Daniel Bernoulli's Modal Decomposition

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Spectral Audio Signal Processing

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The Telharmonium

Spectrum analysis of sound is analogous to decomposing white light into its component colors by means of a prism — Click for https://ccrma.stanford.edu/~jos/mdft/Example_Applications_DFT.html

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The Fourier transform of a signal is a frequency-domain function — Click for https://ccrma.stanford.edu/~jos/mdft/

A periodic signal is a signal that forever repeats itself. — Click for http://en.wikibooks.org/wiki/Signals_and_Systems/Periodic_Signals

Every periodic signal has a fundamental frequency given by the inverse of the period. If the period is in units of seconds, then the fundamental frequency is in units of Hertz (cycles per second). According to Fourier theory, every periodic signal can be expressed as a sum of sinusoids at frequencies given by integer multiples of the fundamental frequency. For integers greater than 1, these frequencies are called harmonic frequencies, and the sinusoids at harmonic frequencies are typically called harmonics. — Click for https://www.physicsclassroom.com/class/sound/Lesson-4/Fundamental-Frequency-and-Harmonics

Pitch is the perceived fundamental frequency of a sound. — Click for http://en.wikipedia.org/wiki/Pitch_(music)

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A sinusoidal model for sound approximates each tonal component of the sound as a sum of slowly varying sinusoids. For tonal sounds such as from vibrating strings or wind instruments (including voiced speech), a sinusoidal model can provide a compact, high-fidelity representation. In addition to providing an intuitive, malleable representation for sound, sinusoidal models are also used in advanced audio compression. — Click for https://ccrma.stanford.edu/~jos/sasp/Sinusoidal_Modeling_Sound.html

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Intermodulation Distortion is created when a signal containing two or more sinusoidal frequency components is passed through a nonlinear system. In general, the output signal contains many new sinusoidal components at frequencies obtained by adding and subtracting the frequencies present in the input signal. In the simplest case of two input frequencies f1 and f2. The output contains may contain component at the sum-frequency f1+f2 and difference-frequency f1-f2 (and f2-f1). Multiplying two sinusoids together and applying the trig identity cos(a+b) = cos(a)cos(b) - sin(a)sin(b) can be used to illustrate this. — Click for http://en.wikipedia.org/wiki/Intermodulation_distortion

A sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any 'pure' vibration, such as that of an ideal tuning fork or mass-spring system. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

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According to Hooke's law, the force F exerted by a spring stretched a distance x from its rest position is F=-kx. — Click for http://en.wikipedia.org/wiki/Spring_%28device%29

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If the mass of a harmonic oscillator is excited, the system responds by vibrating back and forth. Whenever the mass is at its rest position, the system contains only kinetic energy. Conversely, whenever the mass velocity is zero, the system contains only potential energy. — Click for http://ccrma.stanford.edu/realsimple/lab_inst/Background.html

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Spectral Audio Modeling History

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Daniel Bernoulli's Modal Decomposition

The history of spectral modeling of sound arguably begins with Daniel Bernoulli, who first believed (in the 1733-1742 time frame) that any acoustic vibration could be expressed as a superposition of ``simple modes'' (sinusoidal vibrations) [52].^G.1Bernoulli was able to show this precisely for the case of identical masses interconnected by springs to form a discrete approximation to an ideal string. Leonard Euler first wrote down (1749) a mathematical expression of Bernoulli's insight for a special case of the continuous vibrating string:

$\displaystyle y = \sum_{k=1}^\infty a_k \sin(k \pi x/L) \cos(k \pi \nu t)$

(G.1)

Euler's paper was primarily in response to d'Alembert's landmark derivation (1747) of the traveling-wave picture of vibrating strings [266], which showed that vibrating string motion could assume essentially any twice-differentiable shape. Euler did not consider Bernoulli's sinusoidal superposition to be mathematically complete because clearly a trigonometric sum could not represent an arbitrary string shape.^G.2D'Alembert did not believe that a superposition of sinusoids could be ``physical'' in part because he imagined what we would now call ``intermodulation distortion'' due to a high-frequency component ``riding on'' a lower-frequency component (as happens nowadays in loudspeakers).^G.3 Arguments over the generality of the sinusoidal expansion of string vibration persisted for decades, embroiling also Lagrange, and were not fully settled until Fourier theory itself was wrestled into submission by the development of distribution theory, measure theory, and the notion of convergence ``almost everywhere.''

Note that organ builders had already for centuries built machines for performing a kind of ``additive synthesis'' by gating various ranks of pipes using ``stops'' as in pipe organs found today. However, the waveforms mixed together were not sinusoids, and were not regarded as mixtures of sinusoids. Theories of sound at that time, based on the ideas of Galileo, Mersenne, and Sauveur, et al. [52], were based on time-domain pulse-train analysis. That is, an elementary tone at a fixed pitch (and fundamental frequency) was a periodic pulse train, with the pulse-shape being non-critical. Musical consonance was associated with pulse-train coincidence--not frequency-domain separation. Bernoulli clearly suspected the spectrum analysis function integral to hearing as well as color perception [52, p. 359], but the concept of the ear as a spectrum analyzer is generally attributed to Helmholtz (1863) [293].

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Daniel Bernoulli's Modal Decomposition

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University