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Physical Audio Signal Processing
    Equivalence of Digital Waveguide and Finite Difference Schemes
       State Space Formulation
          DW State Space Model
             DW Displacement Inputs

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DW Displacement Inputs

We define general DW inputs as follows:

$\displaystyle y^{+}_{n,m}$ $\displaystyle =$ $\displaystyle y^{+}_{n-1,m-1} + (\underline{\gamma}^{+}_m)^T \underline{\upsilon}(n)$ (P.33)
$\displaystyle y^{-}_{n,m}$ $\displaystyle =$ $\displaystyle y^{-}_{n-1,m+1} + (\underline{\gamma}^{-}_m)^T \underline{\upsilon}(n)$ (P.34)

The $ m$th $ 2q\times2$ block of the input matrix $ {\mathbf{B}_W}$ driving state components $ [y^{+}_{n+2,m},y^{-}_{n+2,m}]^T$ and multiplying $ [\underline{\upsilon}(n+2)^T,\underline{\upsilon}(n+1)^T]^T$ is then given by

$\displaystyle \left({\mathbf{B}_W}\right)_m = \left[\! \begin{array}{cc} (\unde... ...ma}^{-}_m)^T & (\underline{\gamma}^{-}_{m+1})^T \end{array} \!\right]. \protect$ (P.35)

Typically, input signals are injected equally to the left and right along the string, in which case

$\displaystyle \underline{\gamma}^{+}_m = \underline{\gamma}^{-}_m \isdef \underline{\gamma}_m. $

Physically, this corresponds to applied forces at a single, non-moving, string position over time. The state update with this simplification appears as

\begin{displaymath} \underbrace{\left[\! \begin{array}{c} \vdots\\ y^{+}_{n+2,m... ...ine{\upsilon}(n+1) \end{array}\!\right]}_{\underline{u}(n+2)}. \end{displaymath}
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