Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Sponsor

Industry's highest performing at the lowest power DSPs now as low as $5.00*
Start development today!
*volume pricing for 10ku

Chapters

See Also

Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Acoustic Modeling with Digital Delay
       Lossy Acoustic Propagation
          Frequency-Dependent Air-Absorption Filtering

Chapter Contents:

Search Physical Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

Frequency-Dependent Air-Absorption Filtering

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

$\displaystyle z^{-1}\leftarrow 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

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

in the frequency domain, and

$\displaystyle y(n) = \underbrace{g\ast g\ast \dots \ast g \, \ast }_{\hbox{$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

$\displaystyle 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$.

Previous: Exponentially Decaying Traveling Waves
Next: Dispersive Traveling Waves

Order a Hardcopy of Physical Audio Signal Processing

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )