Delay Loop Expansion

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Physical Audio Signal Processing

Modal Expansion

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General Filter Design Methods

For a tone which posesses a perfectly harmonic overtone series, the nth harmonic is generally defined to oscillate at nf₀Hz, where f₀ is the fundamental frequency. — Click for http://en.wikipedia.org/wiki/Harmonic

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Psychoacoustics is the science of the human perception of sound. — Click for http://en.wikipedia.org/wiki/Psychoacoustics

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

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Traveling waves propagate (i.e. travel) through space. — Click for http://ccrma.stanford.edu/realsimple/travelingwaves

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

A comb filter is made from a delay line using a linear combination of its input and output signals. — Click for https://ccrma.stanford.edu/~jos/pasp/Comb_Filters.html

A delay line is used to delay a signal in time. Delay lines for digital signals are typically implemented in software using a circular buffer. — Click for https://ccrma.stanford.edu/~jos/pasp/Delay_Lines.html

A resonator is a recursive filter that boosts signal amplitude at a particular frequency. The simplest resonator is made using a two-pole filter. The impulse response of the two-pole resonator is an exponentially decaying sinusoidal oscillation, where the zero-crossing rate is (twice) the resonance frequency. — Click for https://ccrma.stanford.edu/~jos/filters/Two_Pole.html

The BiQuad is a ubiquitous filter in audio signal processing, so named because its transfer function H(z)=B(z)/A(z) has a quadratic numerator polynomial B(z) and a quadratic denominator polynomial A(z). — Click for https://ccrma.stanford.edu/~jos/filters/BiQuad_Section.html

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Physical Audio Signal Processing

Modal Expansion

State Space Approach to Modal Expansions

General Filter Design Methods

Delay Loop Expansion

When a subset of the resonating modes is nearly harmonically tuned, it can be much more computationally efficient to use a filtered delay loop (see §2.6.5) to generate an entire quasi-harmonic series of modes rather than using a biquad for each modal peak [443]. In this case, the resonator model becomes

$\displaystyle H(z) \eqsp \sum_{k=1}^N \frac{a_k}{1 - H_k(z) z^{-N_k}},$

where

is the length of the delay line in the

th comb filter, and

is a low-order filter which can be used to adjust finely the amplitudes and frequencies of the resonances of the

th comb filter [432]. Recall (Chapter 6) that a single filtered delay loop efficiently models a distributed 1D propagation medium such as a vibrating string or acoustic tube. More abstractly, a superposition of such quasi-harmonic mode series can provide a computationally efficient psychoacoustic equivalent approximation to arbitrary collections of modes in the range of human hearing.

Note that when is close to instead of , primarily only odd harmonic resonances are produced, as has been used in modeling the clarinet [435].

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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

Delay Loop Expansion

``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