Mode Density Requirement

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Choice of Delay Lengths

Mean Free Path

Prime Power Delay-Line Lengths

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

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

T60 is the time (in seconds) to decay by 60 dB, e.g., from 0 dB to -60 dB. It is a typical 'cut-off time' for impulse responses that decay exponentially. — Click for https://ccrma.stanford.edu/~jos/mdft/Audio_Decay_Time_T60.html

In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html

Reverberation is the sound created by many echoes blending together, such as you hear when you clap your hands together in an acoustically 'live' room. — Click for https://ccrma.stanford.edu/~jos/pasp/Artificial_Reverberation.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

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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

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Choice of Delay Lengths

Mean Free Path

Prime Power Delay-Line Lengths

Mode Density Requirement

A guide for the sum of the delay-line lengths is the desired mode density. The sum of delay-line lengths in a lossless FDN is simply the order of the system :

$\displaystyle M \isdef \sum_{i=1}^N M_i\qquad\hbox{(FDN order)}$

The order increases slightly when lowpass filters are introduced after the delay lines to achieve a specific reverberation time at low and high frequencies (as described in the next subsection).

Since the order of a system equals the number of poles, we have that is the number of poles on the unit circle in the lossless prototype. If the modes were uniformly distributed, the mode density would be modes per Hz. Schroeder and Logan [420,421] suggest that, for a reverberation time of 1 second, a mode density of 0.15 modes per Hz is adequate. Since the mode widths are inversely proportional to reverberation time, the mode density for a reverberation time of 2 seconds should be 0.3 modes per Hz, etc. In summary, for a sufficient mode density in the frequency domain, Schroeder's formula is

$\displaystyle M \geq 0.15\, t_{60}\, f_s$

For a sampling rate of 50 kHz and a reverberation time ( $t_{60}$ ) equal to 1 second, we obtain $M\geq 7500$ .

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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

Mode Density Requirement

``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