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Physical Audio Signal Processing
    Digital Waveguide Theory
       The Digital Waveguide Oscillator
          Eigenstructure

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Eigenstructure

Starting with the defining equation for an eigenvector $ \underline{e}$ and its corresponding eigenvalue $ \lambda$,

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

we get, using Eq.$ \,$(C.137),

$\displaystyle \left[\begin{array}{cc} gc & c-1 \\ [2pt] gc+g & c \end{array}\ri... ...n{array}{c} {\lambda_i} \\ [2pt] {\lambda_i}\eta_i \end{array}\right]. \protect$ (C.140)

We normalized the first element of $ \underline{e}_i$ to 1 since $ g\underline{e}_i$ is an eigenvector whenever $ \underline{e}_i$ is. (If there is a missing solution because its first element happens to be zero, we can repeat the analysis normalizing the second element to 1 instead.)

Equation (C.141) gives us two equations in two unknowns:

$\displaystyle gc+\eta_i(c-1)$ $\displaystyle =$ $\displaystyle {\lambda_i} \protect$ (C.141)
$\displaystyle g(1+c) +c\eta_i$ $\displaystyle =$ $\displaystyle {\lambda_i}\eta_i$ (C.142)

Substituting the first into the second to eliminate $ {\lambda_i}$, we get

\begin{eqnarray*} g+gc+c\eta_i &=& [gc+\eta_i(c-1)]\eta_i = gc\eta_i + \eta_i^2 ... ...{g\left(\frac{1+c}{1-c}\right) - \frac{c^2(1-g)^2}{4(1-c)^2}}. \end{eqnarray*}

As $ g$ approaches $ 1$ (no damping), we obtain

$\displaystyle \eta_i = \pm j\sqrt{\frac{1+c}{1-c}} \qquad \hbox{(when $g=1$)}. $

Thus, we have found both eigenvectors:

\begin{eqnarray*} \underline{e}_1&=&\left[\begin{array}{c} 1 \\ [2pt] \eta \end{... ...t{g\left(\frac{1+c}{1-c}\right) - \frac{c^2(1-g)^2}{4(1-c)^2}} \end{eqnarray*}

They are linearly independent provided $ \eta\neq0$. In the undamped case ($ g=1$), this holds whenever $ c\neq -1$. The eigenvectors are finite when $ c\neq 1$. Thus, the nominal range for $ c$ is the interval $ c\in(-1,1)$.

We can now use Eq.$ \,$(C.142) to find the eigenvalues:

\begin{eqnarray*} {\lambda_i}&=& gc+ \eta_i(c-1)\\ &=& gc+ \frac{(1-g)c}{2}\pm ... ... \pm j\sqrt{g(1-c^2) - \left[\frac{c(1-g)}{2}\right]^2} \protect \end{eqnarray*}


Subsections
Previous: State-Space Analysis
Next: Damping and Tuning Parameters

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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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