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Chapters

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

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State-Space Analysis

We will now use state-space analysisJ.4[460] to determine Equations (J.6-J.9).

From Equations (J.1-J.5),

$\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

$\displaystyle \left[\begin{array}{c} x_1(n+1) \\ [2pt] x_2(n+1) \end{array}\rig... ...bf{A} \left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \end{array}\right] \protect$ (J.10)

or, in vector notation,
$\displaystyle \underline{x}(n+1)$ $\displaystyle =$ $\displaystyle \mathbf{A}\underline{x}(n) + \mathbf{B}u(n)$ (J.11)
$\displaystyle y(n)$ $\displaystyle =$ $\displaystyle \mathbf{C}\underline{x}(n)$ (J.12)

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