Preface

This was never intended to be a book-long piece. My original wish upon starting to write it was to introduce into hearing the “paraxial” dispersion equation—perhaps in an article format. Naively, I was convinced that the beauty of the equation—or rather, the beauty in the analogy to vision—would be self-evident and immediately applicable. But as I was trying to come up with a way to introduce these ideas, it had quickly become clear that there is no straightforward way to motivate them at the present state of hearing science. At least, not without taking many steps back and venture into several scientific corners that have somehow escaped rigorous treatment. Wherever I looked, it had seemed as if some aspects of the science froze in time and have left me with an intuition about acoustic waves that would have been excellent at the turn of the previous century. It has taken considerably more research to be able to assort all the pieces that would be required to motivate the need for a dispersion equation in hearing. Some of these pieces were in plain sight, while some are in vogue in contemporary research, and yet others have been long forgotten. The price for picking up these pieces may be considered steep, though: let go of static frequencies and stationary signals, embrace coherence as a fundamental descriptor of sound, and learn to accept that the auditory system must know what it is doing much better than what we would like to think it should want to do.

Therefore, this is an invitation for the daring reader. I expect that a fair share of the topics that are touched upon will elicit resistance in some of the readers. Nevertheless, some of the topics—mainly in the introductory chapters—may appear timely and not nearly as controversial as the more advanced chapters. In more than one occasion I have resorted to indirect and unconfirmed methods that can associate the new ideas with an unwelcome, yet unavoidable, air of speculation. Also, the very broad scope of the present theory may simply be overwhelming, as there should be numerous points of contention that may be deemed worthy of rebuttal. This is all well, as far as I am concerned, as long as the ideas will be seriously considered and stir a long-needed discussion in the community.

My quest for a hearing theory has been initially triggered by a simmering dissatisfaction with the traditional presentation of auditory science. It had struck me as a bewildering collection of facts, which had to be learned by rote instead of through internalization of deeper concepts. Many results had to be empirically measured rather than derived based on higher-level principles. There did not seem to be a theory that is general enough to offer the necessary predictive power that may spare a direct experiment. Quite the contrary—predictions often seem to be proven wrong by experiment, which had made it difficult to develop an intuition for the inner workings of the ear and its inherent logic.

This is why stumbling across the temporal imaging theory in optics seemed like a possible key to my unrest. This theory is based on the space-time duality principle as formulated by Sergeĭ Aleksandrovich Akhmanov et al. (1968) and Akhmanov et al. (1969) and was later developed into a temporal imaging theory by Brian Kolner and Moshe Nazarathy (1989), and greatly elaborated by Kolner (1994a) and his subsequent work. Here was an elegant theory that is completely analogous to the classical single-lens imaging theory from optics, only with the dimensions reversed, as the temporal envelope is used instead of spatial envelope (expressing the distribution of light of the optical object and image). Knowing that vision rests on spatial imaging that is neatly formulated using the paraxial equation and a double Fourier transform, there was an immediate allure of having a paraxial equation and a double Fourier transform expressed in time and frequency coordinates that can, rather organically, represent hearing.

Alas, the basic elements in the temporal imaging theory are group-velocity dispersion, time-lens curvature, and aperture time. What do these concepts have to do with hearing? Everything, it seems. But the reasoning behind it, which is the main subject of this treatise, has taken me the better part of the last four years to arrive at. Apart from my own time-consuming ignorance of many of the associated disciplines—some of them are routinely alluded to in auditory research—there appeared to be several gaps in the fundamentals of acoustics and hearing science, which had to be retraced and patched up in order to be able to tackle the idea of imaging with a degree of rigor that I thought the topic deserves.

There were two principal “culprits” that underpin the gaps in auditory science. The first one is the over-reliance on pure tones—a mathematically degenerate signal with no curvature, which carries little-to-no information and is not encountered in nature. This deficiency is addressed in several chapters that adopt the complex envelope and constant carrier formulation as the most general representation of waves, signals, objects, images, and communication functions. In turn, it opens the door for unifying the auditory concepts of temporal envelope and temporal fine structure with mathematically related concepts in acoustics, optics, and communication.

The second “culprit” is acoustic coherence theory, or the lack thereof. As a scalar wave theory, linear acoustics of plane waves is completely compatible with scalar wave theory in optics, which is also where classical coherence theory was developed. The main developments in coherence theory gathered momentum in the 1950s, at a point in which acoustics and optics may have been practiced by different scholars. Acoustics, and hearing too, imported a mélange of coherence-related concepts from several disciplines—each with its own jargon—that hardly coalesce into a consistent understanding. The two chapters about hearing-relevant coherence theory, while not adding anything new to the science, are a first attempt to unify and revive these ideas in a manner that is consistent with wave physics, room acoustics, hearing (phase locking), communication engineering, and neuroscience. Optical coherence theory provides a bridge that can be applied in sound, using Fourier optics, alongside some of the most insightful tools from imaging. The proposed amalgamation of the different coherence theories attempt to connect concepts of coherence with synchronization that manifest both in the mechanical and in the neural parts of the auditory system, and is thought to generally characterize perception throughout the brain.

With the availability of these introductory chapters, the motivation for the temporal imaging theory should be in place. I have put substantial effort in exploring some of the potential implications of temporal imaging—temporal modulation transfer functions, aberrations, accommodation, and dispersive hearing impairments. Thus, it is my hope that interested readers will be able to follow the wildly different approach to hearing that is presented in the advanced chapters of this work, despite the effort that it may require. While I cannot foresee the correctness of some of the hypotheses put forth, I will feel greatly rewarded to know that these ideas will have influenced future researchers in solving some of the more persistent challenges in our understanding of hearing and hearing impairments.

References

Akhmanov, SA, Chirkin, AS, Drabovich, KN, Kovrigin, AI, Khokhlov, RV, and Sukhorukov, AP. Nonstationary nonlinear optical effects and ultrashort light pulse formation. IEEE Journal of Quantum Electronics, QE-4 (10): 598–605, 1968.

Akhmanov, SA, Sukhorukov, AP, and Chirkin, AS. Nonstationary phenomena and space-time analogy in nonlinear physics. Soviet Physics JETP, 28 (4): 748–757, 1969.

Kolner, Brian H. Space-time duality and the theory of temporal imaging. IEEE Journal of Quantum Electronics, 30 (8): 1951–1963, 1994a.