General Loss Simulation

The substitution

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

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

Example (delay line):

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

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

Frequency-Dependent Losses:

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

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

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

in the frequency domain, and iterated convolution

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

in the time domain