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
    Digital Waveguide Theory
       Non-Cylindrical Acoustic Tubes
          Cylinder with Conical Cap
             Poles at

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?

  

Poles at

We know from the above that the denominator of the cone reflectance has at least one root at $ s=0$. In this subsection we investigate this ``dc behavior'' of the cone more thoroughly.

A hasty analysis based on the reflection and transmission filters in Equations (C.154) and (C.155) might conclude that the reflectance of the conical cap converges to $ -1$ at dc, since $ R(0)=-1$ and $ T(0)=0$. However, this would be incorrect. Instead, it is necessary to take the limit as $ \omega\to0$ of the complete conical cap reflectance $ R_J(s)$:

$\displaystyle R_J(s) = \frac{1 - e^{-2s} - 2s e^{-2s}}{2s - 1 + e^{-2s}}$ (C.165)

We already discovered a root at $ s=0$ in the denominator in the context of the preceding stability proof. However, note that the numerator also goes to zero at $ s=0$. This indicates a pole-zero cancellation at dc. To find the reflectance at dc, we may use L'Hospital's rule to obtain

$\displaystyle R_J(0) = \lim_{s\to0} \frac{N^\prime(s)}{D^\prime(s)} = \lim_{s\to 0}\frac{4s e^{-2s}}{2-2e^{-2s}}$ (C.166)

and once again the limit is an indeterminate $ 0/0$ form. We therefore apply L'Hospital's rule again to obtain

$\displaystyle R_J(0) = \lim_{s\to0} \frac{N^{\prime\prime}(s)}{D^{\prime\prime}(s)} = \lim_{s\to 0}\frac{(4-8s) e^{-2s}}{4e^{-2s}} = +1$ (C.167)

Thus, two poles and zeros cancel at dc, and the dc reflectance is $ +1$, not $ -1$ as an analysis based only on the scattering filters would indicate. From a physical point of view, it makes more sense that the cone should ``look like'' a simple rigid termination of the cylinder at dc, since its length becomes small compared with the wavelength in the limit.

Another method of showing this result is to form a Taylor series expansion of the numerator and denominator:

$\displaystyle N(s)$ $\displaystyle =$ $\displaystyle 2 {s^2} - {{8 {s^3}}\over 3} + 2 {s^4} + \cdots$ (C.168)
$\displaystyle D(s)$ $\displaystyle =$ $\displaystyle 2 {s^2} - {{4 {s^3}}\over 3} + {{2 {s^4}}\over 3} + \cdots$ (C.169)

Both series begin with the term $ 2s^2$ which means both the numerator and denominator have two roots at $ s=0$. Hence, again the conclusion is two pole-zero cancellations at dc. The series for the conical cap reflectance can be shown to be

$\displaystyle R_J(s) = 1 - {{2 s}\over 3} + {{2 {s^2}}\over 9} - {{4 {s^3}}\over {135}} - {{2 {s^4}}\over {405}} + \cdots$ (C.170)

which approaches $ +1$ as $ s\to0$.

An alternative analysis of this issue is given by Benade in [37].

Previous: Reflectance Magnitude
Next: Finite-Difference Schemes

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? )