The Ear as an Analyzer of Instantaneous Frequency
Abstract Generated abstract
The paper examines how the human ear processes sound information, arguing that the auditory system may function primarily as an analyzer of instantaneous frequency rather than as a conventional spectral analyzer. Drawing on known properties of the cochlear basilar membrane and on evidence that clipped or compressed speech can remain intelligible despite removal of amplitude-envelope information, it emphasizes the informational role of phase-derived frequency variation. The author interprets the approximately linear relation between excitation position on the basilar membrane and sound frequency as analogous to an ideal differentiating element, so that variations in instantaneous frequency are isolated and transmitted by the auditory nerve. The proposed model also offers an explanation for the low quality factor of basilar membrane resonators, which would reduce transient distortions for frequency-varying signals while allowing only rough spectral analysis.
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THE EAR AS AN ANALYZER OF INSTANTANEOUS FREQUENCY
N. F. KUCHER
(Presented by the scientific seminar of the Department of Computer Engineering)
A large number of works are currently devoted to the problem of processing sound information by the human ear. A fundamental work in this area is Békésy’s monograph [1], devoted to the physical investigation of the human auditory apparatus. Among the basic parameters of the human auditory analyzer are the amplitude-frequency and coordinate-frequency characteristics of the basilar membrane of the cochlea, shown in Fig. 1.
Fig. 1. Amplitude-frequency and coordinate-frequency characteristics of the basilar membrane
As can be seen from Fig. 1, the basilar membrane is a set of resonators with low quality factor, arranged along the length of the cochlea; moreover, the distance from the stapes to the point of the membrane with the greatest amplitude of oscillation determines the frequency of the resonator. It is believed [2] that, with the aid of the basilar membrane, a person performs spectral analysis of the sound signal, while with the aid of neurons the quality factor of the membrane resonators is increased.
Let us represent the sound signal as a function of time
\[ u(t) = A(t)\cos\varphi(t), \]
where:
\(A(t)\) is the envelope of the amplitude of the sound signal,
\(\omega(t)=\varphi'(t)\) is the envelope of the instantaneous frequency of the sound signal.
In [3] and [4] it is shown that a speech signal does not lose intelligibility even when the envelope \(A(t)\) is destroyed, that is, the speech signal takes the form:
\[ u_1(t)=\operatorname{sign}\cos\varphi(t) \text{ — clipped speech;} \]
\[ u_2(t)=\cos\varphi(t) \text{ — compressed speech.} \]
Consequently, the predominant part of the information about a sound is contained in the variation of the function \(\omega(t)=\varphi'(t)\), that is, in the variation of the instantaneous frequency of the signal. The human auditory apparatus apparently analyzes first of all the instantaneous frequency of the sound signal \(u(t)\).
Fig. 2. Coordinate-frequency characteristic of the basilar membrane
If one looks at Fig. 1, it may be observed that the resonance maxima of the basilar membrane and the distance from the stapes to the point of the membrane with the greatest oscillation amplitude are related to one another by an approximately linear dependence. In [5] there is a graph, reproduced in Fig. 2, relating the distance \(X\) from the stapes to the point of the membrane with the greatest oscillation amplitude and the frequency of the exciting oscillations. This dependence is linear. Consequently, the transfer function of the ear at the level of the basilar membrane is \(k\omega=c\omega\), that is, it depends linearly on frequency. A four-terminal network with such a transfer function is an ideal differentiating element.
Therefore the signal \(u(t)=\cos\varphi(t)\) is analyzed by the ear in such a way that the variation of its instantaneous frequency is singled out:
\[ [u(t)]'=[\cos\varphi(t)]'=\omega(t)\sin\varphi(t). \]
The coordinates of the excitation points of the basilar membrane, corresponding to the instantaneous frequency of the exciting signal, are read by the auditory nerve and transmitted to the brain for further analysis.
The model of the auditory analyzer presented here also explains fairly well the low quality factor of the resonators of the basilar membrane, which is due to the fact that distortions caused by transient processes, when a signal of variable frequency acts on the cochlea, in a resonator with
with low \(Q\)-factor are minimal. Nevertheless, the coordinate of the point of greatest excitation of the basilar membrane is exactly proportional to the frequency of the exciting sound signal. Spectral analysis by resonators with low \(Q\)-factor, however, is carried out very roughly.
REFERENCES
- Békésy G. “Experiments in hearing,” New York, 1960.
- V. K. Labutin, A. P. Molchanov. Hearing and Signal Analysis, “Energiya,” Moscow–Leningrad, 1967.
- Yu. G. Rostovtsev. On the possibilities of using amplitude-limitation speech signals in communication systems. Elektrosvyaz’, No. 6, 1958.
- P. Marku, J. Dagg. New methods of speech transmission. In: Information Theory, ed. V. I. Siforov, IL, Moscow, 1957.
- H. Fletcher, “On the Dynamics of the Cochlea,” JASA, 1951, vol. 23, p. 637.