Mode Extraction Techniques

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

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

Linear Prediction is an important signal modeling tool which captures correlation information in the form of prediction coefficients. In digital audio signal processing, linear prediction computes a superior spectral envelope that emphasizes spectral peaks. — Click for http://en.wikipedia.org/wiki/Linear_prediction

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/

The amplitude response of an LTI filter is simply the magnitude of the (complex) frequency response — Click for https://ccrma.stanford.edu/~jos/filters/Amplitude_Response_I_I.html

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

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

The impulse signal is the shortest pulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). The impulse is typically denoted by the Greek delta symbol δ and is often called the 'Dirac delta function', after Paul Dirac who pioneered its use (in addition to being a lead contributor in the development of Quantum Mechanics). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

The size of the passband, typically expressed in Hz. — Click for http://en.wikipedia.org/wiki/Bandpass

The impulse response of a system is its output signal in response to the impulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.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

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Mode Extraction Techniques

The goal of resonator factoring is to identify and remove the least-damped resonant modes of the impulse response. In principle, this means ascertaining the precise resonance frequencies and bandwidths associated with each of the narrowest ``peaks'' in the resonator frequency response, and dividing them out via inverse filtering, so they can be implemented separately as resonators in series. If, in addition, the amplitude and phase of a resonance peak are accurately measurable in the complex frequency response, the mode can be removed by complex spectral subtraction (equivalent to subtracting the impulse-response of the resonant mode from the total impulse response); in this case, the parametric modes are implemented in a parallel bank as in [66]. However, in the parallel case, the residual impulse response is not readily commuted with the string.

In the inverse-filtering case, the factored resonator components are in cascade (series) so that the damped modes left behind may be commuted with the string and incorporated in the excitation table by convolving the residual impulse response with the desired string excitation signal. In the parallel case, the damped modes do not commute with the string since doing so would require somehow canceling them in the parallel filter sections. In principle, the string would have to be duplicated so that one instance can be driven by the residual signal with no body resonances at all, while the other is connected to the parallel resonator bank and driven only by the natural string excitation without any commuting of string and resonator. Since duplicating the string is unlikely to be cost-effective, the impulse response of the high-frequency modes can be commuted and convolved with the string excitation as in the series case to obtain qualitative results. The error in doing this is that the high-frequency modes are being multiplied by the parallel resonators rather than being added to them.

Various methods are available for estimating the mode parameters for inverse filtering:

Mode Extraction Techniques

``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