State-Space Analysis Example: The Digital Waveguide Oscillator

As an example of state-space analysis, we will use it to determine the frequency of oscillation of the system of Fig.G.3 [90].

**Figure G.3:** The second-order digital waveguide oscillator.
$\begin{figure}\input fig/sswgo.pstex_t \end{figure}$

Note the assignments of unit-delay outputs to state variables and . From the diagram, we see that

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

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 time-update can be written

$\displaystyle \left[\begin{array}{c} x_1(n+1) \\ [2pt] x_2(n+1) \end{array}\rig... ...ay}\right]}_A \left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \end{array}\right]$

or, in vector notation,

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

We have two natural choices of output,

and

$\begin{eqnarray*} y_1(n) &\isdef & x_1(n) = [1, 0] {\underline{x}}(n)\\ y_2(n) &\isdef & x_2(n) = [0, 1] {\underline{x}}(n) \end{eqnarray*}$

A basic fact from linear algebra is that the determinant of a matrix is equal to the product of its eigenvalues. As a quick check, we find that the determinant of is

$\displaystyle \det{A} = c^2 - (c+1)(c-1) = c^2 - (c^2-1) = 1.$

Since an undriven sinusoidal oscillator must not lose energy, and since every lossless state-space system has unit-modulus eigenvalues (consider the modal representation), we expect $\left\vert\det{A}\right\vert=1$ .

Note that ${\underline{x}}(n) = A^n{\underline{x}}(0)$ . If we diagonalize this system to obtain $\tilde{A}=E^{-1}A E$ , where $\tilde{A}=$ diag $[\lambda_1,\lambda_2]$ , and is the matrix of eigenvectors of , then we have

$\displaystyle \underline{{\tilde x}}(n) = \tilde{A}^n\,\underline{{\tilde x}}(0... ...t[\begin{array}{c} {\tilde x}_1(0) \\ [2pt] {\tilde x}_2(0) \end{array}\right]$

where $\underline{{\tilde x}}(n) \isdef E^{-1}{\underline{x}}(n)$ denotes the state vector in these new ``modal coordinates''. Since $\tilde{A}$ is diagonal, the modes are decoupled, and we can write

$\begin{eqnarray*} {\tilde x}_1(n) &=& \lambda_1^n\,{\tilde x}_1(0)\\ {\tilde x}_2(n) &=& \lambda_2^n\,{\tilde x}_2(0). \end{eqnarray*}$

If this system is to generate a real sampled sinusoid at radian frequency $\omega$ , the eigenvalues $\lambda_1$ and $\lambda_2$ must be of the form

$\begin{eqnarray*} \lambda_1 &=& e^{j\omega T}\\ \lambda_2 &=& e^{-j\omega T}, \end{eqnarray*}$

(in either order) where $\omega$ is real, and denotes the sampling interval in seconds.

Thus, we can determine the frequency of oscillation $\omega$ (and verify that the system actually oscillates) by determining the eigenvalues $\lambda_i$ of . Note that, as a prerequisite, it will also be necessary to find two linearly independent eigenvectors of (columns of ).

Subsections

Previous: Jordan Canonical Form
Next: Finding the Eigenstructure of A

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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.

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Introduction to Digital Filters
State Space Filters
State-Space Analysis Example: The Digital Waveguide Oscillator