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

Trapezoidal Rule Frequency Mapping

Let's look at the $ s$ to $ z$ mapping,

$\displaystyle s = \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}
$

on the unit circle, where $ s=j\omega_a$ and $ z=e^{j\omega_d T}$ :

\begin{eqnarray*}
j\omega_a = \frac{2}{T}\frac{1-e^{-j\omega_d T}}{1+e^{-j\omega_d T}} &=&
j\frac{2}{T}\tan(\omega_d T/2)
\end{eqnarray*}

or

$\displaystyle \zbox{\frac{\omega_a T}{2} \;=\;\tan\left(\frac{\omega_d T}{2}\right)}
$

In general, the trapezoid rule is a second-order accurate approximation to a derivative, in the limit of small $ T$ (i.e., near dc). Here, it is third-order accurate along the unit circle at dc.


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

Download DigitizingNewton.pdf
Download DigitizingNewton_2up.pdf
Download DigitizingNewton_4up.pdf

``Introduction to Physical Signal Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2020-06-27 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]