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
       FDNs as Digital Waveguide Networks
          Normalized Scattering

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?

  

Normalized Scattering

For ideal numerical scaling in the $ L_2$ sense, we may choose to propagate normalized waves which lead to normalized scattering junctions analogous to those encountered in normalized ladder filters [297]. Normalized waves may be either normalized pressure $ \tilde{p}_j^+ = p_j^+\sqrt{\Gamma_i}$ or normalized velocity $ \tilde{v}_j^+ = v_j^+/\sqrt{\Gamma_i}$. Since the signal power associated with a traveling wave is simply $ {\cal P_j^+} = (\tilde{p}_j^+)^2 = (\tilde{v}_j^+)^2$, they may also be called root-power waves [432]. Appendix C develops this topic in more detail.

The scattering matrix for normalized pressure waves is given by

$\displaystyle \tilde{\mathbf{A}}= \left[ \begin{array}{llll} \frac{2 \Gamma_{1}... ..._{2}}}{\Gamma_J} & \dots & \frac{2 \Gamma_{n}}{\Gamma_J} -1 \end{array} \right]$ (C.122)

The normalized scattering matrix can be expressed as a negative Householder reflection

$\displaystyle \tilde{\mathbf{A}}= \frac{2}{ \vert\vert\,\tilde{{\bm \Gamma}}\,\vert\vert ^2}\tilde{{\bm \Gamma}}\tilde{{\bm \Gamma}}^T-\mathbf{I}$ (C.123)

where $ \tilde{{\bm \Gamma}}^T= [\sqrt{\Gamma_1},\ldots,\sqrt{\Gamma_N}]$, and $ \Gamma_i$ is the wave admittance in the $ i$th waveguide branch. To eliminate the sign inversion, the reflections at the far end of each waveguide can be chosen as -1 instead of 1. The geometric interpretation of (C.124) is that the incoming pressure waves are reflected about the vector $ \tilde{{\bm \Gamma}}$. Unnormalized scattering junctions can be expressed in the form of an ``oblique'' Householder reflection $ \mathbf{A}= 2\mathbf{1}{\bm \Gamma}^T/\left<\mathbf{1},{{\bm \Gamma}}\right>-\mathbf{I}$, where $ \mathbf{1}^T=[1,\ldots,1]$ and $ {\bm \Gamma}^T= [\Gamma_1,\ldots,\Gamma_N]$.

Previous: Lossless Scattering
Next: General Conditions for Losslessness

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? )