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Education in Twins and Their Parents Across Birth Cohorts Over 100 years: An Individual-Level Pooled Analysis of 42-Twin Cohorts
- Karri Silventoinen, Aline Jelenkovic, Antti Latvala, Reijo Sund, Yoshie Yokoyama, Vilhelmina Ullemar, Catarina Almqvist, Catherine A. Derom, Robert F. Vlietinck, Ruth J. F. Loos, Christian Kandler, Chika Honda, Fujio Inui, Yoshinori Iwatani, Mikio Watanabe, Esther Rebato, Maria A. Stazi, Corrado Fagnani, Sonia Brescianini, Yoon-Mi Hur, Hoe-Uk Jeong, Tessa L. Cutler, John L. Hopper, Andreas Busjahn, Kimberly J. Saudino, Fuling Ji, Feng Ning, Zengchang Pang, Richard J. Rose, Markku Koskenvuo, Kauko Heikkilä, Wendy Cozen, Amie E. Hwang, Thomas M. Mack, Sisira H. Siribaddana, Matthew Hotopf, Athula Sumathipala, Fruhling Rijsdijk, Joohon Sung, Jina Kim, Jooyeon Lee, Sooji Lee, Tracy L. Nelson, Keith E. Whitfield, Qihua Tan, Dongfeng Zhang, Clare H. Llewellyn, Abigail Fisher, S. Alexandra Burt, Kelly L. Klump, Ariel Knafo-Noam, David Mankuta, Lior Abramson, Sarah E. Medland, Nicholas G. Martin, Grant W. Montgomery, Patrik K. E. Magnusson, Nancy L. Pedersen, Anna K. Dahl Aslan, Robin P. Corley, Brooke M. Huibregtse, Sevgi Y. Öncel, Fazil Aliev, Robert F. Krueger, Matt McGue, Shandell Pahlen, Gonneke Willemsen, Meike Bartels, Catharina E. M. van Beijsterveldt, Judy L. Silberg, Lindon J. Eaves, Hermine H. Maes, Jennifer R. Harris, Ingunn Brandt, Thomas S. Nilsen, Finn Rasmussen, Per Tynelius, Laura A. Baker, Catherine Tuvblad, Juan R. Ordoñana, Juan F. Sánchez-Romera, Lucia Colodro-Conde, Margaret Gatz, David A. Butler, Paul Lichtenstein, Jack H. Goldberg, K. Paige Harden, Elliot M. Tucker-Drob, Glen E. Duncan, Dedra Buchwald, Adam D. Tarnoki, David L. Tarnoki, Carol E. Franz, William S. Kremen, Michael J. Lyons, José A. Maia, Duarte L. Freitas, Eric Turkheimer, Thorkild I. A. Sørensen, Dorret I. Boomsma, Jaakko Kaprio
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- Twin Research and Human Genetics / Volume 20 / Issue 5 / October 2017
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- 04 October 2017, pp. 395-405
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Whether monozygotic (MZ) and dizygotic (DZ) twins differ from each other in a variety of phenotypes is important for genetic twin modeling and for inferences made from twin studies in general. We analyzed whether there were differences in individual, maternal and paternal education between MZ and DZ twins in a large pooled dataset. Information was gathered on individual education for 218,362 adult twins from 27 twin cohorts (53% females; 39% MZ twins), and on maternal and paternal education for 147,315 and 143,056 twins respectively, from 28 twin cohorts (52% females; 38% MZ twins). Together, we had information on individual or parental education from 42 twin cohorts representing 19 countries. The original education classifications were transformed to education years and analyzed using linear regression models. Overall, MZ males had 0.26 (95% CI [0.21, 0.31]) years and MZ females 0.17 (95% CI [0.12, 0.21]) years longer education than DZ twins. The zygosity difference became smaller in more recent birth cohorts for both males and females. Parental education was somewhat longer for fathers of DZ twins in cohorts born in 1990–1999 (0.16 years, 95% CI [0.08, 0.25]) and 2000 or later (0.11 years, 95% CI [0.00, 0.22]), compared with fathers of MZ twins. The results show that the years of both individual and parental education are largely similar in MZ and DZ twins. We suggest that the socio-economic differences between MZ and DZ twins are so small that inferences based upon genetic modeling of twin data are not affected.
21 - The Auditory Image
- from Part IV - The Auditory Nervous System
- Richard F. Lyon
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Author Index
- Richard F. Lyon
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9 - Gammatone and Related Filters
- from Part II - Systems Theory for Hearing
- Richard F. Lyon
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Part IV - The Auditory Nervous System
- Richard F. Lyon
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Part IV Dedication: J. C. R. Licklider
This part is dedicated to the memory of Joseph Carl Robnett Licklider (1915–1990). “Lick” is best known as one of the “fathers of the Internet” (Poole et al., 2005), based on his ARPA leadership and his writings such as “Man–Computer Symbiosis” and “The Computer as a Communication Device.”
But before he was a computer network and systems guy, Lick was an auditory psychologist and modeler (November, 2012). His work on pitch perception, as represented in the “duplex theory,” is the basis for much recent work in hearing, including my own, connecting the output of the cochlea to perception and neural processing of complex sounds.
I had the pleasure of meeting Lick just once, in 1984 at a Navy-sponsored workshop on “Artificial Intelligence and Bionics.” I think he was a little surprised to see his duplex theory coming back as a practical computational approach, three decades after he came up with it. It has become even more practical since then, thanks partly to his computer innovations.
In this part, we discuss the levels of processing in the auditory nervous system. We develop the idea of auditory images, of the sort that are thought to be extracted by brainstem and midbrain for projection to auditory cortex.
We start where the last part left off, with the “cable” for the telephone theory of hearing, the auditory nerve, which transmits the vibrations as detected by hair cells in the cochlea to the first stop in the brainstem, the cochlear nucleus.
Several kinds of processing in the cochlear nucleus support both binaural hearing and the extraction of properties such as pitch and timbre that can be monoaural or binaural. We cover the extraction of such properties into the stabilized auditory image, a basis for sound representation in machine hearing systems as well as a model of representations in the inferior colliculus of the midbrain. We cover binaural spatial processing and the brainstem's olivary complex. We finish this part with a discussion of auditory scene analysis as the main aim of the auditory brain, and some ideas for how such analysis might be done in the thalamus and cortex to finally “extract meaning.”
27 - Musical Melody Matching
- from Part V - Learning and Applications
- Richard F. Lyon
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I hope my critics will excuse me if I conclude from the opposite nature of their objections that I have struck out nearly the right path. As to my Theory of Consonance, I must claim it to be a mere systematisation of observed facts (with the exception of the functions of the cochlea of the ear, which is moreover an hypothesis that may be entirely dispensed with). But I consider it a mistake to make the Theory of Consonance the essential foundation of the Theory of Music, and, I had thought that this opinion was clearly enough expressed in my book. The essential basis of Music is Melody.
—On the Sensations of Tone, Hermann Ludwig F. Helmholtz (1870)This chapter draws on material from the 2012 paper “The Intervalgram: An Audio Feature for Large-scale Melody Recognition” by Thomas C. Walters, David A. Ross, and Richard F. Lyon (Walters et al., 2013).
In this chapter, we review a system for representing the melodic content of short pieces of audio using a novel chroma-based representation known as the “intervalgram,” which is a summary of the local pattern of musical intervals in a segment of music. We introduced chroma as pitch within an octave in Section 4.7. The intervalgram is based on a chroma representation derived from the pitchogram, or temporal profile of the stabilized auditory image. Each intervalgram frame is made locally key invariant by means of a “soft” pitch transposition to a local reference. Intervalgrams are generated for a piece of music using multiple overlapping windows. These sets of intervalgrams are used as the basis of a system for detection of identical melodies across a database of music. Using a dynamic-programming-like approach for comparisons between a reference and the song database, performance was evaluated on the dataset. A first test of an intervalgram-based system on this dataset yields a precision at top-1 of 53.8%, with a precision–recall curve that shows very high precision up to moderate recall, suggesting that the intervalgram is adept at identifying the easier-to-match cover songs in the dataset with high robustness. The intervalgram is designed to support locality-sensitive hashing, such that an index lookup from each single intervalgram feature has a moderate probability of retrieving a match, with relatively few false matches.
Part II - Systems Theory for Hearing
- Richard F. Lyon
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Part II Dedication: Charlie Molnar
This part is dedicated to the memory of Charles E. Molnar (1935–1996). Charlie is mostly known outside the hearing field for his invention, with Wesley A. Clark, of the LINC—the “laboratory instrument computer”—which was according to many the first personal computer; and for his work in asynchronous and self-timed computer circuits, which is what he was working on when he died in 1996. He was a super generous guy, always willing to discuss and advise, and our talks about hearing and circuit design were very important to me. His “system of nonlinear differential equations modeling basilar-membrane motion” (Kim, Molnar, and Pfeiffer, 1973) was probably the first example of a great way to integrate nonlinearity into a filter-cascade model of the cochlea.
In this part, we develop the mathematical and engineering basics needed to model the ear.
We start with a review of linear systems theory, the body of knowledge that allows the design and construction of efficient and flexible filters. Even for readers very familiar with linear systems theory, a reading of this chapter should be a useful refresher and an introduction to our terminology and approach.
After a chapter on the discrete-time version of linear system theory, we apply the theory to resonant filters and elaborated resonant filters such as the gammatone family. Then we extend into nonlinear systems, with a whole chapter on automatic gain control.
Finally we discuss wave propagation in distributed systems, and how we model it with linear systems of the sort that will lead to good machine models.
20 - Auditory Nerve and Cochlear Nucleus
- from Part IV - The Auditory Nervous System
- Richard F. Lyon
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I experimented in this way, and eventually found that I could send as many as 352 impulses per second along the nerve of a rabbit and get a note from the muscle of the pitch of 352 vibrations per second … but when I tried by more rapid stimulation of the nerve to get a higher note from the muscle, I failed. … Now, am I to conclude that, because I failed to get a higher note than one of 352 vibrations from the muscle, it is not possible to send more than 352 vibrations per second along a nerve? By no means …
—“A lecture on the sense of hearing,” Rutherford (1887)The auditory nerve (AN), originating in the spiral ganglion in the cochlea, carries the output signals of the cochlea's IHCs into the same-side (ipsilateral) cochlear nucleus (CN) in the brainstem, just a few centimeters away, as shown in Figure 20.1. To a large extent, the inputs and outputs of the CN tell us what information the brain is getting from the ear, and what the important first steps are in processing that information.
Observations of physiological behavior of the CN support our emphasis on the use of fine temporal structure in sound representation, as in the Fletcher (1930) “space– time pattern theory” and the Wever and Bray (1930b) “volley theory” that we reviewed in Chapter 2. The CN's inputs and outputs are events, synchronized to sound waveform structure in a very precise and robust way in some parts of the CN, but less synchronized in other parts, serving different subsequent processing pathways.
In the CN, pathways diverge in support of a number of processing functions in parallel, variously specialized for binaural processing, for periodicity detection, and for other monaural feature extraction. The tonotopic organization of the auditory nerve is maintained as a spatial dimension, and various subsequent brain areas use another spatial dimension to organize the extracted features into what we call auditory images, of various sorts.
2 - Theories of Hearing
- from Part I - Sound Analysis and Representation Overview
- Richard F. Lyon
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In respect of the theory of hearing, it seems to me that we need fewer theories and more theorizing. Of theories, focused upon some new finding and seeking to align the entire body of auditory fact with the new principle, we have more than a plenty.
—“Auditory theory with special reference to intensity, volume, and localization,” Edwin G. Boring (1926)The principle of diversity suggests that a simple description of the auditory process may not be possible because the process may not be simple. Theories that appear at first thought to be alternatives may in fact supplement one another.
—“Place mechanisms of auditory frequency analysis,” William H. Huggins and Licklider (1951)Many theories and models have influenced thinking in this field; here we survey some of these, including those modern theories on which we base machine hearing systems.
A “New” Theory of Hearing
Books and papers entitled “A New Theory of Hearing” or something to that effect were once almost commonplace (Rutherford, 1887; Hurst, 1895; Ewald, 1899; Meyer, 1899; Békésy, 1928; Fletcher, 1930; Wever and Bray, 1930b; Wever, 1949). Like many ideas from a few generations back, some of these theories seem a bit quaint from our modern perspective. But in many cases they really did represent some of the most insightful scientific thinking and freshest experimental observations of their times. We review some of these ideas here, emphasizing those that left a lasting mark on our thinking about how hearing works.
Hermann von Helmholtz's Tonempfindungen (Helmholtz, 1863) presented the first major influential theory of hearing. His theory that structures in the cochlea vibrate sympathetically, each place resonating with its own narrow range of frequencies to stimulate a specific nerve, was the foundation for the long-lasting concept of the ear as a frequency analyzer. The arrangement of nerves in the cochlea was associated with individual just-distinguishable tone frequencies, adapting Müller's doctrine of specific nerve energies (Müller, 1838) and applying Fourier's finding that any periodic signal is equal to a sum of sinusoids of harmonically related frequencies (Fourier, 1822).
Part I - Sound Analysis and Representation Overview
- Richard F. Lyon
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Part I Dedication: John Pierce
This part is dedicated to the memory of John Robinson Pierce (1910–2002). John was a dear friend and mentor for many years, beginning in my undergraduate years at Caltech. He gave me a summer job doing lab work on electronic musical instruments, and then on digital codecs that led to my first journal article. He persuaded his colleagues at Bell Labs to take me on as an intern, even after they had objected to my “less than an A in some important subjects.” I owe my knowledge of digital signal processing to this great start with the early researchers and practitioners there. Pierce's work with George Zweig and Richard Lipes at Caltech, after I had left, became one of the most important influences on my thinking in hearing: the wave analysis that led to my filter-cascade approach to modeling the cochlea (Zweig, Lipes, and Pierce, 1976).
Pierce was better known for his work outside of hearing: from his early work in traveling-wave tubes and communication satellites at Bell Labs, his coining of the word transistor, his chief technologist role at the Jet Propulsion Laboratory, his science fiction writing under the pen name J. J. Coupling, through his enormous influence on computer music starting at Bell Labs and continuing at Stanford's Center for Computer Research in Music and Acoustics (CCRMA) in the 1980s and 1990s. His regular attendance at CCRMA's weekly hearing seminar provided a huge benefit to many of us in the hearing field. He continued to conduct and publish hearing research at Stanford even in his 80s, for example providing clarity on important issues in pitch perception (Pierce, 1991).
In Part I, we survey our concept of what the machine hearing field is, and how it relates to conventional acoustic approaches to sound processing and to a range of theories of hearing. We include a brief overview of human hearing from the conventional psychoacoustics and physiology points of view, which provide the data and some of the models that we build on.
19 - The AGC Loop Filter
- from Part III - The Auditory Periphery
- Richard F. Lyon
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… the output (BM displacement or velocity) varies much less than the stapes input displacement or velocity, for frequencies near the best frequency. The significance of this important finding will become clearer as we proceed, but, in my opinion, it is a precursor to an automatic gain control system which seems to be built into the cochlear filters.
—“Cochlear modeling—1980,” Jont B. Allen (1981)The CARFAC's AGC Loop
Allen (1979, 1981) was among the first to describe cochlear mechanics (as opposed to neural response) as having an automatic gain control (AGC) functionality, based on modeling Rhode's observations of nonlinear mechanical response (Rhode, 1971). Kim (1984) gets much more explicit, defining the roles of the inner- and outer-hair-cell subsystems, and of the medial olivo-cochlear (MOC) efferents in the cochlea's integrated nonlinear system:
The function of the large medial OC neurons is to exert a gain control upon the biomechanics of the organ of Corti by reducing the amount of mechanical energy released from the OHCs via a synaptically mediated regulation of the membrane potential and conductance of the OHCs.
Unlike Allen's concept described in the chapter quote above, my original coupled AGC (Lyon, 1982) was not “built into the cochlear filters,” but it was otherwise not far from the concepts of Allen and Kim and our present models. That is, it was a multiplicative coupled multichannel gain control that followed a linear filterbank, rather than achieving gain variation by varying the filter damping factors.
In this chapter, we describe how the CARFAC models the MOC feedback, and more local feedback, as an AGC loop filter feeding the modeled IHCs (detectors) back to OHCs (gain effectors) described in the previous two chapters.
Compared to the simple AGC model developed in Chapter 11, the AGC loop filter in the CARFAC has several aspects that make it interesting. First, it has four one-pole smoothing filters with their outputs combined, to span a range of time constants instead of having a single characteristic time constant or corner frequency. Second, each of these four filter stages has coupling between neighboring channels—and, in a binaural or multimicrophone version, also between the two or more ears.
6 - Introduction to Linear Systems
- from Part II - Systems Theory for Hearing
- Richard F. Lyon
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The fact that representation of waveforms as a sum of sine waves is useful in the elucidation of human hearing indicates that something involved in hearing is linear or nearly linear.
—“The nature of musical sound,” John R. Pierce (1999)A basic understanding of linear systems is fundamental to understanding the nature of sound and hearing, including sound sources such as speech and music, sound propagation and mixing, and sound analysis in the inner ear. An understanding of linear systems is also a base on which to build an understanding of important nonlinear aspects of hearing.
In this chapter, we attempt to bring the novice and the expert to a common level of shared understanding about linear systems. The terminology and understanding developed here will support the material in subsequent chapters.
The concepts that we discuss include filters, circuits, impulse responses, frequency responses, convolution, transfer functions (including magnitude, phase, and delay), poles and zeros, transforms, time and frequency domains, eigenfunctions, sinusoids, and complex exponentials. The typical electrical engineering undergraduate education covers all these concepts, but too often some of the important connections between them are neglected. The typical bioengineering or hearing science education may only touch on half of the concepts, sometimes omitting Laplace and Z transforms and the important concept of poles and zeros that we need for our cochlear models. Depending on the level at which one wants to work in human or machine hearing, a deeper understanding of this area can be very helpful.
Linear systems are often encountered in the study of electrical, mechanical, acoustic, and even quantum systems, as well as in the processing of nonphysical data sequences such as stock prices. In a modern engineering or computer science curriculum, linear systems will likely be taught using discrete-time data sequences, processed by digital filters in computer programs. A more traditional approach, at least in electrical engineering (EE), is to teach linear systems using electrical circuits. This approach connects more directly with the continuous-time nature of sound waves and sound processing in the mechanical structures of the ear, so that's where we start.
26 - Sound Search
- from Part V - Learning and Applications
- Richard F. Lyon
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This task aims at identifying the pictures relevant to a few word query, within a large picture collection. Solving such a problem is of particular interest from a user perspective since most people are used to efficiently access large textual corpora through text querying and would like to benefit from a similar interface to search collections of pictures.
—“A discriminative kernel-based model to rank images from text queries,” Grangier and Bengio (2008)This chapter is adapted from “Sound retrieval and ranking using auditory sparse-code representations” by Richard F. Lyon, Martin Rehn, Samy Bengio, Thomas C. Walters, and Gal Chechik (Lyon et al., 2010b).
Our first-reported large-scale application of the machine hearing approach is a sound search system (Lyon et al., 2010b) based directly on the PAMIR image search system described by Grangier and Bengio (2008). These are a form of “document ranking and retrieval from text queries,” for image and sound documents.
While considerable effort has been devoted to speech and music recognition and indexing, the wide range of sounds that people—and machines—may encounter in their everyday life has been far less studied. Such sounds cover a wide variety of objects, actions, events, and communications: from natural ambient sounds, through animal and human vocalizations, to artificial sounds that are abundant in today's environment.
Building an artificial system that processes and classifies many types of sounds poses two major challenges. First, we need to develop efficient algorithms that can learn to classify or rank a large set of different sound categories. Recent developments in machine learning, and particularly progress in large-scale methods (Bottou et al., 2007), provide several efficient algorithms for this task. Second, and sometimes more challenging, we need to develop a representation of sounds that captures the full range of auditory features that humans use to discriminate and identify different sounds, so that machines have a chance to do so as well. Unfortunately, our current understanding of how the plethora of naturally encountered sounds should be represented is still very limited.
15 - The CARFAC Digital Cochlear Model
- from Part III - The Auditory Periphery
- Richard F. Lyon
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The modified transmission-line implementation, like the low-pass filter version, is an active system, with adjustments to the filter Q values changing the filter shapes and gains. … This functional variation of Q with level gives a nearly uniform 2.5:1 compression ratio in the cochlear output for inputs ranging from 0 to 100 dB SPL.
—“Accurate Tuning Curves In a Cochlear Model,” James Kates (1993a)The multiple-output cascade of asymmetric resonators with fast-acting compression (CARFAC) model of the auditory periphery combines concepts from many of the previous chapters, toward the primary goal of providing an efficient sound analyzer to support machine hearing applications.
An important secondary goal of CARFAC is to connect closely enough with known auditory physiology and psychophysics that it can be used to visualize and explain many interesting auditory phenomena. It is not a goal to be physically accurate, well calibrated, or in agreement with every detail of peripheral auditory processing, though it can be used as a starting point for those who have such goals.
Other than my own recent work (Lyon et al., 2010b; Lyon, 2010; Lyon et al., 2010a; Lyon, 2011b,a), the closest digital cochlear models in the literature to CARFAC are the variable-Q cascade–parallel models described by Kates (1993a), which also used cascades of asymmetric resonators specified by their poles and zeros; see the opening chapter quote above. The difference is that his stages include a second filter at each tap, and use a rather different pole–zero pattern, motivated by matching iso-rate neural tuning curves; ours is motivated by matching both psychophysical filters and physiological impulse responses, as discussed in Chapter 13.
Putting the Pieces Together
The CARFAC cochlear model pulls together the bits of knowledge that we surveyed in the previous nine chapters.
The coupled-form asymmetric resonators from Chapter 8 are combined into a filterbank with gammatone-like response as described in Chapter 9, using the cascade architecture motivated by Chapter 12.
11 - Automatic Gain Control
- from Part II - Systems Theory for Hearing
- Richard F. Lyon
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In recent years, devices for the automatic control of gain have increased in importance in various areas of amplifier technology. One class of such devices is based on the following principle: a portion of the output signal current of a valve amplifier is extracted, amplified and fed to a rectifier; the resulting rectified signal voltage is then used to vary the grid voltage of an amplifier valve. In this manner an increase in output power leads to a reduction in gain.
—“On the Dynamics of Automatic Gain Controllers,” Karl Küpfmüller (1928)I have long viewed the automatic gain control (AGC) function as one of the most important, and tricky, parts of modeling the function of the cochlea (Lyon, 1982, 1990). To understand or design this important level-adaptive function, one must have an appreciation for the dynamics of feedback control, in a highly variable nonlinear context.
In this chapter, I provide the basic background and analysis techniques that our cochlear models will draw on. In particular, I show how the use of output amplitude to control the damping factors of cascaded resonators can be modeled as a robust feedback control system that compresses a wide input dynamic range into a narrower output dynamic range, by examining this approach in the context of a fairly general single-channel AGC formulation.
Input–Output Level Compression
Systems that use feedback from a detected output level to adjust their own parameters to keep the output level from varying too much are called automatic gain control systems. Such systems are inherently nonlinear, with a compressive input–output function: when the input changes by some factor, the output level changes by a factor closer to 1. AGC has long been used and analyzed in wireless communication systems (Wheeler, 1928; Küpfmüller, 1928), including television (De Forest, 1942), and is an idea that has long inspired corresponding models in biological systems, including vision and hearing (Rose, 1948, 1973; Smith and Zwislocki, 1975; Allen, 1979).
1 - Introduction
- from Part I - Sound Analysis and Representation Overview
- Richard F. Lyon
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… things inanimate have mov'd,
And, as with living Souls, have been inform'd,
By Magick Numbers and persuasive Sound.
—William Congreve (1697) The Mourning BrideThe ear is a most complex and beautiful organ. It is the most perfect
acoustic, or hearing instrument, with which we are acquainted,
and the ingenuity and skill of man would be in vain exercised to imitate it.
—John Frost (1838), The Class Book of Nature: Comprising Lessons on the Universe, the Three Kingdoms of Nature, and the Form and Structure of the Human BodyWould it truly be in vain to exercise our ingenuity to imitate the ear? It would have been, in the 1800s—but now we are beginning to do so, using the “magick” of numbers. Machines imitating the ear already perform useful services for us: answering our queries, telling us what music is playing, locating gunshots, and more. By imitating ears more faithfully, we will be able to make machines hear even better. The goal of this book is to teach readers how to do so.
Understanding how humans hear is the primary strategy in designing machines that hear. Like the study of vision, the study of human hearing is ancient, and has enjoyed impressive advances in the last few centuries. The idea of machines that can see and hear also dates back more than a century, though the computational power to build such machines has become available only in recent decades. It is now, as they say in the computer business, a simple matter of programming. Well, not quite—there is still work to be done to firm up our understanding of sound analysis in the ear, and yet more to be done to understand the enormous capabilities of the human brain, and to abstract these understandings to better support machine hearing. So let's get started.
Humans tend to take hearing for granted. We are so aware of what's going on around us, largely by extracting information from sound, yet so unable to describe or appreciate how we do it. Can we make machines do as well at interpreting their world, and ours, through sound? We can, if we leverage scientific knowledge of how humans process sound.
Frontmatter
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Subject Index
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12 - Waves in Distributed Systems
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- Richard F. Lyon
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The movement of waves down the basilar membrane is analogous to the propagation of light waves in a medium of continuously changing index of refraction. While the velocity of light varies as it travels through the substance, substantial reflections will not occur as long as the index of refraction changes slowly enough.
—“The cochlear compromise,” Zweig, Lipes, and Pierce (1976)The cochlea is not a system of lumped elements, but rather a distributed system. Distributed systems can be linear or nonlinear; before we tackle the cochlea, we need to study the simpler linear case. Linear system theory is still applicable to distributed systems, as it is to systems of lumped elements—but it gets a bit more complicated because the transfer functions are not as simple as ratios of polynomials, and because signals are functions of location, not just of time.
The spatially distributed state of a distributed system is typically described as a wave, a function of continuous time and space, rather than in terms of inputs and outputs. We can conceptually look at the system response (transfer function or impulse response or frequency response) at an infinite number of outputs at a continuum of locations, or at a finite set of outputs at discrete locations, with the wave at some unique location as input.
Systems of lumped elements, having a finite number of degrees of freedom, are described by ordinary differential equations that relate the rate of change of each state variable to the state and inputs of the system. In a distributed system, the motion (displacement, velocity, pressure, current, voltage, or whatever) of every point along a continuum of one or more dimensions is part of the state, so such systems are described by partial differential equations, in which rates of change with respect to both time and space are involved. In this chapter, we do not involve ourselves much with partial differential equations directly, but rather discuss what their solutions tell us about waves.
Plate section
- Richard F. Lyon
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- Book:
- Human and Machine Hearing
- Published online:
- 28 April 2017
- Print publication:
- 02 May 2017, pp 569-576
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