Sign in

Search Online Books

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

Physical Audio Signal Processing
    Digital Waveguide Theory
       Scattering at Impedance Changes
          One-Multiply Scattering Junctions

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?

One-Multiply Scattering Junctions

By factoring out in each equation of (C.60), we can write

$\displaystyle f^{{+}}_i(t)$	$\displaystyle =$	$\displaystyle f^{{+}}_{i-1}(t-T) + f_{{\Delta}}(t)$
$\displaystyle f^{{-}}_{i-1}(t+T)$	$\displaystyle =$	$\displaystyle f^{{-}}_i(t) + f_{{\Delta}}(t)$	(C.62)

where

$\displaystyle f_{{\Delta}}(t) \isdef k_i(t)\left[f^{{+}}_{i-1}(t-T) - f^{{-}}_i(t) \right]$

(C.63)

Thus, only one multiplication is actually necessary to compute the transmitted and reflected waves from the incoming waves in the Kelly-Lochbaum junction. This computation is shown in Fig.C.21, and it is known as the one-multiply scattering junction [297].

**Figure C.21:** The one-multiply scattering junction.
$\includegraphics[scale=0.9]{eps/Fom}$

Another one-multiply form is obtained by organizing (C.60) as

$\displaystyle f^{{+}}_i(t)$	$\displaystyle =$	$\displaystyle f^{{-}}_i(t) + \alpha_i(t)\tilde{f_d}(t)$
$\displaystyle f^{{-}}_{i-1}(t+T)$	$\displaystyle =$	$\displaystyle f^{{+}}_i(t) - \tilde{f_d}(t)$	(C.64)

where

$\displaystyle \alpha_i(t)$	$\displaystyle \isdef$	$\displaystyle 1+k_i(t)$
$\displaystyle \tilde{f_d}(t)$	$\displaystyle \isdef$	$\displaystyle f^{{+}}_{i-1}(t-T) - f^{{-}}_i(t).$	(C.65)

As in the previous case, only one multiplication and three additions are required per junction. This one-multiply form generalizes more readily to junctions of more than two waveguides, as we'll see in a later section.

A scattering junction well known in the LPC speech literature but not described here is the so-called two-multiply junction [297] (requiring also two additions). This omission is because the two-multiply junction is not valid as a general, local, physical modeling building block. Its derivation is tied to the reflectively terminated, cascade waveguide chain. In cases where it applies, however, it can be the implementation of choice; for example, in DSP chips having a fast multiply-add instruction, it may be possible to implement the inner loop of the two-multiply, two-add scattering junction using only two instructions.

Previous: Kelly-Lochbaum Scattering Junctions
Next: Normalized Scattering Junctions

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? )