maintained -- as in our vocal cords when we are talking or singing, for example. Yost says that this latter situation of a continuous driving force is the most common in the real world. Thinking about the effects of driving forces requires considering the relationship between the two vibrating objects -- the driving object and the driven object. The driving vibration is vibrating at some given frequency, but the driven object has a natural resonance to a particular frequency which will determine the effectiveness of the driving object. Resonance is determined by mass and by its stiffness and only if the two objects are matched in resonant frequency does the maximum transfer of energy occurs. If the resonance frequencies do not match then the transfer of energy is "impeded", and the concept describing this effect is called "impedance." The formula for the resonant frequency is given as equation 5.1:

The quantitative details of this formula are hard to imagine in the absence of realistic measures of mass and stiffness for interesting objects (such as the ear). Qualitatively, however, the meaning is clear: stiff materials have a high resonant frequency, massive materials have a low resonant frequency. If you look back to pages 21 and 34 you can see equations for free vibration that also show this same relationship, the square root of the quantity (stiffness/mass). If you go back to the basic definition of impedance in chapter 3, page 30, you see that mass and stiffness have opposite effects on impedance, as in the next formula.

The Xm and the Xs in this formula are called "reactance" and they depend on frequency. As frequency goes up then the Xm construct goes up, but the Xs construct goes down, and you could imagine finding a particular frequency at which the difference between X m and Xs was relatively small: this is the resonant frequency of this object.

Note also in this equation that the R squared part, which is a "resistance" thing such as friction or air pressure is not frequency dependent. But resistance has another interesting property. Back on page 19 Yost introduced the concept of "damping" which is the gradual loss of amplitude over time - damping depends on resistance: if there were no resistance, then an initial impulse, to a pendulum, say, would go on forever. Now here on page 53 Yost provides another effect of resistance. Note that the resonant frequency is really a "range of frequencies" which may be very restricted, or may be very broad. This extent of the range is a function of resistance -- if there is a lot of resistance, then the range is broad, and without any resistance the range is very small, and would show a sharp tip at the resonant frequency on the input-output curve for driving vs. driven energy.

But this situation describes an interesting tradeoff. If there were no resistance, then the response in the driven object would last indefinitely, which means that information about the time of arrival of the driving force would be lost, but you would know precisely what the frequency is. But if the resistance were large then the reverberation time would be small and so timing would be informative, but the information about frequency would be lost This is an important tradeoff in the auditory system. In general if the system is going to maximize frequency discrimination then it will do a bad job on temporal resolution, and vice versa. Because understanding speech requires an appreciation of both the frequency and the temporal domain, the nervous system has to somehow resolve this incompatibility: potentially you might imagine some part of the system particularly sensitive to frequency, another part particularly sensitive to timing.

Yost points back to Chapter 3 to recall that strings and tubes have resonance, for example, that there is resonance in a closed tube (best if the length of the tube is twice the wavelength of the driving frequency, and remember that the resonant frequency of a tube open at one end is the (speed of sound)/(4 tube lengths). This has implications for the sensitivity of our ears to sounds of different frequencies, and insensitivity to other frequencies. This insensitivity lead to the concept of "filters" -- in some sense, the opposite of the resonator, in that a filter is useful to the extent that it does not pass on certain frequencies -- namely, those frequencies that are off the resonant frequency

Commercial filters have four different actions: they can pass all of the frequencies below some cutoff - a low pass filter; or above some cutoff -- a high pass filter; or between two cutoffs -- a band pass filter; or everything except within two cutoffs -- a band reject or notch filter. (See Figure 5.2). But filters do not have sharp edges, though they may be more or less sharp -- so they may "roll-off" at 10 dB per octave; or 24 dB per octave, etc. (Look at Table 5.1 to see harmonics and octaves described.) They can be used to shape signals (as in Figure 5.4) or to measure signals, and do a rough Fourier analysis (as in Figure 5.3). The important idea here is that musical instruments and the human voice do Figure 5.4 sorts of things, and the ear does a Figure 5.3 kind of thing. Filters do something else though -- they change the relative phase of the input, and this is frequency dependent: low frequencies are delayed by the equivalent of the mass of the filter (because of inertia) more so than high frequencies. You could imagine that this has the potential to make it difficult for the nervous system to tell whether two sinusoids happened at the same time or not, if one is delayed by a phase shift.

The last section of this short chapter discusses in more detail the concepts of non-linearity. Filters are in general linear systems in which the input frequencies can be modified in phase or amplitude at the output, but new frequencies are not added. But many physical systems add frequencies, which are usually related to the input frequencies by simple multiples consisting of harmonics of the input (nF, where n is a simple integer, first, second, third harmonics etc.); or summation and difference tones (combination tones, f1 + f2, f1 - f2 etc.). This is called "distortion" and the question is whether the ear and the auditory system more generally produce distortion. Also, it has been suggested that there may be certain neural mechanisms that act to reduce distortion and so provide a clearer auditory image at the central auditory processors that is available at the periphery: this is an important issue in understanding the central mechanisms of auditory function, and has not yet been successfully analyzed. Distortion products like this are also used in a hearing test that uses "evoked acoustic emissions", which are produced by the outer hair cells on the basilar membrane. The emissions are frequency specific and one distortion production is relatively strong, as 2f1 — f2. So one finds an echo for f1 and f2 and also for (2f1 — f2), which provides information on 3 frequencies for one stimulus presentation.

There is one concept in the supplement to note, having to do with the frequency acuity of a filter, or its bandwidth. The cutoff frequencies are generally specified as the "half-power" points, which are 3 dB down, and so the issue is, how to describe the width of the filter between the two cutoffs. Comparing across filters it is usual to consider the bandwidth with respect to the magnitude of the center frequency. So a bandwidth of 100 Hz is not very acute if the center frequency is 100 Hz, but it may be very good if the center frequency is 1,000 Hz. The "quality" of a filter is identified as Q:

so in the first case Q = 100/100 = 1.0; and in the second, Q = 1000/100 = 10. What is the significance of this for audition? It is that auditory filters and auditory nerves can also be categorized by their quality, and under some conditions, such as hearing impairment and age, one can find that the filtering quality of the ear is affected. Linear filters that are very sharp have good frequency resolution but poor temporal resolution, and linear filters that are very broad have good temporal resolution. However, this inverse relationship is not found in the old or hearing impaired, indicating that the engineering analogies are not always successful.