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Physical Audio Signal Processing
    Digital Waveguide Theory
       The Digital Waveguide Oscillator
          State-Space Analysis

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State-Space Analysis

We will now use state-space analysisC.15[449] to determine Equations (C.133-C.136).

From Equations (C.128-C.132),

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

and

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

In matrix form, the state time-update can be written

$\displaystyle \left[\begin{array}{c} x_1(n+1) \\ [2pt] x_2(n+1) \end{array}\rig... ...bf{A} \left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \end{array}\right] \protect$ (C.136)

or, in vector notation,
$\displaystyle \underline{x}(n+1)$ $\displaystyle =$ $\displaystyle \mathbf{A}\underline{x}(n) + \mathbf{B}u(n)$ (C.137)
$\displaystyle y(n)$ $\displaystyle =$ $\displaystyle \mathbf{C}\underline{x}(n)$ (C.138)

where we have introduced an input signal $ u(n)$, which sums into the state vector via the $ 2\times 1$ (or $ 2\times N_u$) vector $ \mathbf{B}$. The output signal is defined as the $ 1\times 2$ vector $ \mathbf{C}$ times the state vector $ \underline{x}(n)$. Multiple outputs may be defined by choosing $ \mathbf{C}$ to be an $ N_y \times 2$ matrix.

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 $ A$ is

$\displaystyle \det{\mathbf{A}} = c^2g - g(c+1)(c-1) = c^2g - g(c^2-1) = c^2g - gc^2+g = g. \protect$ (C.139)

When the eigenvalues $ {\lambda_i}$ of $ \mathbf{A}$ (system poles) are complex, then they must form a complex conjugate pair (since $ \mathbf{A}$ is real), and we have $ \det{\mathbf{A}} = \vert{\lambda_i}\vert^2 = g$. Therefore, the system is stable if and only if $ \vert g\vert<1$. When making a digital sinusoidal oscillator from the system impulse response, we have $ \vert g\vert=1$, and the system can be said to be ``marginally stable''. 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$, which occurs for $ g=1$.

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

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

where $ \tilde{\underline{x}}(n) \isdef \mathbf{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 $ T$ 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 $ A$. Note that, as a prerequisite, it will also be necessary to find two linearly independent eigenvectors of $ A$ (columns of $ \mathbf{E}$).

Previous: Digital Waveguide Resonator
Next: Eigenstructure

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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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