Finite Difference Methods

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Physical Audio Signal Processing

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From Wikipedia: Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound — Click for https://en.wikipedia.org/wiki/Acoustics

In science and technology, a physical model allows us to simulate or visualize something about the thing it represents. — Click for http://en.wikipedia.org/wiki/Model_(physical)

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A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

dc stands for direct current in electrical engineering and signal processing (as opposed to 'alternating current'). The mean (average value) of a signal may be called its dc component. Similarly, frequency zero is called the dc frequency, because a sinusoid with zero frequency is a constant signal. — Click for https://ccrma.stanford.edu/~jos/filters/Mathematical_Sine_Wave_Analysis.html

The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

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A digital filter can be visualized as a 'black box' that accepts a sequence of numbers and emits a new sequence of numbers. In digital audio signal processing applications, such number sequences usually represent sounds. For example, digital filters are used to implement graphic equalizers and other digital audio effects. — Click for https://ccrma.stanford.edu/~jos/filters/

A signal is said to be oversampled when the sampling frequency exceeds the signal bandwidth by some finite margin. For example, 2x oversampling means that the sampling rate is double what it needs to be to avoid aliasing in the frequency domain. In audio, the 88.2 kHz is often called 2x oversampling, because the typical audio sampling rate (for compact disks [CDs]) is 44.1 kHz. Mathematically, 2x oversampling is twice the minimum non-aliasing sampling rate for any given signal. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theorem.html

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Finite Difference Methods

One of the first applications of digital computers to numerical simulation of physical systems was the so-called finite difference approach [484]. The general procedure is to replace derivatives by finite differences, and there are many variations on how this can be done. For example, the first partial derivative with respect to time in Eq.(A.1) may be approximated by

$\displaystyle {\dot y}(t,x) \isdefs \frac{\partial}{\partial t} y(t,x) \;\approx\; \frac{y(t,x) - y(t-T,x)}{T} \approx \frac{y(t+T,x) - y(t-T,x)}{2T}.$

(A.3)

The second form, called a centered finite difference, has the advantage of not introducing a time delay, but at the expense of requiring an extra factor of two oversampling for a given accuracy in its magnitude response (when viewed as a digital filter).

Finite differences are at least as old as Taylor series, since,

$\displaystyle y(t-T)= y(t) - T{\dot y}(t) + \cdots \;\Rightarrow\; {\dot y}(t) \approx \frac{y(t)-y(t-T)}{T}.$

(A.4)

Interestingly, the general Taylor series, published in 1715 by Taylor in his book Methodus Incrementorum Directa et Inversa, was known more than forty years earlier to James Gregory (1668), somewhat earlier to Jean Bernoulli, and to some extent even before 1550 in India [65, pp. 422,469].

Finite differences were used to construct the earliest known digital models of vibrating strings by Pierre Ruiz and Lejaren Hiller ca. 1971 [195].

Finite-difference methods have not historically been aimed at real-time simulation, and they are generally used with very large sampling rates compared with the ``band of interest''. On the order of ten-times oversampling is needed to obtain reasonably accurate simulation across the entire audio band when using classical finite difference methods. In the finite-difference method literature, accuracy is usually only considered at dc, which is inaudible. Since finite difference models are usually linear, time-invariant digital filters, it is straightforward to improve them by filter-design methods, optimizing perceived audio error over the entire audio band. Such improved (and optionally extended) coefficients can then be used to construct a refined, indirectly estimated, partial differential equation.

Other offline (slower than real time) computational physical modeling methods include finite element [483] and boundary element methods. Such offline simulations can be valuable as a source of ``virtual experiments'' which enable the testing and calibration of faster algorithms. As an example, a detailed simulation of guitar acoustics, employing both finite difference and finite element modeling approaches, is reported in [109].

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Finite Difference Methods

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University