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State-Space Analysis Example:
The Digital Waveguide Oscillator

Let's use state-space analysis to determine the frequency of oscillation of the following system:

\epsfig{file=eps/sswgo.eps,width=\textwidth }
The second-order digital waveguide oscillator.

Note the assignments of unit-delay outputs to state variables $ x_1(n)$ and $ x_2(n)$ .

We have

$\displaystyle x_1(n+1) = c[x_1(n) + x_2(n)] - x_2(n) = c\,x_1(n) + (c-1) x_2(n)
$

and

$\displaystyle x_2(n+1) = x_1(n) + c[x_1(n) + x_2(n)] = (1+c) x_1(n) + c\,x_2(n)
$

In matrix form, the state transition can be written as

$\displaystyle \left[\begin{array}{c} x_1(n+1) \\ [2pt] x_2(n+1) \end{array}\right] = \underbrace{\left[\begin{array}{cc} c & c-1 \\ [2pt] c+1 & c \end{array}\right]}_\mathbf{A}\left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \end{array}\right]
$

or, in vector notation,

$\displaystyle \underline{x}(n+1) = \mathbf{A}\, \underline{x}(n)
$

The poles of the system are given by the eigenvalues of $ \mathbf{A}$ , which are the roots of its characteristic polynomial. That is, we solve

$\displaystyle \vert{\lambda_i}\mathbf{I}- \mathbf{A}\vert = 0
$

for $ {\lambda_i}$ , $ i=1,2,\ldots,N$ , or, for our $ N=2$ problem,

$\displaystyle 0 = \left\vert \begin{array}{cc} {\lambda_i}- c & 1-c \\ [2pt] -c-1 & {\lambda_i}-c \end{array} \right\vert =
({\lambda_i}- c)^2 + (1-c)(1+c) = {\lambda_i}^2 - 2{\lambda_i}c + 1
$

Using the quadratic formula, the two solutions are found to be

$\displaystyle {\lambda_i}= c \pm \sqrt{c^2-1} = c\pm j \sqrt{1-c^2}
$

Defining $ c=\cos(\theta)$ , we obtain the simple formula

$\displaystyle {\lambda_i}= \cos(\theta) \pm j \sin(\theta) = \zbox{e^{\pm j\theta}}
$

It is now clear that the system is a real sinusoidal oscillator for $ -1\le c \le 1$ , oscillating at normalized radian frequency $ \omega_c T
\isdeftext \theta \isdeftext \arccos(c) \in [-\pi,\pi]$ .

We determined the frequency of oscillation $ \omega_c T$ from the eigenvalues $ {\lambda_i}$ of $ \mathbf{A}$ . To study this system further, we can diagonalize $ \mathbf{A}$ . For that we need the eigenvectors as well as the eigenvalues.



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``Introduction to State Space Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2015-04-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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