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General Loss Simulation

The substitution

$\displaystyle z^{-1}\leftarrow gz^{-1}

in any transfer function contracts all poles by the factor $ g$ .

Example (delay line):

$\displaystyle H(z) = z^{-M} \;\rightarrow\; g^M z^{-M}

Thus, the contraction factor $ g$ can be interpreted as the per-sample propagation loss factor.

Frequency-Dependent Losses:

$\displaystyle z^{-1}\leftarrow G(z)z^{-1}, \quad \left\vert G(e^{j\omega T})\right\vert\leq1

$ G(z)$ can be considered the filtering per sample in the propagation medium. A lossy delay line is thus described by

$\displaystyle Y(z) = G^M(z) z^{-M}X(z)

in the frequency domain, and iterated convolution

$\displaystyle y(n) = \underbrace{g\ast g\ast \dots \ast g \ast }_{\hbox{$M$\ times}} x(n-M)

in the time domain

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``Computational Acoustic Modeling with Digital Delay'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2020-02-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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