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

Eigenstructure of A

Starting with

$\displaystyle A\underline{e}_i= {\lambda_i}\underline{e}_i,\quad i=1,2
$

we get

$\displaystyle \left[\begin{array}{cc} c & c-1 \\ [2pt] c+1 & c \end{array}\right] \left[\begin{array}{c} 1 \\ [2pt] \eta_i \end{array}\right] = \left[\begin{array}{c} {\lambda_i} \\ [2pt] {\lambda_i}\eta_i \end{array}\right].
$

We now have an explicit formula for the frequency of oscillation:

$\displaystyle \omega = \theta/T = f_s\arccos(c),
$

where $ f_s$ denotes the sampling rate. Or,

$\displaystyle \fbox{$\displaystyle c = \cos(\omega T)$}
$

The coefficient range $ c\in(-1,1)$ corresponds to frequencies $ f \in
(-f_s/2,f_s/2)$ .

We have shown that the example system oscillates sinusoidally at any desired digital frequency $ \omega$ when $ c=\cos(\omega T)$ , where $ T$ denotes the sampling interval.


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

Download StateSpace.pdf
Download StateSpace_2up.pdf
Download StateSpace_4up.pdf

``Introduction to State Space Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2015-03-29 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]