Frequency-Dependent Air-Absorption Filtering

More generally, frequency-dependent air absorption can be modeled using the substitution

$$z^{-1}\rightarrow G(z)z^{-1}$$

where $G(z)$ denotes the filtering per sample in the propagation medium. Since air absorption cannot amplify a wave at any frequency, we have $\left \vert G(e^{j\omega T})\right \vert\leq 1$ . A lossy delay line for plane-wave simulation is thus described by

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

in the frequency domain, and

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

in the time domain, where `$\ast$ ' denotes convolution, and $g(n)$ is the impulse response of the per-sample loss filter $G(z)$ . The effect of $G(z)$ on the poles of the system is discussed in §3.7.4.

For spherical waves, the loss due to spherical spreading is of the form

$$Y(z) \propto \frac{G^M(z) z^{-M}}{r}X(z)$$

where $r$ is the distance from $X$ to $Y$ . We see that the spherical spreading loss factor is ``hyperbolic'' in the propagation distance $r$ , while air absorption is exponential in $r$ .