Next  |  Prev  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

The Bilinear Transform

A class of bilinear transforms map the entire $ j\omega$ axis in the $ s$ plane exactly once to the unit circle in the $ z$ plane:

$\displaystyle s = c \frac{1-z^{-1}}{1+z^{-1}}
$

The real constant $ c>0$ allows one nonzero frequency (at $ s=j\omega_a$) to map exactly to any desired digital frequency (at $ z=e^{j\omega_dT}$). All other frequencies are warped:

$\displaystyle j\omega_a = c \frac{1-e^{-j\omega_d T}}{1+e^{-j\omega_d T}}
= jc\tan\left(\frac{\omega_d T}{2}\right)
$


Next  |  Prev  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Download SMAC03S.pdf
Download SMAC03S_2up.pdf

``Recent Developments in Musical Sound Synthesis Based on a Physical Model'', by Julius O. Smith III, (Stockholm Musical Acoustics Conference (SMAC-03), August 6--9, 2003).
Copyright © 2006-02-19 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]