Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Ads

Chapters

See Also

Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Introduction to Lumped Models
       ``Traveling Waves'' in Lumped Systems
          General Reflectance

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?

  

General Reflectance

Let $ R(s)$ denote a general impedance. Then the wave variable decomposition in (L.8) gives

$\displaystyle F(s)$ $\displaystyle =$ $\displaystyle R(s) V(s)$ (L.14)
$\displaystyle \,\,\Rightarrow\,\,F^{+}(s) + F^{-}(s)$ $\displaystyle =$ $\displaystyle R(s) \left[V^{+}(s) + V^{-}(s)\right]$ (L.15)
  $\displaystyle =$ $\displaystyle R(s) \left[\frac{F^{+}(s) - F^{-}(s)}{R_0}\right]$ (L.16)
$\displaystyle \,\,\Rightarrow\,\,F^{-}(s) \left[\frac{R(s)}{R_0}+1\right]$ $\displaystyle =$ $\displaystyle F^{+}(s) \left[\frac{R(s)}{R_0}-1\right]$ (L.17)
$\displaystyle \,\,\Rightarrow\,\,F^{-}(s)$ $\displaystyle =$ $\displaystyle F^{+}(s) \left[\frac{R(s)-R_0}{R(s)+R_0}\right]$ (L.18)
  $\displaystyle \isdef$ $\displaystyle F^{+}(s) S(s)$ (L.19)

Formally, $ S(s)$ is the reflectance of impedance $ R(s)$ relative to $ R_0$. For example, if a transmission line with characteristic impedance $ R_0$ were terminated in a lumped impedance $ R(s)$, the reflection transfer function at the termination would be $ S(s)$. The interpretation of $ S(s)$ as a reflectance is shown as a wave flow diagram in Fig. L.22c.

Figure L.22: Three different types of diagram for a basic impedance relation: a) Impedance diagram. b) System block diagram. c) Wave flow diagram.
\includegraphics[width=\twidth]{eps/lreflectance}

We are working with reflectance for force waves. Using the elementary relations (L.8), i.e., $ F^{+}(s) = R_0V^{+}(s)$ and $ F^{-}(s) = -R_0V^{-}(s)$, we immediately obtain the corresponding velocity-wave reflectance

$\displaystyle \frac{V^{-}(s)}{V^{+}(s)} = \frac{-F^{-}(s)/R_0}{F^{+}(s)/R_0} = - \frac{F^{-}(s)}{F^{+}(s)} = - S(s) $

Order a Hardcopy of Physical Audio Signal Processing

Previous: Physical Interpretation of Reflection and Transmission in Lumped Systems
Next: Passive Impedances
written by 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? )